**A Mathematical Model for Single Crystal Cylindrical Tube Growth by the Edge-Defined Film-Fed Growth (EFG) Technique**

Loredana Tanasie and Stefan Balint *West University of Timisoara Romania* 

## **1. Introduction**

90 Crystallization – Science and Technology

Conference Proceedings Vol. 1252 , pp. 578-585 , ISBN 978-0-7354-0799-2 Gungor, M. N. (1989) A statistically significant experimental technique for investigating micro-segregation in cast alloys, *Metall. Trans. A*, 20A, pp. 2529-2538. Heger, J., Stetina, J., Kavicka, F., et al. (2002). Pilot calculation of the temperature field of the

Kavicka, F., Stetina, J., Sekanina, B., et al. (2007). The optimization of a concasting

Kavicka, F, Dobrovska, J., Sekanina, B., et al. (2010). A Numerical Model of the Temperature

Conference Proceedings , Vol. 1252 , pp. 571-577 ISBN 978-0-7354-0799-2 Kraft, T., Chang, Y. A. (1997) Predicting microstructure and microsegregation in

Kurz, W., Fisher, D.J. (1986). *Fundamentals of Solidification*. Trans Tech Publications,

Rappaz, M. (1989). Modelling of microstructure formation in solidification processes, *Int.* 

Smrha, L.(1983). *Tuhnutí a krystalizace ocelových ingotů* [*Solidification and crystallisation of steel* 

Stefanescu, D. M.(1995). Methodologies for modelling of solidification microstructure and

Stransky, K., Kavicka, F. , Sekanina, B. et al. (2009). Electromagnetic stirring of the melt of

Stransky, K., Dobrovska, J., Kavicka, F., et al. (2010). Two numerical models of the

Stransky, K., Kavicka, F., Sekanina, B. et al. (2011).The effect of electromagnetic stirring on

concast billets and its importance, *METAL 2009*, Hradec nad Moravici, ISBN 978-

solidification structure of massive ductile cast-iron casting , *Materiali in Tehnologije* / *Materials and Technology* , Vol. 44, Issue 2, (MAR-APR 2010), pp. 93-98 , ISSN

the crystallization of concast billets, *Materiali in Tehnologije /Materials and* 

their capabilities, *ISIJ Int*., 35, pp. 637-650 ISSN 0915-1559

*Technology,* Vol. 45 , Issue 2, pp. 163–166, ISSN 1580-2949

Flemings, M. C. (1974). *Solidification Processing*. McGraw-Hill, New York.

multicomponent alloys, *JOM*, pp. 20-28.

*Mater. Rev*., 34, pp. 93-123 ISSN 0950-6608

Switzerland.

80-87294-04-8

1580-2949

*ingots*] . SNTL, Praha

Vol. 185, Issue 1-3, Sp. Iss. SI, pp. 152-159 , ISSN 0924-0136

2 – Dedicated to professor O. C. Zienkiewicz (1921-2009) Book Series: AIP

ceramic material EUCOR, *Advanced Computational Methods in Heat Transer VII* Book Series: Computational Studies, Vol. 4, pp. 223-232 ISBN 1-85312-906-2

technology by two numerical models , *Journal of Materials Processing Technology* ,

Fi.eld of the Cast and Solidified Ceramic Material , *NUMIFORM 2010*, Vols 1 and 2 - Dedicated to professor O. C. Zienkiewicz (1921-2009) Book Series: AIP

#### **1.1 Crystal growth from the melt by E.F.G. technique**

Modern engineering does not only need crystals of arbitrary shapes but also plate, rod and tube-shaped crystals, i.e., crystals of shapes that allow their use as final products without additional machining. Therefore, the growth of crystals of specified sizes and shapes with controlled defect and impurity structures are required. In the case of crystals grown from the melt, this problem appears to be solved by profiled-container crystallization as in the case of casting. However, this solution is not always possible, for example growing very thin plate-shaped crystals from the melt (to say nothing of more complicated shapes), excludes container application completely.[ Tatarchenko, 1993]

The techniques which allow the shaping of the lateral crystal surface without contact with the container walls are appropriate for the above purpose. In the case of these techniques the shapes and the dimensions of the grown crystals are controlled by the interface and meniscus-shaping capillary force and by the heat- and mass-exchange conditions in the crystal-melt system. The edge-defined film-fed growth (EFG) technique is of this type. Whenever the E.F.G. technique is employed, a shaping device is used (Fig. 1). In the device a capillary channel is manufactured (Fig. 1) in which the melt raises and feeds the growth process. Frequently, a wettable solid body is used to raise the melt column above the shaper, where a thin film is formed. When a wettable body is in contact with the melt, an equilibrium liquid column embracing the surface of the body is formed. The column formation is caused by the capillary forces being present. Such liquid configuration is usually called a meniscus (Fig. 1) and in the E.F.G. technique, its lower boundary (Fig. 1 – point C) is attached to the sharp edge of the shaper.

Let be the temperature of the meniscus upper horizontal section (Fig. 1 – *AB* ) the temperature of the liquid crystallization. So, above the plane of this section, the melt transforms in solid phase. Now set the liquid phase into upward motion with the constant rate, *v*, keeping the position of the phase-transition plane invariable by selection of the heat conditions. When the motion starts, the crystallized position of the meniscus will

A Mathematical Model for Single Crystal

approximation.

Cylindrical Tube Growth by the Edge-Defined Film-Fed Growth (EFG) Technique 93

range of 5 3 5 10 1 10 [m] at rates up to <sup>3</sup> 2 10 [m/s]. In [Erris et al.,1980] a theory of tube growth by the E.F.G. process is developed to show the dependence of the tube wall thickness on the growth variables. The theory concerns the calculations of the shape of the liquid-vapor interface (or meniscus) and of the heat flow in the system. The inner and outer meniscus shapes, (Fig.1), are both calculated from Laplace's capillary equation, in which the pressure difference Δp across a point on meniscus is considered to be *Δp = ρ*◌ּ *g* ◌ּ *Heff* = constant, where *Heff* represents the effective height of the growth interface above the horizontal liquid level in the crucible (Fig.1). According to [Surek et al.,1977], [Swartz et al., 1975], it includes the effects of the viscous flow of the melt in the shaper capillary and in the meniscus film, as well as that of the hydrostatic head. The above approximation for *Δp* is valid for silicon ribbon growth [Surek et al.,1977], [Kalejs et al., 1990], when *Heff >> h*, where *h* is the height of the growth interface above the shaper top (i.e. the meniscus height). Another approximation used in [Erris et al.,1980], concerning the meniscus, is that the inner and outer meniscus shapes are approximated by circular segments. With these relatively tight tolerances concerning the menisci in conjunction with the heat flow calculation in the system, the predictive model developed in [Erris et al.,1980] has been shown to be a useful tool in understanding the feasible limits of wall thickness control. A more precise predictive model would require an increase of the acceptable tolerance range introduced by

Later, this process was scaled up by Kaljes et al. [Kalejs et al., 1990] to grow <sup>2</sup> 15 10 [m] diameter silicon tubes, and the stress behavior in the grown tube was investigated. It has been realized that numerical investigations are necessary for the improvement of the technology. Since the growth system consists of a small die tip ( <sup>3</sup> 1 10 m width) and a thin tube (order of <sup>6</sup> 200 10 [m] wall thickness) the width of the melt/solid interface and meniscus are accordingly very small. Therefore, it is essential to obtain an accurate solution

In [Rajendran et al., 1993] an axisymmetric finite element model of magnetic and thermal field was presented for an inductively heated furnace. Later the same model was used to determine the critical parameters controlling silicon carbide precipitation on the die wall [Rajendran et al., 1994]. Rajendran et al. also developed a three dimensional magnetic induction model for an octagonal E.F.G, system. Recently, in [Roy et al., 2000a], [Roy et al., 2000b], a generic numerical model for an inductively heated large diameter Si tube growth system was reported. In [Sun et al., 2004] a numerical model based on multi-block method and multi-grid technique is developed for induction heating and thermal transport in an E.F.G. system. The model is applied to investigate the growth of large octagon silicon tubes of up to <sup>2</sup> 50 10 m diameter. A 3D dynamic stress model for the growth of hollow silicon polygons is reported in [Behnken et al., 2005]. In [Mackintosh et al., 2006] the challenges fixed in bringing E.F.G. technology into large-scale manufacturing, and ongoing development of furnace designs for growth of tubes for larger wafer production using hexagons with <sup>2</sup> 1 10 m face widths, and wall thicknesses in he range 6 6 250 10 300 10 m is described. In [Kasjanow et al., 2010] the authors present a 3D coupled electromagnetic and thermal modeling of E.F.G. silicon tube growth, successfully

for the temperature and interface position in this tiny region.

validated by experimental tests with industrial installations.

continuously form a solid upward or downward tapering body. In the particular case when the line tangent at the triple point B to the liquid meniscus surface makes a specific angle (angle of growth) with the vertical, the lateral wall of the crystal will be vertical. Thus, the initial body, called the seed, serves to form a meniscus which later on determines the form of the crystallized product, the phase transition position being fixed.

Fig. 1. Prototype tubular crystal growth by E.F.G. method

Based on this description, a conclusion can be drawn that the dimensions and shapes of the specimens being pulled by the E.F.G. technique depend upon the following factors: (i) the shaper geometry; (ii) the pressure of feeding the melt to the shaper; (iii) the crystallization front position; (iv) the seed's shape. The seed's shape is only important for stationary pulling; in this case its cross-section should coincide with the desired product's crosssection. Frequently, especially when complicated profiles are grown, the pulling process is carried out under unstationary conditions by lowering the crystallization surface, which then enhances the dependence of the shapes on the crystal cross-section. With such an approach applied to the pulling process, the dimensions and the shape of the grown crystal are determined by the above-mentioned factors and by the pulling rate-to-crystallization front displacement ratio.[Tatarchenko, 1993].

### **1.2 Background history of tube growth from the melt by E.F.G. method**

The technology of growing tubes can have a significant impact for example on the solar cell technology. The growth of silicon tubes by E.F.G. process was first reported by Erris et al. [Erris et al.,1980]. Tubes were grown with a diameter of <sup>4</sup> 95 10 [m], wall thickness in the

continuously form a solid upward or downward tapering body. In the particular case when the line tangent at the triple point B to the liquid meniscus surface makes a specific angle (angle of growth) with the vertical, the lateral wall of the crystal will be vertical. Thus, the initial body, called the seed, serves to form a meniscus which later on determines the form

Based on this description, a conclusion can be drawn that the dimensions and shapes of the specimens being pulled by the E.F.G. technique depend upon the following factors: (i) the shaper geometry; (ii) the pressure of feeding the melt to the shaper; (iii) the crystallization front position; (iv) the seed's shape. The seed's shape is only important for stationary pulling; in this case its cross-section should coincide with the desired product's crosssection. Frequently, especially when complicated profiles are grown, the pulling process is carried out under unstationary conditions by lowering the crystallization surface, which then enhances the dependence of the shapes on the crystal cross-section. With such an approach applied to the pulling process, the dimensions and the shape of the grown crystal are determined by the above-mentioned factors and by the pulling rate-to-crystallization

Rge

*g*

<sup>A</sup> <sup>B</sup> • •

zi(r)

ri

inner gas flow

re

 *g* hi he *c*

*c*

outer gas flow

ze(r)

• C

tube

Heff

H

r

outer free surface

The technology of growing tubes can have a significant impact for example on the solar cell technology. The growth of silicon tubes by E.F.G. process was first reported by Erris et al. [Erris et al.,1980]. Tubes were grown with a diameter of <sup>4</sup> 95 10 [m], wall thickness in the

**1.2 Background history of tube growth from the melt by E.F.G. method** 

of the crystallized product, the phase transition position being fixed.

z

Fig. 1. Prototype tubular crystal growth by E.F.G. method

inner free surface

melt/solid interface

meniscus melt

shaper

capillary channel

crucible melt

0

Rgi

front displacement ratio.[Tatarchenko, 1993].

range of 5 3 5 10 1 10 [m] at rates up to <sup>3</sup> 2 10 [m/s]. In [Erris et al.,1980] a theory of tube growth by the E.F.G. process is developed to show the dependence of the tube wall thickness on the growth variables. The theory concerns the calculations of the shape of the liquid-vapor interface (or meniscus) and of the heat flow in the system. The inner and outer meniscus shapes, (Fig.1), are both calculated from Laplace's capillary equation, in which the pressure difference Δp across a point on meniscus is considered to be *Δp = ρ*◌ּ *g* ◌ּ *Heff* = constant, where *Heff* represents the effective height of the growth interface above the horizontal liquid level in the crucible (Fig.1). According to [Surek et al.,1977], [Swartz et al., 1975], it includes the effects of the viscous flow of the melt in the shaper capillary and in the meniscus film, as well as that of the hydrostatic head. The above approximation for *Δp* is valid for silicon ribbon growth [Surek et al.,1977], [Kalejs et al., 1990], when *Heff >> h*, where *h* is the height of the growth interface above the shaper top (i.e. the meniscus height). Another approximation used in [Erris et al.,1980], concerning the meniscus, is that the inner and outer meniscus shapes are approximated by circular segments. With these relatively tight tolerances concerning the menisci in conjunction with the heat flow calculation in the system, the predictive model developed in [Erris et al.,1980] has been shown to be a useful tool in understanding the feasible limits of wall thickness control. A more precise predictive model would require an increase of the acceptable tolerance range introduced by approximation.

Later, this process was scaled up by Kaljes et al. [Kalejs et al., 1990] to grow <sup>2</sup> 15 10 [m] diameter silicon tubes, and the stress behavior in the grown tube was investigated. It has been realized that numerical investigations are necessary for the improvement of the technology. Since the growth system consists of a small die tip ( <sup>3</sup> 1 10 m width) and a thin tube (order of <sup>6</sup> 200 10 [m] wall thickness) the width of the melt/solid interface and meniscus are accordingly very small. Therefore, it is essential to obtain an accurate solution for the temperature and interface position in this tiny region.

In [Rajendran et al., 1993] an axisymmetric finite element model of magnetic and thermal field was presented for an inductively heated furnace. Later the same model was used to determine the critical parameters controlling silicon carbide precipitation on the die wall [Rajendran et al., 1994]. Rajendran et al. also developed a three dimensional magnetic induction model for an octagonal E.F.G, system. Recently, in [Roy et al., 2000a], [Roy et al., 2000b], a generic numerical model for an inductively heated large diameter Si tube growth system was reported. In [Sun et al., 2004] a numerical model based on multi-block method and multi-grid technique is developed for induction heating and thermal transport in an E.F.G. system. The model is applied to investigate the growth of large octagon silicon tubes of up to <sup>2</sup> 50 10 m diameter. A 3D dynamic stress model for the growth of hollow silicon polygons is reported in [Behnken et al., 2005]. In [Mackintosh et al., 2006] the challenges fixed in bringing E.F.G. technology into large-scale manufacturing, and ongoing development of furnace designs for growth of tubes for larger wafer production using hexagons with <sup>2</sup> 1 10 m face widths, and wall thicknesses in he range 6 6 250 10 300 10 m is described. In [Kasjanow et al., 2010] the authors present a 3D coupled electromagnetic and thermal modeling of E.F.G. silicon tube growth, successfully validated by experimental tests with industrial installations.

A Mathematical Model for Single Crystal

*dh*

*dt*

In equations (1) 1 and (1) <sup>2</sup> : *v* is the pulling rate,

coordinates (,) *er h* ((,) *ir h* ) and the horizontal *Or* axis (Fig.1 b),

neglected in general, with respect to the hydrostatic pressure

level and the shaper outer (inner) top level (Fig. 1a);

**front h** 

front *h* is:

acceleration.

*<sup>e</sup>*(,, ) *e e r h p* (

Fig. 2. Fluctuations at the triple point

The angle

Cylindrical Tube Growth by the Edge-Defined Film-Fed Growth (EFG) Technique 95

According to [Tatarchenko, 1993] the system of differential equations which governs the evolution of the tube's inner radius *ir* , the outer radius *er* and the level of the crystallization

 

*e i e i*

*<sup>e</sup>*(,, ) *e e r h p* (

 

> <sup>1</sup> *<sup>e</sup> g H* (

*e m <sup>g</sup> ei m <sup>g</sup> <sup>i</sup> pp p g H pp p g H* (2)

*<sup>i</sup>*(,, ) *i i r h p* ) fluctuates due to the fluctuations of: the outer (inner)

(1)

*<sup>i</sup>*(,, ) *i i r h p* ) is the angle

*<sup>g</sup>* is the growth angle (Fig.

<sup>1</sup> *<sup>i</sup> g H* ); *<sup>e</sup>*

*<sup>g</sup> p* ( *<sup>i</sup> <sup>g</sup> p* )

1 is the melt density; *g* is the gravity

 

1 1 2 2

1 1 = ) *e i*

<sup>1</sup> <sup>=</sup> (,,) (,,)

between the tangent line to the outer (inner) meniscus at the three phase point of

1), *<sup>e</sup> p* ( *<sup>i</sup> p* ) is the controllable part of the pressure difference across the free surface given by:

= (

where *pm* is the hydrodynamic pressure in the melt under the free surface, which can be

is the pressure of the gas flow, introduced in order to release the heat from the outer (inner) wall of the tube; *He* ( *Hi* ) is the melt column height between the horizontal crucible melt

radius *er* ( *ir* ), the level *h* of the crystallization front and the outer (inner) pressure *<sup>e</sup> p* ( *<sup>i</sup> p* )

*v G rrh G rrh*

 

**2. The system of differential equations which governs the evolution of the tube's inner radius ri, outer radius re and the level of the crystallization** 

*<sup>e</sup> <sup>e</sup> ee g*

= tan ( , , ) <sup>2</sup>

*<sup>i</sup> <sup>i</sup> ii g*

= tan ( , , ) <sup>2</sup>

*dr v r hp dt*

*dr v r hp dt*

1

The state of the art at 1993-1994 concerning the calculation of the meniscus shape in general in the case of the growth by E.F.G. method is summarized in [Tatarchenko, 1993]. According to [Tatarchenko, 1993], for the general equation describing the surface of a liquid meniscus possessing axial symmetry, there is no complete analysis and solution. For the general equation only numerical integration was carried out for a number of process parameter values that are of practical interest at the moment. The authors of papers [Borodin&Borodin&Sidorov&Petkov, 1999],[Borodin&Borodin&Zhdanov, 1999] consider automated crystal growth processes based on weight sensors and computers. They give an expression for the weight of the meniscus, contacted with a crystal and shaper of arbitrary shape, in which there are two terms related to the hydrodynamic factor. In [Rosolenko et al., 2001] it is shown that the hydrodynamic factor is too small to be considered in the automated crystal growth and it is not clear what equation (of non Laplace type) was considered for the meniscus surface. Finally, in [Yang et al., 2006] the authors present theoretical and numerical study of meniscus dynamics under symmetric and asymmetric configurations. A meniscus dynamics model is developed to consider meniscus shape and its dynamics, heat and mass transfer around the die-top and meniscus. Analysis reveals the correlations between tube thickness, effective melt height, pull-rate, die-top temperature and crystal environmental temperature.

The purpose of this chapter is the mathematical description of the growth process of a single crystal cylindrical tube grown by the edge-defined film-fed growth (EFG) technique. The mathematical model defined by a set of three differential equations governing the evolution of the outer radius and the inner radius of the tube and of the crystallization front level is the one considered in [Tatarchenko, 1993]. This system contains two functions which represent the angle made by the tangent line to the outer (inner) meniscus surface at the three-phase point with the horizontal. The meniscus surface is described mathematically by the solution of the axi-symmetric Young-Laplace differential equation. The analysis of the dependence of solutions of the Young-Laplace differential equation on the pressure difference across the free surface, reveals necessary or sufficient conditions for the existence of solutions which represent convex or concave outer or inner free surfaces of a meniscus. These conditions are expressed in terms of inequalities which are used for the choice of the pressure difference, in order to obtain a single-crystal cylindrical tube with specified sizes.

A numerical procedure for determining the functions appearing in the system of differential equations governing the evolution is presented.

Finally, a procedure is presented for setting the pulling rate, capillary and thermal conditions to grow a cylindrical tube with prior established inner and outer radius. The right hand terms of the system of differential equations serve as tools for setting the above parameters. At the end a numerical simulation of the growth process is presented.

The results presented in this chapter were obtained by the authors and have never been included in a book concerning this topic.

Since the calculus and simulation in this model can be made by a P.C., the information obtained in this way is less expressive than an experiment and can be useful for experiment planing.

The state of the art at 1993-1994 concerning the calculation of the meniscus shape in general in the case of the growth by E.F.G. method is summarized in [Tatarchenko, 1993]. According to [Tatarchenko, 1993], for the general equation describing the surface of a liquid meniscus possessing axial symmetry, there is no complete analysis and solution. For the general equation only numerical integration was carried out for a number of process parameter values that are of practical interest at the moment. The authors of papers [Borodin&Borodin&Sidorov&Petkov, 1999],[Borodin&Borodin&Zhdanov, 1999] consider automated crystal growth processes based on weight sensors and computers. They give an expression for the weight of the meniscus, contacted with a crystal and shaper of arbitrary shape, in which there are two terms related to the hydrodynamic factor. In [Rosolenko et al., 2001] it is shown that the hydrodynamic factor is too small to be considered in the automated crystal growth and it is not clear what equation (of non Laplace type) was considered for the meniscus surface. Finally, in [Yang et al., 2006] the authors present theoretical and numerical study of meniscus dynamics under symmetric and asymmetric configurations. A meniscus dynamics model is developed to consider meniscus shape and its dynamics, heat and mass transfer around the die-top and meniscus. Analysis reveals the correlations between tube thickness, effective melt height, pull-rate, die-top temperature

The purpose of this chapter is the mathematical description of the growth process of a single crystal cylindrical tube grown by the edge-defined film-fed growth (EFG) technique. The mathematical model defined by a set of three differential equations governing the evolution of the outer radius and the inner radius of the tube and of the crystallization front level is the one considered in [Tatarchenko, 1993]. This system contains two functions which represent the angle made by the tangent line to the outer (inner) meniscus surface at the three-phase point with the horizontal. The meniscus surface is described mathematically by the solution of the axi-symmetric Young-Laplace differential equation. The analysis of the dependence of solutions of the Young-Laplace differential equation on the pressure difference across the free surface, reveals necessary or sufficient conditions for the existence of solutions which represent convex or concave outer or inner free surfaces of a meniscus. These conditions are expressed in terms of inequalities which are used for the choice of the pressure difference, in order to obtain a single-crystal cylindrical tube with specified sizes. A numerical procedure for determining the functions appearing in the system of differential

Finally, a procedure is presented for setting the pulling rate, capillary and thermal conditions to grow a cylindrical tube with prior established inner and outer radius. The right hand terms of the system of differential equations serve as tools for setting the above

The results presented in this chapter were obtained by the authors and have never been

Since the calculus and simulation in this model can be made by a P.C., the information obtained in this way is less expressive than an experiment and can be useful for experiment

parameters. At the end a numerical simulation of the growth process is presented.

and crystal environmental temperature.

equations governing the evolution is presented.

included in a book concerning this topic.

planing.

## **2. The system of differential equations which governs the evolution of the tube's inner radius ri, outer radius re and the level of the crystallization front h**

According to [Tatarchenko, 1993] the system of differential equations which governs the evolution of the tube's inner radius *ir* , the outer radius *er* and the level of the crystallization front *h* is:

$$\begin{cases} \frac{dr\_\epsilon}{dt} = -\upsilon \cdot \tan\left[\overline{\alpha}\_\epsilon(r\_\epsilon, h\_\prime p\_\epsilon) - \left(\frac{\pi}{2} - \alpha\_\mathcal{g}\right)\right] \\ \frac{dr\_i}{dt} = \upsilon \cdot \tan\left[\overline{\alpha}\_i(r\_i, h\_\prime p\_i) - \left(\frac{\pi}{2} - \alpha\_\mathcal{g}\right)\right] \\ \frac{dh}{dt} = \upsilon - \frac{1}{\Lambda \cdot \rho\_1} \cdot \left[\lambda\_1 \cdot \mathcal{G}\_1(r\_\epsilon, r\_i, h) - \lambda\_2 \cdot \mathcal{G}\_2(r\_\epsilon, r\_i, h)\right] \end{cases} \tag{1}$$

In equations (1) 1 and (1) <sup>2</sup> : *v* is the pulling rate, *<sup>e</sup>*(,, ) *e e r h p* (*<sup>i</sup>*(,, ) *i i r h p* ) is the angle between the tangent line to the outer (inner) meniscus at the three phase point of coordinates (,) *er h* ((,) *ir h* ) and the horizontal *Or* axis (Fig.1 b), *<sup>g</sup>* is the growth angle (Fig. 1), *<sup>e</sup> p* ( *<sup>i</sup> p* ) is the controllable part of the pressure difference across the free surface given by:

$$\text{l.p.} = p\_m - p\_g^e - \rho\_1 \cdot \text{g} \cdot \text{H}\_e \text{ (} p\_i = p\_m - p\_g^i - \rho\_1 \cdot \text{g} \cdot \text{H}\_i \text{)}\tag{2}$$

where *pm* is the hydrodynamic pressure in the melt under the free surface, which can be neglected in general, with respect to the hydrostatic pressure <sup>1</sup> *<sup>e</sup> g H* ( <sup>1</sup> *<sup>i</sup> g H* ); *<sup>e</sup> <sup>g</sup> p* ( *<sup>i</sup> <sup>g</sup> p* ) is the pressure of the gas flow, introduced in order to release the heat from the outer (inner) wall of the tube; *He* ( *Hi* ) is the melt column height between the horizontal crucible melt level and the shaper outer (inner) top level (Fig. 1a); 1 is the melt density; *g* is the gravity acceleration.

The angle *<sup>e</sup>*(,, ) *e e r h p* ( *<sup>i</sup>*(,, ) *i i r h p* ) fluctuates due to the fluctuations of: the outer (inner) radius *er* ( *ir* ), the level *h* of the crystallization front and the outer (inner) pressure *<sup>e</sup> p* ( *<sup>i</sup> p* )

Fig. 2. Fluctuations at the triple point

A Mathematical Model for Single Crystal

Laplace capillary equation [Finn, 1986]:

**growth** 

 , <sup>2</sup> *gi ge e ge R R*

In the following sections we will show in which way

*r R* and inner radius

equation of the outer free surface is given by:

*zR z R*

which is the Euler equation for the energy functional:

1

*e*

*i*

*r*

*gi*

*R*

*r*

*ge*

*R*

which is the Euler equation for the energy functional

*ge e ge*

( )0

 

Cylindrical Tube Growth by the Edge-Defined Film-Fed Growth (EFG) Technique 97

**3. The choice of the pressure of the gas flow and the melt level in silicon tube** 

In a single crystal tube growth by edge-defined film-fed growth (E.F.G.) technique, in hydrostatic approximation, the free surface of a static meniscus is described by the Young-

> 

 , <sup>2</sup> *gi ge*

*R R*

<sup>2</sup> <sup>2</sup> <sup>1</sup> <sup>2</sup> 1 ' , , 2

Here γ is the melt surface tension, ρ denotes the melt density, g is the gravity acceleration, 1 2 1/ ,1/ *R R* denote the mean normal curvatures of the free surface at a point M of the free surface, z is the coordinate of M with respect to the Oz axis, directed vertically upwards, p is the pressure difference across the free surface. To calculate the outer and inner free surface shape of the static meniscus it is convenient to employ the Young-Laplace eq.(5) in its differential form. This form of the eq.(5) can be obtained as a necessary condition for the minimum of the free energy of the melt column [Finn, 1986].For a tube of outer radius

*i gi*

*r*

*I z z g z p z r dr z r z r*

*e e e ee*

*r*

*I z z g z p z r dr z R z R z r z r* (9)

<sup>2</sup> <sup>2</sup> <sup>1</sup> <sup>2</sup> 1 ' , ( ) 0, <sup>2</sup>

*i gi i gi ii i*

 

 3 2 2 <sup>2</sup> <sup>1</sup> " 1' 1' ' *<sup>e</sup> gz p <sup>z</sup> <sup>z</sup> z z*

1

The axi-symmetric differential equation of the inner free surface is given by:

 3 2 2 <sup>2</sup> <sup>1</sup> " 1' 1' ' *<sup>i</sup> gz p <sup>z</sup> <sup>z</sup> z z*

found starting from the Young-Laplace equation of a capillary surface in equilibrium.

 1 2 1 1

*<sup>e</sup>*(,, ) *e e r h p* and

*<sup>g</sup> <sup>z</sup> <sup>p</sup> R R* (5)

*r R* , the axi-symmetric differential

for [, ] *<sup>e</sup> ge r rR* (6)

for , *gi i r Rr* (8)

(7)

*<sup>i</sup>*(,, ) *i i r h p* can be

In the equation (1) <sup>3</sup> : is the latent melting heat; 1 , 2 are the thermal conductivity coefficients in the melt and the crystal respectively; 1 *<sup>j</sup> <sup>G</sup>* , 2 *<sup>j</sup> G* are the temperature gradients at the interface in the melt *(i=1)* and in the crystal *(i=2)* respectively, given by the formulas:

 1 1 0 1 2 1 1 2 1 11 1 1 <sup>1</sup> <sup>0</sup> 0 *j δ h e i en j j j m en j j j j v k G (r ,r ,h) = T T() <sup>β</sup> <sup>e</sup> SINH(β h) (F ) Bi χ v k T T() kh <sup>δ</sup> SINH(<sup>β</sup> h) <sup>β</sup> COSH(<sup>β</sup> h) k (F ) Bi <sup>χ</sup>* (3) 2 2 2 2 2 2 22 2 2 2 2 <sup>1</sup> ( , , )= (0) ( ) ( ) () () ( ) *j e i m en j j L jj j j j v k G rrh T T kh SINH L F Bi v k SINH L COSH L e k F Bi* (4)

where *i* - the thermal diffusivity coefficient equal to *i i i <sup>c</sup>* , *<sup>i</sup>* - the density, *<sup>i</sup> c* - the heat capacity, *Bi* - the Biot number equal to *i e i <sup>r</sup>* ( *i=1 -* the melt, *i=2 -* the crystal*)*, *k r ii i* - the coefficient of the heat-exchange with environment ( *k <sup>i</sup>* - the convective heat-exchange coefficient and *r <sup>i</sup>* - the linearized radiation heat-exchange coefficient), *Fj* ( *j* 1,2,3 ) the crystal (meniscus) cross – section perimeter – to – its area ratio: *j* 1 and 1 2 = *e F r* , for a thick-walled tube with small inner radius, for which heat is removed from the external surface only, *<sup>j</sup>* 2 and <sup>2</sup> 2 2 <sup>2</sup> <sup>=</sup> *<sup>e</sup> e i <sup>r</sup> <sup>F</sup> r r* for a tube of not to large inner radius for which heat is

removed from the external surface only, *<sup>j</sup>* 3 and <sup>3</sup> <sup>2</sup> <sup>=</sup> *e i F r r* for a tube for which heat is removed from both the outer and inner surfaces([Tatarchenko, 1993], pp. 39-40, 146). *T*0 the melt temperature at the meniscus basis, *Tm* - melting temperature, (0) *Ten* - the environment temperature at = 0 *z* , *k* - the vertical temperature gradient in the furnace, *er* the outer radius of the tube equal to the upper radius of the outer meniscus, *ir* - the inner radius of the tube equal to the upper radius of the inner meniscus, *L* - the tube length and 2 2 <sup>2</sup> = , = ( ) , = 1,2, = 1,2,3 <sup>2</sup> <sup>4</sup> *i ij j i i v v F Bi i j* , *SINH* and *COSH* are the hyperbolic sine

and hyperbolic cosine functions.

Due to the supercooling in this gradients it is assumed that: *T T* <sup>0</sup> *<sup>m</sup>* , *k* 0 , *T T en m* (0) .

at the interface in the melt *(i=1)* and in the crystal *(i=2)* respectively, given by the formulas:

*v k T T() kh <sup>δ</sup> SINH(<sup>β</sup> h) <sup>β</sup> COSH(<sup>β</sup> h) k (F ) Bi <sup>χ</sup>*

*m en j j j*

 

2 22 2 2 2

crystal (meniscus) cross – section perimeter – to – its area ratio: *j* 1 and 1

thick-walled tube with small inner radius, for which heat is removed from the external

removed from both the outer and inner surfaces([Tatarchenko, 1993], pp. 39-40, 146). *T*0 the melt temperature at the meniscus basis, *Tm* - melting temperature, (0) *Ten* - the environment temperature at = 0 *z* , *k* - the vertical temperature gradient in the furnace, *er* the outer radius of the tube equal to the upper radius of the outer meniscus, *ir* - the inner radius of the tube equal to the upper radius of the inner meniscus, *L* - the tube length and

*v v F Bi i j* , *SINH* and *COSH* are the hyperbolic sine

Due to the supercooling in this gradients it is assumed that: *T T* <sup>0</sup> *<sup>m</sup>* , *k* 0 , *T T en m* (0) .

 *i e i*

() () ( )

*v k SINH L COSH L e k*

*jj j j*

*j j*

*SINH L F Bi*

1 1

*j δ h e i en j j j*

1 0 1 2

*v k G (r ,r ,h) = T T() <sup>β</sup> <sup>e</sup> SINH(β h) (F ) Bi χ*

<sup>1</sup> <sup>0</sup>

1

2 2

<sup>1</sup> ( , , )= (0) ( ) ( )

*v k G rrh T T kh*

1 , 

2 1 11 1

*j*

*F Bi*

 *i i i <sup>c</sup>* , 

*<sup>i</sup>* - the linearized radiation heat-exchange coefficient), *Fj* ( *j* 1,2,3 ) the

2 2

*<sup>j</sup> <sup>G</sup>* , 2

2 are the thermal conductivity

(3)

(4)

*<sup>j</sup> G* are the temperature gradients

*<sup>r</sup>* ( *i=1 -* the melt, *i=2 -* the crystal*)*,

for a tube of not to large inner radius for which heat is

*k*

<sup>2</sup> <sup>=</sup> *e i*

*r r*

*F*

2

1

2

*L*

*<sup>i</sup>* - the convective heat-exchange

*<sup>i</sup>* - the density, *<sup>i</sup> c* - the heat

 *k r ii i*

2 = *e*

, for a

*F r*

for a tube for which heat is

In the equation (1) <sup>3</sup> : is the latent melting heat;

0

where

 

*i i*

and hyperbolic cosine functions.

2

2 <sup>2</sup> = , = ( ) , = 1,2, = 1,2,3 <sup>2</sup> <sup>4</sup> *i ij j*

coefficient and

*j*

coefficients in the melt and the crystal respectively; 1

*j*

*i* - the thermal diffusivity coefficient equal to


 <sup>2</sup> 2 2 <sup>2</sup> <sup>=</sup> *<sup>e</sup> e i <sup>r</sup> <sup>F</sup> r r*

removed from the external surface only, *<sup>j</sup>* 3 and <sup>3</sup>

capacity, *Bi* - the Biot number equal to

*r*

surface only, *<sup>j</sup>* 2 and

*e i m en*

In the following sections we will show in which way *<sup>e</sup>*(,, ) *e e r h p* and *<sup>i</sup>*(,, ) *i i r h p* can be found starting from the Young-Laplace equation of a capillary surface in equilibrium.

## **3. The choice of the pressure of the gas flow and the melt level in silicon tube growth**

In a single crystal tube growth by edge-defined film-fed growth (E.F.G.) technique, in hydrostatic approximation, the free surface of a static meniscus is described by the Young-Laplace capillary equation [Finn, 1986]:

$$
\gamma \cdot \left(\frac{1}{R\_1} + \frac{1}{R\_2}\right) = \rho \cdot g \cdot z - p \tag{5}
$$

Here γ is the melt surface tension, ρ denotes the melt density, g is the gravity acceleration, 1 2 1/ ,1/ *R R* denote the mean normal curvatures of the free surface at a point M of the free surface, z is the coordinate of M with respect to the Oz axis, directed vertically upwards, p is the pressure difference across the free surface. To calculate the outer and inner free surface shape of the static meniscus it is convenient to employ the Young-Laplace eq.(5) in its differential form. This form of the eq.(5) can be obtained as a necessary condition for the minimum of the free energy of the melt column [Finn, 1986].For a tube of outer radius *R R R R*

 , <sup>2</sup> *gi ge e ge r R* and inner radius , <sup>2</sup> *gi ge i gi r R* , the axi-symmetric differential

equation of the outer free surface is given by:

$$z'' = \frac{\rho \cdot g \cdot z - p\_{\varepsilon}}{\gamma} \left[1 + \left(z'\right)^{2}\right]^{\frac{3}{2}} - \frac{1}{r} \cdot \left[1 + \left(z'\right)^{2}\right] \cdot z' \quad \text{for} \quad r \in \left[r\_{\varepsilon}, R\_{\text{ge}}\right] \tag{6}$$

which is the Euler equation for the energy functional

$$\begin{aligned} I\_{\epsilon}(\mathbf{z}) &= \bigcap\_{r\_{\epsilon}}^{R\_{\mathcal{B}^{\epsilon}}} \left\{ \gamma \cdot \left[ 1 + \left( z' \right)^{2} \right]^{\frac{1}{2}} + \frac{1}{2} \cdot \rho \cdot g \cdot z^{2} - p\_{\epsilon} \cdot z \right\} \cdot r \cdot dr, \quad z(r\_{\epsilon}) = z\_{\epsilon}(r\_{\epsilon}), \\\ z \left( R\_{\mathcal{B}^{\epsilon}} \right) &= z\_{\epsilon}(R\_{\mathcal{B}^{\epsilon}}) = 0 \end{aligned} \tag{7}$$

The axi-symmetric differential equation of the inner free surface is given by:

$$z'' = \frac{\rho \cdot g \cdot z - p\_i}{\gamma} \left[ 1 + \left( z' \right)^2 \right]^{\frac{3}{2}} - \frac{1}{r} \cdot \left[ 1 + \left( z' \right)^2 \right] \cdot z' \quad \text{for} \quad r \in \left[ R\_{g^i}, r\_i \right] \tag{8}$$

which is the Euler equation for the energy functional:

$$I\_i(z) = \int\_{R\_{ji}}^{r\_i} \left\{ \gamma \cdot \left[1 + \left(z'\right)^2\right]^{\frac{1}{2}} + \frac{1}{2} \cdot \rho \cdot g \cdot z^2 - p \cdot z \right\} \cdot r \cdot dr, \quad z\left(R\_{ji}\right) = z\_i(R\_{ji}) = 0, \quad z\left(r\_i\right) = z\_i\left(r\_i\right) \tag{9}$$

A Mathematical Model for Single Crystal

*RR gige* .

which is in the range ' , *mR mR gi gi* .

The results of the integrations of the system (10) for *z R*( )0 *ge* ,

1 3 4.65 10 *er* [m] is obtained for <sup>1</sup> <sup>2198</sup> *<sup>e</sup> <sup>p</sup>* [Pa].

 

*dz dr*

= <sup>1</sup> 7.2 10 [N/m];

by integrating numerically the following system for *z R*( )0 *ge* ,

tan

*d p gz*

*<sup>e</sup> <sup>e</sup>*

 

*dr r*

equal to <sup>2</sup>

is stable.

= <sup>3</sup> 2.5 10 [kg/m3];

where

1702.52 *<sup>e</sup> p* [Pa] (see Fig. 3):

'

Cylindrical Tube Growth by the Edge-Defined Film-Fed Growth (EFG) Technique 99

If the pressure difference *<sup>i</sup> p* satisfies the inequality (A.1.9) and the value of *<sup>i</sup> p* is close to the value of the right hand term of the inequality (A.1.9) then a static meniscus with convex inner free surface is obtained which is appropriate for the growth of a tube of inner radius

If the pressure difference *<sup>i</sup> p* satisfies inequality (A.1.10) then a static meniscus with convex inner free surface is obtained which is appropriate for the growth of a tube of inner radius

**Theorem 10** (Appendix 1) shows that a static meniscus having a covex inner free surface

appropriate for the growth of a tube of inner radius *ir* situated in the range

For numerical illustrations, the inner radius of the shaper was taken <sup>3</sup> 4.2 10 *Rgi* [m] and outer radius of the shaper was chosen <sup>3</sup> 4.8 10 *Rge* [m] [Eriss, 1980]. Computations were performed in **MathCAD 14**. and for **Si** the following numerical values were considered:

*<sup>g</sup>* =11o; g=9.81[m/s2].

( ) *Rge c* 

tan

*<sup>c</sup>* =30o; 

To create a convex meniscus appropriate for the growth of a tube having the outer radius <sup>1</sup>

equal to 1 3 4.65 10 *er* [m] ( *<sup>n</sup>*<sup>1</sup> 1.03226 ), according to the **Theorem 1** (Appendix 1), *<sup>e</sup> <sup>p</sup>* has to be chosen in the range: 3480.07, 612.35 [Pa]. According to the  **Corollary 3**  (Appendix 1), from this range for the values of *<sup>e</sup> p* smaller than 1702.52 [Pa] the point *er*

1702.52 *<sup>e</sup> p* [Pa] the point *er* for which the above condition is satisfied. This can be made

( ) tan <sup>2</sup> *e e <sup>g</sup> z r* is close to <sup>3</sup> 4.5 10 [m]. Hence, we have to find for

 

cos

*<sup>e</sup> e e <sup>e</sup>*

Since the obtained *er* is <sup>3</sup> 4.609 10 *er* [m], and it is smaller than the desired value 1 3 4.65 10 *er* [m], the value of *<sup>e</sup> <sup>p</sup>* has to be chosen in the range 3480.07, 1702.52 [Pa].

values of *<sup>e</sup> p* in this range, are presented in Fig. 4. This figure shows that the outer radius

1 1

*e*

, <sup>2</sup> *gi ge*

*R R*

*er*

and

(10)

and different

*gi*

*R*

( ) *Rge c* 

In papers [Balint & Balint, 2009b], [Balint&Balint&Tanasie, 2008], [Balint & Tanasie, 2008] , Balint, Tanasie, 2011] some mathematical theorems and corollaries have been rigorously proven regarding the existence of an appropriate meniscus. These results are presented in Appendixes. In the following we will shown in which way the inequalities can be used for creation of the appropriate meniscus.

### **3.1 Convex free surface creation**

In this section, it will be shown in which way the inequalities presented in Appendix 1 can be used for the creation of an appropriate static convex meniscus by the choice of *<sup>e</sup> p* and *<sup>i</sup> p* [Balint, Tanasie, 2011].

Inequalities (A.1.1) establish the range where the pressure difference *<sup>e</sup> p* has to be chosen in order to obtain a static meniscus with convex outer free surface, appropriate for the growth

of a tube of outer radius equal to *Rge n* .

If the pressure difference *<sup>e</sup> <sup>p</sup>* satisfies (A.1.2), then a static meniscus with convex outer free surface is obtained which is appropriate for the growth of a tube of outer radius

$$r\_e \in \left[\frac{\mathcal{R}\_{\rm ge}}{n}, \mathcal{R}\_{\rm ge}\right].$$

If the pressure difference *<sup>e</sup> <sup>p</sup>* satisfies inequality (A.1.4) and the value of *pe <sup>e</sup> <sup>p</sup>* is close to the value of the right hand member of the inequality (A.1.4) then a static meniscus with convex outer free surface is obtained which is appropriate for the growth of a tube of outer radius

equal to <sup>2</sup> *RR gige* .

If the pressure difference *<sup>e</sup> <sup>p</sup>* satisfies inequality (A.1.5), then a static meniscus with convex outer free surface is obtained which is appropriate for the growth of a tube of outer radius in

$$\text{the range}\left[\frac{R\_{\text{ge}}}{n}, \frac{R\_{\text{ge}}}{n'}\right].$$

**Theorem 5** (Appendix 1) shows that a static meniscus having a convex outer free surface,

appropriate for the growth of a tube of outer radius *er* situated in the range , *R R ge ge n n* , is

stable.

Inequalities (A.1.6) establish the range where the pressure difference *<sup>i</sup> p* has to be chosen in order to obtain a static meniscus with convex inner free surface appropriate for the growth of a tube of inner radius equal to *m Rgi* .

If the pressure difference *<sup>i</sup> p* satisfies (A.1.7) then a static meniscus with convex inner free surface is obtained which is appropriate for the growth of a tube of inner radius *<sup>i</sup> Rmr gi* .

In papers [Balint & Balint, 2009b], [Balint&Balint&Tanasie, 2008], [Balint & Tanasie, 2008] , Balint, Tanasie, 2011] some mathematical theorems and corollaries have been rigorously proven regarding the existence of an appropriate meniscus. These results are presented in Appendixes. In the following we will shown in which way the inequalities can be used for

In this section, it will be shown in which way the inequalities presented in Appendix 1 can be used for the creation of an appropriate static convex meniscus by the choice of *<sup>e</sup> p* and

Inequalities (A.1.1) establish the range where the pressure difference *<sup>e</sup> p* has to be chosen in order to obtain a static meniscus with convex outer free surface, appropriate for the growth

surface is obtained which is appropriate for the growth of a tube of outer radius

value of the right hand member of the inequality (A.1.4) then a static meniscus with convex outer free surface is obtained which is appropriate for the growth of a tube of outer radius

outer free surface is obtained which is appropriate for the growth of a tube of outer radius in

**Theorem 5** (Appendix 1) shows that a static meniscus having a convex outer free surface,

Inequalities (A.1.6) establish the range where the pressure difference *<sup>i</sup> p* has to be chosen in order to obtain a static meniscus with convex inner free surface appropriate for the growth

If the pressure difference *<sup>i</sup> p* satisfies (A.1.7) then a static meniscus with convex inner free surface is obtained which is appropriate for the growth of a tube of inner radius *<sup>i</sup> Rmr gi* .

appropriate for the growth of a tube of outer radius *er* situated in the range

satisfies inequality (A.1.4) and the value of *pe*

satisfies (A.1.2), then a static meniscus with convex outer free

satisfies inequality (A.1.5), then a static meniscus with convex

*<sup>e</sup> <sup>p</sup>*

is close to the

 , *R R ge ge n n*

, is

*Rge n* .

*<sup>e</sup> <sup>p</sup>*

*<sup>e</sup> <sup>p</sup>*

*<sup>e</sup> <sup>p</sup>*

creation of the appropriate meniscus.

**3.1 Convex free surface creation** 

of a tube of outer radius equal to

*<sup>i</sup> p* [Balint, Tanasie, 2011].

If the pressure difference

If the pressure difference

If the pressure difference

the range

*RR gige* .

 , *R R ge ge n n* .

of a tube of inner radius equal to *m Rgi* .

 , *ge e ge R r R*

equal to <sup>2</sup>

stable.

*<sup>n</sup>* .

If the pressure difference *<sup>i</sup> p* satisfies the inequality (A.1.9) and the value of *<sup>i</sup> p* is close to the value of the right hand term of the inequality (A.1.9) then a static meniscus with convex inner free surface is obtained which is appropriate for the growth of a tube of inner radius equal to <sup>2</sup> *RR gige* .

If the pressure difference *<sup>i</sup> p* satisfies inequality (A.1.10) then a static meniscus with convex inner free surface is obtained which is appropriate for the growth of a tube of inner radius which is in the range ' , *mR mR gi gi* .

**Theorem 10** (Appendix 1) shows that a static meniscus having a covex inner free surface appropriate for the growth of a tube of inner radius *ir* situated in the range , <sup>2</sup> *gi ge gi R R R* is stable.

For numerical illustrations, the inner radius of the shaper was taken <sup>3</sup> 4.2 10 *Rgi* [m] and outer radius of the shaper was chosen <sup>3</sup> 4.8 10 *Rge* [m] [Eriss, 1980]. Computations were performed in **MathCAD 14**. and for **Si** the following numerical values were considered: = <sup>3</sup> 2.5 10 [kg/m3]; = <sup>1</sup> 7.2 10 [N/m]; *<sup>c</sup>* =30o; *<sup>g</sup>* =11o; g=9.81[m/s2].

To create a convex meniscus appropriate for the growth of a tube having the outer radius <sup>1</sup> *er* equal to 1 3 4.65 10 *er* [m] ( *<sup>n</sup>*<sup>1</sup> 1.03226 ), according to the **Theorem 1** (Appendix 1), *<sup>e</sup> <sup>p</sup>* has to be chosen in the range: 3480.07, 612.35 [Pa]. According to the  **Corollary 3**  (Appendix 1), from this range for the values of *<sup>e</sup> p* smaller than 1702.52 [Pa] the point *er* where ' ( ) tan <sup>2</sup> *e e <sup>g</sup> z r* is close to <sup>3</sup> 4.5 10 [m]. Hence, we have to find for 1702.52 *<sup>e</sup> p* [Pa] the point *er* for which the above condition is satisfied. This can be made by integrating numerically the following system for *z R*( )0 *ge* , ( ) *Rge c* and 1702.52 *<sup>e</sup> p* [Pa] (see Fig. 3):

$$\begin{cases} \frac{d\boldsymbol{z}\_e}{dr} = -\tan\left(\boldsymbol{a}\_e\right) \\ \frac{d\boldsymbol{a}\_e}{dr} = \frac{\boldsymbol{p}\_e - \boldsymbol{\rho} \cdot \mathbf{g} \cdot \boldsymbol{z}\_e}{\boldsymbol{\gamma}} \cdot \frac{1}{\cos\boldsymbol{a}\_e} - \frac{1}{r} \cdot \tan\boldsymbol{a}\_e \end{cases} \tag{10}$$

Since the obtained *er* is <sup>3</sup> 4.609 10 *er* [m], and it is smaller than the desired value 1 3 4.65 10 *er* [m], the value of *<sup>e</sup> <sup>p</sup>* has to be chosen in the range 3480.07, 1702.52 [Pa]. The results of the integrations of the system (10) for *z R*( )0 *ge* , ( ) *Rge c* and different values of *<sup>e</sup> p* in this range, are presented in Fig. 4. This figure shows that the outer radius 1 3 4.65 10 *er* [m] is obtained for <sup>1</sup> <sup>2198</sup> *<sup>e</sup> <sup>p</sup>* [Pa].

A Mathematical Model for Single Crystal

be made by integrating numerically the system:

The results of the integrations of the system (11) for *z R*( )0 *gi* ,

1 2 9.92 10 *Hi* [m] above the crucible melt level. When <sup>2434</sup> *<sup>i</sup>*

the crucible melt level has to be above the shaper's inner top level.

values of *<sup>i</sup> p* in this range are represented in Fig. 4.

 <sup>1</sup> <sup>1</sup> <sup>2434</sup> *<sup>i</sup> Hi <sup>g</sup> <sup>p</sup> g*

 

*dz dr*

*i*

condition ( ) tan

( ) *Rgi c* 

column height is

for *z R*( )0 *gi* ,

0 *<sup>i</sup>*

 236 *i e g g p p* [Pa].

**3.2 Concave free surface creation** 

*<sup>i</sup> p* [Balint&Balint, 2009a].

Cylindrical Tube Growth by the Edge-Defined Film-Fed Growth (EFG) Technique 101

find now for 1945.80 *<sup>i</sup> p* [Pa] the point *ir* where the above condition is achieved. This can

*i i i*

*i*

*dr r*

and 1945.80 *<sup>i</sup> p* [Pa]. (see Fig. 3).

Since the obtained *ir* is <sup>3</sup> 4.390 10 *ir* [m] and it is higher than the desired value 1 3 4.35 10 *ir* [m], we have to choose the value of *<sup>i</sup> <sup>p</sup>* in the range 3723.32 1945.80 [Pa].

This figure shows that the inner radius 1 3 4.35 10 *ir* [m] is obtained for <sup>1</sup> <sup>2434</sup> *<sup>i</sup> <sup>p</sup>* [Pa].

Taking *pm* 0 [Eriss et al., 1980], [Rossolenko et al., 2001], [Yang et al., 2006], the melt

(introduced in the furnace for releasing the heat from the inner wall of the tube). When

*<sup>g</sup> <sup>p</sup>* , then 1 2 9.92 10 *Hi* [m], i.e. the shaper's inner top level has to be with

To create a convex meniscus appropriate for the growth of a tube with the outer radius 1 3 4.65 10 *er* [m] and inner radius 1 3 4.35 10 *ir* [m], when the shaper's inner top is at

the same level as the shaper's outer top, we have to take:

It follows that the pressure of the gas flow, introduced in the furnace for releasing the heat from the inner wall of the tube has to be higher than the pressure of the gas flow, introduced in the furnace for releasing the heat from the outer wall of the tube and we have to take:

In this section, it will be shown in which way the inequalities presented in Appendix 2 can be used for the creation of an appropriate static concave meniscus by the choice of *<sup>e</sup> p* and

, where 0 *<sup>i</sup>*

*d gz p*

tan

<sup>2</sup> *i i <sup>g</sup> z r* is satisfied is close to <sup>3</sup> 4.5 10 [m]. Therefore, we have to

 

cos

1 1

*i*

*i*

( ) *Rgi c* 

*<sup>g</sup> p* is the pressure of the gas flow

*<sup>g</sup> p* , then <sup>1</sup> *Hi* is negative, i.e.

 1 1 <sup>2198</sup> <sup>2434</sup> *e i*

*g g p p g g* .

(11)

and for different

tan

Fig. 3. The results of the integration of systems (10) and (11) for 1702.52 *<sup>e</sup> p* [Pa] and 1945.80 *<sup>i</sup> p* [Pa]

Taking *pm* 0 [Eriss et al., 1980], [Rossolenko et al., 2001], [Yang et al., 2006], the melt column height in this case is <sup>1</sup> <sup>1</sup> <sup>2198</sup> *<sup>e</sup> He <sup>g</sup> <sup>p</sup> g* , where 0 *<sup>e</sup> <sup>g</sup> p* is the pressure of the gas flow (introduced in the furnace for release, the heat from the outer wall of the tube). When 0 *<sup>e</sup> <sup>g</sup> <sup>p</sup>* , then 1 2 8.96 10 *He* [m] i.e. the shaper's outer top level has to be with 1 2 8.96 10 *He* [m] above the crucible melt level.

Fig. 4. The tube outer radius and inner radius versus *<sup>e</sup> p* and *<sup>i</sup> p*

To create a convex meniscus appropriate for the growth of a tube having the inner radius 1 3 4.35 10 *ir* [m] ( *<sup>m</sup>*<sup>1</sup> 1.03571 ), according to the **Theorem 6** (Appendix 1), *<sup>i</sup> <sup>p</sup>* has to be chosen in the range: 3723.32, 847.10 [Pa]. According to the **Corollary 8** (Appendix 1), from this range for the values of *<sup>i</sup> p* smaller than 1945.80 [Pa], the point *ir* where the

Fig. 3. The results of the integration of systems (10) and (11) for 1702.52 *<sup>e</sup> p* [Pa] and

 <sup>1</sup> <sup>1</sup> <sup>2198</sup> *<sup>e</sup> He <sup>g</sup> <sup>p</sup> g*

Fig. 4. The tube outer radius and inner radius versus *<sup>e</sup> p* and *<sup>i</sup> p*

Taking *pm* 0 [Eriss et al., 1980], [Rossolenko et al., 2001], [Yang et al., 2006], the melt

flow (introduced in the furnace for release, the heat from the outer wall of the tube). When

*<sup>g</sup> <sup>p</sup>* , then 1 2 8.96 10 *He* [m] i.e. the shaper's outer top level has to be with

To create a convex meniscus appropriate for the growth of a tube having the inner radius 1 3 4.35 10 *ir* [m] ( *<sup>m</sup>*<sup>1</sup> 1.03571 ), according to the **Theorem 6** (Appendix 1), *<sup>i</sup> <sup>p</sup>* has to be chosen in the range: 3723.32, 847.10 [Pa]. According to the **Corollary 8** (Appendix 1), from this range for the values of *<sup>i</sup> p* smaller than 1945.80 [Pa], the point *ir* where the

, where 0 *<sup>e</sup>*

*<sup>g</sup> p* is the pressure of the gas

1945.80 *<sup>i</sup> p* [Pa]

0 *<sup>e</sup>*

column height in this case is

1 2 8.96 10 *He* [m] above the crucible melt level.

condition ( ) tan <sup>2</sup> *i i <sup>g</sup> z r* is satisfied is close to <sup>3</sup> 4.5 10 [m]. Therefore, we have to find now for 1945.80 *<sup>i</sup> p* [Pa] the point *ir* where the above condition is achieved. This can be made by integrating numerically the system:

$$\begin{cases} \frac{dz\_i}{dr} = \tan \alpha\_i\\ \frac{da\_i}{dr} = \frac{\rho \cdot g \cdot z\_i - p\_i}{\gamma} \cdot \frac{1}{\cos \alpha\_i} - \frac{1}{r} \cdot \tan \alpha\_i \end{cases} \tag{11}$$

for *z R*( )0 *gi* , ( ) *Rgi c* and 1945.80 *<sup>i</sup> p* [Pa]. (see Fig. 3).

Since the obtained *ir* is <sup>3</sup> 4.390 10 *ir* [m] and it is higher than the desired value 1 3 4.35 10 *ir* [m], we have to choose the value of *<sup>i</sup> <sup>p</sup>* in the range 3723.32 1945.80 [Pa]. The results of the integrations of the system (11) for *z R*( )0 *gi* , ( ) *Rgi c* and for different values of *<sup>i</sup> p* in this range are represented in Fig. 4.

This figure shows that the inner radius 1 3 4.35 10 *ir* [m] is obtained for <sup>1</sup> <sup>2434</sup> *<sup>i</sup> <sup>p</sup>* [Pa].

Taking *pm* 0 [Eriss et al., 1980], [Rossolenko et al., 2001], [Yang et al., 2006], the melt column height is <sup>1</sup> <sup>1</sup> <sup>2434</sup> *<sup>i</sup> Hi <sup>g</sup> <sup>p</sup> g* , where 0 *<sup>i</sup> <sup>g</sup> p* is the pressure of the gas flow (introduced in the furnace for releasing the heat from the inner wall of the tube). When 0 *<sup>i</sup> <sup>g</sup> <sup>p</sup>* , then 1 2 9.92 10 *Hi* [m], i.e. the shaper's inner top level has to be with 1 2 9.92 10 *Hi* [m] above the crucible melt level. When <sup>2434</sup> *<sup>i</sup> <sup>g</sup> p* , then <sup>1</sup> *Hi* is negative, i.e. the crucible melt level has to be above the shaper's inner top level.

To create a convex meniscus appropriate for the growth of a tube with the outer radius 1 3 4.65 10 *er* [m] and inner radius 1 3 4.35 10 *ir* [m], when the shaper's inner top is at the same level as the shaper's outer top, we have to take: 1 1 <sup>2198</sup> <sup>2434</sup> *e i g g p p g g* .

It follows that the pressure of the gas flow, introduced in the furnace for releasing the heat from the inner wall of the tube has to be higher than the pressure of the gas flow, introduced in the furnace for releasing the heat from the outer wall of the tube and we have to take: 236 *i e g g p p* [Pa].

## **3.2 Concave free surface creation**

In this section, it will be shown in which way the inequalities presented in Appendix 2 can be used for the creation of an appropriate static concave meniscus by the choice of *<sup>e</sup> p* and *<sup>i</sup> p* [Balint&Balint, 2009a].

A Mathematical Model for Single Crystal

concave meniscus (Fig.6).

height in this case is

0 *<sup>e</sup>*

Cylindrical Tube Growth by the Edge-Defined Film-Fed Growth (EFG) Technique 103

Actually, as it can be seen in the same figure, for *pe* ' 149.7[ ] *Pa* we can also obtain a

second outer radius [108.3 ] <sup>2</sup> <sup>3</sup> *re <sup>m</sup>* , which is not in the desired range

Moreover, the outer free surface of this meniscus is not globally concave; it is a convex-

Taking into account *pm* 0 [Eriss, 1980], [Rossolenko, 2001], [Yang, 2006], the melt column

(introduced in the furnace for releasing the heat from the outer side of the tube wall). When

*<sup>g</sup> <sup>p</sup>* , then ' *He* is negative, [1031.2' ] <sup>3</sup> *He <sup>m</sup>* , i.e. the crucible melt level has to be with

, where 0 *<sup>e</sup>*

 , <sup>2</sup> *gi ge*

*R R*

*<sup>g</sup> p* is the pressure of the gas flow

*ge*

*R* .

Fig. 5. Outer radii *er* versus *<sup>e</sup> p* in the range (134.85; 164.49)[ ] *Pa* .

<sup>3</sup> ' 2.31 10 [ ] *H m <sup>e</sup>* above the shaper top level.

 <sup>1</sup> ' [' ] *<sup>e</sup> He e <sup>g</sup> p p g*

Fig. 6. Non globally concave outer free surface obtained for *pe* ' 149.7[ ] *Pa* .

Inequalities (A.2.1) establish the range in which the pressure difference *<sup>e</sup> p* has to be chosen in order to obtain a static meniscus with concave outer free surface appropriate for the growth of a tube of outer radius equal to *Rge <sup>n</sup>* .

If the pressure difference *<sup>e</sup> <sup>p</sup>* satisfies inequality (A.2.2) then a static meniscus with concave outer free surface is obtained which is appropriate for the growth of a tube of outer radius in the range , *R R ge ge n n* .

**Theorem 13** (Appendix 2) shows that a static meniscus having a concave outer free surface appropriate for the growth of a tube of outer radius , <sup>2</sup> *gi ge e ge R R r R* is stable.

Inequalities (A.2.3) establish the range in which the pressure difference *<sup>i</sup> p* has to be chosen in order to obtain a static meniscus with convex inner free surface appropriate for the growth of a tube of inner radius equal to *m Rgi* .

If the pressure difference *<sup>i</sup> p* satisfies inequality (A.2.4) then a static meniscus with concave inner free surface is obtained which is appropriate for the growth of a tube of inner radius in the range ' , *mR mR gi gi* .

**Theorem 16** (Appendix 2) shows that a static meniscus having a concave inner free surface

appropriate for the growth of a tube of inner radius , <sup>2</sup> *gi ge i gi R R r R* is stable.

Computations were performed for an InSb tube growth: <sup>0</sup> 63.8 *<sup>c</sup>* ; <sup>0</sup> 28.9 *<sup>g</sup>* ; <sup>3</sup> 6582 / *kg m* ; <sup>1</sup> 4.2 10 / *N m* .

If there exists a concave outer free surface, appropriate for the growth of a tube of outer radius 1 3 4.65 10 [ ] *er m* ( *<sup>n</sup>*<sup>1</sup> 1.03226 ), then according to the **Theorem 11** (Appendix 2) this can be obtained for a value of *<sup>e</sup> p* in the range (134.85; 164.49)[ ] *Pa* .

Taking into account the above fact, in order to create a concave outer free surface, appropriate for the growth of a tube of which outer radius is equal to 1 3 4.65 10 [ ] *er m* we have solved the i.v.p. (A.1.3) for different values of *<sup>e</sup> p* in the range (134.85; 164.49)[ ] *Pa* .

More precisely, we have integrated the system (10) for *z Re ge* ( )0 , *z R <sup>e</sup>* '( ) tan *ge c* and different *<sup>e</sup> p* . The obtained outer radii *er* versus *<sup>e</sup> p* are represented in Fig.5, which shows that the desired outer radius ][1065.4 <sup>1</sup> <sup>3</sup> *re <sup>m</sup>* is obtained for *pe* ' 149.7[ ] *Pa* .

Inequalities (A.2.1) establish the range in which the pressure difference *<sup>e</sup> p* has to be chosen in order to obtain a static meniscus with concave outer free surface appropriate for the

outer free surface is obtained which is appropriate for the growth of a tube of outer radius in

**Theorem 13** (Appendix 2) shows that a static meniscus having a concave outer free surface

Inequalities (A.2.3) establish the range in which the pressure difference *<sup>i</sup> p* has to be chosen in order to obtain a static meniscus with convex inner free surface appropriate for the

If the pressure difference *<sup>i</sup> p* satisfies inequality (A.2.4) then a static meniscus with concave inner free surface is obtained which is appropriate for the growth of a tube of inner radius in

**Theorem 16** (Appendix 2) shows that a static meniscus having a concave inner free surface

If there exists a concave outer free surface, appropriate for the growth of a tube of outer radius 1 3 4.65 10 [ ] *er m* ( *<sup>n</sup>*<sup>1</sup> 1.03226 ), then according to the **Theorem 11** (Appendix 2)

Taking into account the above fact, in order to create a concave outer free surface, appropriate for the growth of a tube of which outer radius is equal to 1 3 4.65 10 [ ] *er m* we

different *<sup>e</sup> p* . The obtained outer radii *er* versus *<sup>e</sup> p* are represented in Fig.5, which shows

have solved the i.v.p. (A.1.3) for different values of *<sup>e</sup> p* in the range (134.85; 164.49)[ ] *Pa* .

More precisely, we have integrated the system (10) for *z Re ge* ( )0 , *z R <sup>e</sup>* '( ) tan *ge c*

that the desired outer radius ][1065.4 <sup>1</sup> <sup>3</sup> *re <sup>m</sup>* is obtained for *pe* ' 149.7[ ] *Pa* .

satisfies inequality (A.2.2) then a static meniscus with concave

 , <sup>2</sup> *gi ge e ge R R*

 , <sup>2</sup> *gi ge*

*R R r R* is stable.

> <sup>0</sup> 63.8 *<sup>c</sup>* ;

<sup>0</sup> 28.9 *<sup>g</sup>* ;

and

*i gi*

*r R* is stable.

*Rge <sup>n</sup>* .

growth of a tube of outer radius equal to

*<sup>e</sup> <sup>p</sup>*

appropriate for the growth of a tube of outer radius

growth of a tube of inner radius equal to *m Rgi* .

appropriate for the growth of a tube of inner radius

Computations were performed for an InSb tube growth:

this can be obtained for a value of *<sup>e</sup> p* in the range (134.85; 164.49)[ ] *Pa* .

<sup>1</sup> 4.2 10 / *N m* .

If the pressure difference

 , *R R ge ge n n* .

the range ' , *mR mR gi gi* .

 <sup>3</sup> 6582 / *kg m* ;

the range

Fig. 5. Outer radii *er* versus *<sup>e</sup> p* in the range (134.85; 164.49)[ ] *Pa* .

Actually, as it can be seen in the same figure, for *pe* ' 149.7[ ] *Pa* we can also obtain a second outer radius [108.3 ] <sup>2</sup> <sup>3</sup> *re <sup>m</sup>* , which is not in the desired range , <sup>2</sup> *gi ge ge R R R* . Moreover, the outer free surface of this meniscus is not globally concave; it is a convexconcave meniscus (Fig.6). Taking into account *pm* 0 [Eriss, 1980], [Rossolenko, 2001], [Yang, 2006], the melt column height in this case is <sup>1</sup> ' [' ] *<sup>e</sup> He e <sup>g</sup> p p g* , where 0 *<sup>e</sup> <sup>g</sup> p* is the pressure of the gas flow (introduced in the furnace for releasing the heat from the outer side of the tube wall). When 0 *<sup>e</sup> <sup>g</sup> <sup>p</sup>* , then ' *He* is negative, [1031.2' ] <sup>3</sup> *He <sup>m</sup>* , i.e. the crucible melt level has to be with

<sup>3</sup> ' 2.31 10 [ ] *H m <sup>e</sup>* above the shaper top level.

Fig. 6. Non globally concave outer free surface obtained for *pe* ' 149.7[ ] *Pa* .

A Mathematical Model for Single Crystal

(, , ) *ee e e rhp* (

of the tube, *<sup>e</sup>*

**4. The angles** 

**convex free surface** 

the vertical is the difference

shaper) *z Re ge* ( )0 ;

different values of

With (; , ) *e*

( (; , ) *i i i ci h zr p* )

the function

construction of

*Rge* (( *z Ri gi* ( )0 ,

**technique** 

The angles

The angles

*e c* (*i*

( 

 (; , ) *e e c ce e e r h p* (

Cylindrical Tube Growth by the Edge-Defined Film-Fed Growth (EFG) Technique 105

pressure of the gas flow introduced in the furnace for releasing the heat from the outer wall

**the dynamics of the outer and inner radius of a tube, grown by the E.F.G.** 

outer free surface (inner free surface) of the meniscus at the three phase point, of coordinates , *e e r h* , *i i r h* and the horizontal axis Or. These angles can fluctuate during the growth. The deviation of the tangent to the crystal outer (inner) free surface at the triple point from

(, , ) <sup>2</sup> *ee e e <sup>g</sup> rhp* ,

*<sup>g</sup>* is the growth angle. The deviation can fluctuate also and the outer (inner) radius *er* ( *ir* )

Laplace equation. For this reason for this equation the following strategy is adopted: two conditions are imposed at the outer radius *Rge* of the shaper (inner radius *Rgi* of the

*<sup>c</sup>* ) is a parameter which can fluctuate in a certain range, during the growth. For

*<sup>c</sup>* ) in a given range the solution (; , )

*ii c i ii c i r p z r p* ) is constructed. After that, from (; , )

To the best of our knowledge, there is no algorithm in the literature concerning the

*<sup>g</sup> p* has to be higher than the

(, , ) *ii i i rhp* **for a** 

(, , ) *ii i i rhp* **which appear in system (1) describing** 

(, , ) *ee e e rhp* **and** 

 

(, , ) <sup>2</sup> *ii i i <sup>g</sup> rhp* (Fig. 2), where

*<sup>i</sup> gi c z R* ). In the last condition

*e ce z r p* ( (; , )

 (; , ) *<sup>i</sup>*

*i i ci z r p* ) of the

 ' ( ) tan( )*<sup>e</sup> <sup>e</sup> ge c z R* at

 ' ( ; , ) arctan ( ; , ) *e e ee c e ee c e r p z r p*

*e e e ce h zr p*

*ii c i r p* ) obtaining

(, , ) *ii i i rhp* ) represent the angles between the tangent line to the

(, , ) *ii i i rhp* cannot be obtained directly from the Young-

 ' ( ) tan( )*<sup>i</sup>*

*<sup>c</sup>* ) is expressed as function of *er* , *<sup>e</sup> h* and *<sup>e</sup> p* ( *ir* , *<sup>i</sup> h* and *<sup>i</sup> p* ) .

 (; , ) *<sup>e</sup> ee c e r p* (

(, , ) *ii i i rhp* ) at the level of generality presented here.

*e*

 

 ( , , ) ( ; ( ; , ), ) *<sup>i</sup> ii i i ii ci i i i r h p r rh p p* ).

furnace for releasing the heat from the inner wall of the tube, *<sup>i</sup>*

*<sup>g</sup> p* ; 149.7 16.2 165.9[ ] *i e g g p p Pa* .

**4.1 The procedure for the determination of the angles** 

*i*

*<sup>i</sup> gi c z R* ) at *Rgi* ) is found.

*<sup>e</sup> ge c z R* ( *z Ri gi* ( )0 ,

Young-Laplace equation which satisfies the conditions *z Re ge* ( )0 ;

*i ci z r p* ) the function

*c ci e i r h p* ) is introduced in

is constant when the deviation is constant equal to zero

 ' ( ) tan( )*<sup>e</sup>*

(, , ) *ee e e rhp* and

*e c* (*i*

*e ce z r p* ( (; , )

' ( ; , ) arctan ( ; , ) *i i*

*e c* (*i*

> (; , ) *i i*

 ( , , ) ( ; ( ; , ), ) *<sup>e</sup> ee e e ee c e e e e r h p r rh p p* (

(, , ) *ee e e rhp* (

 ' ( ) tan( )*<sup>i</sup>*

(, , ) *ee e e rhp* **and** 

If there exists a concave inner free surface for the growth of a tube of inner radius

 <sup>3</sup> ' 4.35 10 [ ] *ir m* ( *<sup>m</sup>*<sup>1</sup> 1.03571 ), then according to the **Theorem 14** (Appendix 2), this can be obtained for a value of *<sup>i</sup> p* which is in the range 31.46, 1.07 [ ] *Pa* .

Taking into account the above fact, in order to create a concave inner free surface, appropriate for the growth of a tube whose inner radius is equal to <sup>3</sup> ' 4.35 10 [ ] *ir m* , we have solved the i.v.p. (A.1.8) for different values of *<sup>i</sup> p* in the range 31.46, 1.07 [ ] *Pa* . More precisely, we have integrated the system (11) for *z Ri gi* ( )0 , *z R <sup>i</sup>* '( ) tan *gi c* and different *<sup>i</sup> p* . The obtained inner radii *ir* versus *<sup>i</sup> p* are represented in Fig.7 which shows that the desired inner radius 1 3 4.35 10 [ ] *ir m* is obtained for *pi* ' 16.2[ ] *Pa* .

Fig. 7. Inner radii *ir* versus *<sup>i</sup> p* in the range 31.46, 1.07 [ ] *Pa* .

Taking *pm* 0 [Eriss et al., 1980], [Rossolenko et al., 2001], [Yang et al. 2006], the melt column height in this case is <sup>1</sup> ' [' ]*<sup>i</sup> Hi i <sup>g</sup> p p g* , where 0 *<sup>i</sup> <sup>g</sup> p* is the pressure of the gas flow (introduced in the furnace for releasing the heat from the inner side of the tube wall). When 0 *<sup>i</sup> <sup>g</sup> <sup>p</sup>* , then ' *Hi* is positive, <sup>3</sup> ' 0.25 10 [ ] *H m <sup>i</sup>* , i.e. the crucible melt level has to be with <sup>3</sup> ' 0.25 10 [ ] *H m <sup>i</sup>* under the shaper top level. To create a concave meniscus, appropriate for the growth of a tube with outer radius <sup>3</sup> ' 4.65 10 *er* [m] and inner radius <sup>3</sup> ' 4.35 10 *ir* [m] the melt column heights (with respect to the crucible melt level) have to be <sup>1</sup> <sup>1</sup> 16.2 *<sup>i</sup> Hi <sup>g</sup> <sup>p</sup> g* and <sup>1</sup> <sup>1</sup> 149.7 *<sup>e</sup> He <sup>g</sup> <sup>p</sup> g* . When the shaper's outer top is at the same level as the shaper's inner top, with respect to the crucible melt level, then the relation 1 1 *H H e i* holds . It follows that the pressure of the gas flow, introduced in the

<sup>3</sup> ' 4.35 10 [ ] *ir m* ( *<sup>m</sup>*<sup>1</sup> 1.03571 ), then according to the **Theorem 14** (Appendix 2), this can

Taking into account the above fact, in order to create a concave inner free surface, appropriate for the growth of a tube whose inner radius is equal to <sup>3</sup> ' 4.35 10 [ ] *ir m* , we have solved the i.v.p. (A.1.8) for different values of *<sup>i</sup> p* in the range 31.46, 1.07 [ ] *Pa* .

different *<sup>i</sup> p* . The obtained inner radii *ir* versus *<sup>i</sup> p* are represented in Fig.7 which shows

Taking *pm* 0 [Eriss et al., 1980], [Rossolenko et al., 2001], [Yang et al. 2006], the melt

flow (introduced in the furnace for releasing the heat from the inner side of the tube wall).

with <sup>3</sup> ' 0.25 10 [ ] *H m <sup>i</sup>* under the shaper top level. To create a concave meniscus, appropriate for the growth of a tube with outer radius <sup>3</sup> ' 4.65 10 *er* [m] and inner radius <sup>3</sup> ' 4.35 10 *ir* [m] the melt column heights (with respect to the crucible melt level) have to

*<sup>g</sup> <sup>p</sup>* , then ' *Hi* is positive, <sup>3</sup> ' 0.25 10 [ ] *H m <sup>i</sup>* , i.e. the crucible melt level has to be

 <sup>1</sup> <sup>1</sup> 149.7 *<sup>e</sup> He <sup>g</sup> <sup>p</sup> g*

at the same level as the shaper's inner top, with respect to the crucible melt level, then the relation 1 1 *H H e i* holds . It follows that the pressure of the gas flow, introduced in the

, where 0 *<sup>i</sup>*

*<sup>g</sup> p* is the pressure of the gas

. When the shaper's outer top is

and

More precisely, we have integrated the system (11) for *z Ri gi* ( )0 , *z R <sup>i</sup>* '( ) tan *gi c*

that the desired inner radius 1 3 4.35 10 [ ] *ir m* is obtained for *pi* ' 16.2[ ] *Pa* .

Fig. 7. Inner radii *ir* versus *<sup>i</sup> p* in the range 31.46, 1.07 [ ] *Pa* .

and

 <sup>1</sup> ' [' ]*<sup>i</sup> Hi i <sup>g</sup> p p g*

column height in this case is

 

<sup>1</sup> <sup>1</sup> 16.2 *<sup>i</sup> Hi <sup>g</sup> <sup>p</sup> g*

When 0 *<sup>i</sup>*

be

If there exists a concave inner free surface for the growth of a tube of inner radius

be obtained for a value of *<sup>i</sup> p* which is in the range 31.46, 1.07 [ ] *Pa* .

furnace for releasing the heat from the inner wall of the tube, *<sup>i</sup> <sup>g</sup> p* has to be higher than the pressure of the gas flow introduced in the furnace for releasing the heat from the outer wall of the tube, *<sup>e</sup> <sup>g</sup> p* ; 149.7 16.2 165.9[ ] *i e g g p p Pa* .

#### **4. The angles**  (, , ) *ee e e rhp* **and**  (, , ) *ii i i rhp* **which appear in system (1) describing the dynamics of the outer and inner radius of a tube, grown by the E.F.G. technique**

#### **4.1 The procedure for the determination of the angles**  (, , ) *ee e e rhp* **and**  (, , ) *ii i i rhp* **for a convex free surface**

The angles (, , ) *ee e e rhp* ( (, , ) *ii i i rhp* ) represent the angles between the tangent line to the outer free surface (inner free surface) of the meniscus at the three phase point, of coordinates , *e e r h* , *i i r h* and the horizontal axis Or. These angles can fluctuate during the growth. The deviation of the tangent to the crystal outer (inner) free surface at the triple point from

the vertical is the difference (, , ) <sup>2</sup> *ee e e <sup>g</sup> rhp* , (, , ) <sup>2</sup> *ii i i <sup>g</sup> rhp* (Fig. 2), where

 *<sup>g</sup>* is the growth angle. The deviation can fluctuate also and the outer (inner) radius *er* ( *ir* ) is constant when the deviation is constant equal to zero

The angles (, , ) *ee e e rhp* and (, , ) *ii i i rhp* cannot be obtained directly from the Young-Laplace equation. For this reason for this equation the following strategy is adopted: two conditions are imposed at the outer radius *Rge* of the shaper (inner radius *Rgi* of the shaper) *z Re ge* ( )0 ; ' ( ) tan( )*<sup>e</sup> <sup>e</sup> ge c z R* ( *z Ri gi* ( )0 , ' ( ) tan( )*<sup>i</sup> <sup>i</sup> gi c z R* ). In the last condition *e c* (*i <sup>c</sup>* ) is a parameter which can fluctuate in a certain range, during the growth. For different values of *e c* (*i <sup>c</sup>* ) in a given range the solution (; , ) *e e ce z r p* ( (; , ) *i i ci z r p* ) of the Young-Laplace equation which satisfies the conditions *z Re ge* ( )0 ; ' ( ) tan( )*<sup>e</sup> <sup>e</sup> ge c z R* at *Rge* (( *z Ri gi* ( )0 , ' ( ) tan( )*<sup>i</sup> <sup>i</sup> gi c z R* ) at *Rgi* ) is found.

With (; , ) *e e ce z r p* ( (; , ) *i i ci z r p* ) the function ' ( ; , ) arctan ( ; , ) *e e ee c e ee c e r p z r p* ( ' ( ; , ) arctan ( ; , ) *i i ii c i ii c i r p z r p* ) is constructed. After that, from (; , ) *e e e ce h zr p* ( (; , ) *i i i ci h zr p* ) *e c* (*i <sup>c</sup>* ) is expressed as function of *er* , *<sup>e</sup> h* and *<sup>e</sup> p* ( *ir* , *<sup>i</sup> h* and *<sup>i</sup> p* ) .

 (; , ) *e e c ce e e r h p* ( (; , ) *i i c ci e i r h p* ) is introduced in (; , ) *<sup>e</sup> ee c e r p* ( (; , ) *<sup>i</sup> ii c i r p* ) obtaining the function

$$
\overline{\alpha}\_{\varepsilon}(\mathbf{r}\_{\varepsilon}; h\_{\varepsilon}, p\_{\varepsilon}) = \alpha\_{\varepsilon}(\mathbf{r}\_{\varepsilon}; \alpha\_{\varepsilon}^{\varepsilon}(\mathbf{r}\_{\varepsilon}; h\_{\varepsilon}, p\_{\varepsilon}), p\_{\varepsilon}) \qquad \left(\overline{\alpha}\_{i}(\mathbf{r}\_{i}; h\_{i}, p\_{i}) = \alpha\_{i}(\mathbf{r}\_{i}; \alpha\_{\varepsilon}^{i}(\mathbf{r}\_{i}; h\_{i}, p\_{i}), p\_{i})\right).
$$

To the best of our knowledge, there is no algorithm in the literature concerning the construction of (, , ) *ee e e rhp* ((, , ) *ii i i rhp* ) at the level of generality presented here.

A Mathematical Model for Single Crystal

**Step 10.** Fitting the data *<sup>e</sup>*

 *i e* 0 0 *c c* and

0

**Step 3.** For *<sup>i</sup> <sup>p</sup>* the range , *<sup>i</sup> <sup>i</sup>*

*i c* a range

2

**Step 2.** For

is considered.

**Step 4.** In the range

**Step 5.** In the range , *<sup>i</sup> <sup>i</sup>*

**Step 6.** In a given range

**Step 7.** For a given *<sup>k</sup>*

of *j* values of

 (; , ) *iq <sup>k</sup> i i ci r p* .

*E m*

every 

 

**Step 1.** For

**Step 9.** The values ( ; ,)

For the same reason as in the case of

following numerical procedure was conceived:

 0 0 1 2 , , *i iii Em Em c c* where:

> 

Cylindrical Tube Growth by the Edge-Defined Film-Fed Growth (EFG) Technique 107

*<sup>e</sup>* , the function

(, , ) *ee e e rhp* for the construction of

 

1 2 , , *i ii i Em Em c c* holds.

 

2 2

*iq k i i ci z zr p* and

*c i <sup>i</sup> <sup>g</sup>* , a set

*<sup>c</sup>* are chosen.

*<sup>c</sup>* , *q* 1,*l* the solution of the system (11) which

*<sup>i</sup> Rgi c* is found numerically obtaining the

, *<sup>i</sup> <sup>i</sup> <sup>c</sup> <sup>c</sup>* is determined such that 0

 

*p Em p Em* (15)

*i*

*i*

 

1 2 *<sup>j</sup> <sup>i</sup> <sup>i</sup> i i <sup>i</sup>* .

*iq*

*i i i i*

 *ee e e r h*, , *p* is found.

an *m m* 1, is determined such that

 

 

2 *<sup>i</sup> <sup>i</sup> <sup>i</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup> <sup>g</sup>* and for

(14)

(, , ) *i ii i i rhp* the

*eek eq*

*<sup>k</sup> <sup>q</sup> <sup>s</sup> h <sup>k</sup>*

*m*

 

sin

1 1

0

*i*

<sup>2</sup> (, ) cos cos 1

0 0 0

*i c g i i i*

*p p* defined by:

, *<sup>i</sup> <sup>i</sup> <sup>c</sup> <sup>c</sup>* a set of *l* different values of

, *<sup>i</sup> <sup>i</sup>* , possessing the property

> *iq*

 1 2 , , sup , inf , *<sup>i</sup> <sup>i</sup> <sup>i</sup> <sup>i</sup> <sup>i</sup> <sup>i</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup>*

*c c <sup>i</sup> <sup>i</sup>*

*p p* a set of *n* different values of *<sup>i</sup> p* are chosen.

( )

functions (profiles curves Ref. [Tatarchenko, 1993]): (; , )

*c c g*

*mR R*

0

*E m gR m m R*

*gi i c*

*gi*

 

, *<sup>i</sup> <sup>i</sup> <sup>i</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup>* , the inequality

 

*c g i i*

*m R*

> 

 

 

> 

*i* are chosen:

*<sup>i</sup> p* , *k n* 1, and

satisfies the conditions: *z Ri gi* ( ) 0,

 

  *<sup>k</sup> <sup>q</sup> <sup>s</sup> r <sup>e</sup>*

*<sup>k</sup> <sup>q</sup> s ek <sup>q</sup> sc e h zr p* are found.

*<sup>e</sup> p* and

 1, <sup>2</sup>

*ge gi gi*

*R*

 

<sup>2</sup> , sin 1 tan <sup>2</sup> <sup>1</sup>

*c g gi g*

*gi gi*

*R R*

*s*

Due to the nonlinearity, the above described procedure can't be realized analytically. This is the reason why for the construction of the function (; , ) *ee c e r p* in [Balint&Tanasie, 2010] the following numerical procedure was conceived:

**Step 1.** For a given *e*0 *<sup>c</sup>* ; <sup>0</sup> 0, <sup>2</sup> *<sup>e</sup> <sup>c</sup> <sup>g</sup>* and 2 *ge ge gi R n R R* an *n n* 1, is found such that 0 0 1 2 , , *e e ee En En c c* where:

$$\begin{split} E\_1^{\varepsilon}(n', \alpha\_c^{\varepsilon 0}) &= -\gamma \cdot \frac{\pi \Big\langle \frac{\pi}{2} \Big( \alpha\_c^{\varepsilon 0} + \alpha\_{\mathcal{g}} \Big)}{R\_{\mathcal{g}^{\varepsilon}}} \cdot \frac{n'}{n'-1} \cdot \sin \alpha\_{\mathcal{g}} + \rho\_1 \cdot \mathcal{g} \cdot R\_{\mathcal{g}^{\varepsilon}} \cdot \frac{n'-1}{n'} \cdot \tan \left( \frac{\pi}{2} \Big/ -\alpha\_{\mathcal{g}} \right) + \\ &+ \frac{\gamma}{R\_{\mathcal{g}^{\varepsilon}}} \cdot n' \cdot \cos \alpha\_{\mathcal{g}} \\ E\_2^{\varepsilon}(n, \alpha\_c^{\varepsilon 0}) &= -\gamma \cdot \frac{\pi \Big/ - \left( \alpha\_c^{\varepsilon 0} + \alpha\_{\mathcal{g}} \right)}{R\_{\mathcal{g}^{\varepsilon}}} \cdot \frac{n}{n-1} \cdot \cos \alpha\_c^{\varepsilon 0} + \frac{\gamma}{R\_{\mathcal{g}^{\varepsilon}}} \cdot \sin \alpha\_c^{\varepsilon 0} \end{split} \tag{12}$$

**Step 2.** For *e c* a range , *<sup>e</sup> <sup>e</sup> <sup>c</sup> <sup>c</sup>* is determined such that 0 0 2 *<sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup> <sup>g</sup>* and for every , *<sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup>* the inequality 1 2 , , *e ee e En En c c* holds. **Step 3.** For *<sup>e</sup> <sup>p</sup>* the range , *<sup>e</sup> <sup>e</sup> p p* defined by:

$$\underline{p}\_{\epsilon} = \sup\_{a\_{\epsilon}^{\epsilon} \in \left[\underline{a}\_{\epsilon}^{\epsilon}, \overline{a}\_{\epsilon}^{\epsilon}\right]} E\_1^{\epsilon} \left(n', \alpha\_{\epsilon}^{\epsilon}\right) \qquad \qquad \overline{p}\_{\epsilon} = \inf\_{a\_{\epsilon}^{\epsilon} \in \left[\underline{a}\_{\epsilon}^{\epsilon}, \overline{a}\_{\epsilon}^{\epsilon}\right]} E\_2^{\epsilon} \left(n, \alpha\_{\epsilon}^{\epsilon}\right) \tag{13}$$

is considered.


**Step 9.** The values ( ; ,) *eek eq <sup>k</sup> <sup>q</sup> s ek <sup>q</sup> sc e h zr p* are found.

**Step 10.** Fitting the data *<sup>e</sup> <sup>k</sup> <sup>q</sup> <sup>s</sup> r <sup>e</sup> <sup>k</sup> <sup>q</sup> <sup>s</sup> h <sup>k</sup> <sup>e</sup> p* and *s <sup>e</sup>* , the function *ee e e r h*, , *p* is found.

For the same reason as in the case of (, , ) *ee e e rhp* for the construction of (, , ) *i ii i i rhp* the following numerical procedure was conceived:

**Step 1.** For *i e* 0 0 *c c* and 1, <sup>2</sup> *ge gi gi R R m R* an *m m* 1, is determined such that 0 0 1 2 , , *i iii Em Em c c* where: 0 0 1 1 0 0 0 0 2 <sup>2</sup> , sin 1 tan <sup>2</sup> <sup>1</sup> sin <sup>2</sup> (, ) cos cos 1 *i c g i i c g gi g gi i c gi i c g i i i c c g gi gi E m gR m m R m R E m mR R* (14)

**Step 2.** For *i c* a range , *<sup>i</sup> <sup>i</sup> <sup>c</sup> <sup>c</sup>* is determined such that 0 2 *<sup>i</sup> <sup>i</sup> <sup>i</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup> <sup>g</sup>* and for every , *<sup>i</sup> <sup>i</sup> <sup>i</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup>* , the inequality 1 2 , , *i ii i Em Em c c* holds. **Step 3.** For *<sup>i</sup> <sup>p</sup>* the range , *<sup>i</sup> <sup>i</sup> p p* defined by:

$$\underline{p}\_{i} = \sup\_{\alpha\_{c}^{i} \in \left[\underline{\underline{a}}\_{c}^{i}, \overline{\underline{a}}\_{c}^{i}\right]} E\_{1}^{i}(m', \alpha\_{c}^{i}) \qquad \qquad \overline{p}\_{i} = \inf\_{\alpha\_{c}^{i} \in \left[\underline{\underline{a}}\_{c}^{i}, \overline{\underline{a}}\_{c}^{i}\right]} E\_{2}^{i}(m, \alpha\_{c}^{i}) \tag{15}$$

is considered.

106 Crystallization – Science and Technology

Due to the nonlinearity, the above described procedure can't be realized analytically. This is

<sup>2</sup> <sup>1</sup> , sin tan <sup>2</sup> <sup>1</sup>

 

*c g ge g*

, *<sup>e</sup> <sup>e</sup> <sup>c</sup> <sup>c</sup>* is determined such that 0

 

*p En p En* (13)

*e*

 

*e e e e*

 

1 2 , , *e ee e En En c c* holds.

*<sup>c</sup>* is chosen.

*<sup>c</sup>* , *q* 1,*l* the solution of the system (10)

*e*

*eq <sup>e</sup> ge e ge c zR R* is determined numerically

 ( ; ,) , *e ks eq <sup>e</sup> e kqs c e e <sup>e</sup> r p* , *k m* 1, , *<sup>q</sup>* 1,*l* are

 0 

*<sup>c</sup> <sup>g</sup>* and

 

*n*

2 *ge ge gi R*

*R R*

(; , ) *ee c e r p* in [Balint&Tanasie, 2010]

an *n n* 1, is found such

 

2 *<sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup> <sup>g</sup>* and for

 

*c e <sup>e</sup> <sup>g</sup>* a

2 2

*eq k*

*e e ce z zr p*

(12)

the reason why for the construction of the function

 0 0 1 2 , , *e e ee En En c c* where:

*c g e e*

*<sup>c</sup>* ;

 

0

*e*

1 1

*ge*

 

cos

0

, *<sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup>* the inequality

*e*

*p p* defined by:

, *<sup>e</sup> <sup>e</sup> <sup>c</sup> <sup>c</sup>* a set of *l* different values of

 1 2 , , sup , inf , *<sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup>*

*c c <sup>e</sup> <sup>e</sup>*

*p p* a set of *m* different values of *<sup>e</sup> p* is chosen.

 

obtaining the functions (profiles curves Refs [Tatarchenko, 1993]): (; , )

1 2 *<sup>j</sup> <sup>e</sup> <sup>e</sup> ee e* .

> 

, *<sup>e</sup> <sup>e</sup>* possessing the property

*eq*

*n n E n g R R n <sup>n</sup>*

<sup>2</sup> (, ) cos sin

*c g e e e e c c c*

0 0 0

1

*Rn R*

*ge ge*

 

*g*

 

<sup>0</sup> 0, <sup>2</sup> *<sup>e</sup>*

the following numerical procedure was conceived:

*e*0

 

*R*

 

> 

 

> 

*<sup>k</sup> <sup>q</sup> <sup>s</sup> r* for which

*<sup>e</sup>* is chosen:

*<sup>e</sup> p* , *k m* 1, and

corresponding to the conditions: ( ) 0, ( )

*n*

 

*<sup>n</sup> E n*

*ge*

 

**Step 1.** For a given

that

0

*e c* a range

**Step 3.** For *<sup>e</sup> <sup>p</sup>* the range , *<sup>e</sup> <sup>e</sup>*

every 

2

**Step 2.** For

is considered.

**Step 4.** In the range

**Step 5.** In the range , *<sup>e</sup> <sup>e</sup>*

**Step 6.** In a given range

**Step 7.** For a given *<sup>k</sup>*

and

**Step 8.** The values *<sup>e</sup>*

set of *j* values of

determined.

 (; , ) *eq <sup>k</sup> e e ce r p* . **Step 4.** In the range , *<sup>i</sup> <sup>i</sup> <sup>c</sup> <sup>c</sup>* a set of *l* different values of *i <sup>c</sup>* are chosen.

**Step 5.** In the range , *<sup>i</sup> <sup>i</sup> p p* a set of *n* different values of *<sup>i</sup> p* are chosen.


A Mathematical Model for Single Crystal

*e*0

 

0

*e*

0

*e*

 

*<sup>n</sup> E n*

 

**Step 3.** For *<sup>e</sup> <sup>p</sup>* the range , *<sup>e</sup> <sup>e</sup>*

 *i e* 0 0

*i*

2

**Step 2.** For

*E m*

  **concave free surface** 

surface (when

**Step 1.** For a given

that

*e c* a range

2

**Step 2.** For

is considered.

**Step 1.** For

**4.2 The procedure for the determination of the angles** 

The numerical procedure for the construction of the function

*<sup>c</sup>* ;

 <sup>0</sup> 

 0 0 1 2 , , *ee e e En En c c* where:

1 1

*c g e e e*

 

 

> 

For the inner free surface we have to make:

 

0

*i*

0

*i*

 

 

 

*c* a range

0 0

*c c* and

 0 0 1 2 , ', *iii i Em Em c c* where:

1 1

( 1)

 

*m*

2 (, ) cos cos

 

1

1

*Rn R*

*p p* defined by:

, , sup , inf , *<sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup>*

1, <sup>2</sup>

0 0 0

*mR R*

the inequality

, , sup , inf ', *<sup>i</sup> <sup>i</sup> <sup>i</sup> <sup>i</sup> <sup>i</sup> <sup>i</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup>*

 

*c g i i i*

*gi gi*

, *<sup>i</sup> <sup>i</sup> <sup>c</sup> <sup>c</sup>* is determined such that

 

1 2

2 ( ', ) cos sin ' 1

*c g i i i i c c c*

*c c g ge ge*

the inequality

, 2 2 *<sup>e</sup>*

Cylindrical Tube Growth by the Edge-Defined Film-Fed Growth (EFG) Technique 109

**Step 3**. present the some differences. For the outer free surface we have to consider:

 

*c g* and

 

*n n E n g R <sup>n</sup>*

, *<sup>e</sup> <sup>e</sup> <sup>c</sup> <sup>c</sup>* is determined such that

 

 1 2 , , sup , inf , *<sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup>*

> *ge gi gi*

*R*

0 0

*E m gR m m R m R*

*c g gi c g*

*ii i i E m c c E m* holds.

2 , sin 1 tan cos

 

*ge gi*

*R R*

*c c <sup>e</sup> <sup>e</sup>*

1 2

0 0 0

*R n n R*

<sup>1</sup> <sup>2</sup> , sin tan sin

*c g e e e e c g gec c ge ge*

*e e e e E n c c E n* holds.

 

*p En p En* (17)

*e e e e*

2 *ge ge gi R*

*R R*

 

<sup>2</sup> *<sup>c</sup> <sup>g</sup>* ) is similar to those applied for a convex free surface. Only **Step 1-** 

*n*

(, , ) *ee e e r h p* **and** 

(, , ) *ee e e r h p* for a concave free

an *n n* 1, is found such

 0

 

an *m m* 1, is determined such that

 

2 2

 0

*<sup>i</sup> <sup>i</sup> <sup>i</sup> <sup>c</sup> g c <sup>c</sup>* and

(18)

2 2 *e e e g c <sup>c</sup> <sup>c</sup>* and

(, , ) *iii rhp* **for a** 

(16)


**Step 10.** The function *ii i i r h*, , *p* is found by fitting the data *<sup>i</sup> <sup>k</sup> <sup>q</sup> <sup>s</sup> r <sup>i</sup> <sup>k</sup> <sup>q</sup> <sup>s</sup> h <sup>k</sup> <sup>i</sup> p* and *s i* .

For the case of a silicon tube and the outer free surface the function:

$$\overline{\alpha}\_{\varepsilon}(r\_{\varepsilon}, h\_{\varepsilon}, p\_{\varepsilon}) = \frac{a\_1(p\_{\varepsilon}) + a\_2\left(p\_{\varepsilon}\right) \cdot r\_{\varepsilon} + a\_3\left(p\_{\varepsilon}\right) \cdot r\_{\varepsilon}^2 + a\_4\left(p\_{\varepsilon}\right) \cdot h\_{\varepsilon} + a\_5\left(p\_{\varepsilon}\right) \cdot h\_{\varepsilon}^2 + a\_6\left(p\_{\varepsilon}\right) \cdot h\_{\varepsilon}^3}{1 + a\_7\left(p\_{\varepsilon}\right) \cdot r\_{\varepsilon} + a\_8\left(p\_{\varepsilon}\right) \cdot h\_{\varepsilon} + a\_9\left(p\_{\varepsilon}\right) \cdot h\_{\varepsilon}^2 + a\_{10}\left(p\_{\varepsilon}\right) \cdot h\_{\varepsilon}^3}$$

with: 1 e ( ) 0.400314546-0.00494101 p *<sup>e</sup> a p* 2 e ( ) -374.789455 2.353132518 p *<sup>e</sup> a p* 3 e ( ) 56161.27414-244.013259 p *<sup>e</sup> a p* 4 e ( ) 393.2446055 0.010192507 p *<sup>e</sup> a p* 5 e ( ) 91143.39025-555.405995 p *<sup>e</sup> a p* 6 e ( ) -1097500000-1168100 p *<sup>e</sup> a p* 7 e ( ) -200.489376 0.001998751 p *<sup>e</sup> a p* 8 e ( ) 114.0391478 0.002922594 p *<sup>e</sup> a p* 9 e ( ) 83485.77784-477.770821 p *<sup>e</sup> a p* <sup>10</sup> <sup>e</sup> ( ) 415910000 476372.6048 p *<sup>e</sup> a p*

was obtained and for the inner free surface the function:

$$\varpi\_i(r\_i; h\_i, p\_i) = \frac{b\_1(p\_i) + b\_2\left(p\_i\right) \cdot r\_i + b\_3\left(p\_i\right) \cdot r\_i^2 + b\_4\left(p\_i\right) \cdot h\_i + b\_5\left(p\_i\right) \cdot h\_i^2 + b\_6\left(p\_i\right) \cdot h\_i^3}{1 + b\_7\left(p\_i\right) \cdot r\_i + b\_8\left(p\_i\right) \cdot h\_i + b\_9\left(p\_i\right) \cdot h\_i^2 + b\_{10}\left(p\_i\right) \cdot h\_i^3}$$

with 1 i ( ) 0.102617985 0.004931999 p *<sup>i</sup> b p* 2 i ( ) -172.7251-2.34898717 p *<sup>i</sup> b p* 3 i ( ) 29313.52855 277.5275727 p *<sup>i</sup> b p* 4 i ( ) 444.327355 0.367481455 p *<sup>i</sup> b p* 5 i ( ) -2687700-817.408936 p *<sup>i</sup> b p* 6 i ( ) 5760710000 3810000 p *<sup>i</sup> b p* 7 i ( ) -263.39243-0.00872161 p *<sup>i</sup> b p* 8 i ( ) 707.770183 0.334375383 p *<sup>i</sup> b p* 9 i ( ) -2709500-696.727273 p *<sup>i</sup> b p* <sup>10</sup> <sup>i</sup> ( ) 3090180000 1348660 p *<sup>i</sup> b p*

$$\begin{aligned} b\_2(p\_i) &= -172.7251 \text{-} 2.34898717 \cdot \mathbf{p\_i} \\ b\_4(p\_i) &= 444.327355 + 0.367481455 \cdot \mathbf{p\_i} \\ b\_6(p\_i) &= 5760710000 + 3810000 \cdot \mathbf{p\_i} \\ b\_8(p\_i) &= 707.770183 + 0.334375383 \cdot \mathbf{p\_i} \\ b\_{10}(p\_i) &= 3090180000 + 1348660 \cdot \mathbf{p\_i} \end{aligned}$$

was obtained. For *pe* 2000*Pa* and *pi* 2242 *Pa* the functions (, , ) *ee e e rhp* and (, , ) *ii i i rhp* are represented in Fig. 8.:

Fig. 8. The graphics of (, , ) *ee e e r h p* ( 2000 [ ] *<sup>e</sup> p Pa* ) and (, , ) *iii rhp* ( 2242 [ ] *<sup>i</sup> p Pa* )

#### **4.2 The procedure for the determination of the angles**  (, , ) *ee e e r h p* **and** (, , ) *iii rhp* **for a concave free surface**

The numerical procedure for the construction of the function (, , ) *ee e e r h p* for a concave free surface (when <sup>2</sup> *<sup>c</sup> <sup>g</sup>* ) is similar to those applied for a convex free surface. Only **Step 1- Step 3**. present the some differences. For the outer free surface we have to consider:

$$\begin{aligned} \textbf{Step 1.} \quad \text{For a given } \boldsymbol{a}\_{\boldsymbol{c}}^{\boldsymbol{c}0} \text{; } \boldsymbol{a}\_{\boldsymbol{c}}^{\boldsymbol{c}0} \text{; } \boldsymbol{a}\_{\boldsymbol{c}}^{\boldsymbol{c}0} \in \left( \pi\_{\mathcal{J}}^{\boldsymbol{\ell}} - \boldsymbol{a}\_{\boldsymbol{g}^{\boldsymbol{\ell}}} \pi\_{\mathcal{J}}^{\boldsymbol{\ell}} \right) \text{ and } \boldsymbol{n} = \frac{2 \cdot \mathsf{R}\_{\mathcal{g}^{\boldsymbol{\ell}}}}{\mathsf{R}\_{\mathcal{g}^{\boldsymbol{\ell}}} + \mathsf{R}\_{\mathcal{g}^{\boldsymbol{\ell}}}} \text{ and } \boldsymbol{n}' \in \{1, n\} \text{ is found such that} \\ \textbf{At } \boldsymbol{E}\_{1}^{\boldsymbol{\ell}}(\boldsymbol{n}, \boldsymbol{a}\_{\boldsymbol{c}}^{\boldsymbol{\ell}0}) \in \boldsymbol{E}\_{2}^{\boldsymbol{\ell}}(\boldsymbol{n}', \boldsymbol{a}\_{\boldsymbol{c}}^{\boldsymbol{\ell}0}) \text{ where:} \end{aligned}$$

$$\begin{split} E\_1^{\varepsilon}(n, \alpha\_c^{\varepsilon 0}) &= \gamma \cdot \frac{\alpha\_c^{\varepsilon 0} + \alpha\_g - \frac{\pi}{\gamma} \mathcal{Q}}{R\_{g\varepsilon}} \cdot \frac{n}{n-1} \cdot \sin \alpha\_g + \rho\_1 \cdot g \cdot R\_{g\varepsilon} \cdot \frac{n-1}{n} \cdot \tan \alpha\_c^{\varepsilon 0} + \frac{\mathcal{Y}}{R\_{g\varepsilon}} \cdot n \cdot \sin \alpha\_c^{\varepsilon 0} \\ E\_2^{\varepsilon}(n', \alpha\_c^{\varepsilon 0}) &= \gamma \cdot \frac{\alpha\_c^{\varepsilon 0} + \alpha\_g - \frac{\pi}{\gamma} \mathcal{Q}}{R\_{g\varepsilon}} \cdot \frac{n'}{n'-1} \cdot \cos \alpha\_c^{\varepsilon 0} + \frac{\mathcal{Y}}{R\_{g\varepsilon}} \cdot \cos \alpha\_g \end{split} \tag{16}$$

**Step 2.** For *e c* a range , *<sup>e</sup> <sup>e</sup> <sup>c</sup> <sup>c</sup>* is determined such that 0 2 2 *e e e g c <sup>c</sup> <sup>c</sup>* and the inequality 1 2 , , sup , inf , *<sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup> e e e e E n c c E n* holds.

**Step 3.** For *<sup>e</sup> <sup>p</sup>* the range , *<sup>e</sup> <sup>e</sup> p p* defined by:

$$\underline{p}\_{\epsilon} = \sup\_{\alpha\_{\epsilon}^{\epsilon} \in \left[\underline{a}\_{\epsilon}^{\epsilon'}, \overline{a}\_{\epsilon}^{\epsilon'}\right]} E\_1^{\epsilon} \left(n, \alpha\_{\epsilon}^{\epsilon}\right) \qquad \qquad \overline{p}\_{\epsilon} = \inf\_{\alpha\_{\epsilon}^{\epsilon} \in \left[\underline{a}\_{\epsilon}^{\epsilon'}, \overline{a}\_{\epsilon}^{\epsilon'}\right]} E\_2^{\epsilon} \left(n', \alpha\_{\epsilon}^{\epsilon}\right) \tag{17}$$

is considered.

108 Crystallization – Science and Technology

 

*ii i i i i i*

1 *<sup>i</sup> i ii ii i i i i i i*

78 9 10

12 3 4 5 6

*e ee ee e e e e e e*

*ee e e e e e e*

78 9 10

12 3 4 5 6

 ( ; ,) , *i ks iq <sup>e</sup> i kqs c i i <sup>e</sup> r p* , *k n* 1, , *<sup>q</sup>* 1,*l* are

*<sup>k</sup> <sup>q</sup> <sup>s</sup> r <sup>i</sup>*

2 2 3

2 2 3

2 3

(, , ) *iii rhp* ( 2242 [ ] *<sup>i</sup> p Pa* )

(, , ) *ee e e rhp* and

*<sup>k</sup> <sup>q</sup> <sup>s</sup> h <sup>k</sup>*

2 3

*<sup>i</sup> p* and

*s i* .

*ii i i r h*, , *p* is found by fitting the data *<sup>i</sup>*

with: 1 e ( ) 0.400314546-0.00494101 p *<sup>e</sup> a p* 2 e ( ) -374.789455 2.353132518 p *<sup>e</sup> a p* 3 e ( ) 56161.27414-244.013259 p *<sup>e</sup> a p* 4 e ( ) 393.2446055 0.010192507 p *<sup>e</sup> a p* 5 e ( ) 91143.39025-555.405995 p *<sup>e</sup> a p* 6 e ( ) -1097500000-1168100 p *<sup>e</sup> a p* 7 e ( ) -200.489376 0.001998751 p *<sup>e</sup> a p* 8 e ( ) 114.0391478 0.002922594 p *<sup>e</sup> a p* 9 e ( ) 83485.77784-477.770821 p *<sup>e</sup> a p* <sup>10</sup> <sup>e</sup> ( ) 415910000 476372.6048 p *<sup>e</sup> a p*

with 1 i ( ) 0.102617985 0.004931999 p *<sup>i</sup> b p* 2 i ( ) -172.7251-2.34898717 p *<sup>i</sup> b p* 3 i ( ) 29313.52855 277.5275727 p *<sup>i</sup> b p* 4 i ( ) 444.327355 0.367481455 p *<sup>i</sup> b p* 5 i ( ) -2687700-817.408936 p *<sup>i</sup> b p* 6 i ( ) 5760710000 3810000 p *<sup>i</sup> b p* 7 i ( ) -263.39243-0.00872161 p *<sup>i</sup> b p* 8 i ( ) 707.770183 0.334375383 p *<sup>i</sup> b p* 9 i ( ) -2709500-696.727273 p *<sup>i</sup> b p* <sup>10</sup> <sup>i</sup> ( ) 3090180000 1348660 p *<sup>i</sup> b p*

was obtained. For *pe* 2000*Pa* and *pi* 2242 *Pa* the functions

(, , ) *ee e e r h p* ( 2000 [ ] *<sup>e</sup> p Pa* ) and

*bp b p r b p r b p h b p h b p h rhp bp r bp h bp h b p h*

*<sup>a</sup> <sup>p</sup> <sup>a</sup> <sup>p</sup> r a <sup>p</sup> r a <sup>p</sup> h a <sup>p</sup> h a <sup>p</sup> <sup>h</sup> rhp ap r ap h ap h a p h*

*<sup>k</sup> <sup>q</sup> <sup>s</sup> r* for which

*iik iq*

For the case of a silicon tube and the outer free surface the function:

*<sup>k</sup> <sup>q</sup> s ik <sup>q</sup> sc i h zr p* are found.

**Step 8.** The values *<sup>i</sup>*

**Step 10.** The function

Fig. 8. The graphics of

; ,

(, , ) *ii i i rhp* are represented in Fig. 8.:

*ii i i*

determined.

*ee e e*

**Step 9.** The values ( ; ,)

( ) , , <sup>1</sup>

was obtained and for the inner free surface the function:

( )

For the inner free surface we have to make:

**Step 1.** For *i e* 0 0 *c c* and 1, <sup>2</sup> *ge gi gi R R m R* an *m m* 1, is determined such that 0 0 1 2 , ', *iii i Em Em c c* where: 0 0 0 1 1 0 0 0 0 2 2 , sin 1 tan cos ( 1) 2 ( ', ) cos sin ' 1 *i c g i i i c g gi c g ge gi i c g i i i i c c c gi gi E m gR m m R m R E m mR R* (18) **Step 2.** For *i c* a range , *<sup>i</sup> <sup>i</sup> <sup>c</sup> <sup>c</sup>* is determined such that 0 2 2 *<sup>i</sup> <sup>i</sup> <sup>i</sup> <sup>c</sup> g c <sup>c</sup>* and the inequality 1 2 , , sup , inf ', *<sup>i</sup> <sup>i</sup> <sup>i</sup> <sup>i</sup> <sup>i</sup> <sup>i</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup> ii i i E m c c E m* holds.

A Mathematical Model for Single Crystal

crystallization front displacement rate

[Tanasie&Balint, 2010].

*<sup>c</sup> v* is given by:

<sup>0</sup> 300 < (0) < < *T TT en m* ;

rate *<sup>j</sup>*

Cylindrical Tube Growth by the Edge-Defined Film-Fed Growth (EFG) Technique 111

In this section it will be shown that the results presented in the above sections can be used for setting the pulling rate, the thermal and capillary conditions in view of an experiment

According to [Tatarchenko, 1993] at the level of the crystallization front *h* the crystallization

 

*<sup>c</sup> e i e i v G rrh G rrh j* (20)

*e i e i v G rrh G rrh j* (21)

*<sup>c</sup> v* is equal to the

<sup>1</sup> <sup>=</sup> ( , , ) ( , , ) , = 1,2,3. *<sup>j</sup> j j*

*dt* , *<sup>j</sup>* = 1,2,3 .

In order to keep the crystallization front level *<sup>j</sup> h* constant, the pulling rate and the thermal

 

When the radii *er* , *ir* and the length *L* of the tube, which has to be grown, are prior given, and *h* is known, then the condition (21) can be regarded as an equation in which the pulling rate *v* is unknown. If this equation has a positive solution *v*, it depends on the following parameters: *h* , (0) *Ten* , *T*<sup>0</sup> , and *k* . The setting of the pulling rate, thermal conditions means

<sup>1</sup> ( , , ) ( , , ) = 0, = 1,2,3 *j j*

 1 2 1 2

**5. Setting the pulling rate, the thermal and capillary conditions** 

The difference between the pulling rate *v* and the crystallization rate *<sup>j</sup>*

*<sup>j</sup> dh*

 1 2 1 2

the choice of *v*, (0) *Ten* , *T*0 and *k* such that the following conditions be satisfied:

*<sup>L</sup>* ;

The setting of the capillary condition means to take the tube radii *er* , *ir* (prior given) and the crystallization front level *<sup>c</sup> h* , determined form (21) (for the above chosen *v*, (0) *Ten* , *T*<sup>0</sup> ,

*<sup>g</sup> p* , *<sup>i</sup>*

*g* ,

 1 =

*e g e* *i p p <sup>H</sup>*

( , , )=

 1 =

*i g i*

 <sup>2</sup> *<sup>i</sup> ici g rh p* (22)

*<sup>g</sup> p* , *He* , *Hi* using (2) with *pm* = 0 or

 *<sup>e</sup>* , 

*g* (23)

*<sup>i</sup>* was build up,

(0) 300 0< < *Ten <sup>k</sup>*

*v* is practically the same for every *L* : *L LL* <sup>0</sup> ( *L*0 = the seed length).

equation (21) has a positive solution *v* in an acceptable range.

*k* ) and find the pressures *<sup>e</sup> p* , *<sup>i</sup> p* solving the followings equations:

 

<sup>2</sup> *<sup>e</sup> ece <sup>g</sup> rhp* and

If the solutions *<sup>e</sup> p* , *<sup>i</sup> p* of this equations are in the range for which

*e p p <sup>H</sup>*

then the values *<sup>e</sup> p* , *<sup>i</sup> p* will be used to set *<sup>e</sup>*

( , , )=

conditions have to satisfy the following conditions:

1

1

**Step 3.** For *<sup>i</sup> <sup>p</sup>* the range , *<sup>i</sup> <sup>i</sup> p p* defined by:

$$\underline{p}\_{i} = \sup\_{a\_{c}^{i} \in \left[\underline{a}\_{c}^{i}, \overline{a}\_{c}^{i}\right]} E\_{1}^{i} \left(m\_{\prime}^{i} \underline{a}\_{c}^{i}\right) \qquad\qquad \overline{p}\_{i} = \inf\_{a\_{c}^{i} \in \left[\underline{a}\_{c}^{i}, \overline{a}\_{c}^{i}\right]} E\_{2}^{i} \left(m\_{\prime}^{i} \underline{a}\_{c}^{i}\right) \tag{19}$$

is considered.

In the case of of the InSb tube considered in section 2.2, for the outer free surface the function:

 2 2 3 12 3 4 5 6 2 3 78 9 10 ( ) , , <sup>1</sup> *e ee ee e e e e e e ee e e ee e e e e e e <sup>a</sup> <sup>p</sup> <sup>a</sup> <sup>p</sup> r a <sup>p</sup> r a <sup>p</sup> h a <sup>p</sup> h a <sup>p</sup> <sup>h</sup> rhp ap r ap h ap h a p h* with 1 ( ) = 16.95005004 0.1803412 *e e a p p* 2 ( ) = -6991.91018 76.11771837 *e e a p p* 3( ) = 721251.8226 8031.90955 *e e a p p* 4( ) = 82.08229042 1.41357516 *e e a p p* 5( ) = 820223.151 11947.58366 *e e a p p* 6( ) = 75949100 3370800 *e e a p p*

was obtained. For the inner free surface the function:

$$\sqrt{\varpi}\_i \left( r\_i; h\_i, p\_i \right) = \frac{b\_1(p\_i) + b\_2\left(p\_i\right) \cdot r\_i + b\_3\left(p\_i\right) \cdot h\_i + b\_4\left(p\_i\right) \cdot h\_i^2 + b\_5\left(p\_i\right) \cdot h\_i^3}{1 + b\_6\left(p\_i\right) \cdot r\_i + b\_7\left(p\_i\right) \cdot r\_i^2 + b\_8\left(p\_i\right) \cdot h\_i + b\_9\left(p\_i\right) \cdot h\_i^2}$$

 <sup>7</sup> ( ) = 207.085329 0.00600066 *e e a p p* 8( ) = 0.10447655 0.82800845 *e e a p p* 9( ) = 544970.373 8774.44 *e e a p p* 10 ( ) = 237870000 663676.426 *e e a p p*

$$\begin{array}{ll}\text{with} & b\_{1}(p\_{i}) = -0.03323515 + 0.0000618675 \cdot \mathbf{p}\_{i} & b\_{2}(p\_{i}) = 7.907785322.0.01471419 \cdot \mathbf{p}\_{i} \\ & b\_{3}(p\_{i}) = -10.9920301 + 0.019065783 \cdot \mathbf{p}\_{i} & b\_{4}(p\_{i}) = 81321.53642 + 10.42303955 \cdot \mathbf{p}\_{i} \\ & b\_{5}(p\_{i}) = -6164200 - 109065.464 \cdot \mathbf{p}\_{i} & b\_{6}(p\_{i}) = -478.279159 + 0.004407571 \cdot \mathbf{p}\_{i} \\ & b\_{7}(p\_{i}) = 57185.4894 \cdot 1.04591888 \cdot \mathbf{p}\_{i} & b\_{8}(p\_{i}) = -7.40209299 + 0.012781953 \cdot \mathbf{p}\_{i} \\ & b\_{9}(p\_{i}) = 57826.05599 + 9.103117035 \cdot \mathbf{p}\_{i} \end{array}$$

$$\begin{aligned} b\_2(p\_i) &= 7.907785322 \text{-} 0.01471419 \cdot \mathbf{p\_i} \\ b\_4(p\_i) &= 81321.53642 + 10.42303935 \cdot \mathbf{p\_i} \\ b\_6(p\_i) &= 478.279159 + 0.004407571 \cdot \mathbf{p\_i} \\ b\_8(p\_i) &= -7.40209299 + 0.012781953 \cdot \mathbf{p\_i} \end{aligned}$$

was obtained. For *pe* 180*Pa* and *pi* 290*Pa* the functions (, , ) *ee e e rhp* and (, , ) *ii i i rhp* are represented in Fig. 9:

Fig. 9. The graphics of (, , ) *ii i i rhp* ( *pi* 290*Pa* ) and (, , ) *ii i i rhp* ( *pe* 180*Pa* )

## **5. Setting the pulling rate, the thermal and capillary conditions**

In this section it will be shown that the results presented in the above sections can be used for setting the pulling rate, the thermal and capillary conditions in view of an experiment [Tanasie&Balint, 2010].

According to [Tatarchenko, 1993] at the level of the crystallization front *h* the crystallization rate *<sup>j</sup> <sup>c</sup> v* is given by:

$$
\omega\_c \mathbf{v}\_c^j = \frac{1}{\Lambda \cdot \rho\_1} \cdot \left[ \mathcal{A}\_1 \cdot \mathbf{G}\_1^j(\mathbf{r}\_{\varepsilon'}, \mathbf{r}\_{i'}, \mathbf{h}) - \mathcal{A}\_2 \cdot \mathbf{G}\_2^j(\mathbf{r}\_{\varepsilon'}, \mathbf{r}\_{i'}, \mathbf{h}) \right], j = 1, 2, 3. \tag{20}
$$

The difference between the pulling rate *v* and the crystallization rate *<sup>j</sup> <sup>c</sup> v* is equal to the

crystallization front displacement rate *<sup>j</sup> dh dt* , *<sup>j</sup>* = 1,2,3 .

110 Crystallization – Science and Technology

 1 2 , , sup , inf ', *<sup>i</sup> <sup>i</sup> <sup>i</sup> <sup>i</sup> <sup>i</sup> <sup>i</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup>*

*c c <sup>i</sup> <sup>i</sup>*

In the case of of the InSb tube considered in section 2.2, for the outer free surface the

with 1 ( ) = 16.95005004 0.1803412 *e e a p p* 2 ( ) = -6991.91018 76.11771837 *e e a p p* 3( ) = 721251.8226 8031.90955 *e e a p p* 4( ) = 82.08229042 1.41357516 *e e a p p* 5( ) = 820223.151 11947.58366 *e e a p p* 6( ) = 75949100 3370800 *e e a p p* <sup>7</sup> ( ) = 207.085329 0.00600066 *e e a p p* 8( ) = 0.10447655 0.82800845 *e e a p p* 9( ) = 544970.373 8774.44 *e e a p p* 10 ( ) = 237870000 663676.426 *e e a p p*

(, , ) *ii i i rhp* ( *pe* 180*Pa* )

(, , ) *ii i i rhp* ( *pi* 290*Pa* ) and

with 1 i ( ) -0.03323515 0.0000618675 p *<sup>i</sup> b p* 2 i ( ) 7.907785322-0.01471419 p *<sup>i</sup> b p* 3 i ( ) -10.9920301 0.019065783 p *<sup>i</sup> b p* 4 i ( ) 81321.53642 10.42303935 p *<sup>i</sup> b p* 5 i ( ) -6164200-109065.464 p *<sup>i</sup> b p* 6 i ( ) -478.279159 0.004407571 p *<sup>i</sup> b p* 7 i ( ) 57185.4894-1.04591888 p *<sup>i</sup> b p* 8 i ( ) -7.40209299 0.012781953 p *<sup>i</sup> b p*

*<sup>b</sup> <sup>p</sup> <sup>b</sup> <sup>p</sup> r b <sup>p</sup> h b <sup>p</sup> h b <sup>p</sup> <sup>h</sup> rhp bp r bp r bp h bp h*

*<sup>a</sup> <sup>p</sup> <sup>a</sup> <sup>p</sup> r a <sup>p</sup> r a <sup>p</sup> h a <sup>p</sup> h a <sup>p</sup> <sup>h</sup> rhp ap r ap h ap h a p h*

 

*i ii i i i i i i*

*ii ii i i i i*

67 8 9

12 3 4 5

*e ee ee e e e e e e*

*ee e e e e e e*

78 9 10

12 3 4 5 6

*p Em p Em* (19)

*i i i i*

2 2 3

2 3

2 3

(, , ) *ee e e rhp* and

2 2

*p p* defined by:

**Step 3.** For *<sup>i</sup> <sup>p</sup>* the range , *<sup>i</sup> <sup>i</sup>*

is considered.

*ee e e*

; ,

9 i ( ) 57826.05599 9.103117035 p *<sup>i</sup> b p*

(, , ) *ii i i rhp* are represented in Fig. 9:

Fig. 9. The graphics of

*ii i i*

function:

 

( ) , , <sup>1</sup>

was obtained. For the inner free surface the function:

( )

was obtained. For *pe* 180*Pa* and *pi* 290*Pa* the functions

1

In order to keep the crystallization front level *<sup>j</sup> h* constant, the pulling rate and the thermal conditions have to satisfy the following conditions:

$$v - \frac{1}{\Lambda \cdot \rho\_1} \cdot \left[\mathcal{A}\_1 \cdot \mathcal{G}\_1^j(\mathbf{r}\_{\epsilon}, \mathbf{r}\_i, h) - \mathcal{A}\_2 \cdot \mathcal{G}\_2^j(\mathbf{r}\_{\epsilon}, \mathbf{r}\_i, h)\right] = 0, j = 1, 2, 3\tag{21}$$

When the radii *er* , *ir* and the length *L* of the tube, which has to be grown, are prior given, and *h* is known, then the condition (21) can be regarded as an equation in which the pulling rate *v* is unknown. If this equation has a positive solution *v*, it depends on the following parameters: *h* , (0) *Ten* , *T*<sup>0</sup> , and *k* . The setting of the pulling rate, thermal conditions means the choice of *v*, (0) *Ten* , *T*0 and *k* such that the following conditions be satisfied:


The setting of the capillary condition means to take the tube radii *er* , *ir* (prior given) and the crystallization front level *<sup>c</sup> h* , determined form (21) (for the above chosen *v*, (0) *Ten* , *T*<sup>0</sup> , *k* ) and find the pressures *<sup>e</sup> p* , *<sup>i</sup> p* solving the followings equations:

 ( , , )= <sup>2</sup> *<sup>e</sup> ece <sup>g</sup> rhp* and ( , , )= <sup>2</sup> *<sup>i</sup> ici g rh p* (22)

If the solutions *<sup>e</sup> p* , *<sup>i</sup> p* of this equations are in the range for which *<sup>e</sup>* , *<sup>i</sup>* was build up, then the values *<sup>e</sup> p* , *<sup>i</sup> p* will be used to set *<sup>e</sup> <sup>g</sup> p* , *<sup>i</sup> <sup>g</sup> p* , *He* , *Hi* using (2) with *pm* = 0 or

$$H\_e = -\frac{p\_g^e + p\_e}{\rho\_1 \cdot g}, \quad H\_i = -\frac{p\_g^i + p\_i}{\rho\_1 \cdot g} \tag{23}$$

A Mathematical Model for Single Crystal

T0 1716.46 2110.76 2933.32 *i e*

k 250.00 500.00 1000.00 *<sup>c</sup>*

*<sup>e</sup> p* -1981.44 -1982.93 -1981.61 *<sup>c</sup>*

that in all cases the Hurwitz condition are satisfied.

height at the start is <sup>4</sup> *h* = 2.14838 10 [m].

Table 1. Possible settings of *v* , T0 , (0) *Ten* , *<sup>e</sup> p* , *<sup>i</sup> p* , *i e*

Table 2. The coefficients *ij a* of the linearized system in the steady states

*<sup>i</sup> p* -2252.02 -2253.49 -2252.18

to verify if the steady state ( *<sup>c</sup>*

*v*

Cylindrical Tube Growth by the Edge-Defined Film-Fed Growth (EFG) Technique 113

> 0, >0 ( ) ( ) ( ) > 0

*a a a aa aa aa aa aaa aaa* (24)

*cc c ee e ece ece ece*

*RR h*

*c cc i ii i ci i ci i ci*

*R Rh*

*g g p p*

 

*ir* , *<sup>c</sup> h* ) is asymptotically stable. This last requirement is

 

)

(25)

*c*

*h h*

*g g p p* -270.58 -270.56 -270.57

*er* -3 4.66000 10 -3 4.65990 10 -3 4.65990 10

*ir* -4 4.33900 10 -4 4.33900 10 -4 4.33899 10

*er* , *<sup>c</sup>*

satisfied if the Hurwitz conditions are satisfied [Tatarchenko, 1993] i.e.:

*a a a aaa aaa*

 11 22 33 11 22 33 31 13 22 11 22 33 31 13 11 22 22 33 11 33 11 22 33 31 13 22

*rhp rhp rhp a v av av*

(,,) (,,) (,, = ==

*Sr r h Sr r h Sr r*

<sup>1</sup> ( , , )= (,,) (,,)

F1 F2 F3 F1 F2 F3

(0) *Ten* 400.00 400.00 400.00 *<sup>c</sup> <sup>h</sup>* -4 2.14380 10 -4 2.14324 10 -4 2.14371 10

<sup>11</sup> *a* <sup>13</sup> *a* <sup>22</sup> *a* <sup>23</sup> *a* <sup>31</sup> *a* <sup>32</sup> *a* <sup>33</sup> *a* F1 -0.20278 -0.29136 -0.1984 0.285445 121.8176 0 -2534.0535 F2 -0.20191 -0.29033 -0.19755 0.284434 27285.12 -27215.77 -42171.2316 F3 -0.20183 -0.29002 -0.19747 0.284131 88368.31 -88368.31 -132203.96

The values of the numbers *ij a* in the considered cases are given in Table 2. It is easy to verify

In Fig. 10 simulations of the silicon tube growth is presented when the seed length is <sup>2</sup> 10 and the radii of the seed are <sup>3</sup> = 4.3389 10 *Ri* [m]; <sup>3</sup> = 4.659 10 *Re* [m]. The meniscus

*e i e i e i*

*Sr r h v G rrh G rrh*

(,,) (,,) (,,) = = <sup>0</sup> = =

*c c c c c c ei c ei c e i*

1 1 2 2

8.32993 10 8.29566 1 <sup>0</sup> 8.29107 1 <sup>0</sup> *H H i e* -2 1.10328 10 -2 1.10320 10 -2 1.10324 10

 

*a aa R R*

31 32 33

1

*e i*

*e i*

11 12 13

*e i*

21 22 23

*rhp rhp rhp av av a v*

(,,) (,,) (,,) = =0 <sup>=</sup>

For the growth of a silicon tube with convex profile curves the following numerical data will be used: <sup>3</sup> <sup>1</sup> = 2.5 10 [kg/m3]; <sup>3</sup> <sup>2</sup> = 2.3 10 [kg/m3]; *Tm* = 1683 [K]; <sup>1</sup> = 60 [W/m K]; <sup>2</sup> = 21.6 [W/m K]; <sup>6</sup> = 1.81 10 [J/kg]; 1 1 1 1 = *c* ; 2 2 2 2 = *c* ; *c*<sup>1</sup> = 913 [J/kg K]; *c*<sup>2</sup> = 703 [J/kg K]; <sup>1</sup> = 7300.42 [K]; <sup>2</sup> = 2822.58 [K]; <sup>3</sup> = 4.2 10 *Rgi* [m]; <sup>3</sup> = 4.8 10 *Rge* [m];

<sup>3</sup> = 4.339 10 *<sup>c</sup> Ri* [m]; <sup>3</sup> = 4.66 10 *<sup>c</sup> Re* [m]; *L*<sup>1</sup> = 0.4 [m]; *L*<sup>2</sup> = 0.2 [m]; *L*<sup>3</sup> = 0.1 [m].


Using this input and the values *er* , *h* , *<sup>c</sup> ir* the value of the pulling rates 1 2 40 *vv v* ,,, given by the equation (21) have to be found. If all these values are positive, then: if the average *v* and standard deviation of the set of values of *v*, are acceptable, then the average pulling rate *v* and the initial input thermal conditions can be set, else the initial input thermal conditions have to be reset lowering in general (0) *Ten* and/or increasing *T*<sup>0</sup> .


Following the above steps for the considered silicon tube growth, some of the computed possible settings, are presented in Table 1.

For the above settings the growth process stability analysis is made through the system of nonlinear ordinary differential equations (1) which governs the evolution of *er* , *ir* , *h* for the established settings. It means to verify first of all that the desired *<sup>c</sup> er* , *<sup>c</sup> ir* and the obtained *<sup>c</sup> h* is a steady state of (1).

Furthermore to verify if at the start *<sup>c</sup> er* , *<sup>c</sup> ir* , *<sup>c</sup> h* are perturbed (i.e. the seed sizes are different from *<sup>c</sup> er* , *<sup>c</sup> ir* ) after a period of transition the values *<sup>c</sup> er* , *<sup>c</sup> ir* , *<sup>c</sup> h* are recovered. In other words,

For the growth of a silicon tube with convex profile curves the following numerical data will be

<sup>2</sup> = 2.3 10 [kg/m3]; *Tm* = 1683 [K];

<sup>2</sup> = 2822.58 [K]; <sup>3</sup> = 4.2 10 *Rgi* [m]; <sup>3</sup> = 4.8 10 *Rge* [m];

**Step 1.** A stable static outer meniscus is chosen, whose characteristic parameters *er* , *h* , *<sup>e</sup> p*

**Step 2.** An initial input for *T*<sup>0</sup> , (0) *Ten* and *k* has to be chosen. For *T*<sup>0</sup> , the start can be

by the equation (21) have to be found. If all these values are positive, then: if the average *v*

rate *v* and the initial input thermal conditions can be set, else the initial input thermal

**Step 3.** Consider *v* , (0) *Ten* , *T*<sup>0</sup> , *k* obtained above and solve equation (22) for these values

Following the above steps for the considered silicon tube growth, some of the computed

For the above settings the growth process stability analysis is made through the system of nonlinear ordinary differential equations (1) which governs the evolution of *er* , *ir* , *h* for the

*er* , *<sup>c</sup>*

conditions have to be reset lowering in general (0) *Ten* and/or increasing *T*<sup>0</sup> .

*er* , *<sup>c</sup>*

*g g p p* for = 0 *H H i e* (in the case of a closed crucible).

established settings. It means to verify first of all that the desired *<sup>c</sup>*

*ir* ) after a period of transition the values *<sup>c</sup>*

*er* , *<sup>c</sup>*

2

2 2 = *c*

case considered here such a static meniscus is obtained for *pe* = 1980 [Pa] and its characteristic parameters are: <sup>3</sup> = 4.660112250074 10 *er* [m] and

*T T* <sup>0</sup> = 1 *<sup>m</sup>* . Concerning (0) *Ten* and *k* the start can be *T T en m* (0) = 1 and

*<sup>e</sup>*(,, ) *e e r h p* is valid and for which *er* is close to *<sup>c</sup>*

*ir* the value of the pulling rates 1 2 40 *vv v* ,,, given

*ir* , *<sup>c</sup> h* in equation (23) and solve this equations

*g g p p* (in the case of an open crucible) or find

*er* , *<sup>c</sup>*

*ir* , *<sup>c</sup> h* are recovered. In other words,

*ir* , *<sup>c</sup> h* are perturbed (i.e. the seed sizes are different

*ir* and the obtained *<sup>c</sup> h*

of the set of values of *v*, are acceptable, then the average pulling

*i i r r* (the desired radii) and *h* unknown. Denote by *<sup>c</sup> h* the

<sup>1</sup> = 60 [W/m K];

; *c*<sup>1</sup> = 913 [J/kg K]; *c*<sup>2</sup> = 703 [J/kg K];

<sup>2</sup> = 21.6

*er* . In the

used: 

<sup>3</sup>

<sup>1</sup> = 7300.42 [K];

<sup>1</sup> = 2.5 10 [kg/m3];

are in the range where

(0) 300 <sup>=</sup> *Ten <sup>k</sup>*

choosing = *<sup>c</sup>*

finding *<sup>e</sup> p* , *<sup>i</sup> p* .

*i e*

is a steady state of (1).

*er* , *<sup>c</sup>*

from *<sup>c</sup>*

and standard deviation

<sup>4</sup> *h* = 2.14370857185 10 [m].

*<sup>L</sup>* .

Using this input and the values *er* , *h* , *<sup>c</sup>*

*e e r r* and = *<sup>c</sup>*

obtained solution. Replace *<sup>c</sup>*

**Step 4.** Using *<sup>e</sup> p* , *<sup>i</sup> p* find *H H i e* , for =*e i*

possible settings, are presented in Table 1.

Furthermore to verify if at the start *<sup>c</sup>*

[W/m K]; <sup>6</sup> = 1.81 10 [J/kg];

<sup>3</sup>

1

1

1 1 = *c*

<sup>3</sup> = 4.339 10 *<sup>c</sup> Ri* [m]; <sup>3</sup> = 4.66 10 *<sup>c</sup> Re* [m]; *L*<sup>1</sup> = 0.4 [m]; *L*<sup>2</sup> = 0.2 [m]; *L*<sup>3</sup> = 0.1 [m].

;

2

to verify if the steady state ( *<sup>c</sup> er* , *<sup>c</sup> ir* , *<sup>c</sup> h* ) is asymptotically stable. This last requirement is satisfied if the Hurwitz conditions are satisfied [Tatarchenko, 1993] i.e.:

$$-a\_{11} - a\_{22} - a\_{33} \ge 0,\\ -a\_{11}a\_{22}a\_{33} + a\_{31}a\_{13}a\_{22} \ge 0$$

$$(-a\_{11} - a\_{22} - a\_{33}) \cdot (-a\_{31}a\_{13} + a\_{11}a\_{22} + a\_{22}a\_{33} + a\_{11}a\_{33}) \cdot (-a\_{11}a\_{22}a\_{33} + a\_{31}a\_{13}a\_{22}) \ge 0\tag{24}$$

 11 12 13 21 22 23 31 32 33 (,,) (,,) (,,) = =0 <sup>=</sup> (,,) (,,) (,,) = = <sup>0</sup> = = (,,) (,,) (,, = == *cc c ee e ece ece ece e i c cc i ii i ci i ci i ci e i c c c c c c ei c ei c e i e i rhp rhp rhp av av a v RR h rhp rhp rhp a v av av R Rh Sr r h Sr r h Sr r a aa R R* 1 1 2 2 1 ) <sup>1</sup> ( , , )= (,,) (,,) *c e i e i e i h h Sr r h v G rrh G rrh* (25)


Table 1. Possible settings of *v* , T0 , (0) *Ten* , *<sup>e</sup> p* , *<sup>i</sup> p* , *i e g g p p*


Table 2. The coefficients *ij a* of the linearized system in the steady states

The values of the numbers *ij a* in the considered cases are given in Table 2. It is easy to verify that in all cases the Hurwitz condition are satisfied.

In Fig. 10 simulations of the silicon tube growth is presented when the seed length is <sup>2</sup> 10 and the radii of the seed are <sup>3</sup> = 4.3389 10 *Ri* [m]; <sup>3</sup> = 4.659 10 *Re* [m]. The meniscus height at the start is <sup>4</sup> *h* = 2.14838 10 [m].

A Mathematical Model for Single Crystal

<sup>2</sup> *<sup>e</sup> <sup>g</sup> z r* , *z R*' tan *ge c*

*c g*

*R*

**Theorem 2.** Let be n such that

then there exists

then there exists

*n*

*ge*

 

*ge*

cos

*n*

2 cos sin 1

*ge ge*

 

 ' tan 

Cylindrical Tube Growth by the Edge-Defined Film-Fed Growth (EFG) Technique 115

**Definition 1.** A solution *z zx* of the eq. (6) describes the outer free surface of a static

meniscus on the interval , *<sup>e</sup> ge r R* if it possesses the following properties: <sup>0</sup> *ge z R* ,

**Theorem 1.** If there exists a solution of the eq. (6), which describes a convex outer free

 

*c c e*

*c g ge*

*R n n*

 2 1 *ge*

*R R*

*c g*

 

*gi ge R*

*gz p R R z z z z r R r*

*e c c ge ge n*

2 cos sin 1

*n gR n*

<sup>1</sup> <sup>2</sup> sin tan <sup>2</sup> <sup>1</sup>

and *z r* is strictly decreasing on [ , ] *<sup>e</sup> ge r R* . The

*g g*

. If *<sup>e</sup> p* satisfies the inequality:

 

*<sup>p</sup> Rn R* (A.1.2)

such that the solution of the initial value problem:

*<sup>p</sup> RR R* (A.1.4)

 

*R R ge gi* ) such that the solution of the i.v.p.

*gi ge <sup>e</sup> ge*

 2

*e R n*

*<sup>r</sup>* , *<sup>e</sup> <sup>p</sup>* satisfy:

 

(A.1.1)

(A.1.3)

described outer free surface is convex on [ , ] *<sup>e</sup> ge r R* if " 0 [, ] *<sup>e</sup> ge z r r rR* .

surface of a static meniscus on the closed interval[, ] *<sup>e</sup> ge r R* , then for *ge*

*<sup>p</sup> Rn R*

0, ' tan

*ge ge c*

**Corollary 3.** If for *<sup>e</sup> p* the following inequality holds:

, *ge e ge R r R n*

*zR z R*

*g*

*n*

*r R* (close to

 , <sup>2</sup> *ge gi e ge R R*

on the interval [, ] *<sup>e</sup> ge r R* describes the convex outer free surface of a static meniscus.

 

(12) on the interval [, ] *<sup>e</sup> ge r R* describes a convex outer free surface of a static meniscus.

 ( ) <sup>2</sup> <sup>2</sup> cos sin *c g e c c ge gi ge*

3 <sup>2</sup> <sup>2</sup> <sup>1</sup> <sup>2</sup> " 1 ' 1 ( ') ' for <sup>2</sup>

Fig. 10. The evolution of the outer radius, the inner radius of the tube and the meniscus height obtained integrating numerically the system (1) for *F R* <sup>1</sup> =2/ *<sup>e</sup>* ,

<sup>4</sup> *<sup>v</sup>* 1.6585917365 10 [m/s], *T*<sup>0</sup> = 1714.81 [K] and (0) = 400[ ] *T K en*

#### **6. Conclusions**

Knowing the material constants (density, heat conductivity, etc), the size of the single crystal tube which will be grown from that material, the size of the shaper which will be used and the cooling gas temperature at the entrance, it is possible to predict values of pulling rate, temperature at the meniscus basis, cooling gas temperature at the exit, vertical temperature gradient in the furnace, inner and outer walls cooling gas pressure differences, melt column height differences, crystallization front level, which can be used for a stable growth.

According to the model the predicted values are not unique i.e. there are several possibility to obtain a tube with prior given size from a given material using the same shaper. So, even if in our computation the material and the size of the shaper and tube is the same as in [Eriss] experiment, the computed data given in Table 1 can be different from that used in the real experiment. For this reason our purpose is not rely to compare the computed results with the experimental data. Moreover we want to reveal that a tube of prior given size can be obtained by different settings and the model permit to compute such settings. The choice of a specific setting is the practical crystal grower decision. The model provide possible settings and can be helpful in a new experiment planning.

Concerning the limits of the model it is clear that it is limited in applicability, as all models. The main limits are those introduced by approximations made in equations defining the model.

#### **7. Appendix 1. Inequalities for single crystal tube growth by E.F.G. technique - Convex outer and inner free surface**

Consider the differential equation (6) for 2 *gi ge e ge R R r R* , *c* , *g* such that 0, <sup>2</sup> *<sup>g</sup>* , 0 <sup>2</sup> *<sup>c</sup> <sup>g</sup>* .

Fig. 10. The evolution of the outer radius, the inner radius of the tube and the meniscus

Knowing the material constants (density, heat conductivity, etc), the size of the single crystal tube which will be grown from that material, the size of the shaper which will be used and the cooling gas temperature at the entrance, it is possible to predict values of pulling rate, temperature at the meniscus basis, cooling gas temperature at the exit, vertical temperature gradient in the furnace, inner and outer walls cooling gas pressure differences, melt column

According to the model the predicted values are not unique i.e. there are several possibility to obtain a tube with prior given size from a given material using the same shaper. So, even if in our computation the material and the size of the shaper and tube is the same as in [Eriss] experiment, the computed data given in Table 1 can be different from that used in the real experiment. For this reason our purpose is not rely to compare the computed results with the experimental data. Moreover we want to reveal that a tube of prior given size can be obtained by different settings and the model permit to compute such settings. The choice of a specific setting is the practical crystal grower decision. The model provide possible

Concerning the limits of the model it is clear that it is limited in applicability, as all models. The main limits are those introduced by approximations made in equations defining the

**7. Appendix 1. Inequalities for single crystal tube growth by E.F.G. technique** 

2 *gi ge*

*R R*

*e ge*

*r R* ,

*c* ,*g*  such that 0,

<sup>2</sup> *<sup>g</sup>* ,

height differences, crystallization front level, which can be used for a stable growth.

height obtained integrating numerically the system (1) for *F R* <sup>1</sup> =2/ *<sup>e</sup>* , <sup>4</sup> *<sup>v</sup>* 1.6585917365 10 [m/s], *T*<sup>0</sup> = 1714.81 [K] and (0) = 400[ ] *T K en*

settings and can be helpful in a new experiment planning.

**- Convex outer and inner free surface** 

Consider the differential equation (6) for

**6. Conclusions** 

model.

0 

<sup>2</sup> *<sup>c</sup> <sup>g</sup>* . **Definition 1.** A solution *z zx* of the eq. (6) describes the outer free surface of a static meniscus on the interval , *<sup>e</sup> ge r R* if it possesses the following properties: <sup>0</sup> *ge z R* , ' tan <sup>2</sup> *<sup>e</sup> <sup>g</sup> z r* , *z R*' tan *ge c* and *z r* is strictly decreasing on [ , ] *<sup>e</sup> ge r R* . The described outer free surface is convex on [ , ] *<sup>e</sup> ge r R* if " 0 [, ] *<sup>e</sup> ge z r r rR* .

**Theorem 1.** If there exists a solution of the eq. (6), which describes a convex outer free surface of a static meniscus on the closed interval[, ] *<sup>e</sup> ge r R* , then for *ge e R n <sup>r</sup>* , *<sup>e</sup> <sup>p</sup>* satisfy:

 2 cos sin 1 <sup>1</sup> <sup>2</sup> sin tan <sup>2</sup> <sup>1</sup> cos *c g c c e ge ge c g ge g g ge g ge n <sup>p</sup> Rn R n gR n R n n n R* (A.1.1)

**Theorem 2.** Let be n such that 2 1 *ge gi ge R n R R* . If *<sup>e</sup> p* satisfies the inequality:

$$p\_e < -\gamma \cdot \frac{\pi \not{p}\_2 - \left(a\_c + a\_g\right)}{R\_{\text{g\textnu}}} \cdot \frac{n}{n-1} \cdot \cos a\_c + \frac{\gamma}{R\_{\text{g\textnu}}} \cdot \sin a\_c \tag{A.1.2}$$

then there exists , *ge e ge R r R n* such that the solution of the initial value problem:

$$\begin{cases} z'' = \frac{\rho \cdot g \cdot z - p\_e}{\gamma} \cdot \left[ 1 + \left( z' \right)^2 \right]^{\frac{3}{2}} - \frac{1}{r} \cdot \left[ 1 + \left( z' \right)^2 \right] \cdot z' \text{ for } \frac{R\_{g\bar{t}} + R\_{g\bar{e}}}{2} < r \le R\_{g\varepsilon} \\ z \left( R\_{g\varepsilon} \right) = 0, \ z' \left( R\_{g\varepsilon} \right) = -\tan \alpha\_c \end{cases} \tag{A.1.3}$$

on the interval [, ] *<sup>e</sup> ge r R* describes the convex outer free surface of a static meniscus.

**Corollary 3.** If for *<sup>e</sup> p* the following inequality holds:

$$p\_e < -2 \cdot \gamma \cdot \frac{\pi \not{p}\_2 - (a\_c + a\_g)}{R\_{g\varepsilon} - R\_{g i}} \cdot \cos a\_c + \frac{\gamma}{R\_{g\varepsilon}} \cdot \sin a\_c \tag{A.1.4}$$

then there exists , <sup>2</sup> *ge gi e ge R R r R* (close to 2 *R R ge gi* ) such that the solution of the i.v.p. (12) on the interval [, ] *<sup>e</sup> ge r R* describes a convex outer free surface of a static meniscus.

A Mathematical Model for Single Crystal

then there exists

**Corollary 9.** If for

of the melt column (9).

0 

 

Consider the equation (6) for

2 2 *g c* .

*i gi*

 

0, ' tan

*gi gi c*

**Corollary 8.** If for *<sup>i</sup> p* the following inequality holds,

*zR z R*

Cylindrical Tube Growth by the Edge-Defined Film-Fed Growth (EFG) Technique 117

then there exists [, ] *<sup>i</sup> gi gi r R mR* , such that the solution of the initial value problem:

3

*z z z z Rr r*

on the interval [ ,] *R r gi i* describes the convex inner free surface of a static meniscus.

 

(A.1.8) on the interval [ , ] *R r gi i* describes a convex inner free surface of a static meniscus.

 

1 *c g i c g*

*<sup>p</sup> mR R*

then there exists *ir* in the interval [' , ] *mR mR gi gi* such that the solution of the i.v.p. (A.1.8)

**Theorem 10.** If a solution *z zr* 1 1 of the eq. (8) describes a convex inner free surface of a static meniscus on the interval[ ,] *R r gi i* , then it is a weak minimum for the energy functional

**8. Appendix 2. Inequalities for single crystal tube growth by E.F.G. technique** 

 0< < < 2 *gi ge gi e ge R R R r R* ,

 

on the interval [ , ] *R r gi i* describes a convex inner free surface of a static meniscus.

 ( ) <sup>2</sup> sin ' 1 tan sin <sup>2</sup> ' 1 '

 ( ) <sup>2</sup> <sup>2</sup> cos cos *c g i c g ge gi gi*

<sup>2</sup> <sup>2</sup> <sup>1</sup> <sup>2</sup> " 1 ' 1 ( ') ' for <sup>2</sup>

*gi ge i*

> > > *c* ,

*gi gi*

*g gi g c*

> 

*<sup>p</sup> RR R* (A.1.9)

 2

*<sup>R</sup>* the following inequalities hold

*gi gi gR m m R m R* (A.1.10)

( ) <sup>2</sup> cos cos

*gz p R R*

*gi*

 

*R R gi ge* ) such that the solution of the i.v.p.

 *<sup>g</sup>* such that 0,

<sup>2</sup> *<sup>g</sup>* ,

(A.1.8)

*R R*

2 *gi ge gi*

 

 , <sup>2</sup> *gi ge*

1 '

*m m*

*c g*

**- Concave outer and inner free surface** 

*R R r R* (close to

**Corollary 4.** If for 2 1 ' *ge gi ge R n n R R* the following inequalities holds:

 ( ) ' ' <sup>1</sup> <sup>2</sup> sin tan 'cos <sup>2</sup> ' 1 ' ( ) <sup>2</sup> cos sin 1 *c g g ge g g ge ge c g e c c ge ge n n g R <sup>n</sup> R n n R n <sup>p</sup> Rn R* (A.1.5)

then there exists , *ge ge e R R r n n* such that the solution of the i.v.p. (A.1.3) on the interval [, ] *<sup>e</sup> ge r R* describes a convex outer free surface of a static meniscus.

**Theorem 5.** If a solution *z zr* 1 1 of the eq. (6) describes a convex outer free surface of a static meniscus on the interval [ , ] *<sup>e</sup> ge r R* , then it is a weak minimum for the energy functional of the melt column (7).

**Definition 2.** A solution *z zx* of the eq.(8) describes the inner free surface of a static meniscus on the interval , *R r gi i* , 2 *gi ge gi i R R R r* if it possesses the following properties: *z R*' tan *gi c* , ' tan <sup>2</sup> *<sup>i</sup> <sup>g</sup> z r* , <sup>0</sup> *gi z R* and *z r* is strictly increasing on [ , ] *R r gi i* . The described inner free surface is convex on [ , ] *R r gi i* if *z r* " 0 , gi [R , ]*<sup>i</sup> r r* .

**Theorem 6.** If there exists a solution of the eq. (8), which describes a convex inner free surface of a static meniscus on the closed interval [ , ] *R r gi i* and *<sup>i</sup> gi r mR* with

$$1 < m < \frac{R\_{\text{g}^i} + R\_{\text{g}^e}}{2 \cdot R\_{\text{g}^i}}, \text{ then the following inequalities hold:}$$

$$\begin{split} & -\gamma \cdot \frac{\pi \not{p}\_{2} - \left(\boldsymbol{\alpha}\_{c} + \boldsymbol{\alpha}\_{g}\right)}{\left(m-1\right) \cdot \boldsymbol{R}\_{\mathcal{g}^{i}}} \cdot \cos \boldsymbol{a}\_{c} - \frac{\gamma}{\boldsymbol{R}\_{\mathcal{g}^{i}}} \cdot \cos \boldsymbol{a}\_{g} \leq p\_{i} \\ & \leq -\gamma \cdot \frac{\pi \not{p}\_{2} - \left(\boldsymbol{\alpha}\_{c} + \boldsymbol{\alpha}\_{g}\right)}{\left(m-1\right) \cdot \boldsymbol{R}\_{\mathcal{g}^{i}}} \cdot \sin \boldsymbol{a}\_{g} + \rho \cdot \boldsymbol{g} \cdot \boldsymbol{R}\_{\mathcal{g}^{i}} \cdot \left(m-1\right) \cdot \tan \left(\frac{\pi \not{p}\_{2} - \boldsymbol{a}\_{g}}{\gamma}\right) - \frac{\gamma}{m \cdot \boldsymbol{R}\_{\mathcal{g}^{i}}} \cdot \sin \boldsymbol{a}\_{c} \end{split} \tag{A.1.6}$$

**Theorem 7.** Let m be such that <sup>1</sup> 2 *gi ge gi R R m R* . If *<sup>i</sup> p* satisfies the inequality:

$$p\_i < -\gamma \cdot \frac{\pi \not{p}\_{\mathbf{Z}} - \left(a\_c + a\_{\mathbf{g}}\right)}{(m-1) \cdot R\_{\mathbf{g}i}} \cdot \cos a\_c + \frac{\gamma}{R\_{\mathbf{g}i}} \cdot \cos a\_{\mathbf{g}} \tag{A.1.7}$$

then there exists [, ] *<sup>i</sup> gi gi r R mR* , such that the solution of the initial value problem:

$$\begin{cases} z'' = \frac{\rho \cdot g \cdot z - p\_i}{\nu} \cdot \left[ 1 + \left( z' \right)^2 \right]^{\frac{3}{2}} - \frac{1}{r} \cdot \left[ 1 + \left( z' \right)^2 \right] \cdot z' \text{ for } R\_{g i} < r \le \frac{R\_{g i} + R\_{g \epsilon}}{2} \\ z \left( R\_{g i} \right) = 0, \text{ } z' \left( R\_{g i} \right) = \tan \alpha\_c \end{cases} \tag{A.1.8}$$

on the interval [ ,] *R r gi i* describes the convex inner free surface of a static meniscus.

**Corollary 8.** If for *<sup>i</sup> p* the following inequality holds,

116 Crystallization – Science and Technology

*R R* the following inequalities holds:

( ) ' ' <sup>1</sup> <sup>2</sup> sin tan 'cos <sup>2</sup> ' 1 '

*ge ge*

*n*

**Theorem 5.** If a solution *z zr* 1 1 of the eq. (6) describes a convex outer free surface of a static meniscus on the interval [ , ] *<sup>e</sup> ge r R* , then it is a weak minimum for the energy functional

**Definition 2.** A solution *z zx* of the eq.(8) describes the inner free surface of a static

*gi i*

on [ , ] *R r gi i* . The described inner free surface is convex on [ , ] *R r gi i* if *z r* " 0 , gi [R , ]*<sup>i</sup> r r* .

**Theorem 6.** If there exists a solution of the eq. (8), which describes a convex inner free surface of a static meniscus on the closed interval [ , ] *R r gi i* and *<sup>i</sup> gi r mR* with

<sup>2</sup> sin 1 tan sin <sup>2</sup> <sup>1</sup>

*gi gi*

*gi gi*

2 cos cos

*c g i*

 2

*gi ge*

*R R*

*e c c ge ge*

*R n n R*

( ) <sup>2</sup> cos sin 1

*R r* if it possesses the following

(A.1.6)

<sup>2</sup> *<sup>i</sup> <sup>g</sup> z r* , <sup>0</sup> *gi z R* and *z r* is strictly increasing

*g gi g c*

*<sup>p</sup> mR R* (A.1.7)

. If *<sup>i</sup> p* satisfies the inequality:

 

such that the solution of the i.v.p. (A.1.3) on the interval

*g ge g g*

 

*n n g R <sup>n</sup>*

 

(A.1.5)

 2 1 ' *ge*

, *ge ge*

meniscus on the interval , *R r gi i* ,

 , ' tan 

*<sup>R</sup>* , then the following inequalities hold:

 

1

*c g*

 

 *c g*

 

 <sup>1</sup> 2 *gi ge gi*

*m*

*R*

 

1 *c g i c g*

*R R*

2 cos cos

*<sup>p</sup> mR R*

*gR m m R m R*

*gi gi*

*n n*

*R R*

*gi ge R*

> 

*<sup>p</sup> Rn R*

*c g*

 

[, ] *<sup>e</sup> ge r R* describes a convex outer free surface of a static meniscus.

**Corollary 4.** If for

then there exists

of the melt column (7).

properties: *z R*' tan *gi c*

**Theorem 7.** Let m be such that

*R R*

 <sup>1</sup> 2 *gi ge gi*

*m*

*e*

*r*

 

*c g*

 *n n*

$$p\_i < -2 \cdot \gamma \cdot \frac{\pi \int\_{\mathcal{D}} - (a\_c + a\_g)}{R\_{g\varepsilon} - R\_{g i}} \cdot \cos a\_c - \frac{\gamma}{R\_{g i}} \cdot \cos a\_g \tag{A.1.9}$$

then there exists , <sup>2</sup> *gi ge i gi R R r R* (close to 2 *R R gi ge* ) such that the solution of the i.v.p. (A.1.8) on the interval [ , ] *R r gi i* describes a convex inner free surface of a static meniscus.

**Corollary 9.** If for 1 ' 2 *gi ge gi R R m m <sup>R</sup>* the following inequalities hold

$$-\gamma \cdot \frac{\pi\_{\mathfrak{D}}^{\prime} - (a\_{\varepsilon} + a\_{\mathfrak{g}})}{\left(m^{\prime} - 1\right) \cdot R\_{\mathfrak{g}^{\prime}}} \cdot \sin a\_{\mathfrak{g}} + \rho \cdot \mathfrak{g} \cdot R\_{\mathfrak{g}^{\prime}} \cdot \left(m^{\prime} - 1\right) \cdot \tan\left(\frac{\pi}{2} - a\_{\mathfrak{g}}\right) - \frac{\gamma}{m^{\prime} \cdot R\_{\mathfrak{g}^{\prime}}} \cdot \sin a\_{\varepsilon} \quad \text{(A.1.10)}$$

$$1 < p\_i < -\gamma \cdot \frac{\frac{\pi \gamma}{2} - (a\_c + a\_{\S})}{\left(m - 1\right) \cdot R\_{\S^i}} \cdot \cos \alpha\_c - \frac{\gamma}{R\_{\S^i}} \cdot \cos \alpha\_{\S^i}$$

then there exists *ir* in the interval [' , ] *mR mR gi gi* such that the solution of the i.v.p. (A.1.8) on the interval [ , ] *R r gi i* describes a convex inner free surface of a static meniscus.

**Theorem 10.** If a solution *z zr* 1 1 of the eq. (8) describes a convex inner free surface of a static meniscus on the interval[ ,] *R r gi i* , then it is a weak minimum for the energy functional of the melt column (9).

## **8. Appendix 2. Inequalities for single crystal tube growth by E.F.G. technique - Concave outer and inner free surface**

Consider the equation (6) for 0< < < 2 *gi ge gi e ge R R R r R* , *c* , *<sup>g</sup>* such that 0, <sup>2</sup> *<sup>g</sup>* , 0 2 2 *g c* .

A Mathematical Model for Single Crystal

energy functional of the melt column (9).

732106, pp.1-28

Vol. 4, pp.277-296.

0

4043-4053

Aerospace Vol. 2, pp. 53-70.

**9. References** 

 

Crystal Growth, Vol. 310, pp.382-390

.Crystal Growth, Vol. 198/199, pp.215-219.

*process*, Journal of .Crystal Growth, Vol. 50, pp.200-211.

Growth, Vol. 198/199, pp.220-226.

*ge*

*c g*

Cylindrical Tube Growth by the Edge-Defined Film-Fed Growth (EFG) Technique 119

 

<sup>1</sup> <sup>2</sup> cos < < cos sin

*c g g i c c*

*g gi c*

 

(A.2.4)

0

(, ; ) *ee e e rhp and* 

 

*m gR m R*

1

<sup>1</sup> <sup>2</sup> sin ( 1) tan <sup>1</sup>

*<sup>p</sup> m R m R <sup>R</sup>*

then there exists *ir* in the interval , *m R mR gi gi* and a concave solution of the eq. (8).

**Theorem 16.** A concave solution ( ) *<sup>i</sup> z r* of the equation (8) is a weak minimum of the free

St. Balint, A.M. Balint (2009), *On the creation of the stable drop-like static meniscus, appropriate for* 

St.Balint, A.M.Balint, L.Tanasie (2008) - *The effect of the pressure on the static meniscus shape in* 

St. Balint, L.Tanasie (2008), *Nonlinear boundary value problems for second order differential* 

St.Balint, L.Tanasie(2011), *Some problems concerning the evaluation of the shape and size of the* 

St.Balint, L.Tanasie (2011), *The choice of the pressure of the gas flow and the melt level in silicon tube growth,* Mathematics in Engineering, Science and Aerospace, Vol. 4, pp. H.Behnken, A.Seidl and D.Franke (2005), *A 3 D dynamic stress model for the growth of hollow* 

A.V.Borodin, V.A.Borodin, V.V.Sidorov and I.S.Petkov (1999), *Influence of growth process* 

A.V.Borodin, V.A.Borodin and A.V.Zhdanov (1999), *Simulation of the pressure distribution in* 

L.Erris, R.W.Stormont, T.Surek, A.S.Taylor (1980), *The growth of silicon tubes by the EFG* 

*parameters on weight sensor readings in the Stepanov (EFG) technique*, Journal of

*the melt for sapphire ribbon growth by the Stepanov (EFG) technique*, Journal of .Crystal

*silicon polygons*, Journal of .Crystal Growth, Vol 275, pp. e375-e380.

St.Balint, L.Tanasie (2010), *A procedure for the determination of the angles* 

Problems in Engineering, vol. 2009, Article ID:348538 (2009), pp 1-22 St. Balint, A.M. Balint (2009), *Inequalities for single crystal tube growth by edge-defined film-fed* 

*the growth of a single crystal tube with prior specified inner and outer radii*, Mathematical

*(E.F.G.) technique* , Journal of Inequalities and Applications, vol.2009, Article ID:

*the case of tube growth by edge-defined film-fed growth (E.F.G.) method*, Journal of

equations describing concave equilibrium capillary surfaces, Nonlinear Studies 15,

*meniscus occurring in silicon tube growth* - Mathematics in Engineering, Science and

(, ; ) *iiii rhp which appears in the nonlinear system of differential equations describing the dynamics of the outer and inner radius of a tube, grown by the edge-defined film-fed growth (EFG) technique*, Nonlinear Analysis: Real World Applications, Vol. 11(Issue 5), pp

*gi gi gi*

**Definition 3.** The outer free surface is concave on [ , ] *<sup>e</sup> ge r R* if *z r* " 0 , [, ] *<sup>e</sup> ge r rR* .

**Theorem 11.** If there exists a concave solution = () *e e z zr* of the equation (6) then = *ge e R n r* and *<sup>e</sup> p* satisfy the following inequalities:

$$\begin{split} & \frac{n}{n-1} \cdot \boldsymbol{\gamma} \cdot \frac{\boldsymbol{a}\_c + \boldsymbol{a}\_g - \frac{\pi}{\mathsf{P}\_2}}{R\_{g\boldsymbol{e}}} \cdot \cos \boldsymbol{a}\_c + \frac{\boldsymbol{\gamma}}{R\_{g\boldsymbol{e}}} \cdot \cos \boldsymbol{a}\_g \leq p\_c \leq \frac{n}{n-1} \cdot \boldsymbol{\gamma} \cdot \frac{\boldsymbol{a}\_c + \boldsymbol{a}\_g - \pi \Big\prime\_2}{R\_{g\boldsymbol{e}}} \cdot \sin \boldsymbol{a}\_g \\ & + \frac{n-1}{n} \cdot \boldsymbol{\rho} \cdot \boldsymbol{g} \cdot \tan \boldsymbol{a}\_c + \frac{n \cdot \boldsymbol{\gamma}}{R\_{g\boldsymbol{e}}} \cdot \sin \boldsymbol{a}\_c \end{split} \tag{A.2.1}$$

**Theorem 12.** If for 2 1< < < *ge ge gi R n n R R* and *<sup>e</sup> p* the inequalities hold:

$$\begin{split} \frac{n}{n-1} \cdot \gamma \cdot \frac{\boldsymbol{a}\_c + \boldsymbol{a}\_g - \frac{\pi}{\gamma} \mathsf{I}\_2}{R\_{\mathcal{g}^\varepsilon}} \cdot \sin \boldsymbol{a}\_g + \frac{n-1}{n} \cdot \boldsymbol{\rho} \cdot \boldsymbol{g} \cdot \boldsymbol{R}\_{\mathcal{g}^\varepsilon} \cdot \tan \boldsymbol{a}\_c + \frac{n \cdot \gamma}{R\_{\mathcal{g}^\varepsilon}} \cdot \sin \boldsymbol{a}\_c \leqslant p\_\varepsilon < \\ & \frac{n'}{n'-1} \cdot \boldsymbol{\gamma} \cdot \frac{\boldsymbol{a}\_c + \boldsymbol{a}\_g - \frac{\pi}{\gamma} \mathsf{I}\_2}{R\_{\mathcal{g}^\varepsilon}} \cdot \cos \boldsymbol{a}\_c + \frac{\gamma}{R\_{\mathcal{g}^\varepsilon}} \cdot \cos \boldsymbol{a}\_\mathcal{g}. \end{split} \tag{A.2.2}$$

then there exists , *ge ge e R R r n n* and a concave solution of the equation (6).

**Theorem 13.** A concave solution ( ) *<sup>e</sup> z r* of the equation (6) is a weak minimum of the free energy functional of the melt column (7).

Consider now the differential equation (8) for 0< < < < 2 *gi ge gi i ge R R Rr R* and *c* , *<sup>g</sup>* such that 0 2 2 *g c* , 0, <sup>2</sup> *<sup>g</sup>* .

**Theorem 14.** If there exists a concave solution = () *i i z zr* of the equation (8) then = *<sup>i</sup> gi r m R* and *<sup>i</sup> p* satisfies the following inequalities:

$$\begin{split} \frac{1}{m-1} \cdot \boldsymbol{\gamma} \cdot \frac{\boldsymbol{a}\_c + \boldsymbol{a}\_g - \frac{\pi}{\mathsf{P}\_2}}{R\_{g\boldsymbol{i}}} \cdot \cos \boldsymbol{a}\_c - \frac{\boldsymbol{\gamma}}{R\_{g\boldsymbol{i}}} \cdot \sin \boldsymbol{a}\_c \leq p\_i \leq \frac{1}{m-1} \cdot \boldsymbol{\gamma} \cdot \frac{\boldsymbol{a}\_c + \boldsymbol{a}\_g - \pi \Big\prime\_2}{R\_{g\boldsymbol{i}}} \cdot \sin \boldsymbol{a}\_g \\ + (m-1) \cdot \boldsymbol{\rho} \cdot \boldsymbol{g} \cdot \boldsymbol{R}\_{g\boldsymbol{i}} \cdot \tan \boldsymbol{a}\_c - \frac{\boldsymbol{\gamma}}{m \cdot \boldsymbol{R}\_{g\boldsymbol{i}}} \cdot \cos \boldsymbol{a}\_g \end{split} \tag{A.2.3}$$

**Theorem 15.** If for 2 1< < < *ge ge gi R m m R R* and for *<sup>i</sup> p* the following inequalities hold:

$$\begin{split} & \frac{1}{m-1} \cdot \boldsymbol{\gamma} \cdot \frac{\boldsymbol{a}\_{c} + \boldsymbol{a}\_{\boldsymbol{\mathcal{S}}} - \boldsymbol{\pi} \prime \prime}{R\_{\boldsymbol{\mathcal{S}}^{\boldsymbol{\varepsilon}}}} \cdot \sin \boldsymbol{a}\_{\boldsymbol{\mathcal{S}}} + (m-1) \cdot \boldsymbol{\rho} \cdot \boldsymbol{g} \cdot \boldsymbol{R}\_{\boldsymbol{\mathcal{S}}^{\boldsymbol{i}}} \tan \boldsymbol{a}\_{c} - \\ & \frac{\boldsymbol{\gamma}}{m \cdot \boldsymbol{R}\_{\boldsymbol{\mathcal{S}}^{\boldsymbol{i}}}} \cdot \cos \boldsymbol{a}\_{\boldsymbol{\mathcal{S}}} \leqslant p\_{i} < \frac{1}{m'-1} \cdot \boldsymbol{\gamma} \cdot \frac{\boldsymbol{a}\_{c} + \boldsymbol{a}\_{\boldsymbol{\mathcal{S}}} - \boldsymbol{\pi} \prime \prime}{R\_{\boldsymbol{\mathcal{S}}^{\boldsymbol{i}}}} \cdot \cos \boldsymbol{a}\_{c} + \frac{\boldsymbol{\gamma}}{R\_{\boldsymbol{\mathcal{S}}^{\boldsymbol{i}}}} \cdot \sin \boldsymbol{a}\_{c} \end{split} \tag{A.2.4}$$

then there exists *ir* in the interval , *m R mR gi gi* and a concave solution of the eq. (8).

**Theorem 16.** A concave solution ( ) *<sup>i</sup> z r* of the equation (8) is a weak minimum of the free energy functional of the melt column (9).

## **9. References**

*e R n*

*r*

(A.2.1)

(A.2.2)

*gi r*

(A.2.3)

*m R*

 

> 

 

 

*ge ge*

2 *gi ge gi i ge R R Rr R* and

> 

 

and for *<sup>i</sup> p* the following inequalities hold:

*m gR m R*

*c c i g*

 
> 

*gi c g gi*

*c g*

 

> *c* ,*<sup>g</sup>* such

 

> 

 

*c g e g*

2 2 cos cos sin

and *<sup>e</sup> p* the inequalities hold:

<sup>1</sup> <sup>2</sup> sin tan sin < <

 

and a concave solution of the equation (6).

2 cos cos . 1

**Theorem 13.** A concave solution ( ) *<sup>e</sup> z r* of the equation (6) is a weak minimum of the free

**Theorem 14.** If there exists a concave solution = () *i i z zr* of the equation (8) then = *<sup>i</sup>*

 

*gi gi gi*

1 1 2 2 cos sin sin

*c g c g*

( 1) tan cos

*<sup>p</sup> mR R mR*

1 1

2

*ge gi R*

*R R*

1< < < *ge*

*m m*

*ge ge*

*n*

*n nn g R <sup>p</sup> nR n R*

 

*nR R*

*g ge c c e*

*c g*

0< < < <

118 Crystallization – Science and Technology

**Theorem 11.** If there exists a concave solution = () *e e z zr* of the equation (6) then = *ge*

 

*ge ge ge*

*c g c g*

**Definition 3.** The outer free surface is concave on [ , ] *<sup>e</sup> ge r R* if *z r* " 0 , [, ] *<sup>e</sup> ge r rR* .

*<sup>p</sup> nR R nR*

 

1 1

*n n*

and *<sup>e</sup> p* satisfy the following inequalities:

*n n <sup>g</sup> n R*

<sup>1</sup> tan sin

*n n*

, *ge ge*

*n n*

*R R*

2 1< < < *ge*

*ge gi R*

*R R*

<sup>2</sup> *<sup>g</sup>* .

*c c ge*

 

**Theorem 12.** If for

 

*c g*

then there exists

that 0 

 

*e*

energy functional of the melt column (7).

Consider now the differential equation (8) for

and *<sup>i</sup> p* satisfies the following inequalities:

 

**Theorem 15.** If for

2 2 *g c* , 0,

*r*

1


**5** 

*UK* 

*Durham University* 

*kT c c* ln / *eq* , where *k* is

, between

**Crystallization in Microemulsions:** 

Sharon Cooper, Oliver Cook and Natasha Loines

**A Generic Route to Thermodynamic Control** 

Crystallization is ubiquitous. It is evident in natural processes such as biomineralization and gem formation, and is important in industrial processes both as a purification step and in the production of materials with specific properties, including drug polymorphs, cocrystals, mesocrystals, quasicrystals, quantum dots and other inorganic nanocrystals. Consequently it is essential to gain greater understanding of the process to be able to elicit more control over

Crystallization occurs from melts that are supercooled, i.e. cooled below their equilibrium melting temperatures, *Teq*. For crystallization from solution, the solutions must be supersaturated, i.e. have solute concentrations above their saturation values, *ceq*, defined as the solute concentration in equilibrium with the macroscopic crystal. The supersaturation is

the parent (melt or solution) and daughter (new crystal) phases. For crystallization from the

supercooling with *T* denoting the temperature. Here it is assumed that *fusH* is invariant

the Boltzmann constant, and *c*/*ceq*, is the ratio of the solute concentration compared to its

The formation of any new phase from a bulk parent phase requires the creation of an interface between the two phases, which requires work. Hence there exists an energy barrier to the formation of the new phase. The process of overcoming this energy barrier is known as nucleation. In crystallization, once nucleation has occurred, crystal growth onto the nuclei proceeds until the supersaturation is relieved. Owing to this nucleation stage, crystallization from the bulk melt or solution is typically under kinetic control, with metastable forms often crystallizing initially in accordance with Ostwald's rule of stages (Ostwald, 1897). In contrast, microemulsions have the unique ability to generically exert thermodynamic control over the crystallization process. This provides significant advantages; the size of the critical nucleus can be estimated with good accuracy under thermodynamic control conditions and importantly, the stable form of a material can be identified and readily

*fusHT T*/ *eq* , where *fusH* is the enthalpy of fusion and *T* = *Teq* -*T*, is the

the driving force for crystallization, being the difference in chemical potential,

between *T* and *Teq*. For an ideal solution, the supersaturation is

saturation value, which is known as the supersaturation ratio.

**1. Introduction** 

its outcome.

melt, 

**and the Estimation of Critical Nucleus Size** 


## **Crystallization in Microemulsions: A Generic Route to Thermodynamic Control and the Estimation of Critical Nucleus Size**

Sharon Cooper, Oliver Cook and Natasha Loines *Durham University UK* 

## **1. Introduction**

120 Crystallization – Science and Technology

R. Finn, *Equilibrium capillary surfaces* (1986), Vol. 284, Grundlehren der mathematischen

I.P.Kalejs, A.A.Menna, R.W.Stormont and I.W.Hudrinson (1990), *Stress in thin hollow silicon* 

H.Kasjanow, A.Nikanorov, B.Nacke, H.Behnken, D.Franke and A.Seidl (2007), 3Dcoupled

B.Mackintosh, A.Seidl, M.Quellette, B.Bathey, D.Yates and J.Kalejs (2006) *Large silicon crystal* 

S.Rajendram, M.Larousse, B.R. Bathey and J.P.Kalejs (1993), *Silicon carbide control in the EFG* 

S.Rajendram, K.Holmes and A.Menna (1994), *Three-dimensional magnetic induction model of an* 

S.N.Rossolenko (2001), *Menisci masses and weights in Stepanov (EFG) technique: ribbon, rod,* 

A.Roy, B.Mackintosh, J.P.Kalejs, Q.S.Chen, H.Zhang and V.Prasad(2000), *A numerical model* 

A.Roy, H.Zhang, V.Prasad, B.Mackintosh, M.Quellette and J.Kalejs (2000), *Growth of large* 

D.Sun, Ch.Wang, H.Zhang, B.Mackintosh, D.Yates and J.Kalejs (2004), *A multi-block method* 

T.Surek, B.Chalmers and A.I.Mlavsky (1977), *The edge film-fed growth of controlled shape* 

J.C.Swartz, T.Surek and B.Chalmers (1975), *The EFG process applied to the growth of silicon* 

L.Tanasie, St.Balint (2010) , *Model based, pulling rate, thermal and capillary conditions setting for silicon tube growth*, Journal of Crystal Growth, vol. 312, pp. 3549-3554 V.A.Tatarchenko (1993), *Shaped crystal growth*, Kluwer Academic Publishers, Dordrecht. B.Yang, L.L.Zheng, B.MacKintosh, D.Yates and J.Kalejs (2006), *Meniscus dynamics and melt* 

*cylinders grown by the edge-defined film-fed growth technique*, Journal of .Crystal

electromagnetic and thermal modeling of EFG silicon tube growth, Journal of

*hollow-tube growth by the edge-defined film-fed growth (EFG) method*, Journal of .Crystal

*octagonal edge-defined film-fed growth system*, Journal of .Crystal Growth, Vol. 137(No

*for inductively heated cylindrical silicon tube growth system*, Journal of Crystal Growth ,

*diameter silicon tube by EFG technique: modeling and experiment*, Journal of .Crystal

*and multi-grid technique for large diameter EFG silicon tube growth*, Journal of .Crystal

*solidification in the EFG silicon tube growth process,* Journal of .Crystal Growth, Vol.

Wissenschaften, Springer, New York, NY, USA.

*system*, Journal of .Crystal Growth, Vol. 128, pp.338-342.

*tube*, Journal of .Crystal Growth, vol. 231, pp.306-315.

*crystals*, Journal of .Crystal Growth, Vol. 42, pp.453-457

*ribbons*, J.Electron.Mater, Vol. 4, pp.255-279.

Growth, Vol 104, pp.14-19.

Growth, Vol 287, pp.428-432.

1-2), pp.77-81.

Vol 211, pp.365-371.

293, pp.509-516.

Growth , Vol 230, pp.224-231.

Growth, Vol 266, pp. 167-174

.Crystal Growth, Vol. 303, pp.175-179.

Crystallization is ubiquitous. It is evident in natural processes such as biomineralization and gem formation, and is important in industrial processes both as a purification step and in the production of materials with specific properties, including drug polymorphs, cocrystals, mesocrystals, quasicrystals, quantum dots and other inorganic nanocrystals. Consequently it is essential to gain greater understanding of the process to be able to elicit more control over its outcome.

Crystallization occurs from melts that are supercooled, i.e. cooled below their equilibrium melting temperatures, *Teq*. For crystallization from solution, the solutions must be supersaturated, i.e. have solute concentrations above their saturation values, *ceq*, defined as the solute concentration in equilibrium with the macroscopic crystal. The supersaturation is the driving force for crystallization, being the difference in chemical potential, , between the parent (melt or solution) and daughter (new crystal) phases. For crystallization from the melt, *fusHT T*/ *eq* , where *fusH* is the enthalpy of fusion and *T* = *Teq* -*T*, is the supercooling with *T* denoting the temperature. Here it is assumed that *fusH* is invariant between *T* and *Teq*. For an ideal solution, the supersaturation is *kT c c* ln / *eq* , where *k* is the Boltzmann constant, and *c*/*ceq*, is the ratio of the solute concentration compared to its saturation value, which is known as the supersaturation ratio.

The formation of any new phase from a bulk parent phase requires the creation of an interface between the two phases, which requires work. Hence there exists an energy barrier to the formation of the new phase. The process of overcoming this energy barrier is known as nucleation. In crystallization, once nucleation has occurred, crystal growth onto the nuclei proceeds until the supersaturation is relieved. Owing to this nucleation stage, crystallization from the bulk melt or solution is typically under kinetic control, with metastable forms often crystallizing initially in accordance with Ostwald's rule of stages (Ostwald, 1897). In contrast, microemulsions have the unique ability to generically exert thermodynamic control over the crystallization process. This provides significant advantages; the size of the critical nucleus can be estimated with good accuracy under thermodynamic control conditions and importantly, the stable form of a material can be identified and readily

Crystallization in Microemulsions:

nucleus with a contact angle,

(3), <sup>3</sup> *f* ( ) (2 3cos cos ) / 4

exponential factor, , instead.

and \*

graph of nucleation rate, *J*, vs. supersaturation,

which gives the onset crystallization temperature, *Tc*.

A Generic Route to Thermodynamic Control and the Estimation of Critical Nucleus Size 123

forms on a foreign surface, however, heterogeneous nucleation occurs. The corresponding heterogeneous nucleation barrier, *Fhet* that arises from forming a cap-shaped critical

3 2 \* \* \*

where: \* *Fhom* denotes the homogeneous nucleation barrier of the system given by equation

heterogeneous and corresponding homogeneous critical nuclei, respectively. Note that equation (4) ignores the entropic free energy contribution arising from the number of surface sites upon which the nucleus may form, as this factor is incorporated in the pre-

Fig. 1. (a) Schematic graph of free energy change, *F* vs. nucleus size, *r*, for a homogeneously nucleating crystal showing the critical nucleus, *r*\*, and stable nucleus, *r*0, sizes. (b) Schematic

The kinetic theory of CNT (Volmer & Weber, 1926; Becker & Döring, 1935) used the nucleation barrier, *F*\*, in an Arrhenius-type equation to derive the nucleation rate, *J* as:

where is the pre-exponential factor accounting for the rate at which the molecules impinge upon, and are incorporated into, the critical nuclei. The form of equation (5) is such that *J* remains negligibly small until the supersaturation reaches a critical value, the Ostwald metastability limit, at which point *J* suddenly and dramatically increases (see Figure 1b). Hence, an onset crystallization temperature, *Tc*, can be identified with this metastability limit, and the corresponding nucleation rate can be set, with little loss in accuracy, to a

CNT is widely adopted because of its simplicity. However this simplicity limits its ability to model real systems. Two main assumptions are: firstly that it considers the nuclei to be spherical with uniform density and a structure equivalent to the bulk phase, and secondly

<sup>16</sup> ( ) <sup>3</sup>

(a) (b)

suitable detection limit for the technique monitoring the crystallization.

*het hom*

*v v F fF*

, on the foreign surface is given by:

2 \*

*c het*

*hom*

(4)

*het v* and \* *v hom* denote the volumes of the

, showing the Ostwald metastability limit

*J* exp \* *F kT* (5)

*v*

produced under ambient conditions. In this chapter we discuss the scientific rationale for this thermodynamic control and provide practical examples.

### **2. Theoretical considerations**

#### **2.1 Classical Nucleation Theory (CNT)**

Nucleation can be modelled most simply using classical nucleation theory (CNT). Gibbs thermodynamic treatment of liquid nucleation from a vapour (Gibbs, 1876, 1878) shows that the free energy change, *F*, involved in producing a spherical liquid nucleus from the vapour is given by:

$$
\Delta F = -n\Delta\mu + \gamma A = -\frac{4\pi r^3 \Delta\mu}{3v\_c} + 4\pi r^2 \gamma \tag{1}
$$

where *n* is the number of molecules in the nucleus, denotes the chemical potential difference between the vapour and the liquid nucleus which defines the supersaturation, denotes the surface tension between the nucleus and the surrounding vapour, *A* denotes the surface area of the nucleus, *r* denotes its radius and *vc* denotes the molecular volume of the new condensed phase, i.e. the liquid. This same thermodynamic treatment is often used for crystallization. For crystallization from bulk solutions at constant pressure, the relevant free energy to use is the Gibbs free energy. For crystallization from microemulsions, however, there is a very small Laplace pressure difference across the microemulsion droplet interface and so the Helmholtz free energy for constant volume systems is the appropriate free energy to employ.

Equation (1) clearly shows that the favourable formation of the bulk new phase in any supersaturated system, given by the first term -*n*, is offset by the unfavourable surface free energy term, *A*, that necessarily arises from creating the new interface. The surface free energy term dominates at smaller nucleus sizes and leads to the nucleation energy barrier (see Figure 1a). In particular, differentiating equation (1) with respect to *r* leads to a maximum at

\*2 /*<sup>c</sup> r v* , (2)

i.e. the well-known Gibbs-Thomson equation, with the nucleation barrier given by:

$$
\Delta \mathbf{F}^\* = \frac{16}{\Im \Delta \mu^2} \pi \boldsymbol{\eta}^3 \boldsymbol{v}\_c^{-2} \,. \tag{3}
$$

The *r*\* nucleus is termed the critical nucleus. It is of pivotal importance in CNT because it determines the size above which it is favourable for the new phase to grow. Nuclei smaller than *r*\* have a greater tendency to dissolve (or melt) than grow, whilst nuclei larger than *r*\* will tend to grow. The *r*\* nucleus has an equal probability of growing or dissolving (melting) and is in an unstable equilibrium with the surrounding solution (or melt). At larger *r*, a stable *r*0 nucleus with *F* = 0 occurs (see Figure 1a).

Equation (3) gives the magnitude of the nucleation barrier when the new phase forms within the bulk parent phase, which is known as homogeneous nucleation. If the new phase

produced under ambient conditions. In this chapter we discuss the scientific rationale for

Nucleation can be modelled most simply using classical nucleation theory (CNT). Gibbs thermodynamic treatment of liquid nucleation from a vapour (Gibbs, 1876, 1878) shows that the free energy change, *F*, involved in producing a spherical liquid nucleus from the

*<sup>r</sup> Fn A <sup>r</sup>*

difference between the vapour and the liquid nucleus which defines the supersaturation,

denotes the surface tension between the nucleus and the surrounding vapour, *A* denotes the surface area of the nucleus, *r* denotes its radius and *vc* denotes the molecular volume of the new condensed phase, i.e. the liquid. This same thermodynamic treatment is often used for crystallization. For crystallization from bulk solutions at constant pressure, the relevant free energy to use is the Gibbs free energy. For crystallization from microemulsions, however, there is a very small Laplace pressure difference across the microemulsion droplet interface and so the Helmholtz free energy for constant volume systems is the appropriate free

Equation (1) clearly shows that the favourable formation of the bulk new phase in any

free energy term, *A*, that necessarily arises from creating the new interface. The surface free energy term dominates at smaller nucleus sizes and leads to the nucleation energy barrier (see Figure 1a). In particular, differentiating equation (1) with respect to *r* leads to a

> \*2 /*<sup>c</sup> r v*

\* 3 2 2 16 3 *F v c*

The *r*\* nucleus is termed the critical nucleus. It is of pivotal importance in CNT because it determines the size above which it is favourable for the new phase to grow. Nuclei smaller than *r*\* have a greater tendency to dissolve (or melt) than grow, whilst nuclei larger than *r*\* will tend to grow. The *r*\* nucleus has an equal probability of growing or dissolving (melting) and is in an unstable equilibrium with the surrounding solution (or melt). At larger *r*, a

Equation (3) gives the magnitude of the nucleation barrier when the new phase forms within the bulk parent phase, which is known as homogeneous nucleation. If the new phase

i.e. the well-known Gibbs-Thomson equation, with the nucleation barrier given by:

  <sup>3</sup> <sup>4</sup> <sup>2</sup> <sup>4</sup> 3 *<sup>c</sup>*

 

(1)

denotes the chemical potential

, is offset by the unfavourable surface

, (2)

. (3)

*v* 

this thermodynamic control and provide practical examples.

where *n* is the number of molecules in the nucleus,

supersaturated system, given by the first term -*n*

stable *r*0 nucleus with *F* = 0 occurs (see Figure 1a).

**2. Theoretical considerations** 

vapour is given by:

energy to employ.

maximum at

**2.1 Classical Nucleation Theory (CNT)** 

forms on a foreign surface, however, heterogeneous nucleation occurs. The corresponding heterogeneous nucleation barrier, *Fhet* that arises from forming a cap-shaped critical nucleus with a contact angle, , on the foreign surface is given by:

$$
\Delta F\_{het}^\* = \frac{16\pi\nu^3 \upsilon\_c^2}{3\Delta\mu^2} f(\theta) = \Delta F\_{hom}^\* \frac{\upsilon\_{het}^\*}{\upsilon\_{hom}^\*} \tag{4}
$$

where: \* *Fhom* denotes the homogeneous nucleation barrier of the system given by equation (3), <sup>3</sup> *f* ( ) (2 3cos cos ) / 4 and \* *het v* and \* *v hom* denote the volumes of the heterogeneous and corresponding homogeneous critical nuclei, respectively. Note that equation (4) ignores the entropic free energy contribution arising from the number of surface sites upon which the nucleus may form, as this factor is incorporated in the preexponential factor, , instead.

Fig. 1. (a) Schematic graph of free energy change, *F* vs. nucleus size, *r*, for a homogeneously nucleating crystal showing the critical nucleus, *r*\*, and stable nucleus, *r*0, sizes. (b) Schematic graph of nucleation rate, *J*, vs. supersaturation, , showing the Ostwald metastability limit which gives the onset crystallization temperature, *Tc*.

The kinetic theory of CNT (Volmer & Weber, 1926; Becker & Döring, 1935) used the nucleation barrier, *F*\*, in an Arrhenius-type equation to derive the nucleation rate, *J* as:

$$J = \Omega \exp\left(-\Delta F \, ^\ast \!/ kT\right) \tag{5}$$

where is the pre-exponential factor accounting for the rate at which the molecules impinge upon, and are incorporated into, the critical nuclei. The form of equation (5) is such that *J* remains negligibly small until the supersaturation reaches a critical value, the Ostwald metastability limit, at which point *J* suddenly and dramatically increases (see Figure 1b). Hence, an onset crystallization temperature, *Tc*, can be identified with this metastability limit, and the corresponding nucleation rate can be set, with little loss in accuracy, to a suitable detection limit for the technique monitoring the crystallization.

CNT is widely adopted because of its simplicity. However this simplicity limits its ability to model real systems. Two main assumptions are: firstly that it considers the nuclei to be spherical with uniform density and a structure equivalent to the bulk phase, and secondly

Crystallization in Microemulsions:

is decreased.

A Generic Route to Thermodynamic Control and the Estimation of Critical Nucleus Size 125

larger. In contrast on a convex substrate, *v*\* is increased compared to the planar case, but

(a) (b)

*T TT trans eq trans* , and substituting in equations (3), (5) and (7), gives:

*eq*

*c p fus eq trans*

> , *Teq*, *vc*,

be obtained from the Gibbs-Thomson equation, and \* *R rx* , respectively.

 

*T*

3 2

*c m*

2 <sup>1</sup> 72 ( ) arccos 1 3 ln /

*k HT J*

where

*v f <sup>W</sup>*

Hence *Tc* and *Tm* can be found with *x,*

*vT f T TT*

This equation has three roots corresponding to (1) The onset crystallization temperature *Tc*, (2) the expected onset melting temperature, *Tm*, for a nucleation-based melting transition, which would be required if surface melting did not occur, and (3) a non-physical root *TT W c eq* 2 / 3 1 cos close to 0 K corresponding to the case where the critical nucleus contains only one molecule and the energy barrier is vanishingly small. *Tc* and *Tm* are given by:

3 2

*trans trans eq*

Fig. 2. (a) Schematic diagram describing nucleation upon a concave substrate of radius, *R.*  The dark grey regions depict the nucleus on the concave surface. (b) Schematic diagram showing that for a given supersaturation, and hence critical nucleus radius, *r*\*, all surfaces

through point P that cross the homogeneous critical nucleus surface produce the same \* *Fhet* value for nucleation, since they all have the same

The onset temperature for the phase transition, e.g. the highest temperature, *Ttrans,* at which crystallization should be observable, can then be found by setting the nucleation rate, *Jtrans*, at *Ttrans* to a suitable detection limit, where *Jtrans* exp \* *F kTtrans* . The pre-exponential factor, , can be considered constant provided the temperature range is narrow, since *F*\* has the greater temperature dependence. Using \* ln( / ) *Fk T T J eq trans trans* , where

32 2

2

*kH J*

3 ln /

 0.5 , 2 cos (3 sin ) <sup>3</sup>

.

*T WW* (9)

, *fusH* and /*Jtrans* as input. *r*\* and *R* can then

 

16 ( ) <sup>0</sup>

*fus trans*

*c eq p*

*<sup>p</sup>* value (Cooper et al, 2008).

. (8)

that the nuclei interfaces are infinitely sharp and have the same interfacial tensions as found at the corresponding planar interfaces. A recent review by Erdemir et al., 2009, details all the assumptions of CNT, and its applicability to different systems. Notably, given that CNT stems from the condensation of a liquid from its vapour, it cannot model two stage nucleation (Vekilov, 2010), where solute molecules organize initially into an amorphous cluster, from which long range crystal order then emerges on cluster rearrangement. Despite these many limitations, CNT is useful to benchmark crystallization experiments because this approach has been so widely adopted. More importantly here, it allows useful insights into the crystallization process that are readily apparent due to its simplicity. In particular, the use of CNT has enabled us to establish that thermodynamic control of crystallization is possible in 3D nanoconfined volumes, as shown below.

#### **2.1.1 Adoption of CNT to curved interfaces**

For crystallization in nanodroplets, the planar substrate of CNT's heterogeneous nucleation formulation is replaced by a highly curved concave substrate. For such curved substrates, the free energy becomes (Cooper et al., 2008; Fletcher, 1958):

$$\Delta \mathbf{F}\_{het} = -\frac{4}{3\upsilon\_c} \pi r^3 \Delta \mu [(f(\theta + \phi) - (\mathcal{R} \n) \ r)^3 f(\phi)] + 2 \left[ 1 - \cos(\theta + \phi) \right] \pi r^2 \gamma - 2 \cos \theta (1 - \cos \phi) \pi R^2 \gamma \tag{6}$$

where is the contact angle between the nucleus and the spherical substrate, is the angle between the spherical substrate and the plane connecting the nucleus edge and <sup>3</sup> *f* ( ) 0.25 2 3cos cos (see Figure 2a). Note that for concave surfaces, corresponding to crystallization within the curved substrate, *R* and are assigned negative values.

The maximum in *Fhet* gives the barrier to nucleation, \* *Fhet* , and again this condition is satisfied when \*2 /*<sup>c</sup> r v* to give:

$$
\Delta F\_{\text{het}}^{\*} = \frac{\Delta F\_{\text{hom}}^{\*}}{2} \left\{ 1 - 3\mathbf{x}^{2} \cos \theta + 2\mathbf{x}^{3} + y(1 + \mathbf{x} \cos \theta - 2\mathbf{x}^{2}) \right\} = \Delta F\_{\text{hom}}^{\*} f(\theta\_{p}) \tag{7}
$$

where *x Rr* / \* , 0.5 <sup>2</sup> *y xx* 2 cos 1 with the positive and negative roots applying to a nucleus on a convex and concave surface, respectively and cos*θ<sup>p</sup> x y* . *<sup>p</sup>* is the angle between the corresponding planar critical nucleus and the plane tangential to the curved substrate surface, as shown in Figure 2a.

Equation (7) shows that at a given temperature, and hence constant *\* Fhom* value, \* *Fhet* depends only upon the *<sup>p</sup>* value. Consequently in Figure 2b, all the spherical substrates depicted result in the same \* *Fhet* value. This can be rationalized as follows. For nucleation on a concave surface, the critical nucleus volume, *v*\*, is reduced compared to the planar case, and hence fewer molecules need to cluster together to form the critical nucleus. However this effect is negated by the greater contact angle, , compared to *<sup>p</sup>*, which means that more work is required to create unit area of the nucleus-substrate interface, and so the mean energy increase on addition of a molecule to the sub-critical nucleus is

that the nuclei interfaces are infinitely sharp and have the same interfacial tensions as found at the corresponding planar interfaces. A recent review by Erdemir et al., 2009, details all the assumptions of CNT, and its applicability to different systems. Notably, given that CNT stems from the condensation of a liquid from its vapour, it cannot model two stage nucleation (Vekilov, 2010), where solute molecules organize initially into an amorphous cluster, from which long range crystal order then emerges on cluster rearrangement. Despite these many limitations, CNT is useful to benchmark crystallization experiments because this approach has been so widely adopted. More importantly here, it allows useful insights into the crystallization process that are readily apparent due to its simplicity. In particular, the use of CNT has enabled us to establish that thermodynamic control of crystallization is possible in 3D nanoconfined volumes, as

For crystallization in nanodroplets, the planar substrate of CNT's heterogeneous nucleation formulation is replaced by a highly curved concave substrate. For such curved substrates,

<sup>4</sup> 33 2 2 [( ( ) ( / ) ( )] 2 1 cos( ) 2 cos 1 cos <sup>3</sup> *het*

between the spherical substrate and the plane connecting the nucleus edge and

The maximum in *Fhet* gives the barrier to nucleation, \* *Fhet* , and again this condition is

 \* \* hom 2 3 2 \* 1 3 cos 2 (1 cos 2 ) ( ) hom <sup>2</sup> *het <sup>p</sup> <sup>F</sup> <sup>F</sup> x xy x x Ff*

between the corresponding planar critical nucleus and the plane tangential to the curved

Equation (7) shows that at a given temperature, and hence constant *\* Fhom* value, \* *Fhet*

depicted result in the same \* *Fhet* value. This can be rationalized as follows. For nucleation on a concave surface, the critical nucleus volume, *v*\*, is reduced compared to the planar case, and hence fewer molecules need to cluster together to form the critical nucleus.

means that more work is required to create unit area of the nucleus-substrate interface, and so the mean energy increase on addition of a molecule to the sub-critical nucleus is

 

(7)

(see Figure 2a). Note that for concave surfaces, corresponding

*<sup>p</sup>* value. Consequently in Figure 2b, all the spherical substrates

are assigned negative values.

with the positive and negative roots applying to

, compared to

   (6)

is the angle

 

*<sup>p</sup>* is the angle

*<sup>p</sup>*, which

*F r f Rr f r R*

is the contact angle between the nucleus and the spherical substrate,

 

shown below.

*c*

satisfied when \*2 /*<sup>c</sup> r v*

depends only upon the

<sup>3</sup> *f* ( ) 0.25 2 3cos cos

 

 

where *x Rr* / \* ,

substrate surface, as shown in Figure 2a.

*v*

where 

**2.1.1 Adoption of CNT to curved interfaces** 

the free energy becomes (Cooper et al., 2008; Fletcher, 1958):

 

to give:

0.5 <sup>2</sup> *y xx* 2 cos 1

However this effect is negated by the greater contact angle,

a nucleus on a convex and concave surface, respectively and cos*θ<sup>p</sup> x y* .

to crystallization within the curved substrate, *R* and

larger. In contrast on a convex substrate, *v*\* is increased compared to the planar case, but is decreased.

Fig. 2. (a) Schematic diagram describing nucleation upon a concave substrate of radius, *R.*  The dark grey regions depict the nucleus on the concave surface. (b) Schematic diagram showing that for a given supersaturation, and hence critical nucleus radius, *r*\*, all surfaces through point P that cross the homogeneous critical nucleus surface produce the same \* *Fhet* value for nucleation, since they all have the same *<sup>p</sup>* value (Cooper et al, 2008).

The onset temperature for the phase transition, e.g. the highest temperature, *Ttrans,* at which crystallization should be observable, can then be found by setting the nucleation rate, *Jtrans*, at *Ttrans* to a suitable detection limit, where *Jtrans* exp \* *F kTtrans* . The pre-exponential factor, , can be considered constant provided the temperature range is narrow, since *F*\* has the greater temperature dependence. Using \* ln( / ) *Fk T T J eq trans trans* , where *T TT trans eq trans* , and substituting in equations (3), (5) and (7), gives:

$$
\Delta T\_{trans}{}^3 - \Delta T\_{trans}{}^2 T\_{eq} + \frac{16\pi \gamma^3 v\_c{}^2 T\_{eq}{}^2 f(\theta\_p)}{3k\Delta\_{fus}H^2 \ln\left(\Omega \,/\ J\_{trans}\right)} = 0 \,\tag{8}
$$

This equation has three roots corresponding to (1) The onset crystallization temperature *Tc*, (2) the expected onset melting temperature, *Tm*, for a nucleation-based melting transition, which would be required if surface melting did not occur, and (3) a non-physical root *TT W c eq* 2 / 3 1 cos close to 0 K corresponding to the case where the critical nucleus contains only one molecule and the energy barrier is vanishingly small. *Tc* and *Tm* are given by:

$$T\_{c,m} = \frac{T\_{eq}}{3} \left\{ 2 + \cos \mathcal{W} \mp \left( 3^{0.5} \sin \mathcal{W} \right) \right\} \tag{9}$$

$$\text{where } \mathcal{W} = \frac{1}{3} \arccos\left[1 - \frac{72\pi\gamma^3 v\_c^2 f(\theta\_p)}{k\Delta\_{fus}H^2 T\_{c\eta} \ln\left(\Omega \,/\ J\_{\text{trans}}\right)}\right].$$

Hence *Tc* and *Tm* can be found with *x,*, *Teq*, *vc*, , *fusH* and /*Jtrans* as input. *r*\* and *R* can then be obtained from the Gibbs-Thomson equation, and \* *R rx* , respectively.

Crystallization in Microemulsions:

performed on the same system.

from equation (10) if *R* and

nucleus is then given by:

*n*

The dependence of *n*\* on

nucleation (

<sup>3</sup> \* 32 | | /81 *n Rv* 

crystallization of the spherical droplet, we find:

*v R*

*v v* 


A Generic Route to Thermodynamic Control and the Estimation of Critical Nucleus Size 127

denoted *Rmin*, corresponds to the system where the total phase transition of the droplet, from all liquid to all crystal, or vice versa, occurs with a free energy change *Ftot* = 0. For droplet sizes smaller than |*Rmin*|, the nucleation-melting and crystallization curves describe systems where *Ftot* > 0, so the phase transformation would not proceed. This means that for sizes below |*Rmin*|, the critical nucleus size can be attained as \* *Fhet* is surmountable, but there is then insufficient material within the confining substrate for *F* to decrease to zero through further nucleus growth, e.g. in Figure 1a, the nucleus cannot grow to a size *r*0. Consequently, below |*Rmin*| the crystallization and nucleation-melting curves must both follow the same curve labelled *Ftot* = 0 in the Figure to ensure a thermodynamically feasible phase transformation occurs. Hence the hysteresis normally observed upon heating and cooling the *same* system would be expected to disappear for phase transformations confined to within volumes with |*R*| |*Rmin*|. Unfortunately, there is a difficulty in verifying this lack of hysteresis experimentally, because this would require the *r*0 nuclei to be constrained to this size. Typically, however, the *r*0 nuclei subsequently grow via collisions with uncrystallized droplets, by oriented attachment of other nuclei, or by Ostwald ripening, and so it is difficult to ensure that subsequent melting and crystallization cycles are indeed

Using the condition that for droplet sizes with |*R*||*Rmin*|, *Ftot*=0 on complete

<sup>3</sup> <sup>3</sup> cos \* cos 2

are known. This is an important finding because determination

. (10)

 

can only be estimated to

. (11)

Here we have retained the convention that the substrate radii, *R*, must take negative values for nucleation on a concave surface. Consequently, the critical nucleus size can be obtained

of *r*\* usually relies on the Gibbs-Thomson equation and the inappropriate application of bulk interfacial tension values to small nuclei. The number of molecules, *n*\*, in the critical

is relatively weak, so that even if

within ~10%, *n*\* is known with good precision if *R* can be measured. For homogeneous

An empirical determination of the *Rmin* value is possible for homogeneous nucleation because the onset crystallization temperature, *Tc*, should be approximately constant for confinements with sizes above |*Rmin*|. Note though that nucleation is a stochastic process, so repeated experiments will show some slight variation but an expected homogeneous nucleation temperature should nevertheless be apparent. For instance, the homogeneous nucleation temperature for ice is -40 C (Wood & Walton, 1970; Clausse et al., 1983).

\* 4 | | 1 4 2 (3 / 4)cos (9 /8)cos \*1 3 2 27 cos *c c* 4 3cos

3 2 4

<sup>3</sup> 0.5 <sup>2</sup>

= 180), equation (10) and (11) simplify to \* 2| |/3 *r R* and

*<sup>c</sup>* , so experimental measurement of *R* directly gives *r*\* and *n*\* provided

*<sup>c</sup> <sup>v</sup> R r* 

From equation (7), we find that for a constant contact angle, , the energy barrier to nucleation is smaller for a concave surface than a convex one, and that the reduction in energy increases as |*x*| increases. The onset crystallization temperatures, *Tc,* obtained from equation (9) are therefore correspondingly higher for concave surfaces. Figure 3 shows the expected *Tc* as a function of |*R*| for the case of ice crystallization on a concave surface with a contact angle, , between the crystal nucleus and substrate of (a) 180, i.e. the homogeneous nucleation case and (b) 100. The onset melting temperatures, *Tm*, expected for the same systems (i.e. with supplementary contact angles between the melt-nucleus and substrate of (a) 0 and (b) 80) in the absence of surface melting are also shown in this Figure. This melting is denoted nucleation-melting. The *Tm* and *Tc* curves meet at |*Rmin*| and at smaller concave radii, the melting curve falls below the crystallization one, which is clearly non-physical. Hence, equation (9) cannot model crystallization in 3D nanoconfinements smaller than |*Rmin*|. This demonstrates a fundamental limitation of the theory, and shows that a key factor necessary for crystallization has been ignored.

Fig. 3. The predicted ice onset crystallization temperatures, *Tc*, (filled diamonds) and nucleation-melting temperatures, *Tm*, (dashes), which would occur in the absence of surface melting, as a function of substrate radius, |*R*|, for crystallization within a spherical substrate with (a) = 180 and (b) =100. The ice *Tc* have been determined using reasonable values (Pruppacher, 1995; Bartell , 1998; Speedy, 1987) of *vc* = 3.26 10-29 m3, = 20 mN m-1, *fusH* = 4060 J mol-1 and /*Jtrans* = 1018 for homogeneous nucleation and values of *vc* = 3.26 10-29 m3, = 22 mN m-1, *fusH* = 5000 J mol-1 and /*Jtrans* = 1015 for the heterogeneous nucleation case. Note that for |*R*| < |*Rmin*|, the ice *Tc* and *Tm* are both given by the curve labelled *Ftot* = 0, as the transition temperature is now determined by the condition that the nucleus grows to a size *r*0.

#### **2.2 Phase transitions within nanoconfined volumes (Cooper et al., 2008)**

#### **2.2.1 Crystallization from the melt**

The phase transition temperature in equation (9) is determined only by the ability to surmount the nucleation barrier. The thermodynamic feasibility of the transition, i.e. whether the new phase is stable with *F* 0, however, is not considered. In fact, the crossing point of the crystallization and nucleation-melting curves, occurring at a droplet size

nucleation is smaller for a concave surface than a convex one, and that the reduction in energy increases as |*x*| increases. The onset crystallization temperatures, *Tc,* obtained from equation (9) are therefore correspondingly higher for concave surfaces. Figure 3 shows the expected *Tc* as a function of |*R*| for the case of ice crystallization on a concave surface with

homogeneous nucleation case and (b) 100. The onset melting temperatures, *Tm*, expected for the same systems (i.e. with supplementary contact angles between the melt-nucleus and substrate of (a) 0 and (b) 80) in the absence of surface melting are also shown in this Figure. This melting is denoted nucleation-melting. The *Tm* and *Tc* curves meet at |*Rmin*| and at smaller concave radii, the melting curve falls below the crystallization one, which is clearly non-physical. Hence, equation (9) cannot model crystallization in 3D nanoconfinements smaller than |*Rmin*|. This demonstrates a fundamental limitation of the

theory, and shows that a key factor necessary for crystallization has been ignored.

(a) (b)

**2.2 Phase transitions within nanoconfined volumes (Cooper et al., 2008)** 

Fig. 3. The predicted ice onset crystallization temperatures, *Tc*, (filled diamonds) and nucleation-melting temperatures, *Tm*, (dashes), which would occur in the absence of surface

melting, as a function of substrate radius, |*R*|, for crystallization within a spherical

reasonable values (Pruppacher, 1995; Bartell , 1998; Speedy, 1987) of *vc* = 3.26 10-29 m3,

= 20 mN m-1, *fusH* = 4060 J mol-1 and /*Jtrans* = 1018 for homogeneous nucleation and values

heterogeneous nucleation case. Note that for |*R*| < |*Rmin*|, the ice *Tc* and *Tm* are both given by the curve labelled *Ftot* = 0, as the transition temperature is now determined by the

The phase transition temperature in equation (9) is determined only by the ability to surmount the nucleation barrier. The thermodynamic feasibility of the transition, i.e. whether the new phase is stable with *F* 0, however, is not considered. In fact, the crossing point of the crystallization and nucleation-melting curves, occurring at a droplet size

= 22 mN m-1, *fusH* = 5000 J mol-1 and /*Jtrans* = 1015 for the

, between the crystal nucleus and substrate of (a) 180, i.e. the

=100. The ice *Tc* have been determined using

, the energy barrier to

From equation (7), we find that for a constant contact angle,

a contact angle,

substrate with (a)

of *vc* = 3.26 10-29 m3,

condition that the nucleus grows to a size *r*0.

**2.2.1 Crystallization from the melt** 

= 180 and (b)

denoted *Rmin*, corresponds to the system where the total phase transition of the droplet, from all liquid to all crystal, or vice versa, occurs with a free energy change *Ftot* = 0. For droplet sizes smaller than |*Rmin*|, the nucleation-melting and crystallization curves describe systems where *Ftot* > 0, so the phase transformation would not proceed. This means that for sizes below |*Rmin*|, the critical nucleus size can be attained as \* *Fhet* is surmountable, but there is then insufficient material within the confining substrate for *F* to decrease to zero through further nucleus growth, e.g. in Figure 1a, the nucleus cannot grow to a size *r*0. Consequently, below |*Rmin*| the crystallization and nucleation-melting curves must both follow the same curve labelled *Ftot* = 0 in the Figure to ensure a thermodynamically feasible phase transformation occurs. Hence the hysteresis normally observed upon heating and cooling the *same* system would be expected to disappear for phase transformations confined to within volumes with |*R*| |*Rmin*|. Unfortunately, there is a difficulty in verifying this lack of hysteresis experimentally, because this would require the *r*0 nuclei to be constrained to this size. Typically, however, the *r*0 nuclei subsequently grow via collisions with uncrystallized droplets, by oriented attachment of other nuclei, or by Ostwald ripening, and so it is difficult to ensure that subsequent melting and crystallization cycles are indeed performed on the same system.

Using the condition that for droplet sizes with |*R*||*Rmin*|, *Ftot*=0 on complete crystallization of the spherical droplet, we find:

$$R = \frac{3v\_c\gamma}{\Delta\mu} \cos\theta = \frac{3}{2}r \,\, ^\*\cos\theta \,\, \, \, \tag{10}$$

Here we have retained the convention that the substrate radii, *R*, must take negative values for nucleation on a concave surface. Consequently, the critical nucleus size can be obtained from equation (10) if *R* and are known. This is an important finding because determination of *r*\* usually relies on the Gibbs-Thomson equation and the inappropriate application of bulk interfacial tension values to small nuclei. The number of molecules, *n*\*, in the critical nucleus is then given by:

$$n^\* = \frac{\upsilon^\*}{\upsilon\_c} = \frac{4\pi \left| R \right|^3}{3\upsilon\_c} \left| \frac{1}{2} - \frac{4}{27\cos^3\theta} \left| 1 - \frac{2 - (3/4)\cos^2\theta + (9/8)\cos^4\theta}{\left(4 - 3\cos^2\theta\right)^{0.5}} \right| \right| \tag{11}$$

The dependence of *n*\* on is relatively weak, so that even if can only be estimated to within ~10%, *n*\* is known with good precision if *R* can be measured. For homogeneous nucleation ( = 180), equation (10) and (11) simplify to \* 2| |/3 *r R* and <sup>3</sup> \* 32 | | /81 *n Rv <sup>c</sup>* , so experimental measurement of *R* directly gives *r*\* and *n*\* provided |*R*||*Rmin*|. So we just need to find the value of |*Rmin*|.

An empirical determination of the *Rmin* value is possible for homogeneous nucleation because the onset crystallization temperature, *Tc*, should be approximately constant for confinements with sizes above |*Rmin*|. Note though that nucleation is a stochastic process, so repeated experiments will show some slight variation but an expected homogeneous nucleation temperature should nevertheless be apparent. For instance, the homogeneous nucleation temperature for ice is -40 C (Wood & Walton, 1970; Clausse et al., 1983).

Crystallization in Microemulsions:

nucleus grows. *r*\* and \*

and \* *Fmin* both given by:

where

(16) is solvable with *x*,

1

2

3

and

≤ *Teq*, *Tc* is then given by:

and

16 3 ln /

*vf T <sup>X</sup>*

3ln /

0

3

3

0.5 <sup>2</sup>

3

*TX Y eq <sup>Z</sup> Teq <sup>z</sup>* 

*kJ H*

32 2

 

*c p eq trans fus*

 

, *c*0, *Teq*, *vc*,

where: 0.5 <sup>2</sup>

0.5 <sup>2</sup>

cos 12 3 16

32 2

3

2 2

 

*c eq p*

solution at concentration *c*\* surrounding the critical nucleus, *r*\*.

2

*c eq*

3

*c*

*TX Y eq Z Z Teq <sup>z</sup>* 

*TX Y eq Z Z Teq <sup>z</sup>* 

3 4 *eq*

*T*

cos 3 sin 48 3 3 16

cos 3 sin 48 3 3 16

*trans fus c c <sup>c</sup> vT x v v <sup>Y</sup>*

<sup>32</sup> \*1 \* ln 1 ln 1

*J H Vv c v c V*

0.5

0.5

,

Equation (16) reduces to equation (8), the melt crystallization case, when *Y* = 0. Equation

A Generic Route to Thermodynamic Control and the Estimation of Critical Nucleus Size 129

volume of the crystalline species and *c*0 denotes the initial solute concentration when *n* = 0. The first two terms comprise those expected from classical nucleation theory for crystallization from unconfined volumes, whilst the third term provides the correction due to the supersaturation depletion as the nucleus grows. The free energy difference, *F*, now exhibits a maximum, \* *<sup>F</sup>* , corresponding to the critical nucleus radius, \* *<sup>r</sup>* , *and a minimum*, \* *Fmin* , at a larger nucleus radius, \*

16 ( ) \*1 \* \* ln 1 ln 1

where *v*\* denotes the nucleus volume when *r* = *r*\*, with the subscript *min* used to distinguish the minimum value from the maximum, and *Teq* denotes the saturation temperature for the

As before, the onset crystallization temperature is found from \* / ln( / ) *F kT J <sup>c</sup> trans* ,

with assumed constant. This leads to a quartic equation (Cooper et al., 2008):

*H T Vv c v c V*

*fus c c c vT f v v <sup>F</sup> NkT*

*min r* , owing to the decrease in the supersaturation as the

, (15)

*min r* are both given by the usual Gibbs-Thomson equation with \* *F*

0 0

4 3 () 0 *T T T X Y T TY c eq c c eq* (16)

, *fusH* and /*Jtrans* as input. For the typical case where *Tc*

*T zzz* (17)

,

,

.

0 0

0.5 0.5 0.5 123

Consequently, |R*min*| is readily identifiable as the droplet size at which *Tc* begins to decrease with decreasing |*R*|, provided the system nucleates homogeneously. For heterogeneous nucleation, is also likely to be a function of *R*, so an empirical determination is more difficult. Instead, the theoretical *Rmin* value can be used, which is obtained as follows.

Substituting for *r*\* in equation (10) using the Gibbs-Thompson equation, \*2 / 2 / *<sup>c</sup> c eq fus trans r v vT H T* , we find that for |*R*| ≤ |*Rmin*|, *Tc* and T*m* are given by:

$$T\_c = T\_m = T\_{eq} \left[ 1 + \frac{3\gamma v\_c \cos\theta}{|R|\ln\Delta\_{fus}H} \right]. \tag{12}$$

Equation (12) has previously been identified (Couchman & Jesser, 1977) as giving the minimum possible melting temperature of a small particle. Our analysis shows that it also gives the maximum possible freezing temperature of a confined object (Vanfleet & Mochel, 1995; Enustun et al., 1990). *Rmin* is then readily obtained by substituting in equation (9), since the *Tc* and *Tm* curves from equation (9) meet at *Rmin*. Hence:

$$R\_{min} = \frac{9\gamma v\_c \cos\theta}{\Delta\_{fus}H\left\{1 - \cos\mathcal{W} + \left(3^{0.5}\sin\mathcal{W}\right)\right\}}\tag{13}$$

Finding reliable *Rmin* values from equation (13) requires knowledge of , and *fusH*, which is problematic since the use of bulk , and *fusH* values for such small systems is likely to introduce unquantifiable errors. Consequently, the preferred methodology is that of using homogeneously nucleating systems to identify |*Rmin*| from the confinement size below which the onset crystallization temperatures, *Tc*, start to decrease. Then critical nucleus sizes can be reliably found for all sizes below |*Rmin*| using \* 2| |/3 *r R* and <sup>3</sup> \* 32 | | /81 *n Rv <sup>c</sup>* . Fortunately, crystallization in microemulsions often proceeds via homogeneous nucleation, making these the system of choice.

#### **2.2.2 Crystallization from nanoconfined solution**

Our crystallization model can be extended to crystallization of solutes from nanoconfined solutions, though here the situation is complicated by the decrease in supersaturation that arises as the nucleus grows. In particular, by adopting a classical homogeneous nucleation approach for crystallization from an ideal solution in a spherical confining volume, *V*, the free energy change, *F*, to produce a nuclei containing *n* molecules would be given by (Cooper et al., 2008; Reguera et al., 2003) :

$$\Delta F = -n\Delta\mu + \gamma A + NkT \left\{ \ln \left( 1 - \frac{v}{Vv\_c c\_0} \right) - \frac{1}{v\_c c\_0} \ln \left( 1 - \frac{v}{V} \right) \right\} \tag{14}$$

where *kT c c* ln / *eq* denotes the supersaturation at that nucleus size, with *c N n V v c v Vv c v V* 0 0 1 *<sup>c</sup>* 11 / , and *A* denote the interfacial tension and surface area, respectively, at the nucleus-solution interface, *N* is the initial number of solute molecules when *n* = 0, *v* denotes the nucleus volume, *vc* denotes the molecular volume of the crystalline species and *c*0 denotes the initial solute concentration when *n* = 0. The first two terms comprise those expected from classical nucleation theory for crystallization from unconfined volumes, whilst the third term provides the correction due to the supersaturation depletion as the nucleus grows. The free energy difference, *F*, now exhibits a maximum, \* *<sup>F</sup>* , corresponding to the critical nucleus radius, \* *<sup>r</sup>* , *and a minimum*, \* *Fmin* , at a larger nucleus radius, \* *min r* , owing to the decrease in the supersaturation as the nucleus grows. *r*\* and \* *min r* are both given by the usual Gibbs-Thomson equation with \* *F* and \* *Fmin* both given by:

$$\Delta F^\* = \frac{16\pi \eta^3 \upsilon\_c^2 T\_{eq} ^2 f(\theta\_p)}{3\Delta\_{\text{fus}} H^2 \Delta T\_c^2} + \text{NkT} \left\{ \ln \left( 1 - \frac{\upsilon^\*}{V \upsilon\_c c\_0} \right) - \frac{1}{\upsilon\_c c\_0} \ln \left( 1 - \frac{\upsilon^\*}{V} \right) \right\} \tag{15}$$

where *v*\* denotes the nucleus volume when *r* = *r*\*, with the subscript *min* used to distinguish the minimum value from the maximum, and *Teq* denotes the saturation temperature for the solution at concentration *c*\* surrounding the critical nucleus, *r*\*.

As before, the onset crystallization temperature is found from \* / ln( / ) *F kT J <sup>c</sup> trans* , with assumed constant. This leads to a quartic equation (Cooper et al., 2008):

$$
\Delta T\_c^4 - T\_{eq} \Delta T\_c^3 + (X - Y) \Delta T\_c + T\_{eq} Y = 0 \tag{16}
$$

where 16 3 ln / *vf T <sup>X</sup> kJ H* 

32 2

 

*c p eq trans fus*

128 Crystallization – Science and Technology

Consequently, |R*min*| is readily identifiable as the droplet size at which *Tc* begins to decrease with decreasing |*R*|, provided the system nucleates homogeneously. For heterogeneous

Substituting for *r*\* in equation (10) using the Gibbs-Thompson equation,

*<sup>c</sup> c eq fus trans r v vT H T* , we find that for |*R*| ≤ |*Rmin*|, *Tc* and T*m* are given by:

Equation (12) has previously been identified (Couchman & Jesser, 1977) as giving the minimum possible melting temperature of a small particle. Our analysis shows that it also gives the maximum possible freezing temperature of a confined object (Vanfleet & Mochel, 1995; Enustun et al., 1990). *Rmin* is then readily obtained by substituting in equation (9), since

difficult. Instead, the theoretical *Rmin* value can be used, which is obtained as follows.

*<sup>c</sup> c m eq*

*<sup>v</sup> TT T*

*<sup>c</sup> min fus <sup>v</sup> <sup>R</sup>*

Finding reliable *Rmin* values from equation (13) requires knowledge of

, 

 

the *Tc* and *Tm* curves from equation (9) meet at *Rmin*. Hence:

is also likely to be a function of *R*, so an empirical determination is more

3 cos <sup>1</sup> | |

*fus*

. (12)

(13)

and *fusH*, which

*<sup>c</sup>* .

(14)

and *A* denote the interfacial tension

, 

and *fusH* values for such small systems is likely to

*R H*

0.5

 

*HW W*

9 cos 1 cos 3 sin

introduce unquantifiable errors. Consequently, the preferred methodology is that of using homogeneously nucleating systems to identify |*Rmin*| from the confinement size below which the onset crystallization temperatures, *Tc*, start to decrease. Then critical nucleus sizes can be reliably found for all sizes below |*Rmin*| using \* 2| |/3 *r R* and <sup>3</sup> \* 32 | | /81 *n Rv*

Fortunately, crystallization in microemulsions often proceeds via homogeneous nucleation,

Our crystallization model can be extended to crystallization of solutes from nanoconfined solutions, though here the situation is complicated by the decrease in supersaturation that arises as the nucleus grows. In particular, by adopting a classical homogeneous nucleation approach for crystallization from an ideal solution in a spherical confining volume, *V*, the free energy change, *F*, to produce a nuclei containing *n* molecules would be given by

*v v F n A NkT*

and surface area, respectively, at the nucleus-solution interface, *N* is the initial number of solute molecules when *n* = 0, *v* denotes the nucleus volume, *vc* denotes the molecular

*kT c c* ln / *eq* denotes the supersaturation at that nucleus size, with

0 0 <sup>1</sup> ln 1 ln 1 *c c*

*Vv c v c V*

nucleation,

  

\*2 / 2 /

is problematic since the use of bulk

making these the system of choice.

(Cooper et al., 2008; Reguera et al., 2003) :

where 

**2.2.2 Crystallization from nanoconfined solution** 

 

*c N n V v c v Vv c v V* 0 0 1 *<sup>c</sup>* 11 / ,

$$\text{and} \qquad Y = -\frac{32\pi c\_0}{3\ln\left(\Omega / \int\_{\text{trans}}\right)} \left(\frac{\nu v\_c T\_{eq} \text{x}}{\Delta\_{fus} H}\right)^3 \left\{ \ln\left(1 - \frac{v^\*}{V\nu\_c c\_0}\right) - \frac{1}{\upsilon\_c c\_0} \ln\left(1 - \frac{v^\*}{V}\right) \right\} \cdot \frac{1}{\upsilon\_c}$$

2

Equation (16) reduces to equation (8), the melt crystallization case, when *Y* = 0. Equation (16) is solvable with *x*, , *c*0, *Teq*, *vc*, , *fusH* and /*Jtrans* as input. For the typical case where *Tc* ≤ *Teq*, *Tc* is then given by:

$$T\_c = \frac{3T\_{eq}}{4} + z\_1^{0.5} + z\_2^{0.5} - z\_3^{0.5} \tag{17}$$

$$\text{where: } z\_1 = \left\{-\left[\frac{T\_{eq}\left(X + 3Y\right)}{48}\right]^{0.5} \left(\cos\left(\frac{Z}{3}\right) - 3^{0.5}\sin\left(\frac{Z}{3}\right)\right)\right\} + \frac{T\_{eq}}{16}\text{ \textdegree C}$$

$$z\_2 = \left\{-\left[\frac{T\_{eq}\left(X + 3Y\right)}{48}\right]^{0.5} \left(\cos\left(\frac{Z}{3}\right) + 3^{0.5}\sin\left(\frac{Z}{3}\right)\right)\right\} + \frac{T\_{eq}}{16}\text{ \textdegree C}$$

and

$$z\_3 = \left| \left[ \frac{T\_{eq} \left( X + 3Y \right)}{12} \right]^{0.5} \cos \left( \frac{Z}{3} \right) \right| + \frac{T\_{eq}^2}{16} \text{ .} $$

Crystallization in Microemulsions:

number of molecules, \* *n* ( \*

and *Tc* as input and is given by:

*n*

<sup>3</sup> \* 4 \*3 *n rv <sup>c</sup>* .

macroscopic

3

values. This provides a measure of how bulk values of

which for homogeneous nucleation (

A Generic Route to Thermodynamic Control and the Estimation of Critical Nucleus Size 131

Thus we can work out *r Rx* \* just by measuring *R* and *Tc* values, when |*R*| *≤* |*R*min|. The

4 1 2 2 cos cos 2 cos 2 \* <sup>3</sup> 2 2 <sup>4</sup> *<sup>c</sup> R x x xx x x*

and *fusH* values, provided we know or can estimate

(a) (b)

portion showing the regime controlled by *Ftot* = 0 (Cooper et al., 2008)*.* 

**2.2.3 Thermodynamic control of crystallization** 

Fig. 4. (a) Schematic graph of *F* vs *r* for crystallization in a 3D-nanoconfined solution. (b) Graph showing *Tc*, with confinement radius, |*R*|, for homogeneous nucleation from

solutions of octadecane in dodecane. Open symbols correspond to the regime where \* ln( / ) *F kT J <sup>c</sup> trans* gives *Tc*, filled symbols to the regime where |*R*||*Rmin*| and *Tc* is controlled by \* *F*min = 0. Squares = 0.1 mole fraction of octadecane in dodecane, diamonds =

The above analysis show that for all confinement sizes below |*Rmin*|, crystallization is not governed by the ability to surmount the nucleation barrier, *F*\*, but by the ability to create a

0.25 mole fraction, triangles = 0.5 mole fraction and circles = 0.75 mole fraction. The uppermost curve corresponds to the pure octadecane liquid case, with the thicker line

 

Hence provided |*R*| ≤ |*Rmin*|, we can again determine both *n*\* and *r*\* without reliance on

crystallization temperature, *Tc*, can then be compared with the values predicted from the Gibbs-Thomson and ideal solubility equations using the experimentally found *R*, *x*, and *ceq*

for solute crystallization in nanosystems. Figure 4a shows a schematic graph of *F* vs *r*, whist Figure 4b shows theoretical calculations using equations (17) and (19) to model the homogeneous nucleation of octadecane from dodecane solutions, illustrating again the decrease in *Tc* that occurs below |*Rmin*|, from which the critical nucleus size can be estimated.

*v x y*

3 3 222 3 4

*<sup>c</sup> v v* ) in the critical nucleus is also obtained from *R*,

= 180, *y x* 1 ) reduces to the expected

, *c*0, *vc*,

, (22)

. The experimental onset

and *fusH* are likely to be perturbed

$$\text{where}\quad Z = \arccos\left\{\frac{2\mathcal{T}^{0.5}\left[T\_{eq}\,^3Y + (X-Y)^2\right]}{2\left[T\_{eq}\,(X+3Y)\right]^{1.5}}\right\}.\dots$$

*r*\* and *R* are then found from the Gibbs-Thomson equation, and \* *R rx* , respectively.

Of the other three solutions to the quartic equation (16), two are non-physical, as they correspond to either crystallization close to 0 K, or the onset crystallization temperature close to *Teq* found when the minimum free energy radius, \* *min r* , is used instead of the maximum value, *r*\*. The remaining solution provides the onset crystallization temperature for rare cases when *Tc* > *Teq,* which could in principle arise for sufficiently soluble species when < 90. In this case, positive values of *x* are used since *r* is negative as well as *R* and

$$T\_c = \frac{\Im T\_{eq}}{4} + z\_1^{0.5} - z\_2^{0.5} + z\_3^{0.5} \,. \tag{18}$$

As with the melt crystallization case, equations (17) and (18) are valid until the confinement size decreases to such an extent that there is insufficient crystallizing material present to ensure an energetically feasible phase transformation. For instance, crystallization would not be possible in a 3D-nanoconfined solution having the *F* vs *r* curve shown in Figure 4a, since \* *Fmin* > 0. For these small nanoconfinements, crystallization becomes feasible when the *minimum energy* \* *Fmin* is set to zero so that \* *min* <sup>0</sup> *r r* , i.e. a stable nucleus can form. We then obtain (Cooper et al., 2008):

$$R = \frac{\gamma f(\theta\_{p,\min})}{kT\_c c\_0 x\_{\min} \,^2 \left[ \ln \left( 1 - \frac{\upsilon\_{\min}^\*}{V \upsilon\_c c\_0} \right) - \frac{1}{\upsilon\_c c\_0} \ln \left( 1 - \frac{\upsilon\_{\min}^\*}{V} \right) \right]} = \frac{2 \gamma \upsilon\_c x\_{\min}}{kT\_c \ln \left( c\_{\min}^\* \,^\* \,/ c\_{eq} \right)} \tag{19}$$

from which:

$$\mathbf{x}\_{\min} = \frac{R}{r\_{\min}^\*} = \left(\frac{\ln(\mathbf{c}\_{\min}^\* / \mathbf{c}\_{\neq 0}) f(\theta\_{p, \min})}{2\upsilon\_c c\_0 \ln\left(1 - \frac{\upsilon\_{\min}^\*}{V \upsilon\_c c\_0}\right) - 2\ln\left(1 - \frac{\upsilon\_{\min}^\*}{V}\right)}\right)^{\frac{1}{3}}\tag{20}$$

where \* *min v* denotes the volume of the \* *min r* nucleus with *F* = 0, / 0 *F r* , and 2 2 *F r* / 0 , and \* *min c* denotes the solute concentration surrounding the \* *min r* nucleus. Equation (20) can be solved iteratively to give \* *x Rr min min* / with inputted values for *c*0, , *vc* and *ceq*, but again, crucially not the or *fusH* values. The *x* value is then given by:

$$\frac{\chi}{1\ln(c^\*/c\_{eq})} = \frac{\chi\_{min}}{1\ln(c^\*\_{min}/c\_{eq})} \,. \tag{21}$$

.

*r*\* and *R* are then found from the Gibbs-Thomson equation, and \* *R rx* , respectively.

Of the other three solutions to the quartic equation (16), two are non-physical, as they correspond to either crystallization close to 0 K, or the onset crystallization temperature

maximum value, *r*\*. The remaining solution provides the onset crystallization temperature for rare cases when *Tc* > *Teq,* which could in principle arise for sufficiently

As with the melt crystallization case, equations (17) and (18) are valid until the confinement size decreases to such an extent that there is insufficient crystallizing material present to ensure an energetically feasible phase transformation. For instance, crystallization would not be possible in a 3D-nanoconfined solution having the *F* vs *r* curve shown in Figure 4a, since \* *Fmin* > 0. For these small nanoconfinements, crystallization becomes feasible when

\* \* \* <sup>2</sup>

( ) 2 <sup>1</sup> ln( / ) ln 1 ln 1

*p min c min min min c min eq*

,

or *fusH* values. The *x* value is then given by:

*Vv c V*

 

*v v kT c c*

< 90. In this case, positive values of *x* are used since *r* is negative

*T zzz* . (18)

*min* <sup>0</sup> *r r* , i.e. a stable nucleus can form. We

(19)

(20)

*min r* nucleus.

, *vc*

1 3

*min r* nucleus with *F* = 0, / 0 *F r* , and

*cc c c* . (21)

0.5 0.5 0.5 123

*min r* , is used instead of the

close to *Teq* found when the minimum free energy radius, \*

1.5

3 4 *eq*

*T*

,

 

0 0

*Vv c v c V*

\*

\* \* \*

*r v v*

Equation (20) can be solved iteratively to give \* *x Rr min min* / with inputted values for *c*0,

\* \* ln( ) ln( )

*x x*

*eq min eq*

2 ln 1 2ln 1

*min min min <sup>c</sup> c*

0

*min c* denotes the solute concentration surrounding the \*

*min*

ln( / ) ( )

 

*min eq p min*

*<sup>f</sup> v x <sup>R</sup>*

*c c*

*R c cf*

0

*v c*

*c*

0.5 3 2

*TY XY*

27 ( )

*TX Y* 

2 3 *eq*

*eq*

the *minimum energy* \* *Fmin* is set to zero so that \*

0

*kT c x*

*c min*

*min*

*min v* denotes the volume of the \*

*x*

then obtain (Cooper et al., 2008):

where

*Z*

soluble species when

as well as *R* and

from which:

where \*

2 2 *F r* / 0 , and \*

and *ceq*, but again, crucially not the

arccos

Thus we can work out *r Rx* \* just by measuring *R* and *Tc* values, when |*R*| *≤* |*R*min|. The number of molecules, \* *n* ( \* *<sup>c</sup> v v* ) in the critical nucleus is also obtained from *R*, , *c*0, *vc*, and *Tc* as input and is given by:

$$m^\* = \frac{4\pi R^3}{3\upsilon\_c x^3} \left\{ \frac{1}{2} - \frac{\mathbf{x}^3}{2} + \frac{2 - 2\mathbf{x}\cos\theta + \mathbf{x}^2 - \mathbf{x}^2 \cos^2\theta - 2\mathbf{x}^3 \cos\theta + 2\mathbf{x}^4}{4y} \right\},\tag{22}$$

which for homogeneous nucleation ( = 180, *y x* 1 ) reduces to the expected <sup>3</sup> \* 4 \*3 *n rv <sup>c</sup>* .

Hence provided |*R*| ≤ |*Rmin*|, we can again determine both *n*\* and *r*\* without reliance on macroscopic and *fusH* values, provided we know or can estimate . The experimental onset crystallization temperature, *Tc*, can then be compared with the values predicted from the Gibbs-Thomson and ideal solubility equations using the experimentally found *R*, *x*, and *ceq* values. This provides a measure of how bulk values of and *fusH* are likely to be perturbed for solute crystallization in nanosystems. Figure 4a shows a schematic graph of *F* vs *r*, whist Figure 4b shows theoretical calculations using equations (17) and (19) to model the homogeneous nucleation of octadecane from dodecane solutions, illustrating again the decrease in *Tc* that occurs below |*Rmin*|, from which the critical nucleus size can be estimated.

Fig. 4. (a) Schematic graph of *F* vs *r* for crystallization in a 3D-nanoconfined solution. (b) Graph showing *Tc*, with confinement radius, |*R*|, for homogeneous nucleation from solutions of octadecane in dodecane. Open symbols correspond to the regime where \* ln( / ) *F kT J <sup>c</sup> trans* gives *Tc*, filled symbols to the regime where |*R*||*Rmin*| and *Tc* is controlled by \* *F*min = 0. Squares = 0.1 mole fraction of octadecane in dodecane, diamonds = 0.25 mole fraction, triangles = 0.5 mole fraction and circles = 0.75 mole fraction. The uppermost curve corresponds to the pure octadecane liquid case, with the thicker line portion showing the regime controlled by *Ftot* = 0 (Cooper et al., 2008)*.* 

#### **2.2.3 Thermodynamic control of crystallization**

The above analysis show that for all confinement sizes below |*Rmin*|, crystallization is not governed by the ability to surmount the nucleation barrier, *F*\*, but by the ability to create a

Crystallization in Microemulsions:

produce a stable nucleus ( \*

microemulsions.

**4. Microemulsions** 

polydispersity of

*R*/*Rmax* ≈ 0.1-0.2, where

tension also ensures that the LaPlace pressure difference, *P R* 2

Thus metastable B becomes the majority product in this case.

A Generic Route to Thermodynamic Control and the Estimation of Critical Nucleus Size 133

since for both \* *F kT* min . In Figure 5b, however, polymorph A can form because it can

nucleation barrier is surmountable, this system will crystallize under thermodynamic control to directly give the stable polymorph A. In the system shown in Figure 5c, however, both polymorph A and B can form stable nuclei; crystallization will tend to be under kinetic control with polymorph B forming at a faster rate due to its lower nucleation energy barrier.

The thermodynamic arguments stated above are valid in the thermodynamic limit, where the system is so large that fluctuations do not significantly contribute to the equilibrium properties of the system. Though, of course, it must be remembered that it is these very fluctuations that enable critical nuclei to form and hence initiate the phase transformation. Consequently, in a system comprising a limited number of nanoconfined solutions, the fluctuations do need to be included to accurately model the equilibrium properties of the system (Reguera et al., 2003). We neglect this statistical thermodynamic description in our simple model of onset crystallization temperatures because the system in which we apply it, namely microemulsions, typically consists of a sufficiently large number of droplets, ~1018 per gram of microemulsion, which makes its additional complexity unwarranted. Moreover our simple model contains the essential features required to show how reliable estimates of critical nucleus sizes and thermodynamic control of crystallization are realizable in

Microemulsions are thermodynamically stable, transparent mixtures of immiscible liquids. Typically they comprise oil droplets in water (an oil-in-water microemulsion) or water droplets in oil (a water-in-oil microemulsion). Bicontinuous microemulsions are also possible, however the absence of a 3D nanoconfined solution in these systems make them less suitable for thermodynamic control of crystallization. In the droplet microemulsions, the droplet size is typically 2-10 nm, with a relatively narrow

deviation and *Rmax* is the modal droplet radius (Eriksson and Ljunggren, 1995). The droplets are stabilized by surfactants, frequently in combination with a co-surfactant, which reside at the droplet interface, reducing the interfacial tension there to ~10-3 mN m-1. This ultralow interfacial tension provides the thermodynamic stability of microemulsions, since the small free energy increase involved in creating the droplet interface is more than offset by the increased entropy of the dispersed phase. Note that this ultralow interfacial

curved droplets is small. When the volume fraction of the dispersed phase becomes so low that its properties differ measurably from its usual bulk properties, the terms "swollen micelles", "swollen micellar solutions", "solubilized micellar solutions" or even simply "micellar solutions" can be used instead of microemulsions for oil-in-water systems, whilst for water-in-oil systems, the same terms but with "inverse" or "reverse" inserted before "micelle" or "micellar" may be used. However, because there is, in general, no sharp transition from a microemulsion containing an isotropic core of

*<sup>R</sup>* is the Gaussian distribution standard

, across the highly

, *Fmin A* 0 ) whereas polymorph B cannot. Hence provided the

thermodynamically feasible new phase , i.e. *totF* 0 for melt crystallization or \* *Fmin* 0 for solution crystallization. This means that crystallization is under thermodynamic, rather than the usual kinetic, control. This is significant because crystallization can then be directed to generically produce the most stable crystalline phase in 3D nanoconfined solutions and liquids. This finding is particularly important for polymorphic compounds.

## **3. Polymorphism**

Polymorphism occurs when a substance can crystallize into more than one crystal structure. Each polymorph of a substance will have differing physical properties e.g. melting points, solubilities, compaction behaviour etc. In the pharmaceutical industry, it is imperative that a drug does not transform post-marketing, as this can affect its bioavailability, and reduce the drug's effectiveness. An infamous example of this occurred for the anti-HIV drug, Ritonavir (Chemburkar et al., 2000). In 1998, 2 years after marketing the drug in the form of soft gelatine capsules and oral solutions, the drug failed dissolution tests; a less soluble, thermodynamically more stable, polymorph had formed. This resulted in the precipitation of the drug and a marked decrease in the dissolution rate of the marketed formulations, reducing its bioavailability. Consequently Ritonavir had to be withdrawn from the market and reformulated, to the cost of several hundred million dollars.

The Ritonavir case highlights that crystallization is typically under kinetic control, with metastable polymorphs often crystallizing initially in accordance with Ostwald's rule of stages (Ostwald, 1897). For pharmaceutical companies, Ostwald's rule is a nemesis, as it means that their strategy of relying on high throughput screening of different crystallization conditions in order to identify stable polymorphs is scientifically flawed and so may not succeed. Consequently the industry remains vulnerable to another Ritonavir-type crisis. If the crystallization can be conducted from 3D-nanoconfined solutions, however, the crystallization can be exerted under thermodynamic control so that the thermodynamically stable polymorph is crystallized directly. In particular, in Figure 5a it is evident that neither polymorph A in red or polymorph B in blue will crystallize from the nanoconfined solution,

Fig. 5. Example graphs of free energy change, *F vs.* nucleus size, *r* for crystallization of a polymorphic system from 3D nanoconfined solutions of monodisperse size and supersaturation. (a) System stabilized due to 3D nanoconfinement, with no observable crystallization. (b) Crystallization is under thermodynamic control with stable polymorph A (red) crystallizing, even though it has the higher nucleation barrier. (c) Crystallization occurs under kinetic control with metastable polymorph B (blue) as the majority product.

since for both \* *F kT* min . In Figure 5b, however, polymorph A can form because it can produce a stable nucleus ( \* , *Fmin A* 0 ) whereas polymorph B cannot. Hence provided the nucleation barrier is surmountable, this system will crystallize under thermodynamic control to directly give the stable polymorph A. In the system shown in Figure 5c, however, both polymorph A and B can form stable nuclei; crystallization will tend to be under kinetic control with polymorph B forming at a faster rate due to its lower nucleation energy barrier. Thus metastable B becomes the majority product in this case.

The thermodynamic arguments stated above are valid in the thermodynamic limit, where the system is so large that fluctuations do not significantly contribute to the equilibrium properties of the system. Though, of course, it must be remembered that it is these very fluctuations that enable critical nuclei to form and hence initiate the phase transformation. Consequently, in a system comprising a limited number of nanoconfined solutions, the fluctuations do need to be included to accurately model the equilibrium properties of the system (Reguera et al., 2003). We neglect this statistical thermodynamic description in our simple model of onset crystallization temperatures because the system in which we apply it, namely microemulsions, typically consists of a sufficiently large number of droplets, ~1018 per gram of microemulsion, which makes its additional complexity unwarranted. Moreover our simple model contains the essential features required to show how reliable estimates of critical nucleus sizes and thermodynamic control of crystallization are realizable in microemulsions.

## **4. Microemulsions**

132 Crystallization – Science and Technology

thermodynamically feasible new phase , i.e. *totF* 0 for melt crystallization or \* *Fmin* 0 for solution crystallization. This means that crystallization is under thermodynamic, rather than the usual kinetic, control. This is significant because crystallization can then be directed to generically produce the most stable crystalline phase in 3D nanoconfined solutions and

Polymorphism occurs when a substance can crystallize into more than one crystal structure. Each polymorph of a substance will have differing physical properties e.g. melting points, solubilities, compaction behaviour etc. In the pharmaceutical industry, it is imperative that a drug does not transform post-marketing, as this can affect its bioavailability, and reduce the drug's effectiveness. An infamous example of this occurred for the anti-HIV drug, Ritonavir (Chemburkar et al., 2000). In 1998, 2 years after marketing the drug in the form of soft gelatine capsules and oral solutions, the drug failed dissolution tests; a less soluble, thermodynamically more stable, polymorph had formed. This resulted in the precipitation of the drug and a marked decrease in the dissolution rate of the marketed formulations, reducing its bioavailability. Consequently Ritonavir had to be withdrawn from the market

The Ritonavir case highlights that crystallization is typically under kinetic control, with metastable polymorphs often crystallizing initially in accordance with Ostwald's rule of stages (Ostwald, 1897). For pharmaceutical companies, Ostwald's rule is a nemesis, as it means that their strategy of relying on high throughput screening of different crystallization conditions in order to identify stable polymorphs is scientifically flawed and so may not succeed. Consequently the industry remains vulnerable to another Ritonavir-type crisis. If the crystallization can be conducted from 3D-nanoconfined solutions, however, the crystallization can be exerted under thermodynamic control so that the thermodynamically stable polymorph is crystallized directly. In particular, in Figure 5a it is evident that neither polymorph A in red or polymorph B in blue will crystallize from the nanoconfined solution,

(a) (b) (c)

polymorphic system from 3D nanoconfined solutions of monodisperse size and supersaturation. (a) System stabilized due to 3D nanoconfinement, with no observable

crystallization. (b) Crystallization is under thermodynamic control with stable polymorph A (red) crystallizing, even though it has the higher nucleation barrier. (c) Crystallization occurs under kinetic control with metastable polymorph B (blue) as the

Fig. 5. Example graphs of free energy change, *F vs.* nucleus size, *r* for crystallization of a

liquids. This finding is particularly important for polymorphic compounds.

and reformulated, to the cost of several hundred million dollars.

**3. Polymorphism** 

majority product.

Microemulsions are thermodynamically stable, transparent mixtures of immiscible liquids. Typically they comprise oil droplets in water (an oil-in-water microemulsion) or water droplets in oil (a water-in-oil microemulsion). Bicontinuous microemulsions are also possible, however the absence of a 3D nanoconfined solution in these systems make them less suitable for thermodynamic control of crystallization. In the droplet microemulsions, the droplet size is typically 2-10 nm, with a relatively narrow polydispersity of *R*/*Rmax* ≈ 0.1-0.2, where *<sup>R</sup>* is the Gaussian distribution standard deviation and *Rmax* is the modal droplet radius (Eriksson and Ljunggren, 1995). The droplets are stabilized by surfactants, frequently in combination with a co-surfactant, which reside at the droplet interface, reducing the interfacial tension there to ~10-3 mN m-1. This ultralow interfacial tension provides the thermodynamic stability of microemulsions, since the small free energy increase involved in creating the droplet interface is more than offset by the increased entropy of the dispersed phase. Note that this ultralow interfacial tension also ensures that the LaPlace pressure difference, *P R* 2 , across the highly curved droplets is small. When the volume fraction of the dispersed phase becomes so low that its properties differ measurably from its usual bulk properties, the terms "swollen micelles", "swollen micellar solutions", "solubilized micellar solutions" or even simply "micellar solutions" can be used instead of microemulsions for oil-in-water systems, whilst for water-in-oil systems, the same terms but with "inverse" or "reverse" inserted before "micelle" or "micellar" may be used. However, because there is, in general, no sharp transition from a microemulsion containing an isotropic core of

Crystallization in Microemulsions:

apparent from Young's equation, cos

A Generic Route to Thermodynamic Control and the Estimation of Critical Nucleus Size 135

Reliable estimates for the critical nucleus size can be found for homogeneous nucleation in microemulsions. The homogeneous nucleation temperature should be approximately constant until the droplet size decreases to below |*Rmin*|, and thereafter the temperature should decrease. In the region where *T*c decreases, the crystallization is controlled by the requirement that *totF* 0, rather than the size of the nucleation energy barrier. Consequently, the critical nucleus size can be estimated by measuring the droplet size |*R*|

homogeneous nucleation is comparatively rare in bulk systems, in microemulsions it is more prevalent for two main reasons. Firstly, the droplets are too small to contain foreign material onto whose surfaces heterogeneous nucleation could arise. Secondly, the ability of the surfactants to induce heterogeneous nucleation is often reduced in microemulsions compared to that at planar interfaces and in emulsions, particularly for crystallization from solution. This is because nuclei formation on the surfactant layer is disfavoured at this ultra low interfacial tension interface, and the high curvature may also hinder any templating mechanism that operates at more planar interfaces. The reduction in adsorption is readily

interfacial tension between the crystallizing species and the surrounding melt/solution, with 1 and 2 denoting the interfacial tensions between the surfactant and immiscible phases, and the surfactant and crystallizing species, respectively. The lowering of 1 that occurs in going from an emulsion to a microemulsion results in a higher contact angle,

Given this, we might expect heterogeneous nucleation to be impeded in microemulsions, and indeed other systems with low interfacial tensions, 1. Such an effect was observed at the phase inversion of an emulsion using Span 80 and Tween 80 surfactants to induce glycine crystallization at the oil-aqueous interface (Nicholson et al., 2005). Similarly, the ability of the nonionic surfactants, Span 80 and Brij 30 to heterogeneously nucleate glycine was negligible in microemulsions containing these mixed surfactants (Chen et al., 2011), whereas they promoted the metastable -glycine nucleation at planar interfaces and in emulsions (Allen et al., 2002; Nicholson et al., 2005). There are literature examples where nonionic surfactants do induce heterogeneous nucleation in microemulsions, though. For instance, ice nucleation was promoted by adding heptacosanol, a long chain alcohol cosurfactant, to AOT microemulsions (Liu et al., 2007). Long chain alcohols can induce ice nucleation at temperatures of -2 C at planar air-water interfaces (Popovitz-Biro et al., 1994) and at -8 C at emulsion interfaces (Jamieson et al., 2005). This nucleating ability was diminished in the microemulsions. Nevertheless, ice crystallization still tended to occur at temperatures in the range of -9 to -30 C depending upon the heptacosanol concentration in the microemulsion droplets (Liu et al., 2007), i.e. much higher than the homogeneous nucleation temperature of -40 C (Wood & Walton, 1970; Clausse et al., 1983). For ionic surfactants, like AOT, heterogeneous nucleation in microemulsions can also occur. The longer range electrostatic interactions of ionic surfactants can induce order without requiring direct adsorption onto the surfactant layer. It is important, therefore, to choose surfactants that do not promote crystallization when placed at the planar air-liquid

 1 2 / where

*<sup>c</sup>* . Whilst

is the contact angle, is the

,

and assuming a spherical nucleus so that \* 2| |/3 *r R* and <sup>3</sup> \* 32 | | /81 *n Rv*

 

and hence reduced adsorption for the crystallizing species.

**4.1 Measurement of the critical nucleus size in microemulsions** 

dispersed phase and a micelle progressively swollen with the dispersed phase, many researchers use the term "microemulsion" to include swollen micelles (or swollen inverse micelles) but not micelles containing no dispersed phase. This is the context in which the term "microemulsion" is used here. In the microemulsions, dissolved solutes may be supersaturated within the discontinuous (dispersed) phase, or the dispersed liquid may be supercooled, so that crystallization in the microemulsions can occur. The solute molecules are assumed to be distributed amongst the microemulsion droplets with a Poisson distribution, so that most droplets will have a supersaturation close to the mean, but a minority will have supersaturations significantly higher than the mean.

Microemulsions are dynamic systems with frequent collisions occurring between the droplets. The most energetic of these collisions cause transient dimers to form, allowing the exchange of interior content between the droplets. This exchange means that microemulsions can act as nanoreactors for creating quantum dots and other inorganic nanoparticles, for example. A recent review (Ganguli et al., 2010) on inorganic nanoparticle formation in microemulsions highlights the progress that has been made in this area from its inception with metal nanoparticle synthesis in 1982 by Boutonnet et al., followed by its use in synthesizing quantum dots (Petit et al. 1990) and metal oxides (see e.g. Zarur & Ying, 2000). The transient dimer mechanism also enables crystallization to proceed in microemulsions whenever a transient dimer forms between a droplet containing a crystal nucleus and a nucleus-free droplet which contains supersaturated solution, since the crystal nucleus can then gain access to this supersaturated solution and thereby grow (see Figure 6). Crystallization of organic compounds from microemulsions was first studied by Füredi-Milhofer et al. in 1999 for the aspartame crystal system, with studies on glycine crystallization (Allen et al., 2002; Yano et al., 2000; Nicholson et al., 2011; Chen et al., 2011) and carbamazepine (Kogan et al., 2007) following. The use of microemulsions in producing both inorganic nanoparticles and macroscopic organic crystals shows that the size of the particulates grown can vary from a few nm to mm, depending upon the nucleation rate, the ability to form stable nuclei, and the extent of surfactant adsorption on the resulting particles. Our interest in microemulsions stems from their intrinsic ability to enable reliable estimates of critical nucleus sizes and to exert generic thermodynamic control over the crystallization process for the first time.

Fig. 6. Schematic diagram illustrating an energetic collision between a microemulsion droplet containing a crystal nucleus and a nucleus-free droplet containing supersaturated solution. The energetic collision results in a transient dimer forming, enabling the nucleus to gain access to more molecules and grow. The crystal nucleus is shown in red and the solute molecules are shown in black. The surfactant molecules stabilizing the microemulsion droplets are depicted as blue circles with tails.

### **4.1 Measurement of the critical nucleus size in microemulsions**

134 Crystallization – Science and Technology

dispersed phase and a micelle progressively swollen with the dispersed phase, many researchers use the term "microemulsion" to include swollen micelles (or swollen inverse micelles) but not micelles containing no dispersed phase. This is the context in which the term "microemulsion" is used here. In the microemulsions, dissolved solutes may be supersaturated within the discontinuous (dispersed) phase, or the dispersed liquid may be supercooled, so that crystallization in the microemulsions can occur. The solute molecules are assumed to be distributed amongst the microemulsion droplets with a Poisson distribution, so that most droplets will have a supersaturation close to the mean,

Microemulsions are dynamic systems with frequent collisions occurring between the droplets. The most energetic of these collisions cause transient dimers to form, allowing the exchange of interior content between the droplets. This exchange means that microemulsions can act as nanoreactors for creating quantum dots and other inorganic nanoparticles, for example. A recent review (Ganguli et al., 2010) on inorganic nanoparticle formation in microemulsions highlights the progress that has been made in this area from its inception with metal nanoparticle synthesis in 1982 by Boutonnet et al., followed by its use in synthesizing quantum dots (Petit et al. 1990) and metal oxides (see e.g. Zarur & Ying, 2000). The transient dimer mechanism also enables crystallization to proceed in microemulsions whenever a transient dimer forms between a droplet containing a crystal nucleus and a nucleus-free droplet which contains supersaturated solution, since the crystal nucleus can then gain access to this supersaturated solution and thereby grow (see Figure 6). Crystallization of organic compounds from microemulsions was first studied by Füredi-Milhofer et al. in 1999 for the aspartame crystal system, with studies on glycine crystallization (Allen et al., 2002; Yano et al., 2000; Nicholson et al., 2011; Chen et al., 2011) and carbamazepine (Kogan et al., 2007) following. The use of microemulsions in producing both inorganic nanoparticles and macroscopic organic crystals shows that the size of the particulates grown can vary from a few nm to mm, depending upon the nucleation rate, the ability to form stable nuclei, and the extent of surfactant adsorption on the resulting particles. Our interest in microemulsions stems from their intrinsic ability to enable reliable estimates of critical nucleus sizes and to exert generic thermodynamic control over the

Fig. 6. Schematic diagram illustrating an energetic collision between a microemulsion droplet containing a crystal nucleus and a nucleus-free droplet containing supersaturated solution. The energetic collision results in a transient dimer forming, enabling the nucleus to gain access to more molecules and grow. The crystal nucleus is shown in red and the solute molecules are shown in black. The surfactant molecules stabilizing the microemulsion

but a minority will have supersaturations significantly higher than the mean.

crystallization process for the first time.

droplets are depicted as blue circles with tails.

Reliable estimates for the critical nucleus size can be found for homogeneous nucleation in microemulsions. The homogeneous nucleation temperature should be approximately constant until the droplet size decreases to below |*Rmin*|, and thereafter the temperature should decrease. In the region where *T*c decreases, the crystallization is controlled by the requirement that *totF* 0, rather than the size of the nucleation energy barrier. Consequently, the critical nucleus size can be estimated by measuring the droplet size |*R*| and assuming a spherical nucleus so that \* 2| |/3 *r R* and <sup>3</sup> \* 32 | | /81 *n Rv <sup>c</sup>* . Whilst homogeneous nucleation is comparatively rare in bulk systems, in microemulsions it is more prevalent for two main reasons. Firstly, the droplets are too small to contain foreign material onto whose surfaces heterogeneous nucleation could arise. Secondly, the ability of the surfactants to induce heterogeneous nucleation is often reduced in microemulsions compared to that at planar interfaces and in emulsions, particularly for crystallization from solution. This is because nuclei formation on the surfactant layer is disfavoured at this ultra low interfacial tension interface, and the high curvature may also hinder any templating mechanism that operates at more planar interfaces. The reduction in adsorption is readily apparent from Young's equation, cos 1 2 / where is the contact angle, is the interfacial tension between the crystallizing species and the surrounding melt/solution, with 1 and 2 denoting the interfacial tensions between the surfactant and immiscible phases, and the surfactant and crystallizing species, respectively. The lowering of 1 that occurs in going from an emulsion to a microemulsion results in a higher contact angle, , and hence reduced adsorption for the crystallizing species.

Given this, we might expect heterogeneous nucleation to be impeded in microemulsions, and indeed other systems with low interfacial tensions, 1. Such an effect was observed at the phase inversion of an emulsion using Span 80 and Tween 80 surfactants to induce glycine crystallization at the oil-aqueous interface (Nicholson et al., 2005). Similarly, the ability of the nonionic surfactants, Span 80 and Brij 30 to heterogeneously nucleate glycine was negligible in microemulsions containing these mixed surfactants (Chen et al., 2011), whereas they promoted the metastable -glycine nucleation at planar interfaces and in emulsions (Allen et al., 2002; Nicholson et al., 2005). There are literature examples where nonionic surfactants do induce heterogeneous nucleation in microemulsions, though. For instance, ice nucleation was promoted by adding heptacosanol, a long chain alcohol cosurfactant, to AOT microemulsions (Liu et al., 2007). Long chain alcohols can induce ice nucleation at temperatures of -2 C at planar air-water interfaces (Popovitz-Biro et al., 1994) and at -8 C at emulsion interfaces (Jamieson et al., 2005). This nucleating ability was diminished in the microemulsions. Nevertheless, ice crystallization still tended to occur at temperatures in the range of -9 to -30 C depending upon the heptacosanol concentration in the microemulsion droplets (Liu et al., 2007), i.e. much higher than the homogeneous nucleation temperature of -40 C (Wood & Walton, 1970; Clausse et al., 1983). For ionic surfactants, like AOT, heterogeneous nucleation in microemulsions can also occur. The longer range electrostatic interactions of ionic surfactants can induce order without requiring direct adsorption onto the surfactant layer. It is important, therefore, to choose surfactants that do not promote crystallization when placed at the planar air-liquid

Crystallization in Microemulsions:

**Ostwald's rule of stages** 

crystallization. An effective strategy is detailed below.

The equilibrium population of the \*

Boltzmann factor, \*

*severely hindered if the population of such* \*

population of the \*

nucleus size, \*

A Generic Route to Thermodynamic Control and the Estimation of Critical Nucleus Size 137

As detailed previously, crystallization within 3D nanoconfined solutions differs fundamentally from bulk crystallization because the limited amount of material within a nanoconfined solution results in the supersaturation decreasing substantially as the nucleus grows, leading to a minimum11,12 in the free energy, \* *F*min , at a post-critical nanometre

solution-crystallized from microemulsions even when they have high nucleation barriers. Hence thermodynamic control of crystallization can be generically achieved for the first time. Note that microemulsions differ in two main ways from the theoretical nanoconfined solutions considered previously in sections 2 and 3. Firstly, transient droplet dimer formation enables the nuclei to grow beyond that dictated by the original droplet size; in fact crystals ranging from nm to mm can be produced. Secondly, microemulsions are polydisperse. There will be a range of droplet sizes and supersaturations in any microemulsion system, which must be considered to enable thermodynamic control of

min *r* nuclei will be very low. In contrast, if the \*

nuclei, (near) stable nuclei. Crystallization in microemulsions proceeds initially via the 3D nanoconfined nuclei gaining access to more material and growing during the energetic droplet collisions that allow transient dimers to form. *This crystallization process will be* 

5a, that the colliding droplets are highly unlikely to contain nuclei. In this case, the supersaturated system is stabilized due to the 3D nanoconfinement. In contrast, the crystallization can proceed readily via this transient droplet dimer mechanism if \* *F*min < *kT*

Fig. 8. A schematic diagram showing how Ostwald's rule of stages can be 'leapfrogged' by crystallizing from microemulsions. Stable polymorph A is in red, and metastable B is in blue. The *F* vs *r* plot corresponds to the case where crystallization is brought under

thermodynamic control so that stable polymorph A crystallizes directly.

energies, \* *F*min < *kT,* they will have a sizeable equilibrium population. We term such \*

min *r* (see Figure 4a). This fact, in particular, allows stable polymorphs to be

min *r* nuclei in the microemulsion depends upon the

min *r nuclei is so low* as for the case depicted in Figure

min *r* nuclei have free

min *r*

min exp / *F kT* . Consequently, if \* *F*min > *kT*, the equilibrium

**4.2 Thermodynamic control of crystallization in microemulsions: Leapfrogging** 

or air-solution interface to ensure that homogeneous nucleation occurs in the microemulsions.

Once a suitable homogeneous nucleating microemulsion system has been found, the onset crystallization temperature, *Tc*, for microemulsions of varying size, *R*, can be found by a suitable technique, such as differential scanning calorimetry (DSC). It can be assumed that the exothermic DSC peak arising from the crystallization of the droplets corresponds to the temperature range in which the majority of droplets can crystallize, so that the mean droplet size can be used to accurately determine *r*\* and *n*\*. The mean droplet size of the microemulsion can be determined from small angle X-ray scattering, or small angle neutron scattering measurements. This methodology allows a simple and accurate measurement of the critical nucleus size, and is particularly useful for crystallization of liquids, or solutes which have a high solubility in the confined phase, so that there is sufficient crystallizable material present within the microemulsion for the exothermic crystallization peak to be observable by DSC. We have recently applied this methodology to ice crystallization in AOT microemulsions (Liu et al., 2007). Figure 7a shows homogeneous ice nucleation in AOT microemulsions, compared to heterogeneous nucleation in Figure 7b where the cosurfactant heptacosanol is added to the AOT microemulsions. The larger error bars in the heterogeneous nucleation case shown in Figure 7b reflect the varying number of heptacosanol molecules in the droplets that cause ice nucleation. For the homogeneous case, the reduction in *Tc* with |*R*| occurs at |*Rmin*| 2 nm in Figure 7a, in good agreement with the theoretical value shown in Figure 3a. From this, the critical nucleus at *Rmin* can be estimated to contain 280 molecules (Liu et al., 2007).

Fig. 7. Variation of observed ice onset crystallization temperatures, *Tc*, with water pool size for microemulsions with (a) AOT and (b) AOT plus heptacosanol The error bars show the standard deviation from three or more measurements. The nucleation is homogeneous in (a) and heterogeneous in (b). In the AOT microemulsions with added heptacosanol, several crystallization peaks were often evident, due to the droplets having differing numbers of the heptacosanol cosurfactant molecules that help nucleate ice. Consequently in (b) the highest *Tc* peak (upper curve) and largest *Tc* peak (lower curve) are just plotted for clarity (Liu et al. 2007).

or air-solution interface to ensure that homogeneous nucleation occurs in the

Once a suitable homogeneous nucleating microemulsion system has been found, the onset crystallization temperature, *Tc*, for microemulsions of varying size, *R*, can be found by a suitable technique, such as differential scanning calorimetry (DSC). It can be assumed that the exothermic DSC peak arising from the crystallization of the droplets corresponds to the temperature range in which the majority of droplets can crystallize, so that the mean droplet size can be used to accurately determine *r*\* and *n*\*. The mean droplet size of the microemulsion can be determined from small angle X-ray scattering, or small angle neutron scattering measurements. This methodology allows a simple and accurate measurement of the critical nucleus size, and is particularly useful for crystallization of liquids, or solutes which have a high solubility in the confined phase, so that there is sufficient crystallizable material present within the microemulsion for the exothermic crystallization peak to be observable by DSC. We have recently applied this methodology to ice crystallization in AOT microemulsions (Liu et al., 2007). Figure 7a shows homogeneous ice nucleation in AOT microemulsions, compared to heterogeneous nucleation in Figure 7b where the cosurfactant heptacosanol is added to the AOT microemulsions. The larger error bars in the heterogeneous nucleation case shown in Figure 7b reflect the varying number of heptacosanol molecules in the droplets that cause ice nucleation. For the homogeneous case, the reduction in *Tc* with |*R*| occurs at |*Rmin*| 2 nm in Figure 7a, in good agreement with the theoretical value shown in Figure 3a. From this, the critical nucleus at *Rmin* can be

microemulsions.

2007).

estimated to contain 280 molecules (Liu et al., 2007).

(a) (b)

Fig. 7. Variation of observed ice onset crystallization temperatures, *Tc*, with water pool size for microemulsions with (a) AOT and (b) AOT plus heptacosanol The error bars show the standard deviation from three or more measurements. The nucleation is homogeneous in (a) and heterogeneous in (b). In the AOT microemulsions with added heptacosanol, several crystallization peaks were often evident, due to the droplets having differing numbers of the heptacosanol cosurfactant molecules that help nucleate ice. Consequently in (b) the highest *Tc* peak (upper curve) and largest *Tc* peak (lower curve) are just plotted for clarity (Liu et al.

#### **4.2 Thermodynamic control of crystallization in microemulsions: Leapfrogging Ostwald's rule of stages**

As detailed previously, crystallization within 3D nanoconfined solutions differs fundamentally from bulk crystallization because the limited amount of material within a nanoconfined solution results in the supersaturation decreasing substantially as the nucleus grows, leading to a minimum11,12 in the free energy, \* *F*min , at a post-critical nanometre nucleus size, \* min *r* (see Figure 4a). This fact, in particular, allows stable polymorphs to be solution-crystallized from microemulsions even when they have high nucleation barriers. Hence thermodynamic control of crystallization can be generically achieved for the first time. Note that microemulsions differ in two main ways from the theoretical nanoconfined solutions considered previously in sections 2 and 3. Firstly, transient droplet dimer formation enables the nuclei to grow beyond that dictated by the original droplet size; in fact crystals ranging from nm to mm can be produced. Secondly, microemulsions are polydisperse. There will be a range of droplet sizes and supersaturations in any microemulsion system, which must be considered to enable thermodynamic control of crystallization. An effective strategy is detailed below.

The equilibrium population of the \* min *r* nuclei in the microemulsion depends upon the Boltzmann factor, \* min exp / *F kT* . Consequently, if \* *F*min > *kT*, the equilibrium population of the \* min *r* nuclei will be very low. In contrast, if the \* min *r* nuclei have free energies, \* *F*min < *kT,* they will have a sizeable equilibrium population. We term such \* min *r* nuclei, (near) stable nuclei. Crystallization in microemulsions proceeds initially via the 3D nanoconfined nuclei gaining access to more material and growing during the energetic droplet collisions that allow transient dimers to form. *This crystallization process will be severely hindered if the population of such* \* min *r nuclei is so low* as for the case depicted in Figure 5a, that the colliding droplets are highly unlikely to contain nuclei. In this case, the supersaturated system is stabilized due to the 3D nanoconfinement. In contrast, the crystallization can proceed readily via this transient droplet dimer mechanism if \* *F*min < *kT*

Fig. 8. A schematic diagram showing how Ostwald's rule of stages can be 'leapfrogged' by crystallizing from microemulsions. Stable polymorph A is in red, and metastable B is in blue. The *F* vs *r* plot corresponds to the case where crystallization is brought under thermodynamic control so that stable polymorph A crystallizes directly.

Crystallization in Microemulsions:

growth rate.

A Generic Route to Thermodynamic Control and the Estimation of Critical Nucleus Size 139

larger than the droplets, subsequent growth can then occur via the following processes; energetic collisions with droplets that allow access to the droplets' interior contents, oriented attachment of other nuclei, and impingement from the (typically minuscule) concentration of their molecules in the continuous phase. Thus, the final crystal size can vary from nm to mm, depending upon the concentration of (near) stable nuclei, the supersaturation and the extent to which surfactant adsorption on the crystallites limits their

In order to test the ability of microemulsions to exert thermodynamic control over crystallization, we chose three problem systems that had well-documented difficulty in obtaining their stable polymorphs: namely glycine, mefenamic acid and ROY (Nicholson et al., 2011). In each case, it was successfully demonstrated that the stable polymorph crystallized directly from the microemulsions under conditions where crystallization was only just possible. The time-scale at which ~mm sized crystals grew ranged from minutes to months. Solvent-mediated transformations to more stable polymorphs can occur in this timeframe, although such transformations would be expected to have a significantly reduced rate in microemulsions owing to the exceedingly low concentration of the polymorph's molecules in the continuous phase. Accordingly, transmission electron microscopy (TEM) was used to show that the nanocrystals crystallized within the first 24 hours in the microemulsions were also of the stable form, thereby proving that the initial crystallization was indeed in this form. TEM was performed on the microemulsions by dropping small aliquots of the microemulsions onto TEM grids, allowing the drops to (mostly) evaporate, and then washing the grids with the microemulsion continuous phase to remove residual surfactant. This left predominantly just the crystallites grown in the microemulsion droplets on the TEM grids. Figure 9 shows TEM images of stable Form I nanocrystals of mefenamic acid grown within 1 day from DMF microemulsions at 8 C

containing 80 mg/ml of mefenamic acid in the DMF (Nicholson & Cooper, 2011).

With increasing supersaturation, metastable polymorphs also crystallized from the microemulsions. The relative supersaturation increase that led to the emergence of metastable forms was highly system dependant, though. For instance, for glycine the stable -polymorph crystallized as the majority polymorph under mean initial supersaturation ratios of 2.0 to 2.3, for mefenamic acid the corresponding range was 4.1 to 5.3, whereas for ROY a much larger range of 10 to 24 was found. The small supersaturation range in which the stable -glycine polymorph crystallized as the majority form reflected the small relative energy difference of 0.2 kJ mol between the - and -polymorphs and the much faster growth rate (~500 times) of the -polymorph in aqueous solutions (Chew et al., 2007). Recall that when crystallization is only just possible in the microemulsions, the formation of (near) stable nuclei will be confined to only the largest droplets with the highest supersaturations. Hence the actual initial supersaturations that are required for crystallization to be just possible are always much higher than the mean initial values. An estimate of this actual initial supersaturation was found in the glycine system as follows. Assuming a Poisson distribution of solute molecules amongst the droplets, then for the glycine system, <10-8 of the droplets formed (near) stable nuclei under conditions where crystallization was only just possible, since the 0.2 kJ mol-1 stability difference between the - and -forms would only lead to thermodynamic preference for the stable -form if the (near) stable nuclei contained ~20-30 molecules (Nicholson et al., 2011). This meant initial supersaturation ratios of ~11-15

because then a sizeable population of droplets will contain a (near) stable nucleus. *Thus, crystallization in microemulsions is governed by the ability to form (near) stable nuclei rather than critical nuclei.* In particular, recalling that there will be a range of droplet sizes and solute concentrations within the microemulsion droplets, thermodynamic control of crystallization can be achieved using the following methodology. The supersaturation in a microemulsion can be increased from one that is stabilized due to 3D nanoconfinement until the following condition is met: only the largest and highest supersaturation droplets, which contain the most material, can form (near) stable nuclei of only the most stable crystal form or polymorph. This situation is exemplified in Figure 8 with \* *F kT* min,*<sup>A</sup>* but \* *F kT* min,*<sup>B</sup>* . Crystallization is then only just possible, and importantly, only the most stable polymorph will crystallize, as this is the only form for which (near) stable nuclei are likely to exist and so grow during transient droplet dimer formation. This generic strategy allows us to 'leapfrog' Ostwald's rule of stages and crystallize stable polymorphs directly.

The strategy detailed above will work provided \* \* *F F* min, min, *<sup>A</sup> <sup>B</sup>* , and (near) equilibrium populations of the \* min *r* nuclei are obtained. \* \* *F F* min, min, *<sup>A</sup> <sup>B</sup>* will typically be true because the stable polymorph has the greater bulk stability and it is the least soluble. Hence \* \* min, min, *<sup>A</sup> <sup>B</sup> r r* as stable polymorph *A* can grow to larger nucleus sizes, with typically lower free energies, than metastable *B* before its supersaturation is depleted. Equilibrium populations of the \* min *r* nuclei will arise provided the \* min *r* nuclei formation and dissolution processes are sufficiently rapid. This depends upon the magnitude of the nucleation barriers, \* *F* , and the dissolution energy barriers, \* *F* - \* *F*min , respectively. It can be ensured that the nucleation barriers to all polymorphs are surmountable by crystallizing from sufficiently small droplets. This is because the substantial supersaturation depletion that arises in a small droplet as the nucleus grows means that very high initial supersaturations with respect to the most stable polymorph are required to enable a (near) stable nuclei to form. Consequently it can reliably be assumed that the initial solutions in these droplets will also be sufficiently supersaturated with respect to all polymorphs that all nucleation barriers are indeed surmountable. The nuclei dissolution barriers, i.e. \* *F* - \* *F*min , will be surmountable when the \* *F* barriers are surmounted and \* *F*min 0 as in Figure 8, since then \* *F* - \* *F*min \* *F* . Hence thermodynamic control will indeed be obtainable if the largest and highest supersaturation droplets have free energy curves corresponding to Figure 8. Ostwald's rule will have been 'leapfrogged' because the high initial supersaturations ensure the nucleation barriers are 'leapt over' to directly give the most stable polymorph. In contrast, the analogous unconfined bulk system would crystallize the metastable polymorph initially, in accordance with Ostwald's rule, owing to its lower nucleation barrier.

Note that the ability of microemulsions to exert thermodynamic control over crystallization is independent of the nucleation path; it depends solely upon the ability to form (near) stable nuclei, rather than their formation pathway. Consequently this ability is generic, applying equally to systems that nucleate via a classical one stage mechanism, and to those where two stages are implicated.

Once formed, the (near) stable nuclei can grow via transient droplet dimer formation until the nuclei become larger than the droplets or the supersaturation is relieved. For crystallites

because then a sizeable population of droplets will contain a (near) stable nucleus. *Thus, crystallization in microemulsions is governed by the ability to form (near) stable nuclei rather than critical nuclei.* In particular, recalling that there will be a range of droplet sizes and solute concentrations within the microemulsion droplets, thermodynamic control of crystallization can be achieved using the following methodology. The supersaturation in a microemulsion can be increased from one that is stabilized due to 3D nanoconfinement until the following condition is met: only the largest and highest supersaturation droplets, which contain the most material, can form (near) stable nuclei of only the most stable crystal form or polymorph. This situation is exemplified in Figure 8 with \* *F kT* min,*<sup>A</sup>* but \* *F kT* min,*<sup>B</sup>* . Crystallization is then only just possible, and importantly, only the most stable polymorph will crystallize, as this is the only form for which (near) stable nuclei are likely to exist and so grow during transient droplet dimer formation. This generic strategy allows us to

The strategy detailed above will work provided \* \* *F F* min, min, *<sup>A</sup> <sup>B</sup>* , and (near) equilibrium

the stable polymorph has the greater bulk stability and it is the least soluble. Hence

min, min, *<sup>A</sup> <sup>B</sup> r r* as stable polymorph *A* can grow to larger nucleus sizes, with typically lower free energies, than metastable *B* before its supersaturation is depleted. Equilibrium

processes are sufficiently rapid. This depends upon the magnitude of the nucleation barriers, \* *F* , and the dissolution energy barriers, \* *F* - \* *F*min , respectively. It can be ensured that the nucleation barriers to all polymorphs are surmountable by crystallizing from sufficiently small droplets. This is because the substantial supersaturation depletion that arises in a small droplet as the nucleus grows means that very high initial supersaturations with respect to the most stable polymorph are required to enable a (near) stable nuclei to form. Consequently it can reliably be assumed that the initial solutions in these droplets will also be sufficiently supersaturated with respect to all polymorphs that all nucleation barriers are indeed surmountable. The nuclei dissolution barriers, i.e. \* *F* - \* *F*min , will be surmountable when the \* *F* barriers are surmounted and \* *F*min 0 as in Figure 8, since then \* *F* - \* *F*min \* *F* . Hence thermodynamic control will indeed be obtainable if the largest and highest supersaturation droplets have free energy curves corresponding to Figure 8. Ostwald's rule will have been 'leapfrogged' because the high initial supersaturations ensure the nucleation barriers are 'leapt over' to directly give the most stable polymorph. In contrast, the analogous unconfined bulk system would crystallize the metastable polymorph initially, in

Note that the ability of microemulsions to exert thermodynamic control over crystallization is independent of the nucleation path; it depends solely upon the ability to form (near) stable nuclei, rather than their formation pathway. Consequently this ability is generic, applying equally to systems that nucleate via a classical one stage mechanism, and to those

Once formed, the (near) stable nuclei can grow via transient droplet dimer formation until the nuclei become larger than the droplets or the supersaturation is relieved. For crystallites

min *r* nuclei are obtained. \* \* *F F* min, min, *<sup>A</sup> <sup>B</sup>* will typically be true because

min *r* nuclei formation and dissolution

'leapfrog' Ostwald's rule of stages and crystallize stable polymorphs directly.

min *r* nuclei will arise provided the \*

accordance with Ostwald's rule, owing to its lower nucleation barrier.

populations of the \*

populations of the \*

where two stages are implicated.

\* \*

larger than the droplets, subsequent growth can then occur via the following processes; energetic collisions with droplets that allow access to the droplets' interior contents, oriented attachment of other nuclei, and impingement from the (typically minuscule) concentration of their molecules in the continuous phase. Thus, the final crystal size can vary from nm to mm, depending upon the concentration of (near) stable nuclei, the supersaturation and the extent to which surfactant adsorption on the crystallites limits their growth rate.

In order to test the ability of microemulsions to exert thermodynamic control over crystallization, we chose three problem systems that had well-documented difficulty in obtaining their stable polymorphs: namely glycine, mefenamic acid and ROY (Nicholson et al., 2011). In each case, it was successfully demonstrated that the stable polymorph crystallized directly from the microemulsions under conditions where crystallization was only just possible. The time-scale at which ~mm sized crystals grew ranged from minutes to months. Solvent-mediated transformations to more stable polymorphs can occur in this timeframe, although such transformations would be expected to have a significantly reduced rate in microemulsions owing to the exceedingly low concentration of the polymorph's molecules in the continuous phase. Accordingly, transmission electron microscopy (TEM) was used to show that the nanocrystals crystallized within the first 24 hours in the microemulsions were also of the stable form, thereby proving that the initial crystallization was indeed in this form. TEM was performed on the microemulsions by dropping small aliquots of the microemulsions onto TEM grids, allowing the drops to (mostly) evaporate, and then washing the grids with the microemulsion continuous phase to remove residual surfactant. This left predominantly just the crystallites grown in the microemulsion droplets on the TEM grids. Figure 9 shows TEM images of stable Form I nanocrystals of mefenamic acid grown within 1 day from DMF microemulsions at 8 C containing 80 mg/ml of mefenamic acid in the DMF (Nicholson & Cooper, 2011).

With increasing supersaturation, metastable polymorphs also crystallized from the microemulsions. The relative supersaturation increase that led to the emergence of metastable forms was highly system dependant, though. For instance, for glycine the stable -polymorph crystallized as the majority polymorph under mean initial supersaturation ratios of 2.0 to 2.3, for mefenamic acid the corresponding range was 4.1 to 5.3, whereas for ROY a much larger range of 10 to 24 was found. The small supersaturation range in which the stable -glycine polymorph crystallized as the majority form reflected the small relative energy difference of 0.2 kJ mol between the - and -polymorphs and the much faster growth rate (~500 times) of the -polymorph in aqueous solutions (Chew et al., 2007). Recall that when crystallization is only just possible in the microemulsions, the formation of (near) stable nuclei will be confined to only the largest droplets with the highest supersaturations. Hence the actual initial supersaturations that are required for crystallization to be just possible are always much higher than the mean initial values. An estimate of this actual initial supersaturation was found in the glycine system as follows. Assuming a Poisson distribution of solute molecules amongst the droplets, then for the glycine system, <10-8 of the droplets formed (near) stable nuclei under conditions where crystallization was only just possible, since the 0.2 kJ mol-1 stability difference between the - and -forms would only lead to thermodynamic preference for the stable -form if the (near) stable nuclei contained ~20-30 molecules (Nicholson et al., 2011). This meant initial supersaturation ratios of ~11-15

Crystallization in Microemulsions:

A Generic Route to Thermodynamic Control and the Estimation of Critical Nucleus Size 141

3. Given that a metastable polymorph can potentially have the lowest \* *F*min in a microemulsion if, for example, it is heterogeneously nucleated by the surfactant, or it is sufficiently stabilized by the surrounding solvent, or an inversion of stability between polymorphs occurs at nm sizes, then this possibility should be checked. This can be done readily by using a different solvent and/or surfactant. In addition, the supersaturation of the microemulsion should be gradually increased from the point at which crystallization is only just possible, until crystals/nanocrystals of more than one

polymorph crystallize. In this way, all low energy polymorphs can be identified. 4. The crystallizable species, or more often the solvent, may be absorbed in the surfactant interfacial layer and so the solute concentration within the interior pool of the microemulsion droplet may differ substantially from the bulk concentration used in making the microemulsion. This possibility must be checked by measuring the solubility of the crystallizing species in the microemulsion, and then determining the mean initial supersaturation ratios accordingly. Accurate solubility measurements require adding powdered material to a microemulsion and leaving for a prolonged

5. Bicontinuous and percolating microemulsions are not suitable systems for determining critical nucleus sizes or obtaining thermodynamic control of crystallization. In bicontinuous microemulsions, the absence of 3D nanoconfinement precludes their use. In percolating microemulsions, the droplets cluster and transiently form much larger droplets in which (near) stable nuclei of metastable polymorphs can form and grow, alongside the (near) stable nuclei of the stable form in the non-clustering droplets. For water-in-oil microemulsions, the absence of percolation and bicontinuous structures can

6. Microemulsions are thermodynamically stable and so form spontaneously upon mixing the constituents. Hence shaking by hand and vortexing are suitable preparation methods. Prolonged sonication should be avoided in case this affects the crystallization. 7. When the supersaturation in the microemulsions is achieved via anti-solvent addition or by a chemical reaction, a mixed microemulsion method should generally be implemented whereby two microemulsions are prepared. A different reactant is in each microemulsion, or for the antisolvent crystallization method, one microemulsion contains the undersaturated crystallizable species in a good solvent and the other microemulsion contains the antisolvent. On mixing the two microemulsions, transient dimer formation between droplets containing different reactants and/or solvents enables a relatively rapid equilibration of interior droplet content to take place, on the timescale of ~s to ms (Ganguli et al., 2010). After this, it can be assumed that the dispersed reactants and solvents are distributed amongst the droplets with a Poisson distribution. Adding the antisolvent or second reactant drop-wise to the microemulsion should be avoided as this can create high concentration fluctuations immediately after the addition (Chen et al., 2011). Alternatively, if one of the reactants is soluble in the continuous phase, a solution of this reactant should be added to the microemulsion

be assumed if the microemulsions show minimal conductivity.

one, and thereby crystallizing in preference to the stable form.

period (weeks) to ensure equilibration.


were necessary for crystallization to be possible in the glycine system. A similar analysis for the mefenamic acid case gives initial supersaturation ratios >15 (Nicholson & Cooper, 2011).

Fig. 9. TEM images of stable Form I nanocrystals of mefenamic acid grown from DMF microemulsions. (a) ~4 nm nanocrystals grown in 12 hours and (b) a Form I nanocrystal grown in 24 hours.

These very high initial supersaturation ratios highlight two key factors governing the solution crystallization of stable polymorphs from microemulsions. Firstly, the substantial supersaturation decrease as a nucleus grows in a droplet means that very high initial supersaturations are needed for a (near) stable nucleus to form. Secondly, these very high initial supersaturations help ensure that the nucleation barriers to all possible polymorphs are surmountable. Hence, the use of microemulsions is the only way to crystallize a stable polymorph that has a very high nucleation barrier in bulk solution.

### **4.3 Practical considerations for choosing microemulsion systems**

In choosing suitable microemulsion systems for achieving thermodynamic control of crystallization to obtain stable polymorphs, or for obtaining reliable estimates of critical nucleus sizes, the following should be considered.


were necessary for crystallization to be possible in the glycine system. A similar analysis for the mefenamic acid case gives initial supersaturation ratios >15 (Nicholson & Cooper, 2011).

(a) (b)

polymorph that has a very high nucleation barrier in bulk solution.

relationships \* 2| |/3 *r R* and <sup>3</sup> \* 32 | | /81 *n Rv*

nucleus sizes, the following should be considered.

**4.3 Practical considerations for choosing microemulsion systems** 

grown in 24 hours.

Fig. 9. TEM images of stable Form I nanocrystals of mefenamic acid grown from DMF microemulsions. (a) ~4 nm nanocrystals grown in 12 hours and (b) a Form I nanocrystal

These very high initial supersaturation ratios highlight two key factors governing the solution crystallization of stable polymorphs from microemulsions. Firstly, the substantial supersaturation decrease as a nucleus grows in a droplet means that very high initial supersaturations are needed for a (near) stable nucleus to form. Secondly, these very high initial supersaturations help ensure that the nucleation barriers to all possible polymorphs are surmountable. Hence, the use of microemulsions is the only way to crystallize a stable

In choosing suitable microemulsion systems for achieving thermodynamic control of crystallization to obtain stable polymorphs, or for obtaining reliable estimates of critical

1. The material to be crystallized should be insoluble, or only sparingly soluble in the microemulsion continuous phase. Since the material will have the same chemical potential at equilibrium in the continuous phase, as in the confined phase, the supersaturated material could potentially crystallize in the continuous phase with attendant loss of thermodynamic control. If the crystallizing material has a very low/negligible solubility in the continuous phase, however, the impingement rate onto a nucleus in this phase is so low that the nucleation rate and subsequent growth of such nuclei will be minimized. Then it can be reliably assumed that the crystallization is initially confined to the dispersed phase so that thermodynamic control is possible. 2. The nucleation should ideally be homogeneous. Choosing surfactants that do not promote crystallization at planar or emulsion interfaces helps ensure this. Homogeneous nucleation enables critical nucleus sizes to be obtained more reliably, since the contact angle of 180 is, of course, invariant with droplet size, *R*, and the

*<sup>c</sup>* are valid for droplet sizes



Crystallization in Microemulsions:

A Generic Route to Thermodynamic Control and the Estimation of Critical Nucleus Size 143

to scattering from the growing rutile particles. The growth of the nanoparticles could be increased by the subsequent addition of water to swell the droplets so that an emulsion formed. Notably, even if the water addition occurred only a few minutes after mixing the TIPO solution with the aqueous HCl microemulsion, good crystallinity rutile particles were still formed. In contrast, when the reaction was carried out in the bulk phase, poor crystallinity/amorphous titania was obtained, demonstrating that the microemulsion stage was crucial for the formation of seed rutile nanocrystals. This general strategy of slowing the reaction rate via limited reactants and/or an appropriate pH range can be used to help introduce, or increase, crystallinity of inorganic nanoparticles obtained from microemulsions.

(a) (b)

(c) Fig. 10. Crystallization of rutile from microemulsions at room temperature and pressure. (a) High resolution electron microscopy image showing a 4 nm nanocrystal grown after 12 hours. (b) Electron microscopy image of the rutile nanocrystals taken after 3 days. (c) Powder X-ray diffraction trace of the rutile nanocrystals before and after calcination.

The use of microemulsions to exert thermodynamic control of crystallization is clearly an advantage whenever stable crystal forms are needed, such as in drug formulations and in obtaining nanocrystals with specific size-dependant properties. However, the much slower growth of crystals in microemulsions, compared to that in bulk solution, may limit

**4.5 Advantages and drawbacks of crystallization in microemulsions** 

containing the second reactant. Here, it is necessary to ensure the reaction proceeds predominantly in the microemulsion droplet, or at the droplet interface, by making sure that the second reactant has a negligible solubility in the continuous phase, whilst the first reactant must be able to partition into the droplet interface and/or interior.

8. For determining critical nucleus sizes, crystallization of liquids or high concentration solutes are preferred so that crystallization peaks are observable in the DSC. The crystallization peak is then associated with the mean droplet size | |*R* , since close to this peak most droplets can form stable nuclei and crystallize. To obtain stable polymorphs, solution-crystallization from microemulsions is preferred, as then the substantial supersaturation decrease as the nuclei grow means that large initial supersaturations are required in order to create (near) stable nuclei and these help ensure the nucleation barriers to all polymorphs are surmountable. To ensure crystallization of stable polymorphs from microemulsions, the crystallization should only just be possible, so that the (near) stable nuclei only form in the largest droplets with the highest supersaturations. Hence the initial supersaturations and number of solute molecules in these droplets will be significantly higher than the mean values.

#### **4.4 Crystallization of inorganic systems in microemulsions**

Recently we have extended the thermodynamic control of crystallization methodology to inorganic polymorphic systems. There is much literature detailing inorganic nanoparticle formation via the mixed microemulsion approach (Ganguli et al, 2010). However, often the nanoparticles obtained are amorphous and so require high temperature and/or high pressures to introduce crystallinity. For instance, many literature examples concerning the formation of titania nanoparticles produce the crystallinity via subsequent calcining (e.g. Fernández-Garc et al., 2007; Fresno et al., 2009) or a combined microemulsion-solvothermal process (e.g. Kong et al., 2011). Many methods also involve continual stirring. This is not necessary for microemulsions since they are thermodynamically stable, and stirring may disrupt any potential thermodynamic control if larger transient droplets are formed from multiple colliding droplets. Our microemulsion methodology enables direct crystallization of the nanocrystals of rutile, the stable form of titania, at room temperature provided the solute concentrations are kept sufficiently low and the crystallization is confined (predominantly) to the dispersed phase. To illustrate this, a microemulsion comprising 1.74 g of cyclohexane, 1 hexanol and Triton X-100 in the volume ratio of 7 : 1.2 : 1.8, and 180 l of 2M HCl as the dispersed phase, was prepared. To this was added a solution of 180 l of titanium isopropoxide (TIPO) in 1.74 g of the cyclohexane, 1-hexanol and Triton X-100 surfactant solution. The TIPO molecules reacted with the water predominantly at the droplet interface so that the resulting titanium dioxide resided mainly in the droplets. The use of 2M HCl slowed down the production of titanium dioxide, preventing gellation, and allowing the crystallization to proceed under thermodynamic control to give the stable rutile phase. TEM after 12 hours confirmed that the nanoparticles of size ~4 nm were crystalline rutile (see Figure 10a). After 3 days the nanocrystals were ~100 nm (see Figure 10b).

Similar microemulsion compositions with surfactant:aqueous mass ratios of 2.5:1 or less and ≤9% by volume of TIPO, produced rutile nanoparticles of good crystallinity. Indeed calcining these particles at 450 C for 18 hours led to only a small increase in crystallinity (see Figure 10c). The microemulsions gradually took on a blue tinge over several days due

first reactant must be able to partition into the droplet interface and/or interior. 8. For determining critical nucleus sizes, crystallization of liquids or high concentration solutes are preferred so that crystallization peaks are observable in the DSC. The crystallization peak is then associated with the mean droplet size | |*R* , since close to this peak most droplets can form stable nuclei and crystallize. To obtain stable polymorphs, solution-crystallization from microemulsions is preferred, as then the substantial supersaturation decrease as the nuclei grow means that large initial supersaturations are required in order to create (near) stable nuclei and these help ensure the nucleation barriers to all polymorphs are surmountable. To ensure crystallization of stable polymorphs from microemulsions, the crystallization should only just be possible, so that the (near) stable nuclei only form in the largest droplets with the highest supersaturations. Hence the initial supersaturations and number of solute molecules in these droplets will be significantly higher than the mean values.

Recently we have extended the thermodynamic control of crystallization methodology to inorganic polymorphic systems. There is much literature detailing inorganic nanoparticle formation via the mixed microemulsion approach (Ganguli et al, 2010). However, often the nanoparticles obtained are amorphous and so require high temperature and/or high pressures to introduce crystallinity. For instance, many literature examples concerning the formation of titania nanoparticles produce the crystallinity via subsequent calcining (e.g. Fernández-Garc et al., 2007; Fresno et al., 2009) or a combined microemulsion-solvothermal process (e.g. Kong et al., 2011). Many methods also involve continual stirring. This is not necessary for microemulsions since they are thermodynamically stable, and stirring may disrupt any potential thermodynamic control if larger transient droplets are formed from multiple colliding droplets. Our microemulsion methodology enables direct crystallization of the nanocrystals of rutile, the stable form of titania, at room temperature provided the solute concentrations are kept sufficiently low and the crystallization is confined (predominantly) to the dispersed phase. To illustrate this, a microemulsion comprising 1.74 g of cyclohexane, 1 hexanol and Triton X-100 in the volume ratio of 7 : 1.2 : 1.8, and 180 l of 2M HCl as the dispersed phase, was prepared. To this was added a solution of 180 l of titanium isopropoxide (TIPO) in 1.74 g of the cyclohexane, 1-hexanol and Triton X-100 surfactant solution. The TIPO molecules reacted with the water predominantly at the droplet interface so that the resulting titanium dioxide resided mainly in the droplets. The use of 2M HCl slowed down the production of titanium dioxide, preventing gellation, and allowing the crystallization to proceed under thermodynamic control to give the stable rutile phase. TEM after 12 hours confirmed that the nanoparticles of size ~4 nm were crystalline rutile (see Figure

Similar microemulsion compositions with surfactant:aqueous mass ratios of 2.5:1 or less and ≤9% by volume of TIPO, produced rutile nanoparticles of good crystallinity. Indeed calcining these particles at 450 C for 18 hours led to only a small increase in crystallinity (see Figure 10c). The microemulsions gradually took on a blue tinge over several days due

**4.4 Crystallization of inorganic systems in microemulsions** 

10a). After 3 days the nanocrystals were ~100 nm (see Figure 10b).

containing the second reactant. Here, it is necessary to ensure the reaction proceeds predominantly in the microemulsion droplet, or at the droplet interface, by making sure that the second reactant has a negligible solubility in the continuous phase, whilst the to scattering from the growing rutile particles. The growth of the nanoparticles could be increased by the subsequent addition of water to swell the droplets so that an emulsion formed. Notably, even if the water addition occurred only a few minutes after mixing the TIPO solution with the aqueous HCl microemulsion, good crystallinity rutile particles were still formed. In contrast, when the reaction was carried out in the bulk phase, poor crystallinity/amorphous titania was obtained, demonstrating that the microemulsion stage was crucial for the formation of seed rutile nanocrystals. This general strategy of slowing the reaction rate via limited reactants and/or an appropriate pH range can be used to help introduce, or increase, crystallinity of inorganic nanoparticles obtained from microemulsions.

Fig. 10. Crystallization of rutile from microemulsions at room temperature and pressure. (a) High resolution electron microscopy image showing a 4 nm nanocrystal grown after 12 hours. (b) Electron microscopy image of the rutile nanocrystals taken after 3 days. (c) Powder X-ray diffraction trace of the rutile nanocrystals before and after calcination.

### **4.5 Advantages and drawbacks of crystallization in microemulsions**

The use of microemulsions to exert thermodynamic control of crystallization is clearly an advantage whenever stable crystal forms are needed, such as in drug formulations and in obtaining nanocrystals with specific size-dependant properties. However, the much slower growth of crystals in microemulsions, compared to that in bulk solution, may limit

Crystallization in Microemulsions:

We thank EPSRC for funding.

pp413-417.

pp4030-4034.

483.

(September 2008), 124715.

**6. Acknowledgement** 

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industrial applications. A strategy whereby the microemulsion is controllably destabilized once the (near) stable nuclei have formed may circumvent this problem. For instance, the addition of more dispersed phase to swell the droplets into an emulsion may prove advantageous, provided the additional solution results in growth only on the existing (near) stable nuclei and nanocrystals, rather than the nucleation of new crystals, since the latter would be produced under kinetic, rather than thermodynamic, control. An effective approach to ensure this would be to induce the supersaturation of the additional dispersed phase slowly only after the emulsion has formed e.g. by cooling or adding a separate microemulsion containing an antisolvent.

## **5. Conclusion**

Microemulsions present a unique opportunity for both the reliable estimate of critical nucleus sizes and the thermodynamic control of crystallization. The 3D droplet nanoconfinement results in crystallization being limited by the ability to form (near) stable nuclei, rather than critical nuclei, under conditions where crystallization is only just possible. This is a direct consequence of the limited amount of material within a droplet. In solution crystallization there is a substantial supersaturation decrease as a nucleus grows in a nanodroplet. The supersaturation decrease means that very high initial supersaturations are required in a droplet to achieve a (near) stable nucleus, thus enabling nucleation barriers to be readily surmountable. Hence solution crystallization from microemulsions is the only methodology known to-date that can generically crystallize stable polymorphs directly, even when they have insurmountable nucleation barriers in bulk solution. The transient dimer formation in microemulsions provides a mechanism for nuclei growth; we find it is possible to grow crystals ranging from nm to mm in size. Crystallization in microemulsions has already been successfully applied to 'leapfrog' Ostwald's rule of stages and directly crystallize the stable polymorphs of three 'problem' organic systems: glycine, mefenamic acid and ROY. The methodology should be of significant use in the pharmaceutical industry, as it provides the first generic method for finding the most stable polymorph for any given drug, thereby preventing another Ritonavir-type crisis. Microemulsions have also been used to synthesis nanocrystals of rutile without requiring a subsequent calcination step. Other inorganic systems that typically produce amorphous nanoparticulates are also likely to benefit from this approach. Its application to protein crystallization may prove problematic, given the larger size of protein molecules, though droplet clustering to fully encase the protein may occur in these systems alleviating this limitation. Future work will investigate this possibility. A disadvantage of the methodology is that once the (near) stable nuclei are generated, their growth is significantly impeded. Initially this is due to their nanoconfinement, and subsequently, when the nanocrystals grow bigger than these droplets, results from the limited concentration of their molecules in the continuous phase. Controlled microemulsion destabilization strategies, such as adding more of the dispersed phase to form an emulsion, may prove a viable route to circumvent this problem. Finally, given that the ultimate crystal size can vary from nm to mm, depending upon the population of (near) stable nuclei and their subsequent growth rates, there is a significant need for greater understanding of how the growth rates can be tuned. Then the use of microemulsions in crystallization would be truly unrivalled in producing both high crystallinity forms and the desired crystal size.

## **6. Acknowledgement**

We thank EPSRC for funding.

## **7. References**

144 Crystallization – Science and Technology

industrial applications. A strategy whereby the microemulsion is controllably destabilized once the (near) stable nuclei have formed may circumvent this problem. For instance, the addition of more dispersed phase to swell the droplets into an emulsion may prove advantageous, provided the additional solution results in growth only on the existing (near) stable nuclei and nanocrystals, rather than the nucleation of new crystals, since the latter would be produced under kinetic, rather than thermodynamic, control. An effective approach to ensure this would be to induce the supersaturation of the additional dispersed phase slowly only after the emulsion has formed e.g. by cooling or adding a separate

Microemulsions present a unique opportunity for both the reliable estimate of critical nucleus sizes and the thermodynamic control of crystallization. The 3D droplet nanoconfinement results in crystallization being limited by the ability to form (near) stable nuclei, rather than critical nuclei, under conditions where crystallization is only just possible. This is a direct consequence of the limited amount of material within a droplet. In solution crystallization there is a substantial supersaturation decrease as a nucleus grows in a nanodroplet. The supersaturation decrease means that very high initial supersaturations are required in a droplet to achieve a (near) stable nucleus, thus enabling nucleation barriers to be readily surmountable. Hence solution crystallization from microemulsions is the only methodology known to-date that can generically crystallize stable polymorphs directly, even when they have insurmountable nucleation barriers in bulk solution. The transient dimer formation in microemulsions provides a mechanism for nuclei growth; we find it is possible to grow crystals ranging from nm to mm in size. Crystallization in microemulsions has already been successfully applied to 'leapfrog' Ostwald's rule of stages and directly crystallize the stable polymorphs of three 'problem' organic systems: glycine, mefenamic acid and ROY. The methodology should be of significant use in the pharmaceutical industry, as it provides the first generic method for finding the most stable polymorph for any given drug, thereby preventing another Ritonavir-type crisis. Microemulsions have also been used to synthesis nanocrystals of rutile without requiring a subsequent calcination step. Other inorganic systems that typically produce amorphous nanoparticulates are also likely to benefit from this approach. Its application to protein crystallization may prove problematic, given the larger size of protein molecules, though droplet clustering to fully encase the protein may occur in these systems alleviating this limitation. Future work will investigate this possibility. A disadvantage of the methodology is that once the (near) stable nuclei are generated, their growth is significantly impeded. Initially this is due to their nanoconfinement, and subsequently, when the nanocrystals grow bigger than these droplets, results from the limited concentration of their molecules in the continuous phase. Controlled microemulsion destabilization strategies, such as adding more of the dispersed phase to form an emulsion, may prove a viable route to circumvent this problem. Finally, given that the ultimate crystal size can vary from nm to mm, depending upon the population of (near) stable nuclei and their subsequent growth rates, there is a significant need for greater understanding of how the growth rates can be tuned. Then the use of microemulsions in crystallization would be truly unrivalled in producing both high

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**5. Conclusion** 


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**6** 

Marek Smolik

 *Poland* 

**Chemical, Physicochemical and Crystal –** 

**Chemical Aspects of Crystallization from** 

*Faculty of Chemistry, Silesian University of Technology, Gliwice* 

**Aqueous Solutions as a Method of Purification** 

This chapter is intended to discuss the effect of chemism of crystallizing and co-crystallizing substances (i.e., their chemical, physicochemical and crystal–chemical properties), as well as some other factors on efficiency of their separation and purification during crystallization

There are three main aims of crystallization (Rojkowski & Synowiec, 1991): creation of the

While using crystallization for purification and separation of various substances, as well as for enrichment of trace amounts of new–found radioactive elements, it was established that (in addition to many others) chemical factors strongly affected the mentioned operations. Mechanisms of trace radioactive elements' co-crystallization and the significance of these factors on their enrichment efficiency were reviewed in some works (Przytycka, 1968;

The influence of factors determining the structures of salts of crystal ionic lattices, (salts considered as ionic coordination compounds) and their ability to isomorphous and isodimorphous mixing on their possibility to crystallization separation was thoroughly discussed by Balarew (1987). Whereas developments concerning inclusions of isomorphous impurities during crystallization from solutions were reviewed by Kirkova et al. (1998).

The discovered settlements were useful in the preliminary assessment of the effectivity of the crystallization method for purification of substances (Kirkova, 1994), for concentration microimpurities (Zolotov & Kuzmin, 1982), for growing of single crystals of specific properties (Byrappa, et al., 1986; Demirskaya, et al. 1989), as well as in explanation of the

In spite of development of solvent extraction and ionic exchange methods, crystallization is still a very attractive method of purification, particularly in the preparation of numerous


purified to the level suitable for HPIS and readily removed after crystallization;

high–purity inorganic substances (HPIS). There are two main reasons for that:

**1. Introduction** 

from aqueous solutions.

Niesmeanov, 1975).

solid phase, forming crystals, purification of substances.

genesis of some minerals (Borneman-Starinkevich, 1975).

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Zarur, A. J. & Ying, J.Y. (2000). Reverse Microemulsion Synthesis of Nanostructured Complex Oxides for Catalytic Combustion. *Nature*, Vol. 403, No. 6765, (January 2000), pp65–67.

## **Chemical, Physicochemical and Crystal – Chemical Aspects of Crystallization from Aqueous Solutions as a Method of Purification**

Marek Smolik *Faculty of Chemistry, Silesian University of Technology, Gliwice Poland* 

## **1. Introduction**

148 Crystallization – Science and Technology

Zarur, A. J. & Ying, J.Y. (2000). Reverse Microemulsion Synthesis of Nanostructured

pp10005-10014.

2000), pp65–67.

Isooctane-AOT Microemulsions. *Langmuir*, Vol. 16, No. 26, (December 2000),

Complex Oxides for Catalytic Combustion. *Nature*, Vol. 403, No. 6765, (January

This chapter is intended to discuss the effect of chemism of crystallizing and co-crystallizing substances (i.e., their chemical, physicochemical and crystal–chemical properties), as well as some other factors on efficiency of their separation and purification during crystallization from aqueous solutions.

There are three main aims of crystallization (Rojkowski & Synowiec, 1991): creation of the solid phase, forming crystals, purification of substances.

While using crystallization for purification and separation of various substances, as well as for enrichment of trace amounts of new–found radioactive elements, it was established that (in addition to many others) chemical factors strongly affected the mentioned operations. Mechanisms of trace radioactive elements' co-crystallization and the significance of these factors on their enrichment efficiency were reviewed in some works (Przytycka, 1968; Niesmeanov, 1975).

The influence of factors determining the structures of salts of crystal ionic lattices, (salts considered as ionic coordination compounds) and their ability to isomorphous and isodimorphous mixing on their possibility to crystallization separation was thoroughly discussed by Balarew (1987). Whereas developments concerning inclusions of isomorphous impurities during crystallization from solutions were reviewed by Kirkova et al. (1998).

The discovered settlements were useful in the preliminary assessment of the effectivity of the crystallization method for purification of substances (Kirkova, 1994), for concentration microimpurities (Zolotov & Kuzmin, 1982), for growing of single crystals of specific properties (Byrappa, et al., 1986; Demirskaya, et al. 1989), as well as in explanation of the genesis of some minerals (Borneman-Starinkevich, 1975).

In spite of development of solvent extraction and ionic exchange methods, crystallization is still a very attractive method of purification, particularly in the preparation of numerous high–purity inorganic substances (HPIS). There are two main reasons for that:


Chemical, Physicochemical and Crystal – Chemical Aspects of

– degree of crystallization of the macrocomponent.

**2.1 Co–crystallization coefficients** 

Taking into account that

co-crystallization coefficient:

(solid solutions) if they are isomorphous or isodimorphous.

substance between two immiscible solvents Kx = Cs/Cr.

D = D2/1 =

*s*

*<sup>n</sup> n n <sup>C</sup> m V m*

*<sup>s</sup> <sup>s</sup> <sup>s</sup> s s*

e'k– relative contents of a microcomponent in the crystal [ppm],

where:

in crystal)

Crystallization from Aqueous Solutions as a Method of Purification 151

Kk– crystal purification coefficient (multiplicity of lowering initial microcomponent contents

However, in reality, in numerous cases a microcomponent is captured by the solid phase, mainly by forming mixed crystals. Micro and macrocomponents form real mixed crystals

Homogeneous partition takes place in equilibrium conditions between the whole mass of a

 *<sup>s</sup> o r X s r n n n K*

where : n – number of moles of the microcomponent in a crystal, no – the whole number of moles of the microcomponent in the system, ms – the mass of a crystal, mr – the mass of

where: Vs and Vr – volumes of the solid phase and the solution, Cs and Cr – concentration of a microcomponent in the solid phase and in the solution it is possible to obtain an equation, identical to the well-known Berthelot–Nernst equation describing the partition of a

During the crystallization from the solution containing two components: macrocomponent (1) and microcomponent (2) the ratio of their partition coefficients defines the equilibrium

*s*

*C C K C C C C K C C C C C C*

Substituting: no and n – number of moles of microcomponent in the whole system and in the solid phase, and zo and z – number of moles of macrocomponent in the whole system and in

*r r*

2 2 2 1 2 1 1 2 1 2 1 1

*r s sr s s r*

as well as

*m m* (2)

*C*

. (3)

 *o ro o <sup>r</sup> <sup>r</sup> r r r*

*nn nn nn*

*m V m*

e'o– initial relative contents of a microcomponent (before crystallization) [ppm],

Cr– contents of the macrocomponent in the mother solution, (its solubility) [%],

Ck– contents of the macrocomponent in the crystal [%], (100–Ck) – crystal humidity [%],

**2.1.1 Homogeneous distribution coefficient D2/1 (Henderson– Kraček, Chlopin)** 

crystal and the mother solution, and is described by the Chlopin equation:

solution, s – density of crystal, r – density of solution, KX – Chlopin constant.


## **2. Crystallization as a method of purification**

The crystallization should, in principle, yield very significant purification of a substance, but for the phenomenon of transport of accompanying impurities into the crystal, which may happen in the following ways presented in the simplified scheme below.

Although a suitable choice of crystallization conditions (supersaturation, rate of crystallization), as well as the ways of separation of crystals from mother solutions (filtration, washing) permits minimizing the capture of impurities derived from the inclusion of the mother solution, occlusion or external adsorption, it is impossible to restrict impurities originated from the capture by the whole volume of the solid phase or internal adsorption.1

The highest effect of purification may be expected when impurities are not captured by the solid phase of crystals but get into crystals as a result of the mother solution's residue, which cannot be removed by filtration2. In this boundary case the efficiency of crystal purification after its separation from the mother solution (without washing) is defined by the equation (Gorshtein, 1969):

$$\frac{1}{K\_k} = \frac{e\_k'}{e\_o'} = \frac{100 - \mathcal{C}\_k}{\mathcal{C}\_k} \cdot \frac{\mathcal{C}\_r}{100 - \mathcal{C}\_r} \cdot \frac{1}{1 - a} \tag{1}$$

<sup>1</sup> Internal adsorption takes place when microcomponents cannot form solid solutions with macrocomponent (Niesmieanov,1975)], it is a rather sparsely occurring phenomenon (Przytycka,1968). 2 During the crystallization without stirring or with not vigorous stirring, big, aggregated crystals (twins, intergrowth) are obtained. The presence of cavities on their surface obstructs the separation of mother solution from these crystals, which results in lowering their purity. Stirring during the crystallization at a considerable concentration of crystals causes rounding of crystals because of abrasion. Then large crystals adopt the form of spheres or ellipsoids, whose separation from mother solution by means of filtration is easier, which leads to higher purity of final product (Matusevich, 1961; Bamforth, 1965).

where:

150 Crystallization – Science and Technology


The crystallization should, in principle, yield very significant purification of a substance, but for the phenomenon of transport of accompanying impurities into the crystal, which may

Although a suitable choice of crystallization conditions (supersaturation, rate of crystallization), as well as the ways of separation of crystals from mother solutions (filtration, washing) permits minimizing the capture of impurities derived from the inclusion of the mother solution, occlusion or external adsorption, it is impossible to restrict impurities originated from the capture by the whole volume of the solid phase or internal adsorption.1

The highest effect of purification may be expected when impurities are not captured by the solid phase of crystals but get into crystals as a result of the mother solution's residue, which cannot be removed by filtration2. In this boundary case the efficiency of crystal purification after its separation from the mother solution (without washing) is defined by the equation

> 1 1 100

1 Internal adsorption takes place when microcomponents cannot form solid solutions with macrocomponent (Niesmieanov,1975)], it is a rather sparsely occurring phenomenon (Przytycka,1968). 2 During the crystallization without stirring or with not vigorous stirring, big, aggregated crystals (twins, intergrowth) are obtained. The presence of cavities on their surface obstructs the separation of mother solution from these crystals, which results in lowering their purity. Stirring during the crystallization at a considerable concentration of crystals causes rounding of crystals because of abrasion. Then large crystals adopt the form of spheres or ellipsoids, whose separation from mother solution by means of filtration is

easier, which leads to higher purity of final product (Matusevich, 1961; Bamforth, 1965).

*k kr k kr o e CC*

*K CC <sup>e</sup>* (1)

100 1

happen in the following ways presented in the simplified scheme below.

contamination.

(Gorshtein, 1969):

**2. Crystallization as a method of purification** 

Kk– crystal purification coefficient (multiplicity of lowering initial microcomponent contents in crystal)

e'k– relative contents of a microcomponent in the crystal [ppm],

e'o– initial relative contents of a microcomponent (before crystallization) [ppm],

Ck– contents of the macrocomponent in the crystal [%], (100–Ck) – crystal humidity [%],

Cr– contents of the macrocomponent in the mother solution, (its solubility) [%],

– degree of crystallization of the macrocomponent.

However, in reality, in numerous cases a microcomponent is captured by the solid phase, mainly by forming mixed crystals. Micro and macrocomponents form real mixed crystals (solid solutions) if they are isomorphous or isodimorphous.

## **2.1 Co–crystallization coefficients**

## **2.1.1 Homogeneous distribution coefficient D2/1 (Henderson– Kraček, Chlopin)**

Homogeneous partition takes place in equilibrium conditions between the whole mass of a crystal and the mother solution, and is described by the Chlopin equation:

$$\frac{m\rho\_s}{m\_s} = K\_X \frac{\left(n\_o - n\right)\rho\_r}{m\_r} \tag{2}$$

where : n – number of moles of the microcomponent in a crystal, no – the whole number of moles of the microcomponent in the system, ms – the mass of a crystal, mr – the mass of solution, s – density of crystal, r – density of solution, KX – Chlopin constant.

$$\text{Taking into account that } \frac{n\rho\_s}{m\_s} = \frac{n}{\underbrace{m\_s}} = \frac{n}{V\_s} = \text{C}\_s \text{ as well as } \frac{(n\_o - n)\rho\_r}{m\_r} = \underbrace{\frac{(n\_o - n)}{m\_r}}\_{\rho\_r} = \frac{(n\_o - n)}{V\_r} = \text{C}\_r$$

where: Vs and Vr – volumes of the solid phase and the solution, Cs and Cr – concentration of a microcomponent in the solid phase and in the solution it is possible to obtain an equation, identical to the well-known Berthelot–Nernst equation describing the partition of a substance between two immiscible solvents Kx = Cs/Cr.

During the crystallization from the solution containing two components: macrocomponent (1) and microcomponent (2) the ratio of their partition coefficients defines the equilibrium co-crystallization coefficient:

$$\mathbf{D} = \mathbf{D}\_{2/1} = \frac{K\_2}{K\_1} = \frac{\left(\frac{\mathbf{C}\_s}{\mathbf{C}\_r}\right)\_2}{\left(\frac{\mathbf{C}\_s}{\mathbf{C}\_r}\right)\_1} = \frac{\left(\frac{\mathbf{C}\_2}{\mathbf{C}\_1}\right)\_s}{\left(\frac{\mathbf{C}\_2}{\mathbf{C}\_1}\right)\_r} = \frac{\mathbf{C}\_{2s} \cdot \mathbf{C}\_{1r}}{\mathbf{C}\_{1s} \cdot \mathbf{C}\_{2r}}.\tag{3}$$

Substituting: no and n – number of moles of microcomponent in the whole system and in the solid phase, and zo and z – number of moles of macrocomponent in the whole system and in

Chemical, Physicochemical and Crystal – Chemical Aspects of

amounts of the microcomponent during the crystallization:

the solid phase (Gorshtein, 1969).

D2/1' D2/1

crystallizations) mass of crystals

**efficiency** 

Crystallization from Aqueous Solutions as a Method of Purification 153

Both homogeneous and heterogeneous partitions are boundary cases of distribution of the microcomponent between the solid phase and the mother solution. Experimental study involving which of both coefficients retains constant value with the increase of the degree of

Homogeneous partition coefficients D2/1 are a convenient measure of crystallization efficiency as a method of purification. In the case of homogeneous partition of microcomponent in the solid phase, the final result of purification (without washing) may be expressed by the formula (Gorshtein, 1969), derived on the basis of the balance of

1 100 1

*k k r k k o r*

' ' '

' ' '

*e D D C C*

After careful washing of the crystals by pure, saturated solution of the macrocomponent, inclusions of the mother solution, as well as microcomponents adsorbed on the surface of

where: KK – multiplicity of lowering initial microcomponent contents (e'0, e'k(p) - initial contents of microcomponent –[ppm] in crystal and in washed crystal after crystallization); D2/1(D'2/1) = es(e'k(p))/e'r ( e's and e'r – contents of micomponent - [ppm] in the solid phase and the mother solution); D2/1 – isomorphous co-crystallization coefficient of microcomponent, D2/1' – adsorption–isomorphous co-crystallization coefficient of microcomponent; expression (D'2/1 – D2/1)/D'2/1 qualifies a relative importance of adsorption in the capture of microcomponent by

Knowing D2/1(D2/1') and using the equations (9) and (10) it is possible to evaluate a number of crystallizations in the conditions of homogeneous partition of the microcomponent (at D2/1(D2/1') =const.) necessary to achieve a desirable degree of purification of crystals. An

Table 1. The number of NiSO4·7H2O crystallizations necessary to achieve 10–fold lowering

*e D*

2/1

( ) 2/1

 

*<sup>K</sup> e D C C <sup>D</sup>* (9)

1 1 100

1

2/1 2/1

2/1 2/1

(10)

The whole yield of purification (mf/m0)·100% = (k)·100%

crystallization gives information on what partition is actually taking place.

the crystal, will be removed. The result of the purification in this case will be:

1

example of such evaluation for NiSO47H2O is presented in Table 1.

Number of crystallizations (k)

0.75 0.75 15 0.003 0.50 0.50 6 1.56 0.25 0.25 3 12.5 0.10 0.10 2 25.0 0.05 0.05 1 50.0

D2/1 (D2/1') (Smolik, 2004), Ck=98%, Cr=50%, =0,50 (50%), m0(mf) – initial (after k

of its initial contents of microcomponent for various levels of coefficient

'

'

*K D e*

*k p k o*

**2.2 Homogeneous distribution coefficients D2/1 as indicators of crystallization** 

the solid phase into equation (3) and suitable rearranging, it is possible to obtain a more convenient Henderson & Kraček equation (Niesmieanov, 1975):

$$D\_{2/1} = \frac{n\left(z\_o - z\right)}{z\left(n\_o - n\right)}\tag{4}$$

Further transformation of this equation gives other, often used practical formulae (Smolik, 2004):

$$D\_{2/1} = \frac{\frac{m}{z}}{\frac{(n\_o - n)}{(n\_o - n)}} = \frac{e\_s'}{e\_r'} \tag{5}$$

and

$$D\_{2/1} = \frac{\frac{n}{n\_o} \left(\frac{z\_o}{z\_o} - \frac{z}{z\_o}\right)}{\frac{z}{z\_o} \left(\frac{n\_o}{n\_o} - \frac{n}{n\_o}\right)} = \frac{\beta \left(1 - \alpha\right)}{\alpha \left(1 - \beta\right)}\tag{6}$$

where = z/zo is the degree of crystallization of macrocomponent**,** = n/no is the degree of cocrystallization of microcomponent, e's and e'r are relative concentrations of microcomponent in the solid phase and in the mother solution, respectively ([ppm] in relation to macrocomponent), D2/1 – homogeneous partition coefficient (co-crystallization coefficient).

#### **2.1.2 Heterogeneous (logarithmic) distribution coefficients (Doerner–Hoskins)**

Logarithmic partition can take place if the equilibrium between the whole mass of crystal does not exist, but only between the surface layer of a crystal and solution. If D2/1 1, the concentration of microcomponent in the solution during the crystallization will be changing continuously. So the microcomponent will distribute in the crystal in a stratified manner ("onion" structure). This process for the elementary layer of the crystal may be described by

the equation parallel to that of Henderson–Kraček: *o o dn n n dz z z* ,

where the meaning of *no, n, zo ,z* is the same as previously described and is the heterogeneous (logarithmic) partition coefficient. The integration of this expression yields the known equations (Doerner & Hoskins, 1925):

$$\lambda \ln \frac{n\_o}{n\_o - n} = \lambda \ln \frac{z\_o}{z\_o - z} \tag{7}$$

or

$$
\lambda = \frac{\log\left(1 - \beta\right)}{\log\left(1 - \alpha\right)}\tag{8}
$$

the solid phase into equation (3) and suitable rearranging, it is possible to obtain a more

Further transformation of this equation gives other, often used practical formulae (Smolik,

 

*n <sup>e</sup> <sup>z</sup> <sup>D</sup> n n e z z*

2/1 *<sup>s</sup> o r*

*o*

 

*o oo o o oo o*

*z n n zn n*

**2.1.2 Heterogeneous (logarithmic) distribution coefficients (Doerner–Hoskins)** 

*n z z nz z*

where = z/zo is the degree of crystallization of macrocomponent**,** = n/no is the degree of cocrystallization of microcomponent, e's and e'r are relative concentrations of microcomponent in the solid phase and in the mother solution, respectively ([ppm] in relation to macrocomponent), D2/1 – homogeneous partition coefficient (co-crystallization coefficient).

Logarithmic partition can take place if the equilibrium between the whole mass of crystal does not exist, but only between the surface layer of a crystal and solution. If D2/1 1, the concentration of microcomponent in the solution during the crystallization will be changing continuously. So the microcomponent will distribute in the crystal in a stratified manner ("onion" structure). This process for the elementary layer of the crystal may be described by

where the meaning of *no, n, zo ,z* is the same as previously described and is the heterogeneous (logarithmic) partition coefficient. The integration of this expression yields

> ln ln *o o o o n z*

> > log 1

 

2/1

*D*

the equation parallel to that of Henderson–Kraček:

the known equations (Doerner & Hoskins, 1925):

 

1 1

*nn zz* , (7)

log 1 (8)

 *o o*

*dn n n dz z z* ,

 2/1 *o o nz z*

*zn n* (4)

(5)

, (6)

convenient Henderson & Kraček equation (Niesmieanov, 1975):

2004):

and

or

*D*

Both homogeneous and heterogeneous partitions are boundary cases of distribution of the microcomponent between the solid phase and the mother solution. Experimental study involving which of both coefficients retains constant value with the increase of the degree of crystallization gives information on what partition is actually taking place.

#### **2.2 Homogeneous distribution coefficients D2/1 as indicators of crystallization efficiency**

Homogeneous partition coefficients D2/1 are a convenient measure of crystallization efficiency as a method of purification. In the case of homogeneous partition of microcomponent in the solid phase, the final result of purification (without washing) may be expressed by the formula (Gorshtein, 1969), derived on the basis of the balance of amounts of the microcomponent during the crystallization:

$$\frac{1}{\dot{C}\_{k}} = \frac{\dot{\mathcal{e}}\_{k}}{\dot{\mathcal{e}}\_{o}} = \frac{D\_{2/1}^{\circ}}{a \, D\_{2/1}^{\circ} + 1 - a} + \frac{100 - \mathcal{C}\_{k}}{\mathcal{C}\_{k}} \cdot \frac{\mathcal{C}\_{r}}{100 - \mathcal{C}\_{r}} \cdot \frac{1 - D\_{2/1}^{\circ}}{a \, D\_{2/1}^{\circ} + 1 - a} \tag{9}$$

After careful washing of the crystals by pure, saturated solution of the macrocomponent, inclusions of the mother solution, as well as microcomponents adsorbed on the surface of the crystal, will be removed. The result of the purification in this case will be:

$$\frac{1}{\mathcal{K}\_k} = \frac{\stackrel{\cdot}{e\_{k(p)}}}{\stackrel{\cdot}{e\_o}} = \frac{D\_{2/1}}{aD\_{2/1} + 1 - a} \tag{10}$$

where: KK – multiplicity of lowering initial microcomponent contents (e'0, e'k(p) - initial contents of microcomponent –[ppm] in crystal and in washed crystal after crystallization); D2/1(D'2/1) = es(e'k(p))/e'r ( e's and e'r – contents of micomponent - [ppm] in the solid phase and the mother solution); D2/1 – isomorphous co-crystallization coefficient of microcomponent, D2/1' – adsorption–isomorphous co-crystallization coefficient of microcomponent; expression (D'2/1 – D2/1)/D'2/1 qualifies a relative importance of adsorption in the capture of microcomponent by the solid phase (Gorshtein, 1969).

Knowing D2/1(D2/1') and using the equations (9) and (10) it is possible to evaluate a number of crystallizations in the conditions of homogeneous partition of the microcomponent (at D2/1(D2/1') =const.) necessary to achieve a desirable degree of purification of crystals. An example of such evaluation for NiSO47H2O is presented in Table 1.


Table 1. The number of NiSO4·7H2O crystallizations necessary to achieve 10–fold lowering of its initial contents of microcomponent for various levels of coefficient D2/1 (D2/1') (Smolik, 2004), Ck=98%, Cr=50%, =0,50 (50%), m0(mf) – initial (after k crystallizations) mass of crystals

Chemical, Physicochemical and Crystal – Chemical Aspects of

0 100 200 300 400 **time [h]**

Crystallization from Aqueous Solutions as a Method of Purification 155

**DM/CoSO4.7H2O**

Fig. 1. Changes of co-crystallization coefficients, D2/1 of M2+ ions as the effect of isothermal levelling of supersaturation during the crystallization of CoSO47H2O at 20 oC (Smolik, 2003)

0,00 0,20 0,40 0,60 0,80 1,00 1,20

**Ni2+**

**Zn2+ Cd2+**

**Fe2+**

Fig. 2. The principle of the long–time stirring method for the determination of the

*n*

Ni2+ 1.22 1.86 1.52 ± 0.06 1.62 ± 0.09 1.57 ± 0.06 Cu2+ 0.08 0.22 0.14 ± 0.02 0.15 ± 0.01 0.14 ± 0.01 Co2+ 0.77 1.89 1.18 ± 0.09 1.20 ± 0.05 1.19 ± 0.04 Fe2+ 0.42 1.44 0.72 ± 0.05 0.79 ± 0.06 0.76 ± 0.04 Mg2+ 0.16 3.47 1.33 ± 0.12 1.40 ± 0.09 1.36 ± 0.07 Mn2+ 0.13 0.28 0.17 ± 0.02 0.20 ± 0.02 0.19 ± 0.02 Table 2. Determination of equilibrium D2/1 coefficients of M2+ ions during the crystallization

*<sup>s</sup> D t*

When selecting values Domax and Domin the highest and the lowest values of D2/1 obtained during crystallization by the first method are usually taken into consideration. The

for Domin for DoMAX

Average D2/1 after long time stirring Average

 

*<sup>s</sup> D t*

*n*

equilibrium D2/1 

**Mg2+ Mn2+**

0 100 200 300 400 **time [h]**

**Cu2+**

*<sup>s</sup> D t*

*n*

equilibrium coefficients D2/1 (Zhelnin & Gorshtein, 1971; Chlopin, 1957)

experiments are carried out in the following way:

Domin DoMAX

Initial D2/1

of ZnSO4·7H2O at 230C (Smolik, 2000a)

Microcomponent M2+

0,00 0,20 0,40 0,60 0,80 1,00 1,20 1,40

**DM/CoSO4.7H2O**

The data presented in Table 1 show that the level of D2/1 (D2/1') is a very important parameter for the evaluation of crystallization efficiency as a method of purification, and therefore, its knowledge is significant in planning the utilization of crystallization in different stages of preparation of high purity substances.

## **2.3 Practical and equilibrium D2/1 coefficients**

However, crystallization processes are usually realized at non–equilibrium conditions and the obtained, in this case practical (effective), partition coefficients (Dp2/1) depend on the ways in which the crystallization is carried out. This dependence may be presented by the following expression (Kirkova et al., 1996):

$$\text{D}\mathbb{M}\_{2/1} = \text{D}\mathbb{M}\_{2/1} \, \Theta(\text{T}, \text{\textquotedblleft}\text{o, } \text{m}\_{2\text{-}} \, \text{m}\_{\text{I}} \, \text{\textquotedblright}\text{.}\tag{11}$$

where Do2/1 – equilibrium co-crystallization coefficient, - imbalance factor, which is a function of temperature(T), supersaturation of the solution (), rate of stirring (), concentration of microcomponent 2 (m2), concentration of other microcomponents (mj) and other factors ().

Since the equilibrium co-crystallization coefficient does not depend on crystallization conditions, it may be compared with various crystallization systems. Hence, it is important to trace the determination methods of such coefficients.

## **2.4 Methods of determination of equilibrium distribution coefficients**

The possibility of achieving equilibrium homogeneous partition of microcomponents between solution and crystal solid phase was proved by Chlopin. Subsequent investigations in this area (Gorshtein, 1969; Chlopin, 1957; Zhelnin & Gorshtein, 1971) elaboration upon several methods to accomplish equilibrium partition of microcomponents in the crystal, among which the method of isothermal decreasing of supersaturation and the method of long–time stirring of crushed crystals in their saturated solution are most often used.

## **2.4.1 The method of isothermal decreasing of supersaturation**

It relies on cooling a saturated solution without stirring to the end temperature of crystallization (so that no crystal would appear) and after that on vigorous stirring at the constant end temperature until a complete removal of supersaturation takes place (usually for 3 – 360 h) (Zhelnin & Gorshtein, 1971; Chlopin, 1957). An example of such determination of D2/1eq is presented in Fig. 1

## **2.4.2 The method of long–time stirring of crushed crystals in their saturated solution**

The equilibrium is reached starting either from the initial concentration ratio of a microcomponent in crystal and in solution exceeding the expected value of its equilibrium coefficient (DoMAX=e'ko("contaminated" crystal)/e'ro("purified" solution)) – achieving equilibrium "from above" or from this ratio lower than the expected value mentioned above (Domin=e"ko("purified crystal)/e"ro("contaminated" solution)) – achieving equilibrium "from below" (Fig. 2).

The data presented in Table 1 show that the level of D2/1 (D2/1') is a very important parameter for the evaluation of crystallization efficiency as a method of purification, and therefore, its knowledge is significant in planning the utilization of crystallization in

However, crystallization processes are usually realized at non–equilibrium conditions and the obtained, in this case practical (effective), partition coefficients (Dp2/1) depend on the ways in which the crystallization is carried out. This dependence may be presented by the

 Dp2/1 = Do2/1 (T, , , m2, mj ) (11) where Do2/1 – equilibrium co-crystallization coefficient, - imbalance factor, which is a function of temperature(T), supersaturation of the solution (), rate of stirring (), concentration of microcomponent 2 (m2), concentration of other microcomponents (mj) and

Since the equilibrium co-crystallization coefficient does not depend on crystallization conditions, it may be compared with various crystallization systems. Hence, it is important

The possibility of achieving equilibrium homogeneous partition of microcomponents between solution and crystal solid phase was proved by Chlopin. Subsequent investigations in this area (Gorshtein, 1969; Chlopin, 1957; Zhelnin & Gorshtein, 1971) elaboration upon several methods to accomplish equilibrium partition of microcomponents in the crystal, among which the method of isothermal decreasing of supersaturation and the method of

It relies on cooling a saturated solution without stirring to the end temperature of crystallization (so that no crystal would appear) and after that on vigorous stirring at the constant end temperature until a complete removal of supersaturation takes place (usually for 3 – 360 h) (Zhelnin & Gorshtein, 1971; Chlopin, 1957). An example of such determination

**2.4.2 The method of long–time stirring of crushed crystals in their saturated solution**  The equilibrium is reached starting either from the initial concentration ratio of a microcomponent in crystal and in solution exceeding the expected value of its equilibrium coefficient (DoMAX=e'ko("contaminated" crystal)/e'ro("purified" solution)) – achieving equilibrium "from above" or from this ratio lower than the expected value mentioned above (Domin=e"ko("purified crystal)/e"ro("contaminated" solution)) – achieving equilibrium "from

long–time stirring of crushed crystals in their saturated solution are most often used.

different stages of preparation of high purity substances.

to trace the determination methods of such coefficients.

**2.4 Methods of determination of equilibrium distribution coefficients** 

**2.4.1 The method of isothermal decreasing of supersaturation** 

**2.3 Practical and equilibrium D2/1 coefficients** 

following expression (Kirkova et al., 1996):

other factors ().

of D2/1eq is presented in Fig. 1

below" (Fig. 2).

Fig. 1. Changes of co-crystallization coefficients, D2/1 of M2+ ions as the effect of isothermal levelling of supersaturation during the crystallization of CoSO47H2O at 20 oC (Smolik, 2003)

Fig. 2. The principle of the long–time stirring method for the determination of the equilibrium coefficients D2/1 (Zhelnin & Gorshtein, 1971; Chlopin, 1957)

When selecting values Domax and Domin the highest and the lowest values of D2/1 obtained during crystallization by the first method are usually taken into consideration. The experiments are carried out in the following way:


Table 2. Determination of equilibrium D2/1 coefficients of M2+ ions during the crystallization of ZnSO4·7H2O at 230C (Smolik, 2000a)

Chemical, Physicochemical and Crystal – Chemical Aspects of

2/1

macrocomponent(1) – microcomponent(2)).

ratio may be expressed by the following equation:

Crystallization from Aqueous Solutions as a Method of Purification 157

B'bEeLl bB'(z1)+ + eE(z2)+ +lL(z3)– (obviously: *bz ez lz* 1 23 0 ) the following general formula may be derived for thermodynamic co-crystallization coefficient Do2/1 (Balarew, 1987; Smolik & Kowalik, 2010):

*m m*

where: m01(m02), m01(m02) – molal solubility ([mol/kg]) of the salt BbEeLl(B'bEeLl)) and mean molal activity coefficient of the salt BbEeLl(B'bEeLl) in its binary saturated solution; m1(m2), m1(m2) – molality and mean molal activity coefficient of the salt BbEeLl(B'bEeLl) in the ternary solution being in equilibrium with Bb(B'b)EeLl) solid solution; x1(x2) – mole fraction of B(B') ion and *f*1(*f*2) – activity coefficient of ion B(B') in this solid solution; oIII – the partial molar Gibbs free energy of the phase transition of the salt B'bEeLl from its structure (II) into the structure (I) of the salt BbEeLl, = b + e + l, R – gas constant, T – temperature [K]

 

 01 01 2 1

Equations (14) and (15) should, in principle, permit calculating exactly the equilibrium partition coefficient D2/1, if molal solubilities and all activity coefficients (in the aqueous and the solid phases), as well as the partial molar Gibbs free energy of the phase transition, were known. However, these data (except for molal solubilities) are rarely available. In contrast to mean molal activity coefficients in binary saturated solutions (m01, m02), as well as those in the ternary solution being in equilibrium with Bb(B'b)EeLl) solid solution (m1, m2) which are sometimes directly accessible or calculable by means of Pitzer equations, activity coefficients in the solid solution (*f1, f2),* as well as the partial molar free energy of the phase transition, are generally unknown (except for very rare individual cases of crystallization systems:

The attempts to estimate D2/1 coefficients by means of simplified equations (taking into account only activity coefficients in the liquid phase) are connected with huge errors, which proves that they result from the lack of the activity coefficients in the solid solution (*f*1, *f*2) as well as the partial molar Gibbs free energy of the phase transition oIII (Smolik, 2004).

In the case where coefficients D2/1 are independent of mixed crystal composition, the ratio *f*2/*f*1 remains constant (Balarew, 1987). Assuming the regular solution approximation this

> 

1 1 2

exp *<sup>f</sup> H H*

2

*cal cal II I*

exp. . . 2 exp. exp. 1 [%] 1 1 exp <sup>100</sup>

*D D D f*

*<sup>m</sup> <sup>f</sup> <sup>D</sup>*

 

 

02 02 1 2 *b b m m m m*

In the other case involving isomorphous substances (1) and (2) oII = 0. Hence:

2/1

*x m <sup>m</sup> <sup>f</sup> <sup>D</sup>*

2 1 01 01 2 1

1 2 02 02 1 2

 

*<sup>o</sup> b b <sup>m</sup> <sup>m</sup> II I*

 

*<sup>m</sup> <sup>f</sup>* (15)

*<sup>D</sup> D f bRT* (16)

*<sup>f</sup> RT* (17)

0

*xm m f bRT* (14)

 

exp

Crushed "contaminated" crystals (crushed to pass a 0.1mm sieve – <0.1mm) are introduced into several beakers together with their saturated "purified" solution. Crushed "purified" crystals (crushed to pass a 0.1mm sieve – <0.1mm) and their "contaminated" saturated solution are introduced to some other beakers. Contents of the beakers are stirred for ~360 h with a magnetic stirrer at constant temperature (**Table 2)**.

## **3. Thermodynamic approach to the calculation of equilibrium D2/1 coefficients**

The possibility of achieving a thermodynamic equilibrium during crystallization from solutions, as well as melts, as proved by Chlopin (1957), permits introducing a thermodynamic partition coefficient. Substituting concentrations of microcomponent (2) and macrocomponent (1) in equation (3) with their activities (a1S, a2s, a1r, a2r) it is possible to obtain an expression for thermodynamic co-crystallization coefficient Do2/1. (Kirkova et al., 1996; Ratner, 1933).

$$D\_{2/1}^{o} = \frac{a\_{2s} \cdot a\_{1r}}{a\_{2r} \cdot a\_{1s}} = \exp\left(-\frac{\Delta\mu\_{2/1}^{o}}{RT}\right) \tag{12}$$

o2/1 = o2 – o1, where o2 and o1 are the changes of standard molar chemical potential of components (2) and (1) respectively during the transition from the liquid phase (r) into the solid phase (s). Two cases should be distinguished here: 1) substance (2) is not isomorphous with substance (1), i.e., it crystallizes in different crystal systems (different space groups); 2) substance (2) is isomorphous with substance (1).

In the first case the formation of mixed crystals by substance (2) with substance (1) may indicate the existence (besides the basic form [II] of microcomponent [2]) of a polymorphous form (I), metastable in suitable conditions, which is isomorphous with the crystal of the host: macrocomponent – substance (1). The transition of the substance (2) of the structure (II) into its metastable form of the structure (I) is connected with the increase of chemical potential: oIII = o,I2s – o,II2s. Then o2 = o,I2s – o2r = o,I2s – o,II2s + o,II2s – o2r = oIII + o,II2s – o2r , where oIII = o,I2s – o,II2s – free partial molar enthalpy of phase transition III of crystals of microcomponent (2) of structure (II) into the structure (I) proper to that of macrocomponent(1). Therefore (Smolik, 2004):

$$\begin{split} \mathbf{D}\_{2,II/1,l}^{o} &= \frac{\mathbf{a}\_{1r}\mathbf{a}\_{2s}}{\mathbf{a}\_{1s}\mathbf{a}\_{2r}} = \frac{\mathbf{a}\_{1r}\mathbf{a}\_{2s}^{\mathrm{I}}}{\mathbf{a}\_{1s}\mathbf{a}\_{2r}} = \exp\left(-\frac{\Delta\mu\_{2}^{o} - \Delta\mu\_{1}^{o}}{RT}\right) = \exp\left(-\frac{\Delta\mu\_{\mathrm{II\rightarrow I}}^{o} + \mu\_{2s}^{o,II} - \mu\_{2r}^{o} + \mu\_{1r}^{o} - \mu\_{1s}^{o}}{RT}\right) = \\ &= \exp\left(\frac{\mu\_{1s}^{o} - \mu\_{1r}^{o}}{RT}\right) \cdot \exp\left(\frac{\mu\_{2r}^{o} - \mu\_{2s}^{o,II}}{RT}\right) \cdot \exp\left(-\frac{\Delta\mu\_{\mathrm{II\rightarrow I}}^{o}}{RT}\right) = \frac{a\_{1r}^{o}}{a\_{2r}^{o}} \cdot \exp\left(-\frac{\Delta\mu\_{\mathrm{II\rightarrow I}}^{o}}{RT}\right) \end{split} \tag{13}$$

If two double salts: BbEeLl (1) i BbEeLl (2) capable of forming solid solutions by the exchange of B ions into B'ones (it is possible to exchange E ions into E' or L into L') dissociate into ions in aqueous solution according to the reaction:

$$\mathbf{B}\_{\mathsf{b}} \mathbf{E}\_{\mathsf{c}} \mathbf{L}\_{\mathsf{l}} \mathsf{T} \mathsf{s} \mathsf{b} \mathbf{B}^{(\mathsf{z}1)\*} + \mathbf{e} \mathsf{E}^{(\mathsf{z}2)\*} + \mathsf{l} \mathsf{L}^{(\mathsf{z}3)\*}$$

and

Crushed "contaminated" crystals (crushed to pass a 0.1mm sieve – <0.1mm) are introduced into several beakers together with their saturated "purified" solution. Crushed "purified" crystals (crushed to pass a 0.1mm sieve – <0.1mm) and their "contaminated" saturated solution are introduced to some other beakers. Contents of the beakers are stirred for ~360 h

**3. Thermodynamic approach to the calculation of equilibrium D2/1 coefficients**  The possibility of achieving a thermodynamic equilibrium during crystallization from solutions, as well as melts, as proved by Chlopin (1957), permits introducing a thermodynamic partition coefficient. Substituting concentrations of microcomponent (2) and macrocomponent (1) in equation (3) with their activities (a1S, a2s, a1r, a2r) it is possible to obtain an expression for thermodynamic co-crystallization coefficient Do2/1. (Kirkova et al.,

o2/1 = o2 – o1, where o2 and o1 are the changes of standard molar chemical potential of components (2) and (1) respectively during the transition from the liquid phase (r) into the solid phase (s). Two cases should be distinguished here: 1) substance (2) is not isomorphous with substance (1), i.e., it crystallizes in different crystal systems (different

In the first case the formation of mixed crystals by substance (2) with substance (1) may indicate the existence (besides the basic form [II] of microcomponent [2]) of a polymorphous form (I), metastable in suitable conditions, which is isomorphous with the crystal of the host: macrocomponent – substance (1). The transition of the substance (2) of the structure (II) into its metastable form of the structure (I) is connected with the increase of chemical potential: oIII = o,I2s – o,II2s. Then o2 = o,I2s – o2r = o,I2s – o,II2s + o,II2s – o2r = oIII + o,II2s – o2r , where oIII = o,I2s – o,II2s – free partial molar enthalpy of phase transition III of crystals of microcomponent (2) of structure (II) into the structure (I) proper to that of

exp

2 1

*r s*

*o s r*

*o o o II <sup>o</sup> <sup>s</sup> II I*

1s 1r 2 2 1

exp exp exp exp RT RT

 

exp exp aa aa RT RT

> 

If two double salts: BbEeLl (1) i BbEeLl (2) capable of forming solid solutions by the exchange of B ions into B'ones (it is possible to exchange E ions into E' or L into L') dissociate into ions

BbEeLl bB(z1)+ + eE(z2)+ +lL(z3)–

*o o II o o o r s II I r II I*

1r 2s 1r 2s 2 1 II I 2 2r 1r 1s

*a a <sup>D</sup>*

2 1 2/1

*o*

2

*o r*

*a RT a R*

I o , o o o

 

 *T*

(13)

*a a RT* (12)

with a magnetic stirrer at constant temperature (**Table 2)**.

2/1

space groups); 2) substance (2) is isomorphous with substance (1).

macrocomponent(1). Therefore (Smolik, 2004):

1s 2r 1s 2r o o ,

in aqueous solution according to the reaction:

aa aa

 

2, /1,

*D*

and

1996; Ratner, 1933).

#### B'bEeLl bB'(z1)+ + eE(z2)+ +lL(z3)–

(obviously: *bz ez lz* 1 23 0 ) the following general formula may be derived for thermodynamic co-crystallization coefficient Do2/1 (Balarew, 1987; Smolik & Kowalik, 2010):

$$D\_{2/1} = \left(\frac{\chi\_2 \cdot m\_1}{\chi\_1 \cdot m\_2}\right) = \left(\frac{m\_{01} \cdot \chi\_{m01}}{m\_{02} \cdot \chi\_{m02}}\right)^{\frac{\nu}{b}} \cdot \left(\frac{\chi\_{m2}}{\chi\_{m1}}\right)^{\frac{\nu}{b}} \cdot \frac{f\_1}{f\_2} \cdot \exp\left(-\frac{\Delta\mu^o\_{\,\,II\rightarrow\,I}}{bRT}\right) \tag{14}$$

where: m01(m02), m01(m02) – molal solubility ([mol/kg]) of the salt BbEeLl(B'bEeLl)) and mean molal activity coefficient of the salt BbEeLl(B'bEeLl) in its binary saturated solution; m1(m2), m1(m2) – molality and mean molal activity coefficient of the salt BbEeLl(B'bEeLl) in the ternary solution being in equilibrium with Bb(B'b)EeLl) solid solution; x1(x2) – mole fraction of B(B') ion and *f*1(*f*2) – activity coefficient of ion B(B') in this solid solution; oIII – the partial molar Gibbs free energy of the phase transition of the salt B'bEeLl from its structure (II) into the structure (I) of the salt BbEeLl, = b + e + l, R – gas constant, T – temperature [K]

In the other case involving isomorphous substances (1) and (2) oII = 0. Hence:

$$\_{1}D\_{2/1} = \left(\frac{m\_{01} \cdot \mathcal{I}\_{m01}}{m\_{02} \cdot \mathcal{I}\_{m02}}\right)^{\frac{\nu}{b}} \cdot \left(\frac{\mathcal{I}\_{m2}}{\mathcal{I}\_{m1}}\right)^{\frac{\nu}{b}} \cdot \frac{f\_1}{f\_2} \tag{15}$$

Equations (14) and (15) should, in principle, permit calculating exactly the equilibrium partition coefficient D2/1, if molal solubilities and all activity coefficients (in the aqueous and the solid phases), as well as the partial molar Gibbs free energy of the phase transition, were known. However, these data (except for molal solubilities) are rarely available. In contrast to mean molal activity coefficients in binary saturated solutions (m01, m02), as well as those in the ternary solution being in equilibrium with Bb(B'b)EeLl) solid solution (m1, m2) which are sometimes directly accessible or calculable by means of Pitzer equations, activity coefficients in the solid solution (*f1, f2),* as well as the partial molar free energy of the phase transition, are generally unknown (except for very rare individual cases of crystallization systems: macrocomponent(1) – microcomponent(2)).

The attempts to estimate D2/1 coefficients by means of simplified equations (taking into account only activity coefficients in the liquid phase) are connected with huge errors, which proves that they result from the lack of the activity coefficients in the solid solution (*f*1, *f*2) as well as the partial molar Gibbs free energy of the phase transition oIII (Smolik, 2004).

$$\frac{\Delta[\%]}{100} = \frac{\left| D\_{\text{exp.}} - D\_{\text{cal.}} \right|}{D\_{\text{exp.}}} = \left| 1 - \frac{D\_{\text{cal.}}}{D\_{\text{exp.}}} \right| = \left| 1 - \frac{f\_2}{f\_1} \cdot \exp\left(\frac{\Delta\mu\_{Il \to il}^0}{bRT}\right) \right| \tag{16}$$

In the case where coefficients D2/1 are independent of mixed crystal composition, the ratio *f*2/*f*1 remains constant (Balarew, 1987). Assuming the regular solution approximation this ratio may be expressed by the following equation:

$$\frac{f\_1}{f\_2} = \exp\left(\frac{\Delta \overline{H\_1} - \Delta \overline{H\_2}}{RT}\right) \tag{17}$$

Chemical, Physicochemical and Crystal – Chemical Aspects of

evaluation of D2/1 level.

**4.1.1 Solubility in water (m0)** 

isomorphous salts (<sup>0</sup>

If m01 > m02 ( m01/m02)

not always fulfilled.

/b)· (m2/m1)(

soluble one grows rich in crystal).

MI

**salts** 

((m01/m02)(

MI

solutions (MI

Crystallization from Aqueous Solutions as a Method of Purification 159

To analyse the influence of the above mentioned factors on co-crystallization coefficients D2/1, it is convenient to use correlation coefficients (xy) in the case of the properties that can be formulated quantitatively. For other properties (qualitative), mean D2/1 values for salts revealing and not revealing may be compared. On the other hand, co-crystallization coefficients may be considered as a measure of mutual solubility of co-crystallizing salts in the solid phase. The famous Latin rule: "Similia similibus solvuntur" (similar substances will dissolve similar substances) may be useful in the prediction of this solubility and the

**4.1 Chemical, physicochemical and crystal–chemical properties of co-crystallizing** 

This is the most important factor affecting D2/1coefficients. For the co-crystallization of

However, this equation is proved true only for non-numerous salts fulfilling the additivity rule (Balarew, 1987). This simplified equation is the basis of the Ruff rule (Ruff et al., 1928):

As it can be seen in Table **4**, despite its simplicity and obviousness, this qualitative rule is

MSO4nH2O 100 62 MCl2nH2O 23 74 M(HCOO)22H2O 37 95 Alums, M(III) 9 100

2MII(SO4)26H2O M(I), M(II) 59 96

MClO3, MClO4, MNO3, M2CrO4 17 59

Table 4. The degree of fulfilling the Ruff rule in some crystallization systems:

macrocomponent – microcomponent (Smolik, 2004)

2/1

*D*

2MII(SO4)2·6H2O or MI

crystallization systems

2SO4 24 54 MX 16 69

Mean molal activity coefficients of some isomorphous double salts, forming ideal solid

inversely proportional to the square root of their molal solubility (Hill et al., 1940). Therefore:

 <sup>2</sup> 01 01 01 02 <sup>1</sup>

*m m m m*

*m m m m*

02 02 02 01 2

*b b <sup>b</sup> m o m o*

mean **72** 

MIII(SO4)2·12H2O) in their binary saturated solutions are

(20)

Kind of co-crystallizing salts Number of considered

/b) (f1/f2)=1)) they are expressed by D2/1=(m01/m02) /b.

II I = 0) forming ideal solid and liquid solutions

/b = D2/1 > 1. (During crystallization of two components, the less

Crystallization systems fulfilling the Ruff rule [%]

where *H H* 1 2 is the difference in the partial molar enthalpies of mixing.

According to Balarew (1987) this is the result of the difference in coordination environment around the two substituting ions, affected by ionic size differences (r/r), metal – ligand bond energy differences () with respect to the enthalpy of mixing (Urusov, 1977), as well as the difference in the energy determined by the crystal field (in the case non Jahn–Teller ions):

$$
\Delta \overline{H}\_1 - \Delta \overline{H}\_2 = w\_1 \cdot f\left(\frac{\Delta r}{r}\right) + w\_2 \cdot \rho(\Delta \varepsilon) + w\_3 \cdot \nu\left(\Delta s\right) + \dots \tag{18}
$$

Hence:

$$D\_{2/1} = \left(\frac{c\_{01} \cdot \mathcal{Y}\_{c01}}{c\_{02} \cdot \mathcal{Y}\_{c02}}\right)^{\nu} \cdot \left(\frac{\mathcal{Y}\_{c2}}{\mathcal{Y}\_{c1}}\right)^{\overline{b}} \cdot \exp\left(-\frac{\Delta\mu\_{\parallel \to l}}{RT}\right) \cdot \exp\frac{w\_1 \cdot f\left(\frac{\Delta r}{r}\right) + w\_2 \cdot \varphi(\Delta x) + w\_3 \cdot \psi'(\Delta s) + \dots}{bRT} \tag{19}$$

where: f, , , … – functions sought for, w1, w2, w3, … – estimated coefficients.

To derive an equation for estimating D2/1 by finding the functions (f, , , …) and coefficients (w1, w2, w3, …), it is necessary to check how D2/1 coefficients depend on various factors.

## **4. The dependence of co–crystallization coefficients, D2/1 on chemical, physicochemical and crystal–chemical properties of co–crystallizing salts and ions**

Equilibrium co-crystallization coefficients are determined in the conditions ensuring that they do not depend on hydrodynamic and kinetic conditions of crystallization. However, they are affected by several factors both "external" (in relation to the co-crystallizing substances) and "internal" (resulting from chemical, physicochemical and crystal–chemical properties of the co-crystallizing substances).

"External" factors have chemical characteristics (kind and composition of the solvent – the liquid phase, the presence of ions or other foreign substances, the presence of complexing agents, acidity (pH) of solution, from which crystallization takes place) or non–chemical ones (e.g., temperature). "Internal" factors are presented in Table **3**.


Table 3. Chemical, physicochemical and crystal–chemical properties of co-crystallizing salts and co-crystallizing ions

To analyse the influence of the above mentioned factors on co-crystallization coefficients D2/1, it is convenient to use correlation coefficients (xy) in the case of the properties that can be formulated quantitatively. For other properties (qualitative), mean D2/1 values for salts revealing and not revealing may be compared. On the other hand, co-crystallization coefficients may be considered as a measure of mutual solubility of co-crystallizing salts in the solid phase. The famous Latin rule: "Similia similibus solvuntur" (similar substances will dissolve similar substances) may be useful in the prediction of this solubility and the evaluation of D2/1 level.

### **4.1 Chemical, physicochemical and crystal–chemical properties of co-crystallizing salts**

## **4.1.1 Solubility in water (m0)**

158 Crystallization – Science and Technology

According to Balarew (1987) this is the result of the difference in coordination environment around the two substituting ions, affected by ionic size differences (r/r), metal – ligand bond energy differences () with respect to the enthalpy of mixing (Urusov, 1977), as well as the difference in the energy determined by the crystal field (in the case non Jahn–Teller ions):

> 1 21 2 3 .... *<sup>r</sup> H H wf w w s*

To derive an equation for estimating D2/1 by finding the functions (f, , , …) and coefficients (w1, w2, w3, …), it is necessary to check how D2/1 coefficients depend on various

Equilibrium co-crystallization coefficients are determined in the conditions ensuring that they do not depend on hydrodynamic and kinetic conditions of crystallization. However, they are affected by several factors both "external" (in relation to the co-crystallizing substances) and "internal" (resulting from chemical, physicochemical and crystal–chemical

"External" factors have chemical characteristics (kind and composition of the solvent – the liquid phase, the presence of ions or other foreign substances, the presence of complexing agents, acidity (pH) of solution, from which crystallization takes place) or non–chemical

> Chemical, physicochemical and crystal–chemical properties of Co-crystallizing salts Co-crystallizing ions

Table 3. Chemical, physicochemical and crystal–chemical properties of co-crystallizing salts

Crystal system (CS) Geometrical factor (ionic radius) (r)

*r*

*<sup>r</sup> wf w w s <sup>c</sup> <sup>r</sup> <sup>D</sup>*

 

exp exp

where: f, , , … – functions sought for, w1, w2, w3, … – estimated coefficients.

**4. The dependence of co–crystallization coefficients, D2/1 on chemical, physicochemical and crystal–chemical properties of co–crystallizing salts** 

Hence:

2/1

factors.

**and ions** 

Number of molecules of crystallization water (n)

and co-crystallizing ions

of salt (3)

01 01 2

02 02 1

*c c*

 

properties of the co-crystallizing substances).

Reciprocal solubility in the solid phase (Cs

The volume of one formal molecule

ones (e.g., temperature). "Internal" factors are presented in Table **3**.

Solubility in water (m0) Charge of cation

 

*b b <sup>c</sup> <sup>c</sup> II I*

*<sup>c</sup> RT bRT* (19)

  (18)

....

 

1 23

Character of chemical bond (electronegativity) ()

Crystal field stabilization energy (CFSE)

MAX) Electronic configuration

Cation hardness (h)

where *H H* 1 2 is the difference in the partial molar enthalpies of mixing.

This is the most important factor affecting D2/1coefficients. For the co-crystallization of isomorphous salts (<sup>0</sup> II I = 0) forming ideal solid and liquid solutions ((m01/m02)( /b)· (m2/m1)( /b) (f1/f2)=1)) they are expressed by D2/1=(m01/m02) /b.

However, this equation is proved true only for non-numerous salts fulfilling the additivity rule (Balarew, 1987). This simplified equation is the basis of the Ruff rule (Ruff et al., 1928):

If m01 > m02 ( m01/m02) /b = D2/1 > 1. (During crystallization of two components, the less soluble one grows rich in crystal).

As it can be seen in Table **4**, despite its simplicity and obviousness, this qualitative rule is not always fulfilled.


Table 4. The degree of fulfilling the Ruff rule in some crystallization systems: macrocomponent – microcomponent (Smolik, 2004)

Mean molal activity coefficients of some isomorphous double salts, forming ideal solid solutions (MI 2MII(SO4)2·6H2O or MI MIII(SO4)2·12H2O) in their binary saturated solutions are inversely proportional to the square root of their molal solubility (Hill et al., 1940). Therefore:

$$D\_{2/1} = \left(\frac{m\_{01} \cdot \gamma\_{m01}}{m\_{02} \cdot \gamma\_{m02}}\right)^{\frac{\nu}{b}} = \left(\frac{m\_{01} \cdot \sqrt{m\_{02}}}{m\_{02} \cdot \sqrt{m\_{01}}}\right)^{\frac{\nu}{b}} = \left(\frac{m\_{o1}}{m\_{o2}}\right)^{\frac{\nu}{2b}}\tag{20}$$

Chemical, Physicochemical and Crystal – Chemical Aspects of

**Mn2+**

**Cd2+**

suitable sulfates (Smolik, 2000b]


**log D2/1**

**4.1.2 Crystal system (CS)** 

Crystallization from Aqueous Solutions as a Method of Purification 161

**Fe2+**

**xy = 0.201**

**Cu2+**

**Zn2+ Mg2+ Co2+**

Fig. 5. The dependence of coefficients, D2/1 of co-crystallization of Cd2+, Mn2+, Fe2+, Cu2+, Mg2+, Co2+ and Zn2+ with NiSO47H2O at 20 oC on the molality of saturated solutions of


**log(m01/m02)**

The last three examples point to the crystal structure of co-crystallizing salts as a very important factor significantly affecting D2/1 coefficients. The dependence of similarity of the crystal structure of macro and microcomponent on mean co-crystallization coefficients in

sulfate (MSO4·nH2O) crystallization systems is presented in Fig. 6 and Fig. 7.

orthorhombic

Fig. 6. The dependence of coefficients D2/1 on the similarity of the crystal system of

NiSO4.7H2O

FeSO4.7H2O

MnSO4.7H2O

monoclinic

CoSO4.7H2O

MnSO4.H2O

CdSO4.8/3H2O

triclinic

MnSO4.5H2O

CuSO4.5H2O

(D2/1)B – mean D2/1 when (CSM) (CSm) (Smolik, 2002a, 2004)

MgSO4.7H2O

ZnSO4.7H2O

0,0 1,0 2,0 3,0 4,0 5,0

**(D2/1)A/(D2/1)B**

macrocomponent (CSM) and microcomponent (CSm) (D2/1)A – mean D2/1 when (CSM) = (CSm)

As it can be seen, mean D2/1 coefficients of microcomponents whose hydrates belong to the same crystal system as the macrocomponent are ~ 3 times (for orthorhombic and triclinic macrocomponents) and ~1.5 times (for monoclinic macrocomponents) greater than those whose hydrates belong to a different crystal system than that of macrocomponent) (Fig. 6).

For double salt dissociating: NiSO4MI 2SO46H2O Ni2+ + 2M+ + 2SO42– + 6H2O ( = 5), it is possible to obtain for M+ ions (b = 2): D2/1 = (m01/m02)1,25 **(Fig. 3)** and for M2+ ions (b = 1): D2/1 = (m01/m02)2,5 **(Fig 4)**.

However, for a similar, but simple salt (NiSO47H2O), an analogous dependence does not exist (Fig. 5) (xy = 0.201 is insignificant).

This is because of the significant differences in the crystal system of proper sulfate hydrates, while all of the investigated double salts are isomorphous, of the same space group (P21/a) and of almost identical unit cell parameters (a, b, c, (their relative standard deviations do not exceed 0.8%)

Fig. 3. The dependence of coefficients, D2/1 of co-crystalliztion of Cs+, K+, Rb+ and Tl+ with NiSO4(NH4)2SO46H2O on the molality of saturated solutions of suitable salts (Smolik, 1998a)

Fig. 4. The dependence of coefficients, D2/1 of co-crystallization of Cd2+, Mn2+, Fe2+, Cu2+, Mg2+, Co2+ and Zn2+ with NiSO4(NH4)2SO46H2O on the molality of saturated solutions of suitable salts (Smolik, 2001)

Fig. 5. The dependence of coefficients, D2/1 of co-crystallization of Cd2+, Mn2+, Fe2+, Cu2+, Mg2+, Co2+ and Zn2+ with NiSO47H2O at 20 oC on the molality of saturated solutions of suitable sulfates (Smolik, 2000b]

## **4.1.2 Crystal system (CS)**

160 Crystallization – Science and Technology

possible to obtain for M+ ions (b = 2): D2/1 = (m01/m02)1,25 **(Fig. 3)** and for M2+ ions (b = 1):

However, for a similar, but simple salt (NiSO47H2O), an analogous dependence does not

This is because of the significant differences in the crystal system of proper sulfate hydrates, while all of the investigated double salts are isomorphous, of the same space group (P21/a) and of almost identical unit cell parameters (a, b, c, (their relative standard deviations do

Fig. 3. The dependence of coefficients, D2/1 of co-crystalliztion of Cs+, K+, Rb+ and Tl+ with NiSO4(NH4)2SO46H2O on the molality of saturated solutions of suitable salts (Smolik,

Fig. 4. The dependence of coefficients, D2/1 of co-crystallization of Cd2+, Mn2+, Fe2+, Cu2+, Mg2+, Co2+ and Zn2+ with NiSO4(NH4)2SO46H2O on the molality of saturated solutions of

2SO46H2O Ni2+ + 2M+ + 2SO4

2– + 6H2O ( = 5), it is

For double salt dissociating: NiSO4MI

exist (Fig. 5) (xy = 0.201 is insignificant).

D2/1 = (m01/m02)2,5 **(Fig 4)**.

not exceed 0.8%)

1998a)

suitable salts (Smolik, 2001)

The last three examples point to the crystal structure of co-crystallizing salts as a very important factor significantly affecting D2/1 coefficients. The dependence of similarity of the crystal structure of macro and microcomponent on mean co-crystallization coefficients in sulfate (MSO4·nH2O) crystallization systems is presented in Fig. 6 and Fig. 7.

Fig. 6. The dependence of coefficients D2/1 on the similarity of the crystal system of macrocomponent (CSM) and microcomponent (CSm) (D2/1)A – mean D2/1 when (CSM) = (CSm) (D2/1)B – mean D2/1 when (CSM) (CSm) (Smolik, 2002a, 2004)

As it can be seen, mean D2/1 coefficients of microcomponents whose hydrates belong to the same crystal system as the macrocomponent are ~ 3 times (for orthorhombic and triclinic macrocomponents) and ~1.5 times (for monoclinic macrocomponents) greater than those whose hydrates belong to a different crystal system than that of macrocomponent) (Fig. 6).

Chemical, Physicochemical and Crystal – Chemical Aspects of

values of macro and microcomponents (Fig. 8 and Fig. 9).

CaSO4.2H2O

(Smolik, 2002b, 2004), xy – correlation of (D2/1)av and n

**monoclinic crystals of CoSO4.7H2O structure** 

**4.1.4 Reciprocal solubility in the solid phase Cs**

0,00 0,05 0,10 0,15 0,20 0,25 0,30

**(D2/1)av**

solubility of CoSO4·7H2O in MgSO4·7H2O

0,00 0,20 0,40 0,60 0,80 1,00 1,20 1,40 1,60

**DMg/CoSO4.7H2O**

Crystallization from Aqueous Solutions as a Method of Purification 163

structures is determined by weak hydrogen bonds), but in lower hydrates oxygen atoms of polyatomic anions enter the coordination environment of the metal ions (causing the formation of chains, closed rings, planar or space networks by vertices–sharing coordination polyhedral) (Balarew, 1987). Generally, mean D2/1 values are higher the more similar are n

Fig. 9. (D2/1)av = *f*(n = nM – nm ) during the crystallization of CdSO48/3H2O at 20 oC

M SO4.5H2O

It is known that the coefficients of co-crystallization of impurities are proportional to their solubilities in the solid phase in the case of crystallization of Ge and Si (Fisher, 1962), as well as several dozen molten metals (Vachobov et al., 1968). However, such regularity has not

Fig. 10. The dependence of DMg/CoSO4.7H2O on mole fraction of MgSO4·7H2O in the solid phase (Oikova et al., 1976) ; Mg/Co – solubility of MgSO4·7H2O in CoSO4·7H2O, Co/Mg –

0 0,2 0,4 0,6 0,8 1 **xMgSO4.7H2O**

**Mg/Co Co/Mg**

**miscibility**

 **gap**

**MAX**

012345

**n**

 **2**

SrSO4

**xy = -0,9983**

**xy = 0,9966** 

M SO4.7H2O

**orthorhombic crystals of MgSO4.7H2O structure**

The mean coefficients D2/1 of microcomponents belonging to the same crystal system as the macrocomponent are the highest, and they drop as the similarity of their crystal structure and that of the macrocomponent decreases (taking into account the following direction of the increase of crystal systems symmetry: triclinic<monoclinic<orthorhombic) (Fig. 7).

Fig. 7. The dependence of coefficients D2/1 on the similarity of crystal systems of macrocomponent (CSM) and microcomponent (CSm) (D2/1)mean – mean D2/1 of microcomponents belonging to the same crystal system (Smolik, 2002a, 2004)

#### **4.1.3 Number of molecules of crystallization water (n)**

During the crystallization of hydrates the number of molecules of crystallization water (n) is an additional factor which may influence co-crystallization coefficients D2/1. It affects the coordination environment of the metal ion, which in the case of hepta or hexahydrates consists of only water molecules (the linkage of coordination octahedra in these crystals'

Fig. 8. D2/1 = *f*(n = nM – nm ) nM(nm) – number of molecules of crystallization water of macrocomponent (microcomponent) (Smolik, 2002a, 2004),(D2/1)A – mean D2/1 when n = 0 (D2/1)B – mean D2/1 when n 0

The mean coefficients D2/1 of microcomponents belonging to the same crystal system as the macrocomponent are the highest, and they drop as the similarity of their crystal structure and that of the macrocomponent decreases (taking into account the following direction of the increase of crystal systems symmetry: triclinic<monoclinic<orthorhombic) (Fig. 7).

Fig. 7. The dependence of coefficients D2/1 on the similarity of crystal systems of macrocomponent (CSM) and microcomponent (CSm) (D2/1)mean – mean D2/1 of microcomponents belonging to the same crystal system (Smolik, 2002a, 2004)

orthorhombic

NiSO4.7H2O

monoclinic

MnSO4.7H2O

monoclinic

MnSO4.H2O

triclinic

MnSO4.5H2O

orthorhombic monoclinic triclinic

During the crystallization of hydrates the number of molecules of crystallization water (n) is an additional factor which may influence co-crystallization coefficients D2/1. It affects the coordination environment of the metal ion, which in the case of hepta or hexahydrates consists of only water molecules (the linkage of coordination octahedra in these crystals'

Fig. 8. D2/1 = *f*(n = nM – nm ) nM(nm) – number of molecules of crystallization water of macrocomponent (microcomponent) (Smolik, 2002a, 2004),(D2/1)A – mean D2/1 when n = 0

FeSO4.7H2O

MnSO4.7H2O

CoSO4.7H2O

orthorhombic

monoclinic triclinic

MnSO4.H2O

CuSO4.5H2O

MnSO4.5H2O

**4.1.3 Number of molecules of crystallization water (n)** 

**(D2/1)/(D2/1)B**

MgSO4.7H2O

ZnSO4.7H2O

NiSO4.7H2O

orthorhombic

MgSO4.7H2O

orthorhombic

ZnSO4.7H2O

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8

**(D2/1)mean**

(D2/1)B – mean D2/1 when n 0

structures is determined by weak hydrogen bonds), but in lower hydrates oxygen atoms of polyatomic anions enter the coordination environment of the metal ions (causing the formation of chains, closed rings, planar or space networks by vertices–sharing coordination polyhedral) (Balarew, 1987). Generally, mean D2/1 values are higher the more similar are n values of macro and microcomponents (Fig. 8 and Fig. 9).

Fig. 9. (D2/1)av = *f*(n = nM – nm ) during the crystallization of CdSO48/3H2O at 20 oC (Smolik, 2002b, 2004), xy – correlation of (D2/1)av and n

#### **4.1.4 Reciprocal solubility in the solid phase Cs MAX**

It is known that the coefficients of co-crystallization of impurities are proportional to their solubilities in the solid phase in the case of crystallization of Ge and Si (Fisher, 1962), as well as several dozen molten metals (Vachobov et al., 1968). However, such regularity has not

Fig. 10. The dependence of DMg/CoSO4.7H2O on mole fraction of MgSO4·7H2O in the solid phase (Oikova et al., 1976) ; Mg/Co – solubility of MgSO4·7H2O in CoSO4·7H2O, Co/Mg – solubility of CoSO4·7H2O in MgSO4·7H2O

Chemical, Physicochemical and Crystal – Chemical Aspects of

structures of low symmetry these distances become equivocal.

~400 – 1000 times lower than those of M+ and M3+ ions (Fig 11b).

**log DM(III)**

**Na K Tl Co Ni Mn Cu Cd Zn Mg Ca Al Cr**


**log DM(I)**

**log D2/1**

**log DM(I)**

**log DM(II)**

(Table 5).


**log D2/1**

**4.2.1 Charge of cation** 

**ions** 

Crystallization from Aqueous Solutions as a Method of Purification 165

According to Urusov (1977) this parameter is better than ionic radius in the evaluation of the effect of geometric factor on D2/1 coefficients, because it takes into account real interatomic distances defined by crystal system and unit cell parameters. However, is unambiguous only in ionic crystals of high (cubic) symmetry and in the case of complicated heterodesmic

The significant effect of this factor has occurred in some acetate crystallization systems

The cation charge is one of the most important factors influencing D2/1 coefficients. Taking into account the earlier mentioned Latin rule "Similia similibus solvuntur" it might be expected that microcomponent ions having the same charge as that of macrocomponent ion should co-crystallize in higher degree than those of different ion charge. In many crystallization systems this rule is fulfilled, e.g., in the case of crystallization of FeSO4·7H2O at 20 oC the mean D2/1 of M2+ ions (of the same charge as the macrocomponent Fe2+) are ~14 (50) times greater than those of M+(M3+) ions (of the charge different from that of macrocomponent) (Fig. 11a), and in the case of Fe(NH4)(SO4)2·12H2O crystallization at 20 oC the mean D2/1 of M2+ ions (of different charge from macrocomponent ions NH4+ and Fe3+) is

(a) (b)


**log DM(I)**

**log DM(II)**

**log D2/1**

**Na K Rb Cs Tl Cu Zn Mg Co Ni Cd Mn Ca Al Cr**

**log DM(III)**

(c) Fig. 11. The effect of ion charge on D2/1 during crystallization of: a) FeSO4·7H2O at 20 oC (Smolik & Lipowska, 1995); b) Fe(NH4) alum at 20 oC; (Smolik, 1995a); c) Na2SO4 at 50 oC; (Smolik, 1998b); log DM(I)(M(II),M(III)) – logarithms of mean D2/1 for M+(M2+, M3+) ions

**K Tl Rb Cs Mg Co Zn Cd Mn Ni Cu Fe Al Cr**

**log DM(II)**

**log DM(III)**

**4.2 Chemical, physicochemical and crystal–chemical properties of co-crystallizing** 

been found yet during the crystallization of salts from aqueous solutions. The term "solubility in the solid phase" is explained in Fig 10. It is the maximal concentration of a hydrate in another hydrate, which does not cause the change in its structure.

The effect of the solubility in the solid phase (Cs MAX) on D2/1 coefficients is presented in Table **5**. As it can be seen in most analysed cases, D2/1 is proportional to the maximal reciprocal solubility in the solid phase (Cs MAX.) which is proved by relatively high and significant correlation coefficients (xy), marked bold.


Table 5. Correlation (xy) of co-crystallization coefficients D2/1 and solubility in the solid phase (Cs MAX) (Smolik, 2004)

#### **4.1.5 The volume of one formal molecule**

The volume of one formal molecule can be calculated, knowing the molar mass of crystallizing salt (compound) and its density, by the following formula 3 24 <sup>10</sup> *Mx d N* **[**Å3] where: Mx – molar mass [g/mole], d – density [g/cm3], N – Avogadro number = 6,022·1023/mole). This is very close to that calculated using unit cell parameters (a, b, c, , , ).


Table 6. The dependence of ln D2/1 ( = ln D2/1 – ln (m01/m02)3) on various functions of during the crystallization of Mg(CH3COO)24H2O (1) and Mn(CH3COO)24H2O (2) at 25 oC

According to Urusov (1977) this parameter is better than ionic radius in the evaluation of the effect of geometric factor on D2/1 coefficients, because it takes into account real interatomic distances defined by crystal system and unit cell parameters. However, is unambiguous only in ionic crystals of high (cubic) symmetry and in the case of complicated heterodesmic structures of low symmetry these distances become equivocal.

The significant effect of this factor has occurred in some acetate crystallization systems (Table 5).

#### **4.2 Chemical, physicochemical and crystal–chemical properties of co-crystallizing ions**

## **4.2.1 Charge of cation**

164 Crystallization – Science and Technology

been found yet during the crystallization of salts from aqueous solutions. The term "solubility in the solid phase" is explained in Fig 10. It is the maximal concentration of a

Table **5**. As it can be seen in most analysed cases, D2/1 is proportional to the maximal

The volume of one formal molecule can be calculated, knowing the molar mass of crystallizing

molar mass [g/mole], d – density [g/cm3], N – Avogadro number = 6,022·1023/mole). This is

Table 6. The dependence of ln D2/1 ( = ln D2/1 – ln (m01/m02)3) on various functions of during the crystallization of Mg(CH3COO)24H2O (1) and Mn(CH3COO)24H2O (2) at 25 oC

Mn2+, Cd2+, Ca2+ Sr2+ ln D2/1 **–0.8657** (Smolik, 2008) Ni2+, Co2+, Mn2+ = lnD2/1 –

 <sup>3</sup> 3

ln(m01/m02)3 **-0.9973** 

 3 24 <sup>10</sup> *Mx*

x

xy

(m01/m02)3 **–0.9987** (Smolik,

2

 

*d N* **[**Å3] where: Mx –

Ref.

2011)

No Macrocomponent Microcomponents xy 1. MgSO4·7H2O Ni2+, Mn2+, Fe2+, Cu2+, Zn2+, Cd2+, Co2+ **0.7635**  2. ZnSO4·7H2O Ni2+, Mn2+, Fe2+, Cu2+, Mg2+, Cd2+, Co2+ **0.8608**  3. NiSO4·7H2O Zn2+, Fe2+, Cu2+, Mg2+, Cd2+, Co2+ **0.8455**  4. CoSO4·7H2O Ni2+, Mn2+, Fe2+, Mg2+, Cd2+, Zn2+ **0.9172**  5. MnSO4·5H2O Zn2+, Cu2+, Mg2+ **0.9971**  6. FeSO4·7H2O Ni2+, Zn2+, Cu2+, Mg2+, Cd2+, Co2+ **0.7983**  8. Ni(NO3)2·6H2O Mn2+,Zn2+, Mg2+, Co2+ 0.7022 9. Ni(NO3)2·6H2O Mn2+, Mg2+,Co2+ **0.9970**  10. NiCl2·6H2O Zn2+, Mn2+, Fe2+, Cu2+, Co2+ **0.9974**  11. K2SO4 Cs+, Tl+, Rb+ 0.1041 Table 5. Correlation (xy) of co-crystallization coefficients D2/1 and solubility in the solid

MAX) on D2/1 coefficients is presented in

MAX.) which is proved by relatively high and

hydrate in another hydrate, which does not cause the change in its structure.

The effect of the solubility in the solid phase (Cs

significant correlation coefficients (xy), marked bold.

reciprocal solubility in the solid phase (Cs

MAX) (Smolik, 2004)

**4.1.5 The volume of one formal molecule** 

Micro-

Ni2+, Cu2+, Co2+, Zn2+,

salt (compound) and its density, by the following formula

very close to that calculated using unit cell parameters (a, b, c, , , ).

components <sup>y</sup>

(2) Mg2+, Co2+, Ni2+ = ln D2/1 – ln

phase (Cs

Macrocomponent

(1)

The cation charge is one of the most important factors influencing D2/1 coefficients. Taking into account the earlier mentioned Latin rule "Similia similibus solvuntur" it might be expected that microcomponent ions having the same charge as that of macrocomponent ion should co-crystallize in higher degree than those of different ion charge. In many crystallization systems this rule is fulfilled, e.g., in the case of crystallization of FeSO4·7H2O at 20 oC the mean D2/1 of M2+ ions (of the same charge as the macrocomponent Fe2+) are ~14 (50) times greater than those of M+(M3+) ions (of the charge different from that of macrocomponent) (Fig. 11a), and in the case of Fe(NH4)(SO4)2·12H2O crystallization at 20 oC the mean D2/1 of M2+ ions (of different charge from macrocomponent ions NH4+ and Fe3+) is ~400 – 1000 times lower than those of M+ and M3+ ions (Fig 11b).

Fig. 11. The effect of ion charge on D2/1 during crystallization of: a) FeSO4·7H2O at 20 oC (Smolik & Lipowska, 1995); b) Fe(NH4) alum at 20 oC; (Smolik, 1995a); c) Na2SO4 at 50 oC; (Smolik, 1998b); log DM(I)(M(II),M(III)) – logarithms of mean D2/1 for M+(M2+, M3+) ions

Chemical, Physicochemical and Crystal – Chemical Aspects of

Mn2+

0,65 0,75 0,85 0,95 1,05 **ionic radius [A]** 

**Mn2+**

Ni2+

0,0 0,2 0,4 0,6 0,8 1,0 1,2

**D2/1**

Cu2+

Co2+ Zn2+

0,65 0,75 0,85 0,95 1,05 **ionic radius [A]** 

Fe2+

Co2+ Zn2+ Fe2+

**Mg2+**

Crystallization from Aqueous Solutions as a Method of Purification 167

**D2/1**

(a) (b)

Cd2+ Ca2+

Cd2+

Ca2+

(c) (d)

Ni2+ , Cu2+, Co2+, Zn2+, Fe2+, Mn2+, Cd2+, Ca2+ **–0.9061\*** 

Ni2+ , Cu2+, Co2+, Zn2+, Fe2+, Mn2+, Cd2+, Ca2+ **–0.8353**  Co2+, Fe2+, Cd2+, Ca2+ (monoclinic) **–0.9948** 

Ni2+ , Cu2+, Co2+, Zn2+, Fe2+, Mn2+, Cd2+, Ca2+ –0.6659 Co2+, Fe2+, Mn2+, Cd2+, Ca2+ (monoclinic) **–0.9949** 

Co2+, Fe2+, Mn2+, Cd2+, Ca2+ (monoclinic) **–0.9992**

Table 7. Correlation coefficients (xy) of ln D2/1 and (r/rM)2 (or r/rM) in some sulfate crystallization systems for all ions or those ions whose sulfate hydrates are monoclinic

Correlation coefficients xy of ln D2/1 and: r/rM (r/rM)2

Ca2+ Cd2+

Cs<sup>+</sup>

Rb+

Mn2+ Fe2+

0,65 0,75 0,85 0,95 1,05 **ionic radius [A]** 

> **NH4 +**

Cr3+ Na+ K<sup>+</sup> Tl<sup>+</sup>

0,4 0,9 1,4 1,9 **ionic radius [A]**

Zn2+

Co2+ Mg2+

**Cu2+**

Ni2+

Al3+

**Fe 3+**

0,00 0,01 0,02 0,03 0,04 0,05 0,06

0

5

10

15

**D2/1**

20

25

Fig. 12. The effect of ionic radius on coefficients D2/1 during the crystallization of : a – MgSO4·7H2O at 25 oC (Smolik, 1999a), b – CuSO4·5H2O at 25 oC (Smolik & Zolotajkin, 1993), c – MnSO4·5H2O at 20 oC (Smolik et al., 1995), d – NH4Fe alum at 20 oC (Smolik, 1995c)

> Microcomponents (M2+)

\* –significant xy (for the confidence level of 0.95) are marked bold (Smolik, 2004)

Macro– component

Ni2+ Mg2+

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8

**D2/1**

Cu2+

Orthorhombic MgSO47H2O

triclinic MnSO45H2O

triclinic CuSO45H2O

However, there are crystallization systems where this simple and evident rule is not fulfilled, e.g., during the crystallization of Na2SO4 at 50 oC (Fig. 11c). In this case the mean D2/1 of M+ ions (of the same charge as the macrocomponent Na+) is the lowest. This is caused by the formation of double, less soluble salts by Na2SO4 with M (II) and M (III) sulfates which, because of their structures, are capable of in–build into Na2SO4 crystal.

## **4.2.2 Geometrical factor (ionic radius (r))**

The geometrical factor, determined by the difference in size of mutually substituting ions, has been considered for a long time as one of the most significant factors affecting the existence of isomorphism. Beginning from the empirical Goldschmidt rule postulating the border of 15% relative difference of ionic radii for the occurrence of isomorphic substitution, various values of this border (e.g., 5%) have been given by other authors. In addition it has occurred that they are dependent on other factors. This parameter was not recommended by Urusov (1977), who preferred to take into consideration the differences in interionic distances in the solid phase. In this way he calculated D2/1 values for M+ ions were strictly consistent with those experimental ones during the crystallization of alkali metal halides from melt (Urusov & Kravchuk, 1976). However, as proved by the same author, correlation of interionc distances in the solid phase and D2/1 values obtained during the crystallization from aqueous solutions occurred as significantly weaker (Urusov, 1980). Moreover, the results of many investigations [(Smolik, 1993, 1995, 1998a, 2003,2007, 2010) indicate that ionic radius can be a useful parameter in the evaluation of co-crystallization coefficients. Some typical dependences of D2/1 coefficients on ionic radii have been presented in Fig. 12 a,b,c,d.

We can observe some types of the dependence of D2/1 coefficients on ionic radius: a) monotonic, hyperbolic drop of D2/1 with the increase of ionic radius (if the case of Cu2+ is ignored, because of the structure of triclinic CuSO4·5H2O significantly departing from the structures of other sulfates); b) the existence of the maximum of D2/1 coefficients for ions, whose radii are closest to the radius of macrocomponent and the monotonic drop of D2/1 as the absolute value of the difference in ionic radii of macrocomponent and microcomponent increases; c) similar type like "b", disturbed in the case of (mainly) Cu2+ because of the almost identical structure of triclinic CuSO4·5H2O and MnSO4·5H2O; d) there are two ranges of higher D2/1 coefficients corresponding to the values of ionic radii very close to those of two macrocomponent ions NH4+ and Fe3+.

Very high correlation coefficients of ln D2/1 and r/rM or (r/rM)2 in some crystallization systems (Table **7**) indicate that these co-crystallization coefficients strongly depend on the similarity of ionic radii of micro and macrocomponents.

Coefficients of Ca2+ co-crystallization with various sulfate hydrates MSO4·nH2O, taking into account slight solubility of CaSO4·2H2O, should be very high. However, because of its large radius rCa2+ , Ca2+ ion cannot in–build into MSO4.nH2O crystals of smaller M2+ ions. Therefore, the ionic radius of the macrocomponent is a more important factor than the solubility determining DCa/MSO4.nH2O values (Fig. 13) (Smolik, 2004).

Thus, ionic radii in the case of crystallization of salts from aqueous solutions can be a convenient and important parameter for the investigation and sometimes evaluation of D2/1

However, there are crystallization systems where this simple and evident rule is not fulfilled, e.g., during the crystallization of Na2SO4 at 50 oC (Fig. 11c). In this case the mean D2/1 of M+ ions (of the same charge as the macrocomponent Na+) is the lowest. This is caused by the formation of double, less soluble salts by Na2SO4 with M (II) and M (III) sulfates which, because of their structures, are capable of in–build into Na2SO4 crystal.

The geometrical factor, determined by the difference in size of mutually substituting ions, has been considered for a long time as one of the most significant factors affecting the existence of isomorphism. Beginning from the empirical Goldschmidt rule postulating the border of 15% relative difference of ionic radii for the occurrence of isomorphic substitution, various values of this border (e.g., 5%) have been given by other authors. In addition it has occurred that they are dependent on other factors. This parameter was not recommended by Urusov (1977), who preferred to take into consideration the differences in interionic distances in the solid phase. In this way he calculated D2/1 values for M+ ions were strictly consistent with those experimental ones during the crystallization of alkali metal halides from melt (Urusov & Kravchuk, 1976). However, as proved by the same author, correlation of interionc distances in the solid phase and D2/1 values obtained during the crystallization from aqueous solutions occurred as significantly weaker (Urusov, 1980). Moreover, the results of many investigations [(Smolik, 1993, 1995, 1998a, 2003,2007, 2010) indicate that ionic radius can be a useful parameter in the evaluation of co-crystallization coefficients. Some typical dependences of D2/1 coefficients on ionic radii have been presented in Fig. 12

We can observe some types of the dependence of D2/1 coefficients on ionic radius: a) monotonic, hyperbolic drop of D2/1 with the increase of ionic radius (if the case of Cu2+ is ignored, because of the structure of triclinic CuSO4·5H2O significantly departing from the structures of other sulfates); b) the existence of the maximum of D2/1 coefficients for ions, whose radii are closest to the radius of macrocomponent and the monotonic drop of D2/1 as the absolute value of the difference in ionic radii of macrocomponent and microcomponent increases; c) similar type like "b", disturbed in the case of (mainly) Cu2+ because of the almost identical structure of triclinic CuSO4·5H2O and MnSO4·5H2O; d) there are two ranges of higher D2/1 coefficients corresponding to the values of ionic radii very close to those of

Very high correlation coefficients of ln D2/1 and r/rM or (r/rM)2 in some crystallization systems (Table **7**) indicate that these co-crystallization coefficients strongly depend on the

Coefficients of Ca2+ co-crystallization with various sulfate hydrates MSO4·nH2O, taking into account slight solubility of CaSO4·2H2O, should be very high. However, because of its large radius rCa2+ , Ca2+ ion cannot in–build into MSO4.nH2O crystals of smaller M2+ ions. Therefore, the ionic radius of the macrocomponent is a more important factor than the

Thus, ionic radii in the case of crystallization of salts from aqueous solutions can be a convenient and important parameter for the investigation and sometimes evaluation of D2/1

+ and Fe3+.

similarity of ionic radii of micro and macrocomponents.

solubility determining DCa/MSO4.nH2O values (Fig. 13) (Smolik, 2004).

**4.2.2 Geometrical factor (ionic radius (r))** 

a,b,c,d.

two macrocomponent ions NH4

Fig. 12. The effect of ionic radius on coefficients D2/1 during the crystallization of : a – MgSO4·7H2O at 25 oC (Smolik, 1999a), b – CuSO4·5H2O at 25 oC (Smolik & Zolotajkin, 1993), c – MnSO4·5H2O at 20 oC (Smolik et al., 1995), d – NH4Fe alum at 20 oC (Smolik, 1995c)


\* –significant xy (for the confidence level of 0.95) are marked bold (Smolik, 2004)

Table 7. Correlation coefficients (xy) of ln D2/1 and (r/rM)2 (or r/rM) in some sulfate crystallization systems for all ions or those ions whose sulfate hydrates are monoclinic

Chemical, Physicochemical and Crystal – Chemical Aspects of

Groups of considered salts

MI

= lnD2/1 – ln(m01/m02)

(D1/2)cal = (m02/m01)

So:

Crystallization from Aqueous Solutions as a Method of Purification 169

as well, apart from oxide ligands originating from water or inorganic anions. However, a kind of dependence of D2/1 coefficients on electronegativity has not been univocally defined and the kind of function () is very important to decide if unilateral isomorphism occurs,

MSO4nH2O 0.7309 0.6377 M(NO3)2nH2O 0.9421 0.6502 MCl2nH2O 0.6258 0.7328 MIMIII(SO4)212H2O, M(I) 0.9188 non-significant MIMIII(SO4)212H2O, M(III) 0.7651 non-significant MIMII(SO4)26H2O, M(II) 0.4921 0.3358 M(HCOO)22H2O non-significant 0.7604

> 2SO4nH2O 0.8005 0.6900 MX 0.8749 0.8745

> > /b and *f*(r) or () in considered groups of salts (Smolik, 2004)

Table 8. Comparison of absolute maximal values of correlation coefficients of functions

/b

The value 1/2 (2/1) is a measure of the extension (or diminution) of the experimental (D1/2)exp ((D2/1)exp) in relation to (D1/2)cal((D2/1)cal) , which may be brought about by the effect of electronegativity. 1/2 and 2/1 coefficients allowing for the solubility of corresponding salts are independent of the structure and the relative difference in interionic distances of cocrystallizing isomorphous salts. Therefore, they are most adequate to evaluate the occurrence of unilateral isomorphism. The results of such an analysis of all isomorphous salts forming hydrates, for which D2/1 (D1/2) coefficients have been available, show the lack

Electronic configuration of Mn+ ions, as well as crystal field stabilization energy in high spin octahedral complexes [ML6], may influence the crystal structure of co-crystallizing salts. According to their electronic configuration these ions may be divided into two groups:

/b; (D2/1)cal = (m01/m02)

ln = ln1/2 – ln2/1 = [ln(D1/2)exp – ln(m02/m01)

of any example of unilateral isomorphism (Smolik, 2004)

**4.2.4 Electronic configuration** 

To evaluate if unilateral isomorphism occurs during low temperature crystallization from aqueous solutions (according to the rule known for a long time in geology [Urusov, 1970] that lithophilic elements are substituted in the solid phase by chalcophilic and siderophilic elements and not to the contrary) several criteria may be applied (Smolik, 2004). The strongest of them in the case of co-crystallization of more (M1) and less (M2) electronegative ions (1>2) looks as follows: >3, where = 1/2/2/1; 1/2 = (D1/2)exp./(D1/2)cal and 2/1 = (D2/1)exp./(D2/1)cal;

maximal values of correlation coefficients xy of function = lnD2/1 – ln(m01/m02)

and *f*(r) and ()

/b] – [ln(D2/1)exp – ln(m01/m02)

/b

/b] > 1.1.

in spite of a buffering action of surrounding hydrate mantle (Kirkova et al., 1996).

coefficients. The comparison of correlation coefficient (*ρxy*) of *π* = [lnD2/1−ln(m01/m02)3][y] and |((rCo2+)3 − (rM2+)3)/(rCo2+)3|[x] (*ρxy* = −0.9174) with analogous correlation coefficient of *π* [y] and |((Co2+)3 –(M2+)3)/(Co2+)3|[x] (*ρxy* = −0.8590) indicates that the latter parameter, preferred by Urusov (1977) to estimate *D* 2*/*1 values, is in the case of Co(CH3COO)2·4H2O crystallization not better than the first one related to ionic radius (Smolik et al., 2007).

Fig. 13. The dependence of ln DCa/MSO4.nH2O on ionic radii of macrocomponent ions (M2+) (Smolik, 2004)

## **4.2.3 Electronegativity of mutually substituting elements ()**

Apart from the geometric factor, the partition coefficients may be affected by the nature (polarity) of the chemical bond of mutually substituting components. The exact quantitative characteristic of the polarity of the chemical bond is given by the integral of overlapping of atomic orbitals, but such data for many isomorphous systems are not available yet. So some authors (Urusov, 1977; Ringwood, 1955) consider the difference in electronegativity of elements as a measure of polarity.

Spectacular examples of a huge effect of this factor on D2/1 are given by (Urusov, 1977) where during the crystallization from the melt in systems NaBr – AgBr, NaCl – AgCl, and NaCl – CuCl, co-crystallization does not take place, although the relative differences of interionic distances of co-crystallizing isomorphous salts are very low or close to zero. The difference in the effective ionic charges of Ag(Cu) and Na in these systems was believed to be responsible for the extremely low miscibility of these systems (Kirkova et al., 1996).

In the case of crystallization of several groups of salts from aqueous solutions, the effect of electronegativity of mutually substituting components on distribution coefficients has been compared with the effect of ionic radius. However, for this comparison function = lnD2/1 – ln(m01/m02) /b has been taken into consideration (it allows for the solubility ratio of co-crystallizing salts). The results are given in Table **8**.

As it can be seen, higher correlation coefficients xy of and () than those of and *f*(r) occur in the case of chlorides (MCl2nH2O) and formates (M(HCOO)22H2O). Generally, more significant correlation coefficients xy occur in the case when in coordination surroundings of appropriate cations Mn+ in the solid phase, halogen or formate ions appear as well, apart from oxide ligands originating from water or inorganic anions. However, a kind of dependence of D2/1 coefficients on electronegativity has not been univocally defined and the kind of function () is very important to decide if unilateral isomorphism occurs, in spite of a buffering action of surrounding hydrate mantle (Kirkova et al., 1996).


Table 8. Comparison of absolute maximal values of correlation coefficients of functions = lnD2/1 – ln(m01/m02) /b and *f*(r) or () in considered groups of salts (Smolik, 2004)

To evaluate if unilateral isomorphism occurs during low temperature crystallization from aqueous solutions (according to the rule known for a long time in geology [Urusov, 1970] that lithophilic elements are substituted in the solid phase by chalcophilic and siderophilic elements and not to the contrary) several criteria may be applied (Smolik, 2004). The strongest of them in the case of co-crystallization of more (M1) and less (M2) electronegative ions (1>2) looks as follows: >3, where = 1/2/2/1; 1/2 = (D1/2)exp./(D1/2)cal and 2/1 = (D2/1)exp./(D2/1)cal; (D1/2)cal = (m02/m01) /b; (D2/1)cal = (m01/m02) /b

So:

168 Crystallization – Science and Technology

coefficients. The comparison of correlation coefficient (*ρxy*) of *π* = [lnD2/1−ln(m01/m02)3][y] and |((rCo2+)3 − (rM2+)3)/(rCo2+)3|[x] (*ρxy* = −0.9174) with analogous correlation coefficient of *π* [y] and |((Co2+)3 –(M2+)3)/(Co2+)3|[x] (*ρxy* = −0.8590) indicates that the latter parameter, preferred by Urusov (1977) to estimate *D* 2*/*1 values, is in the case of Co(CH3COO)2·4H2O

crystallization not better than the first one related to ionic radius (Smolik et al., 2007).

**CdSO4.8/3H2O**

**xy = - 0,9085**

Fig. 13. The dependence of ln DCa/MSO4.nH2O on ionic radii of macrocomponent ions (M2+)

**MnSO4.5H2O**

**FeSO4.7H2O**

**CuSO4.5H2O**

**MgSO4.7H2O NiSO4.7H2O**

**ZnSO4.7H2O CoSO4.7H2O**

0.00 0.10 0.20 0.30 0.40 0.50

**(rCa2+ - rM2+)/rM2+**

Apart from the geometric factor, the partition coefficients may be affected by the nature (polarity) of the chemical bond of mutually substituting components. The exact quantitative characteristic of the polarity of the chemical bond is given by the integral of overlapping of atomic orbitals, but such data for many isomorphous systems are not available yet. So some authors (Urusov, 1977; Ringwood, 1955) consider the difference in electronegativity of

Spectacular examples of a huge effect of this factor on D2/1 are given by (Urusov, 1977) where during the crystallization from the melt in systems NaBr – AgBr, NaCl – AgCl, and NaCl – CuCl, co-crystallization does not take place, although the relative differences of interionic distances of co-crystallizing isomorphous salts are very low or close to zero. The difference in the effective ionic charges of Ag(Cu) and Na in these systems was believed to be responsible for the extremely low miscibility of these systems (Kirkova et al., 1996).

In the case of crystallization of several groups of salts from aqueous solutions, the effect of electronegativity of mutually substituting components on distribution coefficients has been compared with the effect of ionic radius. However, for this comparison function

As it can be seen, higher correlation coefficients xy of and () than those of and *f*(r) occur in the case of chlorides (MCl2nH2O) and formates (M(HCOO)22H2O). Generally, more significant correlation coefficients xy occur in the case when in coordination surroundings of appropriate cations Mn+ in the solid phase, halogen or formate ions appear

/b has been taken into consideration (it allows for the solubility

**4.2.3 Electronegativity of mutually substituting elements ()** 


**ln DCa/MSO4.nH2O**

ratio of co-crystallizing salts). The results are given in Table **8**.

(Smolik, 2004)

elements as a measure of polarity.

= lnD2/1 – ln(m01/m02)

$$\ln \Theta = \ln \theta\_{1/2} - \ln \theta\_{2/1} = \left[ \ln (\mathcal{D}\_{1/2})\_{\exp} - \ln (\mathcal{m}\_{02}/\mathcal{m}\_{01})^{\text{v}} \right] - \left[ \ln (\mathcal{D}\_{2/1})\_{\exp} - \ln (\mathcal{m}\_{01}/\mathcal{m}\_{02})^{\text{v}} \right] \geq 1.1.$$

The value 1/2 (2/1) is a measure of the extension (or diminution) of the experimental (D1/2)exp ((D2/1)exp) in relation to (D1/2)cal((D2/1)cal) , which may be brought about by the effect of electronegativity. 1/2 and 2/1 coefficients allowing for the solubility of corresponding salts are independent of the structure and the relative difference in interionic distances of cocrystallizing isomorphous salts. Therefore, they are most adequate to evaluate the occurrence of unilateral isomorphism. The results of such an analysis of all isomorphous salts forming hydrates, for which D2/1 (D1/2) coefficients have been available, show the lack of any example of unilateral isomorphism (Smolik, 2004)

#### **4.2.4 Electronic configuration**

Electronic configuration of Mn+ ions, as well as crystal field stabilization energy in high spin octahedral complexes [ML6], may influence the crystal structure of co-crystallizing salts. According to their electronic configuration these ions may be divided into two groups:

Chemical, Physicochemical and Crystal – Chemical Aspects of

Lewis base strength of anion (x) (Brown, 1981)

coefficients in considered groups of simple salts (Smolik, 2004)

different degree), whose measure is coefficient D2/1.

/b and s = sMACR – smicr, s , or (s)2.

Group of salt

ln(m01/m02)

Number of crystallization systems

crystallization systems (Smolik, 2004)

coefficients of = ln D2/1– ln (m01/m02)

Crystallization from Aqueous Solutions as a Method of Purification 171

Dc mean

MSO4nH2O 0.50 0.28 0.50 1.79 M(NO3)2nH2O 0.33 0.48 0.75 1.56 MCl2nH2O 1.00 0.02 0.37 18.5 M(HCOO)22H2O 0.50 1.43 6.48 4.53 M(CH3COO)2nH2O 0.55 1.26 8.07 6.40

Table 10. The effect of electron configuration of microcomponents (ions) on their D2/1

The base strength of anions occurring in chloride, acetate and formate systems is generally higher than that for water molecules and equals: 1,00 for Cl–, 0,55 for CH3COO– and 0,50 for HCOO– (Balarew, 1987)]. Due to this, as well as because of lower excess of water in relation to CH3COO– and HCOO– in the solid phase, these anions can compete with water molecules in coordination surrounding cations of macro and microcomponents. The presence of both kinds of ligands (water molecule and anions Cl–, CH3COO– or HCOO–) differing in size and charge) causes stronger deformation of the octahedral surrounding of these cations as compared with the presence of one ligand. This deformation depends on the electron configuration of the cation of both the macrocomponent and microcomponent. Therefore, this factor may influence the ability of mutual substitution of those octahedra (deformed to a

The dependence of coefficients D2/1 on electron configuration is usually connected with their dependence on the crystal field stabilization energy (s), which may be expressed quantitatively for high spin octahedral complexes (most of them occurring in the structures of the considered salts) in kJ/mol or in Dq (where Dq – natural theoretical unit for crystal– field splitting energies (Porterfield, 1993). Thus, it is possible to characterize quantitatively this dependence calculating the correlation coefficients of lnD2/1 or = lnD2/1 –

37 0.4156 –0.1998 –0.1170 **0.7604 0.8486** 

The direct effect of s on ln D2/1 is very slight in most considered groups of salts, but having taken into account the solubility ratio of the co-crystallizing salts, the found correlation

crystallization systems (xy = **0.8486**). In this group of salts, this correlation coefficient of and s is the highest as compared to the ones involving all analysed factors (Table **11**).

Table 11. Comparison of correlation coefficients (xy) of = ln D2/1– ln (m01/m02)3 and functions of some factors affecting co-crystallization coefficients D2/1 in formate

correlation coefficients (xy) of = ln D2/1– ln (m01/m02)3 and *f*(r) *f*() n s

/b and s are relatively high only for formate

Mean coefficients (D2/1) for microcomponents (ions)

*<sup>D</sup>* (y) closed shell

open shell Domean

xy = 0.9723

*o mean C mean*

*D*

**closed shell ions** having the configuration p6 (Mg2+, Ca2+, Sr2+, Ba2+) or d10 (Zn2+, Cd2+), as well as d5, but only when they are in the high spin state (Mn2+). The crystal field stabilization energy (CFSE) of such ions is zero;

#### **open shell ions** having the configuration dn (n ≠ 0, 5, 10), where CFSE ≠ 0.

In the first case the energy of these ionic coordination compounds due to the metal ions would be independent of the spatial orientation of the metal–ligand bonds. For this reason these metal ions permit variations over wide ranges of structural parameters, mainly the structure defining angles (angular deformations) (Balarew, 1987).

In the second case the CFSE depends on the orientation of metal–ligand bonds. Therefore, there are some preferred structures, for which CFSE has a maximum value, and the change in geometry of coordination polyhedron with respect to these preferred structures is related to CFSE losses (Balarew, 1987). However, the amount of the CFSE is only 5 – 10% of the whole bonding energy in the crystals and other factors mentioned previously (ionic radii, their charge, energy of metal – ligand bonds) determine the structure of predominantly ionic crystals (Balarew, 1987). Hence its effect on D2/1 coefficients is rarely observable.


Some examples of the direct influence of electron configuration of ion on D2/1 coefficients are presented in Tables **9-10**.

Table 9. The effect the electron configuration on D2/1 coefficients during the crystallization of NiCl2·6H2O at 20 oC (Smolik, 1999b)

As it can be seen, the direct effect of electronic configuration of microcomponent ions on coefficients D2/1 is most distinct in the case of chloride, formate and acetate crystallization systems, where mean (D2/1)open shell coefficients are several times greater than (D2/1)closed shell ones. This effect is lower in sulfate and nitrate crystallization systems.

The direct effect of the electron configuration of ions depends on the kind of anion of the crystallizing salt. It is slightly perceptible in the case of nitates and sulfates, where, besides water molecules, oxoanions NO3– and SO42– appear. The valence available between oxygen and metal ion, which is a measure of their anion base strength equals: 0,33 and 0,50, respectively, and is very close to that of water molecules (0,40) (Balarew, 1987). Because of a great excess of water both in the liquid phase and in the solid one (particularly in hepta and hexahydrates), these anions cannot compete with water molecules in the bonding of metal ion. So the environment around both metal cations will be formed mainly by water molecules.

**closed shell ions** having the configuration p6 (Mg2+, Ca2+, Sr2+, Ba2+) or d10 (Zn2+, Cd2+), as well as d5, but only when they are in the high spin state (Mn2+). The crystal field

In the first case the energy of these ionic coordination compounds due to the metal ions would be independent of the spatial orientation of the metal–ligand bonds. For this reason these metal ions permit variations over wide ranges of structural parameters, mainly the

In the second case the CFSE depends on the orientation of metal–ligand bonds. Therefore, there are some preferred structures, for which CFSE has a maximum value, and the change in geometry of coordination polyhedron with respect to these preferred structures is related to CFSE losses (Balarew, 1987). However, the amount of the CFSE is only 5 – 10% of the whole bonding energy in the crystals and other factors mentioned previously (ionic radii, their charge, energy of metal – ligand bonds) determine the structure of predominantly ionic

Some examples of the direct influence of electron configuration of ion on D2/1 coefficients

As it can be seen, the direct effect of electronic configuration of microcomponent ions on coefficients D2/1 is most distinct in the case of chloride, formate and acetate crystallization systems, where mean (D2/1)open shell coefficients are several times greater than (D2/1)closed shell

The direct effect of the electron configuration of ions depends on the kind of anion of the crystallizing salt. It is slightly perceptible in the case of nitates and sulfates, where, besides water molecules, oxoanions NO3– and SO42– appear. The valence available between oxygen and metal ion, which is a measure of their anion base strength equals: 0,33 and 0,50, respectively, and is very close to that of water molecules (0,40) (Balarew, 1987). Because of a great excess of water both in the liquid phase and in the solid one (particularly in hepta and hexahydrates), these anions cannot compete with water molecules in the bonding of metal ion. So the environment around both metal cations will

ones. This effect is lower in sulfate and nitrate crystallization systems.

Ion M2+ Electron configuration D2/1 Mg2+ 1s22s2p6 0.009 0.005 Ca2+ 1s22s2p63s2p6 0.022 0.008 Sr2+ 1s22s2p63s2p64s2p6 0.013 0.006 Zn2+ 1s22s2p63s2p6d10 0.014 0.005 Cd2+ 1s22s2p63s2p6d104s2p6d10 0.010 0.006 **Cu2+ 1s22s2p63s2p6d9 0.040 0.007 Mn2+ 1s22s2p63s2p6d5 0.40 0.02 Fe2+ 1s22s2p63s2p6d6 1.70 0.20 Co2+ 1s22s2p63s2p6d7 2.60 0.30**  Table 9. The effect the electron configuration on D2/1 coefficients during the crystallization of

crystals (Balarew, 1987). Hence its effect on D2/1 coefficients is rarely observable.

**open shell ions** having the configuration dn (n ≠ 0, 5, 10), where CFSE ≠ 0.

structure defining angles (angular deformations) (Balarew, 1987).

stabilization energy (CFSE) of such ions is zero;

are presented in Tables **9-10**.

NiCl2·6H2O at 20 oC (Smolik, 1999b)

be formed mainly by water molecules.


Table 10. The effect of electron configuration of microcomponents (ions) on their D2/1 coefficients in considered groups of simple salts (Smolik, 2004)

The base strength of anions occurring in chloride, acetate and formate systems is generally higher than that for water molecules and equals: 1,00 for Cl–, 0,55 for CH3COO– and 0,50 for HCOO– (Balarew, 1987)]. Due to this, as well as because of lower excess of water in relation to CH3COO– and HCOO– in the solid phase, these anions can compete with water molecules in coordination surrounding cations of macro and microcomponents. The presence of both kinds of ligands (water molecule and anions Cl–, CH3COO– or HCOO–) differing in size and charge) causes stronger deformation of the octahedral surrounding of these cations as compared with the presence of one ligand. This deformation depends on the electron configuration of the cation of both the macrocomponent and microcomponent. Therefore, this factor may influence the ability of mutual substitution of those octahedra (deformed to a different degree), whose measure is coefficient D2/1.

The dependence of coefficients D2/1 on electron configuration is usually connected with their dependence on the crystal field stabilization energy (s), which may be expressed quantitatively for high spin octahedral complexes (most of them occurring in the structures of the considered salts) in kJ/mol or in Dq (where Dq – natural theoretical unit for crystal– field splitting energies (Porterfield, 1993). Thus, it is possible to characterize quantitatively this dependence calculating the correlation coefficients of lnD2/1 or = lnD2/1 – ln(m01/m02) /b and s = sMACR – smicr, s , or (s)2.


Table 11. Comparison of correlation coefficients (xy) of = ln D2/1– ln (m01/m02)3 and functions of some factors affecting co-crystallization coefficients D2/1 in formate crystallization systems (Smolik, 2004)

The direct effect of s on ln D2/1 is very slight in most considered groups of salts, but having taken into account the solubility ratio of the co-crystallizing salts, the found correlation coefficients of = ln D2/1– ln (m01/m02) /b and s are relatively high only for formate crystallization systems (xy = **0.8486**). In this group of salts, this correlation coefficient of and s is the highest as compared to the ones involving all analysed factors (Table **11**).

Chemical, Physicochemical and Crystal – Chemical Aspects of

21

8

2/1

2/1

2/1

2/1

9

Cr3+ <sup>9</sup>

02

7

2.5 01

exp

02 1

*m r*

2/1 2 02 2 <sup>1</sup> exp(0.835 0.082) *<sup>m</sup> <sup>D</sup> m r*

02

*m*

<sup>43</sup>

02

K+ <sup>5</sup>

2/1

Table 12. Part 1. Some examples of the estimation of coefficients D2/1 (Smolik, 2004)

*<sup>m</sup> <sup>D</sup> m* *m*

2/1

2/1

Micro components (Mn+) (1)

Co2+, Fe2+, Mn2+, Cd2+

Ni2+, Mg2+, Zn2+

Ni2+, Mg2+, Zn2+, Co2+ Mn2+

Ni2+, Mg2+, Zn2+, Co2+ Cu2+, Mn2+, Cd2+, Ca2+, Sr2+

Na+, K+, Rb+, Cs+, Tl+,

Ni2+, Mg2+, Zn2+, Mn2+, Co2+ Cu2+, Cd2+

Ni2+, Mg2+, Zn2+, Mn2+, Co2+ Cu2+, Cd2+

Na+, K+, Rb+, Cs+, Tl+,

Cs+, Rb+, K+ <sup>9</sup>

Cs+, Rb+,

37

9

2/1

alums Al3+, Fe3+,

+, Rb+}

Macro– component

orthorhombic M'SO47H2O M' = {Mg, Zn, Ni}

orthorhombic M'(NO3)2 6H2O M' = {Zn, Mn}

CoCl2 6H2O

MI 2 MII(SO4)26H2O MI'={NH4

MI 2 MII(SO4)26H2O MI'={Ni, Mg, Cu, Co, Zn, Fe, Mn}

M'(HCOO)22H2O MI'={Ni, Mg, Co, Zn, Fe, Mn, Cd}

2SO4nH2O

={ Na, K, Tl}

MX X={Cl, Br, I}

MNO3 M' ={K, Rb, Cs}

M'

M'

Crystallization from Aqueous Solutions as a Method of Purification 173

k Equation

 2 2 01 1 2

 2 2 01 1 2

 2 2 01 1 2

 3 2 01 1 2

 <sup>3</sup> 2/1 *D* exp 0.0078 1.42 0.32 2.92 *h s* 27.0

0.057 1.65 0.153

*h s*

2

02 1 0.362 exp 25.95 0,0072 *<sup>m</sup> r r <sup>D</sup> m r*

02 2 0.020 exp 25.95 0,0072 *<sup>m</sup> r r <sup>D</sup> m r*

Cu 21.6 2+

02 1 exp 25.95 0,0072 *<sup>m</sup> r r <sup>D</sup> m r*

02 1 exp 55.83 0.143 *<sup>m</sup> r r <sup>D</sup> m r*

exp 16.85 130.2 *<sup>m</sup> <sup>D</sup>*

 1.25 01

> 

3 3 <sup>3</sup> 1 2 01 3 2/1 1

 2/1 <sup>2</sup> 2 <sup>1</sup> *<sup>D</sup>* exp 12.88 4.829 7,684 *r*

 

2 2 01 1 2

exp 132.8 4.38 0.072 *<sup>m</sup> r r D h*

2 <sup>1</sup> *<sup>D</sup>* exp 16.57 349.4 9,209 *r*

1.67 1.37

exp 1.48 10.2 *<sup>m</sup> <sup>D</sup>*

2 01

av [%]

14.8

29.8

9.1

25.9

31.0

22.8 (4)

29.3 (4)

18.9

Significant correlation coefficients of = ln D2/1– ln (m01/m02) /b and s or (s)2 occur in none of the considered groups of salts, which means that does not depend (in them) on the similarity of the CFSE of the macrocomponent and microcomponent ion.

## **4.2.5 Cation hardness (h)**

The concept of cation and anion hardness introduced by Pearson (1963) was utilized by Balarew and co–workers (1984) to solve some crystal–chemical problems. Based on a quantitative definition of hardness (Klopman, 1968) and using his procedure they determined hardness of several open shell cations (Mn2+, Fe2+, Co2+, Ni2+ i Cu2+), which together with the values given by Klopman for anions and other cations they used for the anticipation of a kind of coordination polyhedra in these compounds and their structure. The hardness of several other cations was given by Tepavičarova et al. (1995).

The hardness of cations affecting their coordination surrounding in the case of several ligands of different anion hardness may change the crystal structure of appropriate hydrates causing the formation of pure simple salts, solid solutions or double salts (Balarew, 1987) and therefore, influencing D2/1 coefficients.

Absolute values of the determined correlation coefficients (xy) of ln D2/1 and h are generally low (Smolik, 2004). However, after taking into account the solubility ratio of the cocrystallizing salts (function = ln D2/1 – ln (m01/m02) /b), xy values are significant for sulfates (MSO4·nH2O and M2SO4 ·xH2O), chlorides (MCl2·nH2O) and alkali halides MX (X = Cl–, Br–, I–), particularly in the presence of Tl+ and become the highest ((xy)śr > 0.80) and significant in chloride (MCl2nH2O) and halide (MX) crystallization systems. In each case they are negative, which means that with the increasing similarity of macro and microcomponent (with regard to hardness) values grow.

Low values of xy occur in groups of salts, where coordination surrounding of cations is homogeneous. It is composed of water molecules, which are hard ligands and anions NO3 –, SO42– and CH3COO–, classified as hard bases. In chloride crystallization systems (MCl2nH2O) anions Cl– occur, whose hardness is less than that of water molecules, and in halogen crystallization systems the cation surrounding in the solid phase consists only of chloride, bromide and iodide anions, which are classified as decidedly soft anions. The greatest effect of hardness on co-crystallization coefficients appears here (particularly in the presence of Tl+ ions significantly differing in their hardness from alkali ions).

## **5. Possibility of estimation of D2/1 coefficients basing on the determined dependences**

The determined correlation coefficients of D2/1 or = lnD2/1 – ln(m01/m02) /b and various functions of ionic radii (r), electronegativity (), crystal field stabilization energy (s), hardness of cations (h), number of molecules of crystallization water (n), the volume of one formal molecule of hydrate salt (3) permit to find a kind of functions (f, , , …) of these parameters and to estimate coefficients (w1, w2, w3, …) in the general equation (19) for the evaluation of D2/1 coefficients. Some particular equations of such a general type for the estimation of D2/1 in several groups of crystallization systems (macrocomponent – microcomponents) at average error not exceeding 31% are presented in Table **12**.

none of the considered groups of salts, which means that does not depend (in them) on the

The concept of cation and anion hardness introduced by Pearson (1963) was utilized by Balarew and co–workers (1984) to solve some crystal–chemical problems. Based on a quantitative definition of hardness (Klopman, 1968) and using his procedure they determined hardness of several open shell cations (Mn2+, Fe2+, Co2+, Ni2+ i Cu2+), which together with the values given by Klopman for anions and other cations they used for the anticipation of a kind of coordination polyhedra in these compounds and their structure.

The hardness of cations affecting their coordination surrounding in the case of several ligands of different anion hardness may change the crystal structure of appropriate hydrates causing the formation of pure simple salts, solid solutions or double salts (Balarew, 1987)

Absolute values of the determined correlation coefficients (xy) of ln D2/1 and h are generally low (Smolik, 2004). However, after taking into account the solubility ratio of the co-

sulfates (MSO4·nH2O and M2SO4 ·xH2O), chlorides (MCl2·nH2O) and alkali halides MX (X = Cl–, Br–, I–), particularly in the presence of Tl+ and become the highest ((xy)śr > 0.80) and significant in chloride (MCl2nH2O) and halide (MX) crystallization systems. In each case they are negative, which means that with the increasing similarity of macro and

Low values of xy occur in groups of salts, where coordination surrounding of cations is homogeneous. It is composed of water molecules, which are hard ligands and anions NO3

SO42– and CH3COO–, classified as hard bases. In chloride crystallization systems (MCl2nH2O) anions Cl– occur, whose hardness is less than that of water molecules, and in halogen crystallization systems the cation surrounding in the solid phase consists only of chloride, bromide and iodide anions, which are classified as decidedly soft anions. The greatest effect of hardness on co-crystallization coefficients appears here (particularly in the

presence of Tl+ ions significantly differing in their hardness from alkali ions).

The determined correlation coefficients of D2/1 or = lnD2/1 – ln(m01/m02)

microcomponents) at average error not exceeding 31% are presented in Table **12**.

**5. Possibility of estimation of D2/1 coefficients basing on the determined** 

functions of ionic radii (r), electronegativity (), crystal field stabilization energy (s), hardness of cations (h), number of molecules of crystallization water (n), the volume of one formal molecule of hydrate salt (3) permit to find a kind of functions (f, , , …) of these parameters and to estimate coefficients (w1, w2, w3, …) in the general equation (19) for the evaluation of D2/1 coefficients. Some particular equations of such a general type for the estimation of D2/1 in several groups of crystallization systems (macrocomponent –

/b and s or (s)2 occur in

/b), xy values are significant for

–,

/b and various

Significant correlation coefficients of = ln D2/1– ln (m01/m02)

**4.2.5 Cation hardness (h)** 

**dependences** 

and therefore, influencing D2/1 coefficients.

crystallizing salts (function = ln D2/1 – ln (m01/m02)

microcomponent (with regard to hardness) values grow.

similarity of the CFSE of the macrocomponent and microcomponent ion.

The hardness of several other cations was given by Tepavičarova et al. (1995).


Table 12. Part 1. Some examples of the estimation of coefficients D2/1 (Smolik, 2004)

Chemical, Physicochemical and Crystal – Chemical Aspects of

<sup>100</sup>0.50±

50 20 30 0.42±

60 38 2 0.39±

63 37 0.57±

D2/1 during the crystallization of NiSO47H2O at 25 oC (Smolik, 1984)

of both the macrocomponent (m1) and microcomponent (m2).

 <sup>1</sup> 2 . *<sup>m</sup>*

**6.3 The effect of the presence of complexing agents** 

 

*ML M*

> *ML M*

1

2/1 2/1

*D D*

*k*

1 '

*m*

activity coefficients, so that

the D2/1 value (Chlopin, 1938).

1962):

H2O iso–

Crystallization from Aqueous Solutions as a Method of Purification 175

PrOH CH3OH C2H5OH Et2O Me2CO Mg2+ Co2+ Fe2+ Mn2+ Cu2+

0.04

0.04

0.03

0.05

Table 13. The effect of addition of various organic solvents: iso-propyl alcohol (iso–PrOH), methanol (CH3OH), ethanol (C2H5OH), diethyl ether (Et2O), acetone (Me2CO) on coefficients

Interactions which happen in the aqueous phase may also influence D2/1 coefficients. This effect is formally taken into consideration in equation (19) by the mean activity coefficients

If the action of various factors in aqueous solution causes the same changes in both mean

D2/1 coefficient remains constant (e.g., the addition of HBr during the co-crystallization of Ra2+ with BaBr2 does not affect D2/1 coefficient, likewise the introduction of weak electrolytes (glucose or CH3COONa + CH3COOH) having no common ions with micro and macrocomponent (Ra(NO3)2 i Ba(NO3)2) and not reacting with them also does not change

However, if substances are present in the solution that react in a different way with the macro and microcomponent ions forming slightly dissociated compounds, an essential change of D2/1 coefficients takes place (e.g., the addition of CH3COONa + CH3COOH in the crystallization system Pb(NO3)2 – Ra(NO3)2 – H2O causes bonding a part of Pb2+ ions in slightly dissociated acetate, which leads to the lowering of its mean activity coefficient and

The presence of complexing agents has a significant influence on coefficient, D2/1. In the case of forming complexes by both macrocomponent and microcomponent, the relationship between the value of co-crystallization coefficient in the presence of complexing agent (D2/1)k and that in the case of its absence (D2/1) is expressed by equation (Mikheev et al.,

finally to the increase of radium co-crystallization coefficient (Chlopin, 1938).

**6.2 The effect of the presence of other ions or substances in the liquid phase** 

0.96± 0.07

0.80± 0.06

0.96± 0.07

0.70± 0.06

*const* , then at unchanged properties of the solid phase

, where (') – stability constant of the complex of

0.60± 0.05

0.31± 0.03

0.46± 0.05

0.49± 0.04

0.22± 0.02

0.10± 0.01

0.14± 0.01

0.18± 0.02

0.21± 0.02

0.14± 0.01

0.17± 0.02

0.15± 0.02

Composition of the mother solution [% v/v] Co-crystallization coefficients D2/1


(1) – Mn+ is not microcomponent, when M'=M;

(2) – subscripts "1" and "2" relate to macrocomponent and microcomponent respectively;

$$\text{L}^{\otimes} \text{ and } \eta = 10^8 \sqrt[3]{\frac{M}{D\_x N}} \text{ [Å], where M - molar mass of salt [g/mol].}\\\text{D}\_1-\text{its density [g/cm}^3]. $$

N – Avogadro number [6.022·1023/mol];

(4) – in this case (r1–r2)/r1<0.20 (for the other cases D2/1 < 0.06); k – number of crystallizations systems (macrocomponent)j – (microcomponent)ji in a given group of salt.

Table 12. Part 2. Some examples of the estimation of coefficients D2/1 (Smolik, 2004)

## **6. Methods of lowering D2/1 values as the way of increasing purification efficiency of crystallization**

From the practical point of view it is very interesting to ascertain how to increase the efficiency of crystallization purification of inorganic substances. This is possible when cocrystallization coefficients D2/1 can be lowered. They depend generally, as shown above, on chemical, physicochemical and crystal–chemical properties of co-crystallizing salts and ions. However, there are some previously mentioned "external" factors (such as the kind and composition of the solvent – the liquid phase, the presence of ions or other foreign substances, the presence of complexing agents, acidity [pH] of the solution, from which crystallization takes place, temperature) which may influence these coefficients. Their effect on D2/1 coefficients will be discussed below.

## **6.1 The effect of the kind and composition of the solvent – the liquid phase**

The change of composition of the solvent, from which the crystallization takes place, alternates solubilities of co-crystallizing salts, as well as activity coefficients of all components in the liquid phase and indirectly in the solid phase.

Because of decreased water activity, the formed crystal hydrates have a lower number of molecules of crystallization water of different structures, which may influence the similarity of the crystal structure of macro and microcomponents. All these mentioned factors vary in different directions and to a different degree, and therefore, they may finally cause the change of co-crystallization coefficients. Examples are given in Table **13**.

K+ <sup>3</sup>

*m*

01

02

K+ <sup>4</sup> 2

*m*

2/1

(2) – subscripts "1" and "2" relate to macrocomponent and microcomponent respectively;

*D N* [Å], where M – molar mass of salt [g/mol], Dx – its density [g/cm3],

(4) – in this case (r1–r2)/r1<0.20 (for the other cases D2/1 < 0.06); k – number of crystallizations systems

From the practical point of view it is very interesting to ascertain how to increase the efficiency of crystallization purification of inorganic substances. This is possible when cocrystallization coefficients D2/1 can be lowered. They depend generally, as shown above, on chemical, physicochemical and crystal–chemical properties of co-crystallizing salts and ions. However, there are some previously mentioned "external" factors (such as the kind and composition of the solvent – the liquid phase, the presence of ions or other foreign substances, the presence of complexing agents, acidity [pH] of the solution, from which crystallization takes place, temperature) which may influence these coefficients. Their effect

The change of composition of the solvent, from which the crystallization takes place, alternates solubilities of co-crystallizing salts, as well as activity coefficients of all

Because of decreased water activity, the formed crystal hydrates have a lower number of molecules of crystallization water of different structures, which may influence the similarity of the crystal structure of macro and microcomponents. All these mentioned factors vary in different directions and to a different degree, and therefore, they may finally cause the

Table 12. Part 2. Some examples of the estimation of coefficients D2/1 (Smolik, 2004)

**6. Methods of lowering D2/1 values as the way of increasing purification** 

**6.1 The effect of the kind and composition of the solvent – the liquid phase** 

components in the liquid phase and indirectly in the solid phase.

change of co-crystallization coefficients. Examples are given in Table **13**.

02

2 01

exp 4.282 4.074 *<sup>m</sup> D r*

01 1 2

02 1 exp 7.967 0.424 *<sup>m</sup> r r <sup>D</sup> m r*

<sup>2</sup> 1 2

74.20

2.345 0,185

*h*

*r r*

2

5.3

26.5

5.1

2/1 2

2/1 1

exp

*<sup>m</sup> <sup>D</sup> <sup>r</sup>*

MClO3 M' ={K, Rb}

MClO4 M' ={K, Rb, Cs}

M2CrO4 M' ={K, Rb, Cs}

(3) –<sup>8</sup> 10 <sup>3</sup> Cs+, Rb+,

Cs+, Rb+, K+ <sup>5</sup>

Cs+, Rb+,

(macrocomponent)j – (microcomponent)ji in a given group of salt.

(1) – Mn+ is not microcomponent, when M'=M;

*x M*

N – Avogadro number [6.022·1023/mol];

**efficiency of crystallization** 

on D2/1 coefficients will be discussed below.


Table 13. The effect of addition of various organic solvents: iso-propyl alcohol (iso–PrOH), methanol (CH3OH), ethanol (C2H5OH), diethyl ether (Et2O), acetone (Me2CO) on coefficients D2/1 during the crystallization of NiSO47H2O at 25 oC (Smolik, 1984)

### **6.2 The effect of the presence of other ions or substances in the liquid phase**

Interactions which happen in the aqueous phase may also influence D2/1 coefficients. This effect is formally taken into consideration in equation (19) by the mean activity coefficients of both the macrocomponent (m1) and microcomponent (m2).

If the action of various factors in aqueous solution causes the same changes in both mean

activity coefficients, so that <sup>1</sup> 2 . *<sup>m</sup> m const* , then at unchanged properties of the solid phase

D2/1 coefficient remains constant (e.g., the addition of HBr during the co-crystallization of Ra2+ with BaBr2 does not affect D2/1 coefficient, likewise the introduction of weak electrolytes (glucose or CH3COONa + CH3COOH) having no common ions with micro and macrocomponent (Ra(NO3)2 i Ba(NO3)2) and not reacting with them also does not change the D2/1 value (Chlopin, 1938).

However, if substances are present in the solution that react in a different way with the macro and microcomponent ions forming slightly dissociated compounds, an essential change of D2/1 coefficients takes place (e.g., the addition of CH3COONa + CH3COOH in the crystallization system Pb(NO3)2 – Ra(NO3)2 – H2O causes bonding a part of Pb2+ ions in slightly dissociated acetate, which leads to the lowering of its mean activity coefficient and finally to the increase of radium co-crystallization coefficient (Chlopin, 1938).

#### **6.3 The effect of the presence of complexing agents**

The presence of complexing agents has a significant influence on coefficient, D2/1. In the case of forming complexes by both macrocomponent and microcomponent, the relationship between the value of co-crystallization coefficient in the presence of complexing agent (D2/1)k and that in the case of its absence (D2/1) is expressed by equation (Mikheev et al.,

1962): 2/1 2/1 1 1 ' *k ML M D D ML M* , where (') – stability constant of the complex of

Chemical, Physicochemical and Crystal – Chemical Aspects of

crystallization of NiCl2·6H2O at 25 oC (Smolik, 1999b)

**6.5 The effect of the change of the oxidation state** 

component

at 20 oC Co2+ Fe2+

at 25 oC Al3+ Fe3+

in Table 16.

Crystallized salt temperature

MnSO4·5H2O

CoSO4·7H2O

NH4Al(SO4)2·12H2O

oC (Fig. 12b).

**6.6 The effect of temperature** 

at 20 oC Mn2+

Crystallization from Aqueous Solutions as a Method of Purification 177

Co2+ 2.600.30 2.300.30 1.800.10 Mn2+ 0.460.02 0.200.02 0.210.01 Cu2+ 0.040.01 0.040.01 0.020.01 Fe2+ 1.700.20 1.200.20 0.400.08

0 0.5 5

changing oxidation state Macro– D2/1 Ref.

H2O2

H2O2

NH2OH.H2SO4

1995) Fe3+ < 0.03

Fe 2003) 3+ < 0.03

1995b) Fe2+ < 0.01

1.04 (Smolik

1.20 (Smolik,

0.01 (Smolik,

0.04±

et al.,

Microcomponent Average D2/1 coefficients for HCl concentrations [mol/L]

Table 15. The effect of HCl concentration on D2/1 coefficients of some M2+ ions during the

In some cases it is possible to change easily the oxidation state of the microcomponent or macrocomponent during or before crystallization. Usually this is accompanied by a significant alteration of D2/1 coefficients, which may be utilized for the rise of purification efficiency. Several examples of the change of microcomponent oxidation state are presented

Oxidation state of Factor

Micro– component

Fe2+

Table 16. The effect of the change of oxidation state of microcomponent on D2/1 coefficients

The change of the oxidation state of macrocomponent can be used for the purification of iron salts: crystallization of FeSO4·7H2O at 20 oC permits, with great efficiency, removal of all M+ and M3+ ions (Fig. 12a) and the remaining M2+ ions can be easily removed after transferring FeSO4·7H2O into NH4Fe(SO4)2·12H2O by oxidation and its crystallization at 20

The effect of temperature on D2/1 coefficients is very complex. As temperature increases, the solubilities of the macro and microcomponent (m01, m02), the mean activity coefficients in their binary saturated solutions (m01, m02), the mean activity coefficients in the ternary solution being in equilibrium with their mixed crystal (m1, m2) (temperature affects dehydration of ions, processes of hydrolysis or complex formation in solution (Kirkova et al., 1996), as well as activity coefficients of both in their solid solution (*f*1, *f*2) (temperature influences enthalpy of mixing in the solid phase) change in different directions in various crystallization systems. Therefore, both the increase and drop of co-crystallization coefficient

microcomponent (macrocomponent), [ML] and [M] are the concentrations of macrocomponent complexes and its free ions. Hence if /'>1 (stability of the complex with microcomponent is higher), then Dk2/1 < D2/1 (microcomponent co-crystallizes to a lower degree and vice versa.

An example of the use of a complexing agent to significantly lower D2/1 coefficients is presented in Table 14.

The presence of EDTA4– in stoichiometric amount causes 4 – 180 fold lowering of D2/1 coefficients of M2+ and M3+ ions, because of the formation of [M(EDTA)]2– or [M(EDTA)]– anionic complexes, and in the case of some of them (Cd2+, Mn2+, Ni2+, Al3+ and Cr3+) two– fold increase of EDTA4– excess leads to an additional drop of D2/1 coefficients.


Table 14. The effect of EDTA4– addition on D2/1 coefficients of co-crystallization of M2+ i M3+ ions with Na2SO4 at 50oC. (1998b)

#### **6.4 The effect of the acidity (pH) of the solution, from which crystallization takes place**

Cations of macrocomponent ([M(H2O)x]n+) and microcomponent ([M'(H2O)x']n+) present in the solution, from which crystallization usually takes place, may hydrolyze according to the following equations: [M(H2O)x]n+ + H2O [M(H2O)x–1(OH)](n–1)+ + H3O+ and [M'(H2O)x']n+ + H2O [M'(H2O)x'–1(OH)](n–1)+ + H3O+. The degree of hydrolysis depends on [M(H2O)x]n+ ([M'(H2O)x']n+) cation acidic strength, meant as Brönstedt acid

(Khi = {[M(H2O)x–1(OH)](n–1)+}·{H3O+}/ {[M(H2O)x]n+}. If the difference in Kh of both ions is significant (e.g., Kh2 >> Kh1), ions ([M'(H2O)x'–1(OH)](n–1)+) of different charge than those of macrocomponent ([M(H2O)x]n+ ) are present in the solution, which in–build into crystals of macrocomponent to a lower degree (e.g., DFe(III)/NH4Al alum = 0.038 ± 0.005 in 0,1 M H2SO4 solution and DFe(III)/NH4Al alum = 0.063± 0.009 in 1,0 M H2SO4 solution) (Smolik, 1995b).

In the case of NiCl2·6H2O crystallization with increasing HCl concentration, the lowering of co-crystallization coefficients D2/1 of Co2+, Mn2+, Cu2+ and Fe2+ (Table 15) is caused not only by the rise of acidity of the solution, but also by the formation of chloride complexes at higher Cl– concentrations. A significant decrease of Mn2+ coefficient (DMn) occurs even at 0.5 M HCl, but that of Fe2+ and Co2+ only at 5M HCl.

microcomponent (macrocomponent), [ML] and [M] are the concentrations of macrocomponent complexes and its free ions. Hence if /'>1 (stability of the complex with microcomponent is higher), then Dk2/1 < D2/1 (microcomponent co-crystallizes to a lower

An example of the use of a complexing agent to significantly lower D2/1 coefficients is

The presence of EDTA4– in stoichiometric amount causes 4 – 180 fold lowering of D2/1 coefficients of M2+ and M3+ ions, because of the formation of [M(EDTA)]2– or [M(EDTA)]– anionic complexes, and in the case of some of them (Cd2+, Mn2+, Ni2+, Al3+ and Cr3+) two–

Fe3+ 2.24 0.02 0.02 Co2+ 0.21 0.01 0.01 Zn2+ 0.18 <0.01 <0.01 Cd2+ 1.40 0.10 <0.01 Mn2+ 1.08 0.27 0.07 Ni2+ 0.19 0.01 0.01 Cu2+ 0.28 0.01 0.01 Al3+ 0.46 0.05 0.01 Cr3+ 1.92 0.16 0.03 Table 14. The effect of EDTA4– addition on D2/1 coefficients of co-crystallization of M2+ i M3+

**6.4 The effect of the acidity (pH) of the solution, from which crystallization takes place**  Cations of macrocomponent ([M(H2O)x]n+) and microcomponent ([M'(H2O)x']n+) present in the solution, from which crystallization usually takes place, may hydrolyze according to the following equations: [M(H2O)x]n+ + H2O [M(H2O)x–1(OH)](n–1)+ + H3O+ and [M'(H2O)x']n+ + H2O [M'(H2O)x'–1(OH)](n–1)+ + H3O+. The degree of hydrolysis depends on [M(H2O)x]n+

(Khi = {[M(H2O)x–1(OH)](n–1)+}·{H3O+}/ {[M(H2O)x]n+}. If the difference in Kh of both ions is significant (e.g., Kh2 >> Kh1), ions ([M'(H2O)x'–1(OH)](n–1)+) of different charge than those of macrocomponent ([M(H2O)x]n+ ) are present in the solution, which in–build into crystals of macrocomponent to a lower degree (e.g., DFe(III)/NH4Al alum = 0.038 ± 0.005 in 0,1 M H2SO4 solution and DFe(III)/NH4Al alum = 0.063± 0.009 in 1,0 M H2SO4 solution) (Smolik, 1995b).

In the case of NiCl2·6H2O crystallization with increasing HCl concentration, the lowering of co-crystallization coefficients D2/1 of Co2+, Mn2+, Cu2+ and Fe2+ (Table 15) is caused not only by the rise of acidity of the solution, but also by the formation of chloride complexes at higher Cl– concentrations. A significant decrease of Mn2+ coefficient (DMn) occurs even at 0.5

Co-crystallization coefficients , D2/1 Ratio of the number of moles of EDTA4– to the sum of the number of moles of M2+ and M3+ ions before crystallization 0:1 1:1 2:1

fold increase of EDTA4– excess leads to an additional drop of D2/1 coefficients.

degree and vice versa.

presented in Table 14.

Ions

ions with Na2SO4 at 50oC. (1998b)

([M'(H2O)x']n+) cation acidic strength, meant as Brönstedt acid

M HCl, but that of Fe2+ and Co2+ only at 5M HCl.


Table 15. The effect of HCl concentration on D2/1 coefficients of some M2+ ions during the crystallization of NiCl2·6H2O at 25 oC (Smolik, 1999b)

## **6.5 The effect of the change of the oxidation state**

In some cases it is possible to change easily the oxidation state of the microcomponent or macrocomponent during or before crystallization. Usually this is accompanied by a significant alteration of D2/1 coefficients, which may be utilized for the rise of purification efficiency. Several examples of the change of microcomponent oxidation state are presented in Table 16.


Table 16. The effect of the change of oxidation state of microcomponent on D2/1 coefficients

The change of the oxidation state of macrocomponent can be used for the purification of iron salts: crystallization of FeSO4·7H2O at 20 oC permits, with great efficiency, removal of all M+ and M3+ ions (Fig. 12a) and the remaining M2+ ions can be easily removed after transferring FeSO4·7H2O into NH4Fe(SO4)2·12H2O by oxidation and its crystallization at 20 oC (Fig. 12b).

### **6.6 The effect of temperature**

The effect of temperature on D2/1 coefficients is very complex. As temperature increases, the solubilities of the macro and microcomponent (m01, m02), the mean activity coefficients in their binary saturated solutions (m01, m02), the mean activity coefficients in the ternary solution being in equilibrium with their mixed crystal (m1, m2) (temperature affects dehydration of ions, processes of hydrolysis or complex formation in solution (Kirkova et al., 1996), as well as activity coefficients of both in their solid solution (*f*1, *f*2) (temperature influences enthalpy of mixing in the solid phase) change in different directions in various crystallization systems. Therefore, both the increase and drop of co-crystallization coefficient

Chemical, Physicochemical and Crystal – Chemical Aspects of

**8. References** 

England

Moskva, Russia

a given pair of macrocomponent (1) and microcomponent (2).

Crystallization from Aqueous Solutions as a Method of Purification 179

coefficients. The investigation involving the effect of these factors is of great importance because it makes it possible to evaluate the usefulness of crystallization in the separation of

By means of "external" factors a significant lowering of D2/1coefficients is sometimes

Growing knowledge concerning coefficients D2/1 in new crystallization systems, as well as better understanding of the dependences of these coefficients on different factors, permits evaluating in a progressively better way the possibilities of the crystallization method in new crystallization systems and more effective control with "external" conditions to achieve higher yields of crystallization purification, enrichment of trace amounts of rare, scattered elements for preparative or analytical purposes. In addition it also helps in improving the growing of single crystals of specific properties or explaining the genesis of some minerals.

Balarew, Chr.; Duhlev R. & Spassov D. (1984). Hydrated metal halide structures and the

Balarew, Chr. (1987). Mixed crystals and double salts between metal (II) salt hydrates,

Blamforth, A. W. (1965). *Industrial Crystallization*, Leonard Hill, London 1965, (Great Britain)

Borneman-Starinkevich, I. D. (1975) Calculation of the crystallochemical formula as one of

Chukhrov, B. E. Borutsky & N. N. Mozgova), pp. 125-33 . Nauka, Moskva Brown, I. D. (1981). Bond-valence method: an empirical approach to chemical structure and

Byrappa, K.; Srikantaswamy, S.; Gopalakrishna G. S. & Venkatachalapaty V. (1986)

Chlopin, V. G. (1938). The distribution of electrolytes between solid crystals and liquid

Chlopin, V. G. (1957). Izbrannye Trudy, (Selected Work) Vol. 6, pp. 173, *Izd. AN SSSR*,

Demirskaya, O. V.; Kislomed A.N.; Velikhov Y.N.; Glushova L. V. & Vlasova D. I. (1989)

from aqueous solutions at 25 oC. *Vysokochistye Veshchestva,* Vol. 1, pp. 14-16 Doerner, H. A. & Hoskins, W. M. (1925). Coprecipitation of radium and barium sulfates, *J.* 

Fisher, S. (1962). Correlation between maximum solid solubility and distribution coefficient

Gorshtein, G. I. (1969). In *Methods of obtaining of high-purity inorganic substances* (in Russ.),

Hill, A. E.; Durham, G. S. & Ricci, J .E. (1940). Distribution of isomorphous salts in solubility equilibrium between liquid and solid phases, *J. Am. Chem. Soc*., Vol. 62, pp 2723-32

the methods for the investigation of minerals, In *Izomorfizm Minerallov* (Eds. F. V.

bonding. In *Structure and bonding in crystals*. (Eds. M. O'Keeffe, A. Nawrotsky) Vol.

Influence of admixtures on the crystallization and morphology of AlPO4 crystals. *J.* 

Distribution of impurities during crystallization of potassium dihydrophosphate

HSAB concept, *Crystal Res. Technol.* Vol. 19(11), pp. 1469

*Zeitschrift fuer Kristallographie*., Vol. 181, pp. 35-82

II, p. 1 – 30. New York: Academic Press 1981

phase, *Trudy Rad. Inst. AN SSSR* Vol. 4, pp. 34-79

for impurities in Ge and Si, *J. Appl. Phys*., Vol.33, pp 1615

Izd. Chimia, Sanct Petersburg, Russia, (63-125)

*Mater.Sci.,* Vol. 21, pp. 2202 - 2206

*Am. Chem. Soc.*,Vol.47, pp 662-75

possible, and thus, the improvement of the efficiency of crystallization purification.

may be observed or sometimes the maintenance of its constant value (in the case of compensation of the all mentioned changes). However, the alteration of D2/1 runs generally in a continuous manner as long as there are no phase transitions. If a phase transition in the system takes place, a jump change of D2/1 appears, connected with the transition from isomorphous co-crystallization into the isodimorphous one. The determination of temperatures at which such a jump change of co-crystallization coefficients takes place, permits finding the temperatures of the phase transitions for many hydrate sulfates (Purkayastha & Das, 1972, 1975), as well as predicting the existence (at specific conditions) of hydrates of some salts, as yet unknown (Purkayastha & Das, 1971).


Some examples of the changes of co-crystallization coefficients D2/1 at various temperatures have been presented in Table **17**.

Table 17. The effect of temperature on D2/1 coefficients

The observation of the alterations of D2/1 coefficients with changing temperature sometimes permits finding such ranges of this parameter, where they are low enough that crystallization purification of the macrocomponent from a given microcomponent will be very effective (Purkayastha & Das, 1972; Smolik & Kowalik, 2010, 2011). In such a manner it turned out to be possible to accomplish essential purification of CoSeO4 from almost all M2+ ions (most difficult to remove) solely by the crystallization method (Kowalik et al., 2011).

## **7. Conclusions**

Crystallization of substances from solutions seems still to be a convenient method of their purification, particularly in obtaining of high purity inorganic compounds. The effectiveness of this process depends on the kind of both macrocomponent (1) and microcomponent (2) and can be evaluated by means of co-crystallization coefficient, D2/1 (Henderson – Kraček, Chlopin). These coefficients are affected by conditions applied in the crystallization process, but those which are equilibrium ones, depend exclusively on "internal" and "external" factors.

"Internal" factors (resulting from chemical, physicochemical and crystal-chemical properties of co-crystallizing salts and ions to a significant degree) determine the level of D2/1

coefficients. The investigation involving the effect of these factors is of great importance because it makes it possible to evaluate the usefulness of crystallization in the separation of a given pair of macrocomponent (1) and microcomponent (2).

By means of "external" factors a significant lowering of D2/1coefficients is sometimes possible, and thus, the improvement of the efficiency of crystallization purification.

Growing knowledge concerning coefficients D2/1 in new crystallization systems, as well as better understanding of the dependences of these coefficients on different factors, permits evaluating in a progressively better way the possibilities of the crystallization method in new crystallization systems and more effective control with "external" conditions to achieve higher yields of crystallization purification, enrichment of trace amounts of rare, scattered elements for preparative or analytical purposes. In addition it also helps in improving the growing of single crystals of specific properties or explaining the genesis of some minerals.

## **8. References**

178 Crystallization – Science and Technology

may be observed or sometimes the maintenance of its constant value (in the case of compensation of the all mentioned changes). However, the alteration of D2/1 runs generally in a continuous manner as long as there are no phase transitions. If a phase transition in the system takes place, a jump change of D2/1 appears, connected with the transition from isomorphous co-crystallization into the isodimorphous one. The determination of temperatures at which such a jump change of co-crystallization coefficients takes place, permits finding the temperatures of the phase transitions for many hydrate sulfates (Purkayastha & Das, 1972, 1975), as well as predicting the existence (at specific conditions)

Some examples of the changes of co-crystallization coefficients D2/1 at various temperatures

MnSO4 Cu2+ 20 MnSO4·5H2O (tcl.) CuSeO4·5H2O (tcl.) 1.63 (Smolik,

Ni2+ 25 ZnSeO4·6H2O (tetr.) NiSeO4·6H2O (tetr.) 2.93 40 ZnSeO4·5H2O (tcl.) NiSeO4·6H2O (tetr.) 0.21 Sr(NO3)2 Pb2+ 29 Sr(NO3)2·4H2O(mcl.) Pb(NO3)2 (cub.) 0.66 (Niesmie-

Na2SO4 Fe3+ 25 Na2SO4 (rhomb.) 0.01 (Smolik,

The observation of the alterations of D2/1 coefficients with changing temperature sometimes permits finding such ranges of this parameter, where they are low enough that crystallization purification of the macrocomponent from a given microcomponent will be very effective (Purkayastha & Das, 1972; Smolik & Kowalik, 2010, 2011). In such a manner it turned out to be possible to accomplish essential purification of CoSeO4 from almost all M2+ ions (most difficult to remove) solely by the crystallization method (Kowalik et al., 2011).

Crystallization of substances from solutions seems still to be a convenient method of their purification, particularly in obtaining of high purity inorganic compounds. The effectiveness of this process depends on the kind of both macrocomponent (1) and microcomponent (2) and can be evaluated by means of co-crystallization coefficient, D2/1 (Henderson – Kraček, Chlopin). These coefficients are affected by conditions applied in the crystallization process, but those which are equilibrium ones, depend exclusively on "internal" and "external" factors. "Internal" factors (resulting from chemical, physicochemical and crystal-chemical properties of co-crystallizing salts and ions to a significant degree) determine the level of D2/1

Kind of hydrate (crystal structure) of

25 ZnSeO4·6H2O (tetr.) CuSeO4·5H2O (tcl.) 0.51

40 ZnSeO4·5H2O (tcl.) CuSeO4·5H2O (tcl.) 1.77 50 ZnSeO4·H2O (mcl.) CuSeO4·5H2O (tcl.) 0.12

50 MnSO4·H2O (mcl.) CuSeO4·5H2O (tcl.) 0.15 2004)

34 Sr(NO3)2 (cub.) Pb(NO3)2 (cub.) 3.30 anov, 1975)

50 Na2SO4·10H2O(mcl.) 2.24 1998b)

Macro– D2/1 Ref.

Micro– component

> (Smolik & Kowalik, 2010, 2011)

of hydrates of some salts, as yet unknown (Purkayastha & Das, 1971).

component

have been presented in Table **17**.

Tempe– rature [ oC]

Table 17. The effect of temperature on D2/1 coefficients

ion

Cu2+

(macro– component)

ZnSeO4

**7. Conclusions** 


Chemical, Physicochemical and Crystal – Chemical Aspects of

Inorganic chemistry, Gliwice, pp. 91-101

crystallization. *Austr. J. Chem*. Vol. 52, pp. 425-430

NiSO47H2O, *Polish J. Chem*., Vol. 74, pp. 1447-1461

pp.1322-7

Lublin, Poland

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Smolik, M. (1999a). Partition of microamounts of some M2+ ions during MgSO47H2O

Smolik M. (1999b) Distribution of trace amounts of some M2+ ions during nickel chloride

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Smolik, M. (2001). Distribution of microamounts of M2+ during NiSO4(NH4)2SO46H2O crystallization, *Separation Science and Technology* Vol. 36(13), pp. 2959-2969 Smolik, M. (2002a). Factors influencing cocrystallization coefficients D of trace amounts of

Smolik, M. (2004) (Effect of some chemical, physicochemical and crystal-chemical factors on

Smolik, M. Jakóbik, A. & Trojanowska J. (2007). Distribution of trace amounts of M2+ ions during Co(CH3COO)2•4H2O crystallization, *Sep. Pur. Techn.,* Vol. 54, pp. 272-276 Smolik, M. (2008). Effect of chemical, physiochemical and crystal-chemical factors on co-

Smolik, M. & Kowalik, A. (2010). Co-crystallization of trace amounts of M2+ ions with

Smolik, M. & Kowalik, A. (2011a). Equilibrium coefficients of co-crystallization of M2+ ions

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cocrystallization coefficients D2/1 of trace amounts of metal ions during the crystallization of chosen salts from water solutions (in Polish), *Zesz. Nauk. Pol. Sl.*

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6th Symposium "*Industrial crystallization*" Rudy 1998 (*in Polish*), Institute of


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formation on the separation of elements by cocrystallization process obeying

sulfate-cobalt sulfate-water and magnesium sulfate-zinc sulfate-water systems at 25 oC,

Sofia


**7** 

*South Africa* 

**Recrystallization of Active** 

Nicole Stieger and Wilna Liebenberg

**Pharmaceutical Ingredients** 

**System Relationship between** 

Cubic a = b = c

Tetragonal a = b ≠<sup>c</sup>

Orthorhombic a ≠ b ≠<sup>c</sup>

Hexagonal a = b ≠<sup>c</sup>

Monoclinic a ≠ b ≠<sup>c</sup>

Triclinic a ≠ b ≠<sup>c</sup>

Rhombohedral (or trigonal)

Table 1. Crystal systems and lattice parameters. (Adapted, with permission of Informa

Healthcare, from Rodríguez-Homedo *et al*., 2006.)

**lattice parameters** 

α = β = γ = 90°

α = β = γ = 90°

α = β = γ = 90°

α = β = 90°, γ = 120°

α = γ = 90°, β ≠ 90°

α ≠ β ≠ γ ≠ 90°

a ≠ b ≠ c α = β = γ ≠ 90°

*North-West University, Unit for Drug Research & Development* 

Recrystallization can be described simply as a process whereby a crystalline form of a compound may be obtained from other solid-state forms, being themselves crystalline or amorphous, of the same substance. Recrystallization is the process most often employed for the intermediate separation and last-step purification of solid active pharmaceutical ingredients (APIs) (Shekunov & York, 2000; Tiwary, 2006). Chemical purity is however not the only property of a pharmaceutical active that will affect its performance. Crystal structure (Table 1), crystal habit (Table 2) and particle size all play a part. Polymorphism (different crystal structures of the same substance) affects the physico-chemical properties and stability of an API, whereas crystal habit and particle size mostly affect various indices impacting on dosage form production and performance: particle orientation; flowability; packing and density; surface area; aggregation; compaction; suspension stability; and

dissolution (Blagden *et al*., 2007; Doherty & York, 1988; Tiwary, 2006).

**1. Introduction** 


## **Recrystallization of Active Pharmaceutical Ingredients**

Nicole Stieger and Wilna Liebenberg *North-West University, Unit for Drug Research & Development South Africa* 

## **1. Introduction**

182 Crystallization – Science and Technology

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from a water solution, *Geochimia*, Vol.54, pp. 627-44

Amsterdam, Oxford, New York, Tokyo

*J. Chem.*, Vol. 64, pp. 34-37

*Geochimia*, Vol. 8, pp. 1204-23

*Chem*. 114, 385-91

ZnSeO4·5H2O at 40 °C and ZnSeO4·H2O at 50 °C and their dependences on various physicochemical and crystal-chemical factors, *J. Cryst. Growth,* Vol. 337, pp. 46-51 Smolik, M. & Siepietowski, L. (2011). Some factors influencing distribution coefficients of

trace amounts of M2+ ions during the crystallization of Mn(CH3COO)2·4H2O, *S. Afr.* 

Me+X-CuX2-H2O systems (Me+ = K+, NH4+, Rb+, Cs+; X- = Cl-, Br-), *J. Solid State* 

coefficients of isovalent isomorphous admixtures during crystallization of melts,

distribution coefficient for crystallization purification and the limited solubility of

Recrystallization can be described simply as a process whereby a crystalline form of a compound may be obtained from other solid-state forms, being themselves crystalline or amorphous, of the same substance. Recrystallization is the process most often employed for the intermediate separation and last-step purification of solid active pharmaceutical ingredients (APIs) (Shekunov & York, 2000; Tiwary, 2006). Chemical purity is however not the only property of a pharmaceutical active that will affect its performance. Crystal structure (Table 1), crystal habit (Table 2) and particle size all play a part. Polymorphism (different crystal structures of the same substance) affects the physico-chemical properties and stability of an API, whereas crystal habit and particle size mostly affect various indices impacting on dosage form production and performance: particle orientation; flowability; packing and density; surface area; aggregation; compaction; suspension stability; and dissolution (Blagden *et al*., 2007; Doherty & York, 1988; Tiwary, 2006).


Table 1. Crystal systems and lattice parameters. (Adapted, with permission of Informa Healthcare, from Rodríguez-Homedo *et al*., 2006.)

Recrystallization of Active Pharmaceutical Ingredients 185

The number of solid-state forms that an API could exist in relies on the range of noncovalent interactions and molecular assemblies, the order range, and the balance between entropy and enthalpy that defines the free energy states and processes (Figure 1). When an API exists as more than one solid-state form, thermodynamics control their relative stability and the conditions and direction in which a transformation can occur, whilst kinetics determine how long it will take for a transformation to reach equilibrium. Thermodynamics establish the stability domains of the solid-states, but once a metastable domain is encountered, the kinetic pathways will determine which form will be created and for how

Small changes in recrystallization procedure can influence the crystallization process and may lead to changes in API crystal structure, crystal habit and particle size with subsequent variability of raw material characteristics and dosage form performance. Herein lies the challenge and the opportunity: manufacturers of pharmaceutical actives go to great lengths to ensure production uniformity, but a pharmaceutical researcher might choose to alter

Crystallization is generally thought of as the evolution from solution or melt of the crystalline state (Blagden *et al*., 2007), but a broader definition includes precipitation and solid-state transitions (Shekunov & York, 2000). Crystallization methods can be solvent or non-solvent based and the varied reaction conditions generate different crystal forms (Banga *et al*., 2004). Non-solvent based methods include sublimation, thermal treatment, desolvation of solvates, grinding and crystallization of a melt (Guillory, 1999). Applied in polymorph screening studies, traditional crystallization approaches most often will not yield all possible polymorphs of a given API. Therefore, there is a continuous search for

In solution, crystallization is the creation of a crystalline phase by a process (Figure 2) initiated by molecular aggregation, leading to the formation of nuclei (the smallest possible

The sections that follow will address factors that influence solvent recrystallization in general and will not venture into the specifics of specialized solvent-based production methods like spray-drying, freeze-drying, etc. The reader should keep in mind that although different parameters are discussed under separate headings, they are interactive and not independent of each other. For example: changing the composition of a solvent system will change the level of saturation; changes in temperature alter viscosity and level of saturation;

Each of the parameters discussed can influence polymorphism, and polymorphism in turn affects crystal habit. It is also well-known that each polymorph of an API can exhibit multiple crystal habits, therefore a parameter might influence crystal habit without

innovative methods of manipulating the crystallization process (Rouhi, 2003).

units with defined crystal lattice) and ultimately crystal growth (Banga *et al*., 2004).

**2.1 Parameters that influence nucleation and crystal growth** 

changing the internal structure of the crystals produced (Tiwary, 2006).

long it will survive (Rodríguez-Spong *et al*., 2004).

**2. Solvent based recrystallization** 

agitation may increase temperature; etc.

recrystallization conditions to manipulate API characteristics.


Table 2. Descriptions of common pharmaceutical crystal morphologies. (Adapted, with permission of Informa Healthcare, from Rodríguez-Homedo et al., 2006.)

Fig. 1. Diagram showing the phenomena that govern solid-state transformations. Mechanical, thermal and chemical (solvents, additives, impurities, relative humidity) stresses affect the competition (or reinforcement) among these processes. (Adapted, with permission of Informa Healthcare, from Rodríguez-Homedo *et al*., 2006.)

Plate/platy Flat crystals with similar width and length but thicker than a flake Prismatic/bipyramid Hexagonal crystals with faces parallel to the growth axis; width and thickness greater than acicular and shorter in length

Rosette/spherulite Sphere composed of needles or rods radiating from a common center Tablet/tabular Flat crystal with similar width and length but thicker than a plate Table 2. Descriptions of common pharmaceutical crystal morphologies. (Adapted, with

**Crystal morphology Description** 

Acicular Slender, needle-like crystal Aggregate Mass of adhered crystals Blade Long, thin, flat crystal

Dendritic Tiny crystallites forming a tree-like pattern Equant/cubic Crystal with similar length, width and thickness

Rod Cylindrical crystals elongated along one axis

permission of Informa Healthcare, from Rodríguez-Homedo et al., 2006.)

Fig. 1. Diagram showing the phenomena that govern solid-state transformations. Mechanical, thermal and chemical (solvents, additives, impurities, relative humidity) stresses affect the competition (or reinforcement) among these processes. (Adapted, with

permission of Informa Healthcare, from Rodríguez-Homedo *et al*., 2006.)

Fiber Long, thin needle; longer than acicular Flake/lath Thin, flat crystal similar in width and length The number of solid-state forms that an API could exist in relies on the range of noncovalent interactions and molecular assemblies, the order range, and the balance between entropy and enthalpy that defines the free energy states and processes (Figure 1). When an API exists as more than one solid-state form, thermodynamics control their relative stability and the conditions and direction in which a transformation can occur, whilst kinetics determine how long it will take for a transformation to reach equilibrium. Thermodynamics establish the stability domains of the solid-states, but once a metastable domain is encountered, the kinetic pathways will determine which form will be created and for how long it will survive (Rodríguez-Spong *et al*., 2004).

Small changes in recrystallization procedure can influence the crystallization process and may lead to changes in API crystal structure, crystal habit and particle size with subsequent variability of raw material characteristics and dosage form performance. Herein lies the challenge and the opportunity: manufacturers of pharmaceutical actives go to great lengths to ensure production uniformity, but a pharmaceutical researcher might choose to alter recrystallization conditions to manipulate API characteristics.

Crystallization is generally thought of as the evolution from solution or melt of the crystalline state (Blagden *et al*., 2007), but a broader definition includes precipitation and solid-state transitions (Shekunov & York, 2000). Crystallization methods can be solvent or non-solvent based and the varied reaction conditions generate different crystal forms (Banga *et al*., 2004). Non-solvent based methods include sublimation, thermal treatment, desolvation of solvates, grinding and crystallization of a melt (Guillory, 1999). Applied in polymorph screening studies, traditional crystallization approaches most often will not yield all possible polymorphs of a given API. Therefore, there is a continuous search for innovative methods of manipulating the crystallization process (Rouhi, 2003).

## **2. Solvent based recrystallization**

In solution, crystallization is the creation of a crystalline phase by a process (Figure 2) initiated by molecular aggregation, leading to the formation of nuclei (the smallest possible units with defined crystal lattice) and ultimately crystal growth (Banga *et al*., 2004).

## **2.1 Parameters that influence nucleation and crystal growth**

The sections that follow will address factors that influence solvent recrystallization in general and will not venture into the specifics of specialized solvent-based production methods like spray-drying, freeze-drying, etc. The reader should keep in mind that although different parameters are discussed under separate headings, they are interactive and not independent of each other. For example: changing the composition of a solvent system will change the level of saturation; changes in temperature alter viscosity and level of saturation; agitation may increase temperature; etc.

Each of the parameters discussed can influence polymorphism, and polymorphism in turn affects crystal habit. It is also well-known that each polymorph of an API can exhibit multiple crystal habits, therefore a parameter might influence crystal habit without changing the internal structure of the crystals produced (Tiwary, 2006).

Recrystallization of Active Pharmaceutical Ingredients 187

The main effect of temperature on crystallization from solution is secondary to its influence on the solubility of the solute, and subsequently the degree of saturation of the solution. Although molecular recognition is required for the formation of molecular clusters, and this is dependent on molecular mobility and collision rates, both of which increase at higher temperatures, the molecular mobility in liquids is too high to be the rate-limiting factor in

The relationship between nucleation rate and supersaturation is well known and Figure 3 illustrates that the number of nuclei generated by a high rate of cooling increases exponentially with increasing supersaturation. A high number of nuclei at the outset limits

In cases where the crystallization of a metastable polymorph precedes in-solvent transformation to a more stable polymorph as described by Ostwald's Rule of Stages (Boistelle & Astier, 1988; Ostwald, 1897), the solvent-mediated transformation can be affected by the temperature of the crystallization medium (Stieger *et al*., 2009), because the

Temperature cycling or oscillation is sometimes used to accelerate the effect of postcrystallization Ostwald Ripening in slurries containing just one polymorph of an API. Small particles and rough edges of larger particles dissolve faster during heating periods, followed by their recrystallization onto the existing crystals during cooling. The overall effect is more uniformly shaped particles and a narrower particle size distribution range (Kim *et al*., 2003;

Fig. 3. Nucleation versus supersaturation (Adapted, with permission of John Wiley and

nucleation and crystal growth (Rodríguez-Spong *et al*., 2004).

their growth potential (Tung *et al*., 2009).

Mullin, 2001).

Sons, from Tung *et al*., 2009).

process is thermally activated (Beckman, 2000).

Fig. 2. The course of crystallization and its rate-limiting steps. (Adapted, with permission of Touch Briefings, from Banga *et al*., 2004.)

#### **2.1.1 Concentration and temperature**

The difference in chemical potential between the crystallization solution and the solid phase is the fundamental driving force for crystallization, but it is more convenient to express the driving force in terms of supersaturation (Fujiwara *et al*., 2005). Supersaturation (the difference between solution concentration and saturation concentration at a specific temperature) leads to the creation of metastable (far from equilibrium) liquid states and crystallization provides a means to reduce the free energy of the system to the most stable state (equilibrium) (Rodríguez-Homedo *et al*., 2006). The kinetics of nucleation and crystal growth are strongly dependent on supersaturation (Braatz, 2002; Togkalidou *et al*., 2002). An increase in the degree of supersaturation of a solution leads to a reduction in the size of crystals produced. At high supersaturation nucleation is more rapid than growth, resulting in the precipitation of fine particles (Carstensen *et al*., 1993; Tiwary, 2006). It is therefore important to control the degree of supersaturation during crystallization, because the size, shape and solid-state form of the crystals produced are all influenced by supersaturation (Fujiwara *et al*., 2005).

Supersaturation is typically achieved through processes that either increase the solute concentration (evaporation or dissolution of a metastable solid phase with subsequent transformation to the more stable, but less soluble form) or decrease the solubility of the solute (cooling, addition of an antisolvent, pH change or the addition of ions that participate in precipitation of the solute) or through a combination of these strategies (Fujiwara *et al*., 2005; Rodríguez-Homedo *et al*., 2006).

Fig. 2. The course of crystallization and its rate-limiting steps. (Adapted, with permission of

The difference in chemical potential between the crystallization solution and the solid phase is the fundamental driving force for crystallization, but it is more convenient to express the driving force in terms of supersaturation (Fujiwara *et al*., 2005). Supersaturation (the difference between solution concentration and saturation concentration at a specific temperature) leads to the creation of metastable (far from equilibrium) liquid states and crystallization provides a means to reduce the free energy of the system to the most stable state (equilibrium) (Rodríguez-Homedo *et al*., 2006). The kinetics of nucleation and crystal growth are strongly dependent on supersaturation (Braatz, 2002; Togkalidou *et al*., 2002). An increase in the degree of supersaturation of a solution leads to a reduction in the size of crystals produced. At high supersaturation nucleation is more rapid than growth, resulting in the precipitation of fine particles (Carstensen *et al*., 1993; Tiwary, 2006). It is therefore important to control the degree of supersaturation during crystallization, because the size, shape and solid-state form of the crystals produced are all influenced by supersaturation

Supersaturation is typically achieved through processes that either increase the solute concentration (evaporation or dissolution of a metastable solid phase with subsequent transformation to the more stable, but less soluble form) or decrease the solubility of the solute (cooling, addition of an antisolvent, pH change or the addition of ions that participate in precipitation of the solute) or through a combination of these strategies (Fujiwara *et al*.,

Touch Briefings, from Banga *et al*., 2004.)

**2.1.1 Concentration and temperature** 

(Fujiwara *et al*., 2005).

2005; Rodríguez-Homedo *et al*., 2006).

The main effect of temperature on crystallization from solution is secondary to its influence on the solubility of the solute, and subsequently the degree of saturation of the solution. Although molecular recognition is required for the formation of molecular clusters, and this is dependent on molecular mobility and collision rates, both of which increase at higher temperatures, the molecular mobility in liquids is too high to be the rate-limiting factor in nucleation and crystal growth (Rodríguez-Spong *et al*., 2004).

The relationship between nucleation rate and supersaturation is well known and Figure 3 illustrates that the number of nuclei generated by a high rate of cooling increases exponentially with increasing supersaturation. A high number of nuclei at the outset limits their growth potential (Tung *et al*., 2009).

In cases where the crystallization of a metastable polymorph precedes in-solvent transformation to a more stable polymorph as described by Ostwald's Rule of Stages (Boistelle & Astier, 1988; Ostwald, 1897), the solvent-mediated transformation can be affected by the temperature of the crystallization medium (Stieger *et al*., 2009), because the process is thermally activated (Beckman, 2000).

Temperature cycling or oscillation is sometimes used to accelerate the effect of postcrystallization Ostwald Ripening in slurries containing just one polymorph of an API. Small particles and rough edges of larger particles dissolve faster during heating periods, followed by their recrystallization onto the existing crystals during cooling. The overall effect is more uniformly shaped particles and a narrower particle size distribution range (Kim *et al*., 2003; Mullin, 2001).

Fig. 3. Nucleation versus supersaturation (Adapted, with permission of John Wiley and Sons, from Tung *et al*., 2009).

Recrystallization of Active Pharmaceutical Ingredients 189

where spontaneous nucleation occurs – this is the border of the metastable zone. For seeding to be successful in producing just the required product, the concentration and temperature of the crystallization medium must be strictly controlled within this zone. If the solubility line is crossed to the left, the seed crystals will dissolve and if the metastable zone limit is crossed to the right, spontaneous nucleation will take place and could result in unwanted

Fig. 5. The operating region of seeded industrial batch crystallization is the metastable zone, which is bound by the solubility curve and the metastable limit for the specific API. The concentration-temperature profile (control trajectory) to be used lies within the metastable zone (operating region). (Adapted, with permission of Elsevier, from Fujiwara *et al*., 2005.)

In evaporative crystallization, crystals are sometimes observed to form preferentially near the surface of the solution, due to a higher local concentration of solute. The meniscus of a solution can also have geometry favoring higher evaporation rates, with crystals then

Dependent on the conditions employed, the crystallization of API polymorphs from a solvent may be under kinetic or thermodynamic control. When crystallization of a dimorphic compound (Figure 6) is conducted sufficiently above or below the transition point, in an area defined by the solubility curves of the two polymorphs, the solvent used is immaterial provided that the API solubility is adequate to allow the prescribed concentrations to be reached. Irrespective of the kinetics, the outcome is under total

forming at the contact line with the crystallization vessel (Capes & Cameron, 2007).

**2.1.3 Solvent** 

crystal forms (Beckman, 2000; Fujiwara *et al*., 2005).

## **2.1.2 Interfaces and surfaces**

Nucleation can be either homogeneous or surface catalyzed (Figure 4). Homogeneous nucleation seldom occurs in volumes greater than 100 μl, because solutions contain random impurities that may induce nucleation. Surface catalyzed nucleation can be promoted by surfaces of the crystallizing solute (secondary nucleation) or a surface/interface of different composition than the solute may induce nucleation (heterogeneous nucleation) by decreasing the energy barrier for the formation of nuclei (Rodríguez-Homedo *et al*., 2006).

Fig. 4. Mechanisms for crystal nucleation. (Adapted, with permission of Informa Healthcare, from Rodríguez-Homedo *et al*., 2006.)

Surfaces promoting heterogeneous nucleation may be introduced into the crystallization solution intentionally (as a means of controlling crystal form of the product) (Rodríguez-Spong *et al*., 2004) or unintentionally (dust and other impurities), or they may be an unavoidable part of the process (crystallization vessel, vessel-solution interface and the solution-air interface) (Florence & Attwood, 2006; Kuzmenko *et al*., 2001).

The intentional introduction, into crystallization solutions, of surfaces that catalyze nucleation is known as "seeding". Usually, seeding is performed by introducing crystals of the solute that have the preferred crystal structure one wishes to obtain. Seeding can also be performed using isomorphous substances that differ from the solute (Florence & Attwood, 2006). Seeding techniques can be applied to initiate crystallization; to control particle size – usually when larger crystals with uniform size distribution are required; to avoid encrustation through spontaneous nucleation; to control polymorphic form; and to obtain crystals of high purity, high perfection, desired orientation and sufficient size for crystal structure determination by X-ray diffraction (Beckman, 2000).

When the concentration of an API in the crystallization medium is increased past its solubility curve (Figure 5), whether by cooling or by evaporation of solvent, nucleation does not immediately occur. The solution has to reach a certain concentration-temperature point

Nucleation can be either homogeneous or surface catalyzed (Figure 4). Homogeneous nucleation seldom occurs in volumes greater than 100 μl, because solutions contain random impurities that may induce nucleation. Surface catalyzed nucleation can be promoted by surfaces of the crystallizing solute (secondary nucleation) or a surface/interface of different composition than the solute may induce nucleation (heterogeneous nucleation) by decreasing the energy barrier for the formation of nuclei (Rodríguez-Homedo *et al*., 2006).

Fig. 4. Mechanisms for crystal nucleation. (Adapted, with permission of Informa Healthcare,

Surfaces promoting heterogeneous nucleation may be introduced into the crystallization solution intentionally (as a means of controlling crystal form of the product) (Rodríguez-Spong *et al*., 2004) or unintentionally (dust and other impurities), or they may be an unavoidable part of the process (crystallization vessel, vessel-solution interface and the

The intentional introduction, into crystallization solutions, of surfaces that catalyze nucleation is known as "seeding". Usually, seeding is performed by introducing crystals of the solute that have the preferred crystal structure one wishes to obtain. Seeding can also be performed using isomorphous substances that differ from the solute (Florence & Attwood, 2006). Seeding techniques can be applied to initiate crystallization; to control particle size – usually when larger crystals with uniform size distribution are required; to avoid encrustation through spontaneous nucleation; to control polymorphic form; and to obtain crystals of high purity, high perfection, desired orientation and sufficient size for crystal

When the concentration of an API in the crystallization medium is increased past its solubility curve (Figure 5), whether by cooling or by evaporation of solvent, nucleation does not immediately occur. The solution has to reach a certain concentration-temperature point

solution-air interface) (Florence & Attwood, 2006; Kuzmenko *et al*., 2001).

structure determination by X-ray diffraction (Beckman, 2000).

**2.1.2 Interfaces and surfaces** 

from Rodríguez-Homedo *et al*., 2006.)

where spontaneous nucleation occurs – this is the border of the metastable zone. For seeding to be successful in producing just the required product, the concentration and temperature of the crystallization medium must be strictly controlled within this zone. If the solubility line is crossed to the left, the seed crystals will dissolve and if the metastable zone limit is crossed to the right, spontaneous nucleation will take place and could result in unwanted crystal forms (Beckman, 2000; Fujiwara *et al*., 2005).

Fig. 5. The operating region of seeded industrial batch crystallization is the metastable zone, which is bound by the solubility curve and the metastable limit for the specific API. The concentration-temperature profile (control trajectory) to be used lies within the metastable zone (operating region). (Adapted, with permission of Elsevier, from Fujiwara *et al*., 2005.)

In evaporative crystallization, crystals are sometimes observed to form preferentially near the surface of the solution, due to a higher local concentration of solute. The meniscus of a solution can also have geometry favoring higher evaporation rates, with crystals then forming at the contact line with the crystallization vessel (Capes & Cameron, 2007).

### **2.1.3 Solvent**

Dependent on the conditions employed, the crystallization of API polymorphs from a solvent may be under kinetic or thermodynamic control. When crystallization of a dimorphic compound (Figure 6) is conducted sufficiently above or below the transition point, in an area defined by the solubility curves of the two polymorphs, the solvent used is immaterial provided that the API solubility is adequate to allow the prescribed concentrations to be reached. Irrespective of the kinetics, the outcome is under total

Recrystallization of Active Pharmaceutical Ingredients 191

Fig. 7. Crystal habits arising from growth inhibition at crystal faces. (Adapted, with

Mixing of crystallization solutions and crystal slurries is often necessary, especially in industry, to ensure homogeneity (heat transfer, dispersion of additives, uniformity of crystal suspension, avoidance of settling, etc.). Its effects on nucleation, both primary and secondary, and crystal growth are far reaching and complex. An unagitated solution can, in general, be cooled further before onset of nucleation, because the overall result of mixing is a decrease in the width of the metastable zone. In a mixed solution at constant supersaturation, with no crystals yet present, mixing intensity can reduce induction time – the time elapsed before crystals first appear. Induction time decreases up to a critical speed and then remains unchanged (Karpinski & Wey, 2001; Mullin, 2001; Tung *et al*., 2009).

Mixing actively generates secondary nuclei through crystal-crystal, crystal-vessel and crystal-impeller impacts. A greater number of nuclei generated in agitated systems leads to a decrease in the ultimate crystal size of the product. Mixing also has an intensity dependent effect on the mass transfer rate of growing crystals with a growth limiting outcome (Tung *et* 

It has been found that stirring can reproducibly affect the chiral symmetry of crystallization products. If a substance crystallizes as an equal mixture of dextro and levo crystals when unstirred, its chiral symmetry can be disrupted by stirring (Kondepudi *et al*., 1990; McBride & Carter, 1991). This phenomenon has been attributed to the effect of stirring on secondary

Almost all polymorph screening studies and industrial crystallization processes are performed under ambient pressure, but low pressure is also often applied to increase the

permission of Informa Healthcare, from Carstensen *et al*., 1993.)

**2.1.4 Agitation, mixing and stirring** 

nucleation (Kondepuddi & Sabanayagam, 1994).

*al*., 2009).

**2.1.5 Pressure** 

thermodynamic control. If crystallization takes place outside the area described above (B2, C1, D1, E1 and E2), the choice of solvent may or may not be critical. This will be determined by the relative kinetics of formation, growth, and transformation of the two polymorphs in the various solvents (Threlfall, 2000).

Fig. 6. Polymorphic system of two enantiomorphically related polymorphs I and II. (Transition point X at TX; A-G, initial state of hot, undersaturated solutions; A1-G1, B2 and E2, state of solution at point of initial crystallization. If B is seeded at B1, it behaves as A1.) (Adapted, with permission of the American Chemical Society, from Threlfall, 2000.)

Recrystallization from solvents often leads to the isolation of solvates, in fact, API solvates are very common (Griesser, 2006). Although they have a recognized potential to improve dissolution kinetics (Brittain & Grant, 1999; Haleblian, 1975; Tros de Ilarduya *et al*., 1997), they are rarely selected for further development or dosage form formulation – the only exceptions being hydrates. The main reasons for the rarity of marketed API solvates are their solid-state metastability and the relative toxicity of any included solvent (Douillet *et al*., 2011). It goes without saying that, in the production of solvates, the solvent/s used for recrystallization will be the one/s that could potentially be included in the crystal lattice.

The nature of the crystallization solvent can affect crystal habit, regardless of change in polymorphism (Stieger *et al*., 2010a). The interaction of the solvent at the different crystalsolution interfaces may lead to altered roundness of growing crystal faces and/or edges, changes in crystal growth kinetics, and enhancement or inhibition of crystal growth at certain faces (Figure 7), thereby changing the crystal habit (Stoica *et al*., 2004; Tiwary, 2006).

thermodynamic control. If crystallization takes place outside the area described above (B2, C1, D1, E1 and E2), the choice of solvent may or may not be critical. This will be determined by the relative kinetics of formation, growth, and transformation of the two polymorphs in

Fig. 6. Polymorphic system of two enantiomorphically related polymorphs I and II. (Transition point X at TX; A-G, initial state of hot, undersaturated solutions; A1-G1, B2 and E2, state of solution at point of initial crystallization. If B is seeded at B1, it behaves as A1.) (Adapted, with permission of the American Chemical Society, from Threlfall, 2000.)

Recrystallization from solvents often leads to the isolation of solvates, in fact, API solvates are very common (Griesser, 2006). Although they have a recognized potential to improve dissolution kinetics (Brittain & Grant, 1999; Haleblian, 1975; Tros de Ilarduya *et al*., 1997), they are rarely selected for further development or dosage form formulation – the only exceptions being hydrates. The main reasons for the rarity of marketed API solvates are their solid-state metastability and the relative toxicity of any included solvent (Douillet *et al*., 2011). It goes without saying that, in the production of solvates, the solvent/s used for recrystallization will be the one/s that could potentially be included in the crystal lattice.

The nature of the crystallization solvent can affect crystal habit, regardless of change in polymorphism (Stieger *et al*., 2010a). The interaction of the solvent at the different crystalsolution interfaces may lead to altered roundness of growing crystal faces and/or edges, changes in crystal growth kinetics, and enhancement or inhibition of crystal growth at certain faces (Figure 7), thereby changing the crystal habit (Stoica *et al*., 2004; Tiwary, 2006).

the various solvents (Threlfall, 2000).

Fig. 7. Crystal habits arising from growth inhibition at crystal faces. (Adapted, with permission of Informa Healthcare, from Carstensen *et al*., 1993.)

## **2.1.4 Agitation, mixing and stirring**

Mixing of crystallization solutions and crystal slurries is often necessary, especially in industry, to ensure homogeneity (heat transfer, dispersion of additives, uniformity of crystal suspension, avoidance of settling, etc.). Its effects on nucleation, both primary and secondary, and crystal growth are far reaching and complex. An unagitated solution can, in general, be cooled further before onset of nucleation, because the overall result of mixing is a decrease in the width of the metastable zone. In a mixed solution at constant supersaturation, with no crystals yet present, mixing intensity can reduce induction time – the time elapsed before crystals first appear. Induction time decreases up to a critical speed and then remains unchanged (Karpinski & Wey, 2001; Mullin, 2001; Tung *et al*., 2009).

Mixing actively generates secondary nuclei through crystal-crystal, crystal-vessel and crystal-impeller impacts. A greater number of nuclei generated in agitated systems leads to a decrease in the ultimate crystal size of the product. Mixing also has an intensity dependent effect on the mass transfer rate of growing crystals with a growth limiting outcome (Tung *et al*., 2009).

It has been found that stirring can reproducibly affect the chiral symmetry of crystallization products. If a substance crystallizes as an equal mixture of dextro and levo crystals when unstirred, its chiral symmetry can be disrupted by stirring (Kondepudi *et al*., 1990; McBride & Carter, 1991). This phenomenon has been attributed to the effect of stirring on secondary nucleation (Kondepuddi & Sabanayagam, 1994).

#### **2.1.5 Pressure**

Almost all polymorph screening studies and industrial crystallization processes are performed under ambient pressure, but low pressure is also often applied to increase the

Recrystallization of Active Pharmaceutical Ingredients 193

The addition of salts, polymers or antisolvents can be used to create supersaturation of the solution by decreasing the solute solubility in the crystallization medium. Ions, polymeric molecules, or other substances introduced into the crystallization solvent can also act as impurities for growing crystals. These substances may get adsorbed in the crystal lattice of a growing crystal and disturb the regular and repeating arrangements of the crystal, creating defects and leading to polymorphic modifications. Impurities, and surfactants in particular, can also inhibit crystal growth at certain crystal faces, resulting in crystal habit changes

When selecting suitable solvents for antisolvent crystallization, one should select a pair that is miscible. The solute must be more soluble in one solvent and less soluble in the other (antisolvent). A water-soluble API will most likely be more soluble in a polar solvent and less soluble in a non-polar solvent (Table 4). The opposite holds true for a poorly watersoluble API. Antisolvent crystallization is performed by dissolving the API in the solvent and then gradually adding an antisolvent. This results in a decrease of solute solubility and an increase in supersaturation – much like cooling crystallization (Nonoyama *et al*., 2006; Stieger *et al*., 2010a; Zhou *et al*., 2006). Reverse addition is a variation of antisolvent recrystallization whereby the API solution is added to the antisolvent. The resulting rapid increase in supersaturation leads to swift nucleation and the precipitation of very fine particles (Tung *et al*., 2009). The composition of the crystallization medium can affect both

**Solvent 1 (more polar) Solvent 2 (less polar) Solvent Dielectric constant (ε) Solvent Dielectric constant** 

Water 78.3 Ethanol 24.3 Water 78.3 Acetone 20.7 Methanol 32.6 Dichloromethane 9.08 Ethanol 24.3 Acetone 20.7 Acetone 20.7 Diethyl ether 4.34 Acetone 20.7 Petroleum ether 1.90 Diethyl ether 4.34 Hexane 1.89 Ethyl acetate 6.02 Cyclohexane 1.97 Ethyl acetate 6.02 Petroleum ether 1.90 Dichloromethane 9.08 Petroleum ether 1.90 Toluene 2.38 Petroleum ether 1.90 Table 4. Common solvent-antisolvent pairs. (Adapted, with permission of the author, from

One of the biggest challenges for pharmaceutical researchers is finding all the solid-state forms of an API that can exist at ambient conditions. There is no single method for producing all conceivable forms and a particular polymorph may go undetected for many years. Scientists are continually searching for novel ways of uncovering hidden solid-states.

**(ε)** 

the crystal form and crystal habit of the product (Stieger *et al*., 2010a).

**2.2 Novel strategies and new trends in solvent-based crystallization** 

A few of these methods are briefly discussed below.

**2.1.7 Addition of salts, polymers or antisolvents** 

(Tiwary, 2006).

Skonieczny, 2009.)

rate of solvent evaporation. As with rapid cooling, rapid evaporation leads to higher nucleation rates.

Only recently have researchers turned their attention to high pressure recrystallization from solution as an added dimension in the search for new pharmaceutical polymorphs and solvates. High pressure encourages denser structures in which molecules must pack more efficiently and this means that changes in relative orientations are likely to occur. This gives rise to different themes of molecular interactions, the strengths of which are in turn sensitive to distance and therefore the effects of pressure. The interactions between solute and solvent molecules are also modified by pressure, changing the solubility of a given polymorph or solvate. In some instances, the differences in solubility between two forms might also change, thereby encouraging recrystallization of one form at the expense of another (Fabbiani *et al*., 2004).

### **2.1.6 Moisture and humidity**

When recrystallizing from hygroscopic organic solvents, care should be taken to use only properly dried solvents (Table 3) or newly opened containers from reputable suppliers. If a crystallization process generates a metastable intermediary form, it can absorb water when exposed to moisture and change into a hydrate. Water molecules, because of their small size and multidirectional hydrogen bonding capabilities, are particularly suited to fill structural voids (Manek & Kolling, 2004). If a crystallization system is open to atmospheric conditions, as is likely to be the case with evaporative crystallization, hygroscopic solvents will absorb moisture from the air if present. Should an evaporative crystallization process be sensitive to the presence of moisture, additional measures will have to be put in place to eliminate atmospheric humidity.


\* Never dry halogenated solvents with alkali metals or alkali metal hydrides as this can cause violent explosions.

Table 3. Common drying agents for organic solvents. (Adapted, with permission of Indian Streams Research Journal, from Shinde *et al*., 2011.)

rate of solvent evaporation. As with rapid cooling, rapid evaporation leads to higher

Only recently have researchers turned their attention to high pressure recrystallization from solution as an added dimension in the search for new pharmaceutical polymorphs and solvates. High pressure encourages denser structures in which molecules must pack more efficiently and this means that changes in relative orientations are likely to occur. This gives rise to different themes of molecular interactions, the strengths of which are in turn sensitive to distance and therefore the effects of pressure. The interactions between solute and solvent molecules are also modified by pressure, changing the solubility of a given polymorph or solvate. In some instances, the differences in solubility between two forms might also change, thereby encouraging recrystallization of one form at the expense of another

When recrystallizing from hygroscopic organic solvents, care should be taken to use only properly dried solvents (Table 3) or newly opened containers from reputable suppliers. If a crystallization process generates a metastable intermediary form, it can absorb water when exposed to moisture and change into a hydrate. Water molecules, because of their small size and multidirectional hydrogen bonding capabilities, are particularly suited to fill structural voids (Manek & Kolling, 2004). If a crystallization system is open to atmospheric conditions, as is likely to be the case with evaporative crystallization, hygroscopic solvents will absorb moisture from the air if present. Should an evaporative crystallization process be sensitive to the presence of moisture, additional measures will have to be put in place to eliminate

Alcohols Anhydrous potassium carbonate; anhydrous

pentoxide.

Halogenated solvents\* Calcium hydride; phosphorous pentoxide.

Aldehydes Anhydrous sodium, magnesium or calcium sulphate. Ketones Anhydrous sodium, magnesium or calcium sulphate;

Organic bases (amines) Solid potassium or sodium hydroxide; calcium oxide; barium oxide. Organic acids Anhydrous sodium, magnesium or calcium sulphate.

\* Never dry halogenated solvents with alkali metals or alkali metal hydrides as this can cause violent

Table 3. Common drying agents for organic solvents. (Adapted, with permission of Indian

anhydrous potassium carbonate.

magnesium or calcium sulphate; calcium oxide.

Anhydrous calcium chloride; anhydrous sodium, magnesium or calcium sulphate; phosphorous

Anhydrous calcium chloride; anhydrous calcium sulphate; metallic sodium; phosphorous pentoxide.

nucleation rates.

(Fabbiani *et al*., 2004).

atmospheric humidity.

Saturated and aromatic

Alkyl halides Aryl halides

hydrocarbons Ethers

explosions.

**Organic Solvent Drying Agent** 

Streams Research Journal, from Shinde *et al*., 2011.)

**2.1.6 Moisture and humidity** 

## **2.1.7 Addition of salts, polymers or antisolvents**

The addition of salts, polymers or antisolvents can be used to create supersaturation of the solution by decreasing the solute solubility in the crystallization medium. Ions, polymeric molecules, or other substances introduced into the crystallization solvent can also act as impurities for growing crystals. These substances may get adsorbed in the crystal lattice of a growing crystal and disturb the regular and repeating arrangements of the crystal, creating defects and leading to polymorphic modifications. Impurities, and surfactants in particular, can also inhibit crystal growth at certain crystal faces, resulting in crystal habit changes (Tiwary, 2006).

When selecting suitable solvents for antisolvent crystallization, one should select a pair that is miscible. The solute must be more soluble in one solvent and less soluble in the other (antisolvent). A water-soluble API will most likely be more soluble in a polar solvent and less soluble in a non-polar solvent (Table 4). The opposite holds true for a poorly watersoluble API. Antisolvent crystallization is performed by dissolving the API in the solvent and then gradually adding an antisolvent. This results in a decrease of solute solubility and an increase in supersaturation – much like cooling crystallization (Nonoyama *et al*., 2006; Stieger *et al*., 2010a; Zhou *et al*., 2006). Reverse addition is a variation of antisolvent recrystallization whereby the API solution is added to the antisolvent. The resulting rapid increase in supersaturation leads to swift nucleation and the precipitation of very fine particles (Tung *et al*., 2009). The composition of the crystallization medium can affect both the crystal form and crystal habit of the product (Stieger *et al*., 2010a).


Table 4. Common solvent-antisolvent pairs. (Adapted, with permission of the author, from Skonieczny, 2009.)

#### **2.2 Novel strategies and new trends in solvent-based crystallization**

One of the biggest challenges for pharmaceutical researchers is finding all the solid-state forms of an API that can exist at ambient conditions. There is no single method for producing all conceivable forms and a particular polymorph may go undetected for many years. Scientists are continually searching for novel ways of uncovering hidden solid-states. A few of these methods are briefly discussed below.

Recrystallization of Active Pharmaceutical Ingredients 195

often affects its bioavailability, and therefore also its efficacy (Lipinsky *et al*., 1997). This necessitates a compromise between stability and solubility: we search for a metastable form with improved solubility that is still relatively stable enough to withstand the rigors of

Polymorphism, occurring in a single-component crystalline molecular solid, may display monotropism or enantiotropism or both (Figure 8). In a monotropic system (A and C, B and C), only one polymorph is stable below the melting point and a phase change from the metastable form (A and B) to the stable form (C) is irreversible. For enantiotropic systems (A and B), a reversible phase transition is observed at a definite transition temperature (Tt,A-B), where the free energy curves intersect before the melting point (Tm). At temperatures and pressures below Tt,A-B form A will be stable (lower free energy and solubility), whilst in the temperature and pressure range between Tt,A-B and Tm,B form B is more stable. Polymorphs in an enantiotropic system are referred to as enantiotropes and those in a monotropic system

Fig. 8. Gibbs free energy curves for a hypothetical single-component system that exhibits crystalline and amorphous phase transitions. Monotropic systems (A and C, B and C), enantiotropic system (A and B) with a transition temperature Tt, and an amorphous and super-cooled liquid with a glass transition temperature Tg. Melting points, Tm, for the crystalline phases are shown at the intersection of curves for the crystalline and liquid states.

(Adapted, with permission of the authors, from Shalaev & Zografi, 2002.)

pharmaceutical processing and will yield a product with acceptable shelf-life.

**3.1.1 Transformation of polymorphs** 

are monotropes (Bernstein 2002; Grant 1999).

## **2.2.1 Laser-induced crystallization**

Non-photochemical laser-induced nucleation (NPLIN) is a crystallization technique that can affect both nucleation rate and the crystal form produced. Laser pulses act predominantly on pre-existing molecular clusters by assisting in the organization of pre-nucleating clusters and embryos into nuclei (Figure 2), leading to dramatically increased nucleation rates for supersaturated solutions. It is believed that the plane-polarized light aligns the prenucleating clusters and thereby reduces the entropic barrier to the free energy of activation for critical nucleus formation (Banga *et al*., 2004; Garetz *et al*., 1996; Rodríguez-Spong *et al*., 2004; Zaccaro *et al*., 2001). This method has not yet been applied to pharmaceuticals, but it should produce similar results as for other organic substances.

## **2.2.2 Capillary crystallization**

In order to access metastable forms of an API, high levels of supersaturation are often required. Capillary tubes as recrystallization vessels are ideal for manipulating the metastable zone width through slow evaporation and because the small volumes of solution isolate heterogeneous nucleants, and reduce turbulence and convection. This technique offers the additional advantage that the crystals need not be removed from the capillary tube prior to characterization by single crystal- or powder X-ray diffraction (Banga *et al*., 2004; Rodríguez-Spong *et al*., 2004).

## **2.2.3 Sonocrystallization**

This technique utilizes ultrasound to increase nucleation rate, but it is also an effective means of crystal size reduction that eliminates many of the disadvantages associated with mechanical size reduction. Sonic waves give rise to a phenomenon called cavitation – the formation of bubbles that decrease in size until a critical size is reached, leading to collapse and the formation of cavities. Cavitation provides energy that accelerates the nucleation process (Banga *et al*., 2004; Kim *et al*, 2003).

## **3. Non-solvent based recrystallization**

Much emphasis has been placed on the importance of solvent based recrystallization in the production of different solid-state forms of APIs, but non-solvent based recrystallization is equally important to industry and researchers alike. Recrystallization through solid-state transitions affects not only the production of APIs, but also the stability of the final product. Additionally, physical vapor deposition (PVD) recrystallization offers yet more opportunities for the preparation of polymorphs.

## **3.1 Recrystallization through solid-state transitions**

Polymorphs may be obtained through solid-state transition by recrystallization of metastable polymorphs and amorphous forms, or through desolvation of solvates (including hydrates). No solid-state transition will take place unless it is thermodynamically favored and solids will always tend to transform to their lowest energy state – the most stable form. Stability is of major concern where APIs are concerned, but it is also true that stable forms are less soluble than their metastable counterparts and solubility of an API often affects its bioavailability, and therefore also its efficacy (Lipinsky *et al*., 1997). This necessitates a compromise between stability and solubility: we search for a metastable form with improved solubility that is still relatively stable enough to withstand the rigors of pharmaceutical processing and will yield a product with acceptable shelf-life.

## **3.1.1 Transformation of polymorphs**

194 Crystallization – Science and Technology

Non-photochemical laser-induced nucleation (NPLIN) is a crystallization technique that can affect both nucleation rate and the crystal form produced. Laser pulses act predominantly on pre-existing molecular clusters by assisting in the organization of pre-nucleating clusters and embryos into nuclei (Figure 2), leading to dramatically increased nucleation rates for supersaturated solutions. It is believed that the plane-polarized light aligns the prenucleating clusters and thereby reduces the entropic barrier to the free energy of activation for critical nucleus formation (Banga *et al*., 2004; Garetz *et al*., 1996; Rodríguez-Spong *et al*., 2004; Zaccaro *et al*., 2001). This method has not yet been applied to pharmaceuticals, but it

In order to access metastable forms of an API, high levels of supersaturation are often required. Capillary tubes as recrystallization vessels are ideal for manipulating the metastable zone width through slow evaporation and because the small volumes of solution isolate heterogeneous nucleants, and reduce turbulence and convection. This technique offers the additional advantage that the crystals need not be removed from the capillary tube prior to characterization by single crystal- or powder X-ray diffraction (Banga *et al*.,

This technique utilizes ultrasound to increase nucleation rate, but it is also an effective means of crystal size reduction that eliminates many of the disadvantages associated with mechanical size reduction. Sonic waves give rise to a phenomenon called cavitation – the formation of bubbles that decrease in size until a critical size is reached, leading to collapse and the formation of cavities. Cavitation provides energy that accelerates the nucleation

Much emphasis has been placed on the importance of solvent based recrystallization in the production of different solid-state forms of APIs, but non-solvent based recrystallization is equally important to industry and researchers alike. Recrystallization through solid-state transitions affects not only the production of APIs, but also the stability of the final product. Additionally, physical vapor deposition (PVD) recrystallization offers yet more

Polymorphs may be obtained through solid-state transition by recrystallization of metastable polymorphs and amorphous forms, or through desolvation of solvates (including hydrates). No solid-state transition will take place unless it is thermodynamically favored and solids will always tend to transform to their lowest energy state – the most stable form. Stability is of major concern where APIs are concerned, but it is also true that stable forms are less soluble than their metastable counterparts and solubility of an API

**2.2.1 Laser-induced crystallization** 

**2.2.2 Capillary crystallization** 

2004; Rodríguez-Spong *et al*., 2004).

process (Banga *et al*., 2004; Kim *et al*, 2003).

**3. Non-solvent based recrystallization** 

opportunities for the preparation of polymorphs.

**3.1 Recrystallization through solid-state transitions** 

**2.2.3 Sonocrystallization** 

should produce similar results as for other organic substances.

Polymorphism, occurring in a single-component crystalline molecular solid, may display monotropism or enantiotropism or both (Figure 8). In a monotropic system (A and C, B and C), only one polymorph is stable below the melting point and a phase change from the metastable form (A and B) to the stable form (C) is irreversible. For enantiotropic systems (A and B), a reversible phase transition is observed at a definite transition temperature (Tt,A-B), where the free energy curves intersect before the melting point (Tm). At temperatures and pressures below Tt,A-B form A will be stable (lower free energy and solubility), whilst in the temperature and pressure range between Tt,A-B and Tm,B form B is more stable. Polymorphs in an enantiotropic system are referred to as enantiotropes and those in a monotropic system are monotropes (Bernstein 2002; Grant 1999).

Fig. 8. Gibbs free energy curves for a hypothetical single-component system that exhibits crystalline and amorphous phase transitions. Monotropic systems (A and C, B and C), enantiotropic system (A and B) with a transition temperature Tt, and an amorphous and super-cooled liquid with a glass transition temperature Tg. Melting points, Tm, for the crystalline phases are shown at the intersection of curves for the crystalline and liquid states. (Adapted, with permission of the authors, from Shalaev & Zografi, 2002.)

Recrystallization of Active Pharmaceutical Ingredients 197

Sometimes, a polymorphic transition may mimic the first of these scenarios. Nevirapine's metastable Form IV is isostructural to a series of solvates prepared from primary alcohols (Stieger *et al*., 2010b) except that its structure contains no solvent molecules. When removed from its recrystallization medium, Form IV will spontaneously and rapidly convert to the stable Form I in exactly the same way as the aforementioned solvates (Stieger *et al*., 2009).

Amorphous pharmaceutical solids may be prepared by common pharmaceutical processes including melt quenching, freeze- and spray-drying, milling, wet granulation and desolvation of solvates (Yu, 2001). Glasses are most often produced from a melt of an API. If crystallization does not occur on cooling the melt below the melting point (Tm) of the crystalline phase, a supercooled liquid is obtained. Further cooling induces a change to a glassy state at the glass transition temperature (Tg), accompanied by a dramatic decrease in molecular mobility and heat capacity. In contrast to melting point, the glass transition temperature of a compound can fluctuate with operating conditions (including the rate of cooling/heating) and as a function of the history of the sample. This indicates that the glass

An amorphous or glass form is often an intermediate in the production of crystalline APIs and such processes can be useful for overcoming specific kinds of activation energies. Thermodynamically, amorphous solids are out-of-equilibrium states that contain an excess of stored energy, with reference to the crystalline phases (Figure 9), making them unstable by definition. The excess energy can be released either completely through crystallization or

Crystallization from amorphous solids, as for any crystallization, involves successive nucleation and growth steps. The crystallization rate may be affected by numerous, possibly interdependent, parameters notably including temperature and plasticizers (of which water is one). When the temperature of a supercooled liquid decreases from the melting point to the glass transition temperature, the nucleation rate increases exponentially whereas molecular mobility required for growth decreases exponentially. A maximum crystallization rate therefore occurs between preferred temperatures of nucleation and growth. Cooling rate also affects nucleation, with slow cooling allowing the maintenance of a steady-state nucleation rate. Rapid cooling, on the other hand, prevents full development of nucleation

Amorphous solids offer certain pharmaceutical advantages over their crystalline counterparts. They are more soluble, have higher dissolution rates, and in some cases they may even have better compression characteristics than corresponding crystals. Therefore, we may not necessarily want amorphous APIs to recrystallize. Some pharmaceuticals have a natural tendency to exist as amorphous solids, while others require deliberate prevention of recrystallization to enter, and remain in, the amorphous state. Contemporary research into

transition is a thermal event affected by kinetic factors (Petit & Coquerel, 2006).

partially by means of irreversible relaxation processes (Petit & Coquerel, 2006).

and thereby facilitates glass formation (Petit & Coquerel, 2006; Yu, 2001).

Desolvation of a solvate generally results in one of the following forms (Byrn *et al*., 1999):

 An unsolvated polymorph with a crystal structure different to that of the solvate. An unsolvated polymorph with a crystal structure that is the same as that of the solvate.

An amorphous material that may or may not recrystallize.

**3.1.3 Crystallization from amorphous phases** 

## **3.1.1.1 Effects of pharmaceutical processing**

Unintentional solid-state conversion of polymorphs sometimes occurs upon exposure to the energetics of pharmaceutical processing and a variety of phase conversions are possible when APIs are exposed to milling, wet granulation, oven drying, and compaction (Brittain & Fiese, 1999):

Grinding or milling is often the last step in the production of bulk APIs and it is performed to reduce particle size and improve particle size homogeneity. Milling can impart a significant amount of energy on a solid and could potentially lead to a full or partial polymorphic conversion or generation of an amorphous substance (or at least a degree of amorphous content). Amorphous forms, being metastable, can in turn reconvert to a crystalline state which may differ from the original product.

Wet granulation, used to improve powder flow and blend homogeneity, is a step that often precedes the production of tablets. During this process, the API is exposed to a solvent (water or an organic solvent with low toxicity, like ethanol) and is once more prone to undergoing solvent-mediated transformations. Exposure to humidity can create similar conditions which could lead to a hydrate being formed.

Drying of APIs is typically achieved with heat and moving air. The possible ramifications of temperature change on polymorph stability have been discussed in the previous section. Drying conditions need to be carefully controlled to avoid possible transformations.

Although not generally a common occurrence, compaction (during tablet manufacturing) can potentially cause metastable polymorphs to convert to the stable form. This can be attributed to the energy applied to bring about compaction. The extent of transformation is dependent on the zone of the tablet, the pressure applied, the compression temperature and the particle size of the API powder.

## **3.1.1.2 High-pressure polymorphic transitions**

Solid-state pressure-induced structural changes in molecular crystals can cause polymorphic transitions (Boldyreva, 2003). However, unlike with high-pressure solvent recrystallization (please refer to section 2.1.5) which is experimentally and mechanistically similar, the conversion tends to be only partial and the data are poor. It is thought that even though the application of high pressure to larger organic molecules may thermodynamically favor the adoption of a new polymorphic form, the solid-state has a substantial kinetic barrier to overcome before the molecules are mobile enough to rearrange (Fabbiani *et al*., 2004).

## **3.1.2 Desolvation of hydrates and solvates**

With the exception of hydrates that are often used in pharmaceutical dosage forms, most solvates are the penultimate solid form in the production of APIs (Byrn *et al*., 1999). Solvates may be desolvated by removing the recrystallization solvent and exposing the crystals to air at ambient temperature, leading to a decrease in vapor pressure of the solvent. If a solvate is stable under these conditions, it may be dried under vacuum or in an oven at mild temperatures.

Unintentional solid-state conversion of polymorphs sometimes occurs upon exposure to the energetics of pharmaceutical processing and a variety of phase conversions are possible when APIs are exposed to milling, wet granulation, oven drying, and compaction (Brittain

Grinding or milling is often the last step in the production of bulk APIs and it is performed to reduce particle size and improve particle size homogeneity. Milling can impart a significant amount of energy on a solid and could potentially lead to a full or partial polymorphic conversion or generation of an amorphous substance (or at least a degree of amorphous content). Amorphous forms, being metastable, can in turn reconvert to a

Wet granulation, used to improve powder flow and blend homogeneity, is a step that often precedes the production of tablets. During this process, the API is exposed to a solvent (water or an organic solvent with low toxicity, like ethanol) and is once more prone to undergoing solvent-mediated transformations. Exposure to humidity can create similar

Drying of APIs is typically achieved with heat and moving air. The possible ramifications of temperature change on polymorph stability have been discussed in the previous section.

Although not generally a common occurrence, compaction (during tablet manufacturing) can potentially cause metastable polymorphs to convert to the stable form. This can be attributed to the energy applied to bring about compaction. The extent of transformation is dependent on the zone of the tablet, the pressure applied, the compression temperature and

Solid-state pressure-induced structural changes in molecular crystals can cause polymorphic transitions (Boldyreva, 2003). However, unlike with high-pressure solvent recrystallization (please refer to section 2.1.5) which is experimentally and mechanistically similar, the conversion tends to be only partial and the data are poor. It is thought that even though the application of high pressure to larger organic molecules may thermodynamically favor the adoption of a new polymorphic form, the solid-state has a substantial kinetic barrier to overcome before the molecules are mobile enough to

With the exception of hydrates that are often used in pharmaceutical dosage forms, most solvates are the penultimate solid form in the production of APIs (Byrn *et al*., 1999). Solvates may be desolvated by removing the recrystallization solvent and exposing the crystals to air at ambient temperature, leading to a decrease in vapor pressure of the solvent. If a solvate is stable under these conditions, it may be dried under vacuum or in an oven at mild

Drying conditions need to be carefully controlled to avoid possible transformations.

**3.1.1.1 Effects of pharmaceutical processing** 

crystalline state which may differ from the original product.

conditions which could lead to a hydrate being formed.

the particle size of the API powder.

rearrange (Fabbiani *et al*., 2004).

temperatures.

**3.1.1.2 High-pressure polymorphic transitions** 

**3.1.2 Desolvation of hydrates and solvates** 

& Fiese, 1999):

Desolvation of a solvate generally results in one of the following forms (Byrn *et al*., 1999):


Sometimes, a polymorphic transition may mimic the first of these scenarios. Nevirapine's metastable Form IV is isostructural to a series of solvates prepared from primary alcohols (Stieger *et al*., 2010b) except that its structure contains no solvent molecules. When removed from its recrystallization medium, Form IV will spontaneously and rapidly convert to the stable Form I in exactly the same way as the aforementioned solvates (Stieger *et al*., 2009).

## **3.1.3 Crystallization from amorphous phases**

Amorphous pharmaceutical solids may be prepared by common pharmaceutical processes including melt quenching, freeze- and spray-drying, milling, wet granulation and desolvation of solvates (Yu, 2001). Glasses are most often produced from a melt of an API. If crystallization does not occur on cooling the melt below the melting point (Tm) of the crystalline phase, a supercooled liquid is obtained. Further cooling induces a change to a glassy state at the glass transition temperature (Tg), accompanied by a dramatic decrease in molecular mobility and heat capacity. In contrast to melting point, the glass transition temperature of a compound can fluctuate with operating conditions (including the rate of cooling/heating) and as a function of the history of the sample. This indicates that the glass transition is a thermal event affected by kinetic factors (Petit & Coquerel, 2006).

An amorphous or glass form is often an intermediate in the production of crystalline APIs and such processes can be useful for overcoming specific kinds of activation energies. Thermodynamically, amorphous solids are out-of-equilibrium states that contain an excess of stored energy, with reference to the crystalline phases (Figure 9), making them unstable by definition. The excess energy can be released either completely through crystallization or partially by means of irreversible relaxation processes (Petit & Coquerel, 2006).

Crystallization from amorphous solids, as for any crystallization, involves successive nucleation and growth steps. The crystallization rate may be affected by numerous, possibly interdependent, parameters notably including temperature and plasticizers (of which water is one). When the temperature of a supercooled liquid decreases from the melting point to the glass transition temperature, the nucleation rate increases exponentially whereas molecular mobility required for growth decreases exponentially. A maximum crystallization rate therefore occurs between preferred temperatures of nucleation and growth. Cooling rate also affects nucleation, with slow cooling allowing the maintenance of a steady-state nucleation rate. Rapid cooling, on the other hand, prevents full development of nucleation and thereby facilitates glass formation (Petit & Coquerel, 2006; Yu, 2001).

Amorphous solids offer certain pharmaceutical advantages over their crystalline counterparts. They are more soluble, have higher dissolution rates, and in some cases they may even have better compression characteristics than corresponding crystals. Therefore, we may not necessarily want amorphous APIs to recrystallize. Some pharmaceuticals have a natural tendency to exist as amorphous solids, while others require deliberate prevention of recrystallization to enter, and remain in, the amorphous state. Contemporary research into

Recrystallization of Active Pharmaceutical Ingredients 199

(Sarma *et al*., 2006); caffeine (Griesser *et al*., 1999; Carlton, 2011); carbamazepine (Griesser *et al*., 1999; Zeitler *et al*., 2007); 1,3-dimethyluracil (Sakiyama & Imamura, 1989); ibuprofen (Perlovich *et al*., 2004;); malonamide (Sakiyama & Imamura, 1989); nevirapine (Figure 10) (Stieger & Liebenberg, 2009); stanozolol (Karpinska *et al*., 2011); and theophylline (Fokkens

Fig. 10. Micro-spherical aggregates of nano-crystalline nevirapine prepared by PVD (A), the

Fig. 11. Pressure-temperature diagram of a one-component crystalline solid for which only one solid phase exists. (Adapted, with permission of Dover Publications, from Ricci, 1966.)

individual crystallites of which could be obtained through light grinding (B).

*et al*., 1983; Griesser *et al*., 1999).

the stabilization of amorphous pharmaceuticals focuses on three key areas: (1) the use of additives for the stabilization of labile substances (e.g. proteins and peptides) during processing and storage; (2) prevention of crystallization of excipients that must remain amorphous to perform their intended functions; and (3) determining the appropriate storage conditions for optimal stability of amorphous materials (Yu, 2001).

Fig. 9. Schematic representation of enthalpy or volume variations as a function of temperature for condensed materials. (Adapted, with permission of John Wiley and Sons, from Petit & Coquerel, 2006.)

#### **3.2 Recrystallization through Physical Vapor Deposition**

Physical vapor deposition (PVD) is an atomistic deposition process in which material is vaporized from a solid or liquid source in the form of atoms or molecules and transported in the form of vapor through a low pressure environment to the substrate where it condenses (Mattox, 1998). During the PVD process, molecules unpack from the original crystal lattice and then recrystallize in the new lattice (Byrn *et al*., 1999).

The research into pharmaceutical applications of PVD has, to date, largely been limited to the production of amorphous phases – particularly ones with enhanced stability. Although we have long known that sublimation may be used for the production of polymorphs, comparatively few papers have been published on recrystallization *via* PVD. This is surprising when one considers that approximately two-thirds of organic compounds sublime (Guillory, 1999) and most commercially available APIs are crystalline.

Studies on the sublimation-based recrystallization of the following APIs and organic compounds have been published: anthranilic acid (Carter & Ward, 1994); 9,10 anthraquinone-2-carboxilic acid (Tsai *et al*., 1993); 1,1-bis(4-hydroxyphenyl)cyclohexane

the stabilization of amorphous pharmaceuticals focuses on three key areas: (1) the use of additives for the stabilization of labile substances (e.g. proteins and peptides) during processing and storage; (2) prevention of crystallization of excipients that must remain amorphous to perform their intended functions; and (3) determining the appropriate storage

conditions for optimal stability of amorphous materials (Yu, 2001).

Fig. 9. Schematic representation of enthalpy or volume variations as a function of

**3.2 Recrystallization through Physical Vapor Deposition** 

and then recrystallize in the new lattice (Byrn *et al*., 1999).

from Petit & Coquerel, 2006.)

temperature for condensed materials. (Adapted, with permission of John Wiley and Sons,

Physical vapor deposition (PVD) is an atomistic deposition process in which material is vaporized from a solid or liquid source in the form of atoms or molecules and transported in the form of vapor through a low pressure environment to the substrate where it condenses (Mattox, 1998). During the PVD process, molecules unpack from the original crystal lattice

The research into pharmaceutical applications of PVD has, to date, largely been limited to the production of amorphous phases – particularly ones with enhanced stability. Although we have long known that sublimation may be used for the production of polymorphs, comparatively few papers have been published on recrystallization *via* PVD. This is surprising when one considers that approximately two-thirds of organic compounds

Studies on the sublimation-based recrystallization of the following APIs and organic compounds have been published: anthranilic acid (Carter & Ward, 1994); 9,10 anthraquinone-2-carboxilic acid (Tsai *et al*., 1993); 1,1-bis(4-hydroxyphenyl)cyclohexane

sublime (Guillory, 1999) and most commercially available APIs are crystalline.

(Sarma *et al*., 2006); caffeine (Griesser *et al*., 1999; Carlton, 2011); carbamazepine (Griesser *et al*., 1999; Zeitler *et al*., 2007); 1,3-dimethyluracil (Sakiyama & Imamura, 1989); ibuprofen (Perlovich *et al*., 2004;); malonamide (Sakiyama & Imamura, 1989); nevirapine (Figure 10) (Stieger & Liebenberg, 2009); stanozolol (Karpinska *et al*., 2011); and theophylline (Fokkens *et al*., 1983; Griesser *et al*., 1999).

Fig. 10. Micro-spherical aggregates of nano-crystalline nevirapine prepared by PVD (A), the individual crystallites of which could be obtained through light grinding (B).

Fig. 11. Pressure-temperature diagram of a one-component crystalline solid for which only one solid phase exists. (Adapted, with permission of Dover Publications, from Ricci, 1966.)

Recrystallization of Active Pharmaceutical Ingredients 201

It has, incidentally, been found that recrystallization from the vapor phase can also be affected or controlled by the nature of the substrate (Carter & Ward, 1994) as has been

Polymorphs of active pharmaceutical ingredients can be obtained though recrystallization processes based on solvents, solid-state transitions and vapor deposition. When the possible number of experimental variations on each of these is considered, it is easy to become overwhelmed by the task of comprehensive polymorph screening. Indeed, it is no wonder that some polymorphs go undetected for decades or longer. Various semi-automated and high-throughput methods have been developed to assist in this daunting task. The efforts of researchers laboring towards automation are, however "subverted" by colleagues who continually find novel ways of accessing hidden polymorphs! There is, to date, no single

We thank North-West University (Potchefstroom Campus) and the National Research

Banga, S., Chawla, G. & Bansal, A.K. 2004. New Trends in the Crystallisation of Active Pharmaceutical Ingredients. *Business Briefing: Pharmagenerics*, p. 1-5. Beckman, W. 2000. Seeding the Desired Polymorph: Background, Possibilities, Limitations, and Case Studies. *Organic Process Research & Development*, 4:372-383. Bernstein, J. 2002. *Polymorphism in Molecular Crystals*. Oxford:Clarendon Press. 410 p.

Blagden, N., De Matas, M., Gavan, P.T. & York, P. 2007. Crystal Engineering of Active

Boldyreva, E.V. 2003. High-Pressure-Induced Structural Changes in Molecular Crystals

Boistelle, R. & Astier, J.P. 1988. Crystallization Mechanisms in Solution. *Journal of Crystal* 

Braatz, R.D. 2002. Advanced Control of Crystallization Processes. *Annual Reviews in Control*,

Brittain, H.G. & Fiese, E.F. 1999. *Effects of Pharmaceutical Processing on Drug Polymorphs and* 

Byrn, S.R., Pfeiffer, R.R. & Stowell, J.G. 1999. *Solid-State Chemistry of Drugs*. 2nd edition. West

Pharmaceutical Ingredients to Improve Solubility and Dissolution Rates. *Advanced* 

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shown for solvent recrystallization (Rodríguez-Spong *et al*., 2004).

method that can identify all possible solid forms of an API.

Foundation of South Africa for providing research support.

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**4. Conclusion** 

**5. Acknowledgement** 

**6. References** 

Figure 11 demonstrates how transformations mediated by the vapor phase are highly dependent on vapor pressure and therefore also on temperature (Byrn *et al*., 1999). Vapor of an API can be generated by its liquid phase once melted (evaporation) or by its solid phase (sublimation). The vast majority of pharmaceutical actives are organic compounds and, as such, they are sensitive to heat degradation. It is well-known that many APIs degrade when melted. It follows that pharmaceutical PVD processes should ideally operate at low pressure and temperature to obtain vapor through sublimation, using compounds with sufficient thermal stability.

Figure 12 illustrates how it is possible to obtain different enantiotropic polymorphs through vapor deposition. At a pressure sufficiently low for sublimation to take place the solid API becomes a vapor, at which point it is no longer Solid A or B because polymorphism is a solid-state phenomenon. The temperature at which the vapor phase recrystallizes determines which polymorph is obtained (provided the pressure is sufficiently low). It can generally be assumed that unstable/metastable polymorphs (in this example, Solid A) will form preferentially at lower temperatures and stable polymorphs (Solid B) can be expected at higher temperatures (Guillory, 1999). This is why the temperature, and distance from the solid, of the collection surface is so important in PVD systems.

Fig. 12. Pressure-temperature diagram of a one-component crystalline solid with enantiotropic behavior. (Adapted, with permission of John Wiley and Sons, from Lohani & Grant, 2006.)

It has, incidentally, been found that recrystallization from the vapor phase can also be affected or controlled by the nature of the substrate (Carter & Ward, 1994) as has been shown for solvent recrystallization (Rodríguez-Spong *et al*., 2004).

## **4. Conclusion**

200 Crystallization – Science and Technology

Figure 11 demonstrates how transformations mediated by the vapor phase are highly dependent on vapor pressure and therefore also on temperature (Byrn *et al*., 1999). Vapor of an API can be generated by its liquid phase once melted (evaporation) or by its solid phase (sublimation). The vast majority of pharmaceutical actives are organic compounds and, as such, they are sensitive to heat degradation. It is well-known that many APIs degrade when melted. It follows that pharmaceutical PVD processes should ideally operate at low pressure and temperature to obtain vapor through sublimation, using compounds with sufficient

Figure 12 illustrates how it is possible to obtain different enantiotropic polymorphs through vapor deposition. At a pressure sufficiently low for sublimation to take place the solid API becomes a vapor, at which point it is no longer Solid A or B because polymorphism is a solid-state phenomenon. The temperature at which the vapor phase recrystallizes determines which polymorph is obtained (provided the pressure is sufficiently low). It can generally be assumed that unstable/metastable polymorphs (in this example, Solid A) will form preferentially at lower temperatures and stable polymorphs (Solid B) can be expected at higher temperatures (Guillory, 1999). This is why the temperature, and distance from the

solid, of the collection surface is so important in PVD systems.

Fig. 12. Pressure-temperature diagram of a one-component crystalline solid with

enantiotropic behavior. (Adapted, with permission of John Wiley and Sons, from Lohani &

thermal stability.

Grant, 2006.)

Polymorphs of active pharmaceutical ingredients can be obtained though recrystallization processes based on solvents, solid-state transitions and vapor deposition. When the possible number of experimental variations on each of these is considered, it is easy to become overwhelmed by the task of comprehensive polymorph screening. Indeed, it is no wonder that some polymorphs go undetected for decades or longer. Various semi-automated and high-throughput methods have been developed to assist in this daunting task. The efforts of researchers laboring towards automation are, however "subverted" by colleagues who continually find novel ways of accessing hidden polymorphs! There is, to date, no single method that can identify all possible solid forms of an API.

## **5. Acknowledgement**

We thank North-West University (Potchefstroom Campus) and the National Research Foundation of South Africa for providing research support.

## **6. References**


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**Section 2** 

**Mineral Formation** 

**Applications, Techniques and** 

