**2. Crystallization**

Crystallization of a liquid or an amorphous solid is a complex process involving simultaneous nucleation and growth of crystallites (Yinnon & Uhlmann, 1983). Crystallization is initiated by crystal nucleation. Nucleation may occur spontaneously or it may be induced artificially. It is not always possible, however, to decide whether a system has nucleated of its own accord or whether it has done so under the influence of some external stimulus (Mullin, 2001). The nucleation either occurs without the involvement of a foreign substance in the interior of the parent phase which is called "homogeneous or primary nucleation" or with the contact of the parent phase with a foreign substance that acts as a preferred nucleation site which is called "heterogeneous or secondary nucleation". The nucleation process is followed by the growth of the crystal nuclei to macroscopic dimensions, which is called "crystal growth" (Kalb, 2009).

#### **2.1 Homogeneous nucleation**

Homogeneous nucleation occurs in the interior of the parent phase without the involvement of a foreign substance. At temperatures below a material's melting point (*T*m), the driving force for solidification is the difference in Gibbs free energy (∆*G*) between the liquid and the solid. If we assume that the heat capacities of the liquid and solid are equal, then the molar enthalpy and molar entropy of solidification will each remain constant as a function of temperature, and ∆*G* can be calculated as follows:

$$\text{liquid} \rightarrow \text{solid}$$

$$\Delta G = \Delta H \cdot T\Delta S \tag{1}$$

Note that ∆*H* = -*L*, where *L* is the latent heat of fusion.

$$
\Delta G = \text{-}L + T \left. \frac{L}{T\_m} \right. \tag{2}
$$

$$
\Delta G = \frac{L}{T\_{\text{m}}} \text{ (T - T\_{\text{m}})} \tag{3}
$$

When a spherical particle of solid of radius *r* is formed, the change in Gibbs free energy is the volume of the particle multiplied by the volumetric Gibbs free energy change, ∆*G*v.

$$
\Delta G\_{\rm vol} = \frac{4}{3} \text{Tr} r^3 \Delta G\_{\rm V} \tag{4}
$$

where ∆*G*v is the Gibbs free energy per unit volume,

$$
\Delta G\_{\rm V} = \frac{1}{V} \frac{L}{T\_{\rm m}} (T - T\_{\rm m}) \tag{5}
$$

$$
\Delta \mathbf{G}\_{\text{Vol}} = \frac{4}{3} \text{mr}^3 \frac{1}{V} \frac{\text{L}}{\text{T}\_{\text{in}}} \left( \text{T} - \text{T}\_{\text{m}} \right) \tag{6}
$$

therefore this chapter "Crystallization Kinetics of Amorphous Materials" is crucial to have a

Crystallization of a liquid or an amorphous solid is a complex process involving simultaneous nucleation and growth of crystallites (Yinnon & Uhlmann, 1983). Crystallization is initiated by crystal nucleation. Nucleation may occur spontaneously or it may be induced artificially. It is not always possible, however, to decide whether a system has nucleated of its own accord or whether it has done so under the influence of some external stimulus (Mullin, 2001). The nucleation either occurs without the involvement of a foreign substance in the interior of the parent phase which is called "homogeneous or primary nucleation" or with the contact of the parent phase with a foreign substance that acts as a preferred nucleation site which is called "heterogeneous or secondary nucleation". The nucleation process is followed by the growth of the crystal nuclei to macroscopic

Homogeneous nucleation occurs in the interior of the parent phase without the involvement of a foreign substance. At temperatures below a material's melting point (*T*m), the driving force for solidification is the difference in Gibbs free energy (∆*G*) between the liquid and the solid. If we assume that the heat capacities of the liquid and solid are equal, then the molar enthalpy and molar entropy of solidification will each remain constant as a function of

liquid → solid

∆*G* = ∆*H* - *T*∆*S* (1)

*T*<sup>m</sup>

*V L T*<sup>m</sup>

<sup>3</sup> <sup>π</sup>r3 <sup>1</sup> V L Tm

When a spherical particle of solid of radius *r* is formed, the change in Gibbs free energy is the volume of the particle multiplied by the volumetric Gibbs free energy change, ∆*G*v.

*T*<sup>m</sup>

(2)

(*T* - *T*m) (3)

<sup>3</sup> <sup>π</sup>*r*3∆*G*v (4)

(*T* - *T*m) (5)

(T - Tm) (6)

complete understanding of the crystallization phenomenon.

dimensions, which is called "crystal growth" (Kalb, 2009).

temperature, and ∆*G* can be calculated as follows:

Note that ∆*H* = -*L*, where *L* is the latent heat of fusion.

∆*G* = -*L* +*T <sup>L</sup>*

∆*G* = *<sup>L</sup>*

<sup>∆</sup>*G*vol <sup>=</sup><sup>4</sup>

<sup>∆</sup>*G*<sup>v</sup> <sup>=</sup><sup>1</sup>

<sup>∆</sup>Gvol <sup>=</sup><sup>4</sup>

where ∆*G*v is the Gibbs free energy per unit volume,

**2. Crystallization** 

**2.1 Homogeneous nucleation** 

But when the particle of radius *r* is formed, there is another energy term to be considered, the surface energy. The surface energy of the particle is:

$$
\Delta G\_s = 4\text{nr}r^2\text{yr}\tag{7}
$$

where γ = γs-l the surface energy between solid and liquid.

The sum of the two energy term is:

$$
\Delta \mathbf{G\_t} = 4\pi n^2 \mathbf{\dot{y}} + \frac{4}{3} \text{nr}^3 \Delta \mathbf{G\_V} \tag{8}
$$

The first of these terms involves the increase in energy required to form a new surface. The second term is negative and represents the decrease in Gibbs free energy upon solidification. Because the first is a function of the second power of the radius, and the second a function of the third power of the radius, the sum of the two increases, goes through a maximum, and then decreases (Fig. 4).

Fig. 4. The free energy change associated with homogeneous nucleation of a sphere of radius *r*  (Ragone, 1994)

The radius at which the Gibbs free energy curve is at maximum is called the critical radius *r*\*, for a nucleus of solid in liquid. The driving force of the Gibbs free energy will tend to cause a particle with a smaller radius than *r*\* to decrease in size. This is a particle of subcritical size for nucleation. A viable nucleus is one with radius greater than or equal to *r*\*. The critical Gibbs free energy corresponding to the radius *r*\* is ∆*G*\*. These terms can be shown to be:

Crystallization Kinetics of Amorphous Materials 133

form of a spherical cap (Fig. 5), the surface energy term involves the surface energy of the

The terms involving the interactions between the catalyst surface and the liquid and solid can be expressed in terms of the solid-liquid interfacial energy by noting the relationships

γc-l = γc-s + γs-l (cosθ)

The volumetric Gibbs free energy change is the product of the volume of the cap and Δ*G*v, the specific Gibbs free energy change. That volume, in terms of its radius of the curvature

(2+cosθ�(1-cosθ)

<sup>3</sup> <sup>π</sup>r3∆*G*v*f*(θ�

2 <sup>4</sup> �

2

<sup>3</sup> <sup>π</sup>*r*<sup>3</sup> �

<sup>∆</sup>*G*vol = <sup>4</sup>

*r*\* = - 2γs-l ∆*G*<sup>v</sup> 

\* = <sup>16</sup> 3 πγs-l 3 ∆*G*<sup>v</sup> <sup>2</sup> *f*(θ)

to the volume of the sphere with the same radius of curvature (Ragone, 1994).

It is particularly important to note that the critical radius of curvature, *r*\*, does not change when the nucleation becomes heterogeneous. The critical Gibbs free energy, Δ*G*\*, however, is strongly influenced by the wetting that occurs at the surface of the material that catalyzes the nucleation. A lower values of Δ*G*\* means a lower activation energy to be overcome in nucleation; that is, nucleation takes place more easily. The magnitude of the effect can be appreciated by considering values of *f*(θ). *f*(θ) is the ratio of the volume of the heterogeneous nucleus (the cap)

<sup>+</sup>π*r*2 (1-cos2θ)(γc-s- γc-l) (12)

(13)

<sup>4</sup> � (16)

(14)

(15)

(17)

(18)

catalyst surface as it is coated by the nucleus. The surface energy term is derived as follows:

<sup>∆</sup>*G*surface = 2π*r*<sup>2</sup> �1-cosθ�γs-l

γc-s= solid-catalyst interfacial energy γc-l= liquid-catalyst interfacial energy

where γs-l= solid-liquid interfacial energy

V = <sup>4</sup>

∆*G*

�(�� � �(2+cosθ�(1-cosθ)

where *r* is the radius of the curvature of the nucleus. Then we write

solid-liquid surface = 2π*r*2(1 – cosθ) catalyst-solid surface = π*r*2(1 – cos2θ)

among them.

or

where

and contact angle is:

$$r^\* \text{ when } \left(\frac{\partial \Lambda G\_r}{\partial r}\right)\_T = 0 = 8\text{nr}\,\text{y} + 4\text{nr}r^2\Delta G\_\text{v} \tag{9}$$

$$r^\* = \text{-}\frac{2\eta}{\Delta G\_V} \tag{10}$$

$$
\Delta G^{\ast} = \frac{16}{3} \left. \frac{\text{m} \text{y}^{3}}{\Delta G\_{\text{v}}^{2}} \right. \tag{11}
$$

In practice homogeneous nucleation is rarely encountered in solidification. Instead heterogeneous nucleation occurs at crevices in mould walls, or at impurity particles suspended in the liquid.

#### **2.2 Heterogeneous nucleation**

Usually, foreign phases like container walls and impurities aid in the nucleation process and thereby increase the nucleation rate. In this case, nucleation is called heterogeneous (Kalb, 2009).

Under practical solidification conditions, supercooling of only a few degrees is observed because nuclei of the solid can be formed on surfaces that catalyze solidification, such as inclusions in the material being solidified, the walls of the container in which it is being held, or the surfaces of the casting molds. To catalyze solidification, the nucleus of solid must wet the catalyst to some extent. A nucleus catalyzed on a surface is shown schematically in Fig. 5.

Fig. 5. Catalysis of a nucleus on a surface (Porter & Easterling, 1992)

The analysis of the energies involved in heterogeneous nucleation follows the same method as the one used for homogeneous nucleation. In the case of a heterogeneous nucleus in the form of a spherical cap (Fig. 5), the surface energy term involves the surface energy of the catalyst surface as it is coated by the nucleus.

The surface energy term is derived as follows:

solid-liquid surface = 2π*r*2(1 – cosθ)

catalyst-solid surface = π*r*2(1 – cos2θ)

where *r* is the radius of the curvature of the nucleus. Then we write

$$\Delta G\_{\text{surface}} = 2 \text{m}r^2 \begin{Bmatrix} 1 \text{-} \cos \theta \end{Bmatrix} \chi\_{\text{s-l}} + \text{m}r^2 \begin{Bmatrix} 1 \text{-} \cos^2 \theta \end{Bmatrix} (\chi\_{\text{c-s}} \cdot \chi\_{\text{c-l}}) \tag{12}$$

where γs-l= solid-liquid interfacial energy

γc-s= solid-catalyst interfacial energy

γc-l= liquid-catalyst interfacial energy

The terms involving the interactions between the catalyst surface and the liquid and solid can be expressed in terms of the solid-liquid interfacial energy by noting the relationships among them.

$$\chi\_{\rm c-l} = \chi\_{\rm c-s} + \chi\_{\rm s-l} \text{ (cosß)} \tag{13}$$

The volumetric Gibbs free energy change is the product of the volume of the cap and Δ*G*v, the specific Gibbs free energy change. That volume, in terms of its radius of the curvature and contact angle is:

$$\mathbf{V} = \frac{4}{3} \text{rn}^3 \left\{ \frac{(2 \star \cos \theta)(1 \star \cos \theta)^2}{4} \right\} \tag{14}$$

or

132 Advances in Crystallization Processes

∆*G*<sup>v</sup>

In practice homogeneous nucleation is rarely encountered in solidification. Instead heterogeneous nucleation occurs at crevices in mould walls, or at impurity particles

Usually, foreign phases like container walls and impurities aid in the nucleation process and thereby increase the nucleation rate. In this case, nucleation is called heterogeneous (Kalb,

Under practical solidification conditions, supercooling of only a few degrees is observed because nuclei of the solid can be formed on surfaces that catalyze solidification, such as inclusions in the material being solidified, the walls of the container in which it is being held, or the surfaces of the casting molds. To catalyze solidification, the nucleus of solid must wet the catalyst to some extent. A nucleus catalyzed on a surface is shown

= 0 = 8π*r*γ + 4π*r*2∆*G*v (9)

(10)

2 (11)

 *r*\* when �

Fig. 5. Catalysis of a nucleus on a surface (Porter & Easterling, 1992)

The analysis of the energies involved in heterogeneous nucleation follows the same method as the one used for homogeneous nucleation. In the case of a heterogeneous nucleus in the

 *r*\* = - 2<sup>γ</sup>

suspended in the liquid.

schematically in Fig. 5.

2009).

**2.2 Heterogeneous nucleation** 

∂∆*G*<sup>r</sup> <sup>∂</sup>*<sup>r</sup>* � *T*

> ∆*G*\* = <sup>16</sup> 3 πγ<sup>3</sup> ∆*G*<sup>v</sup>

$$
\Delta \mathbf{G}\_{\rm vol} = \frac{4}{3} \text{mr}^3 \Delta \mathbf{G}\_{\rm v} f(\boldsymbol{\theta}) \tag{15}
$$

where

$$f(\theta) = \begin{cases} \frac{(2+\cos\theta)(1-\cos\theta)^2}{4} \\ \end{cases} \tag{16}$$

$$r \, r^\* = -\frac{2\chi\_{s1}}{\Delta G\_{\rm v}} \tag{17}$$

$$
\Delta G^{\ast} = \frac{16}{3} \left. \frac{m \chi\_{\ast 1}^{\ast}}{\Delta G\_{\text{v}}^{2}} f(\emptyset) \right. \tag{18}
$$

 It is particularly important to note that the critical radius of curvature, *r*\*, does not change when the nucleation becomes heterogeneous. The critical Gibbs free energy, Δ*G*\*, however, is strongly influenced by the wetting that occurs at the surface of the material that catalyzes the nucleation. A lower values of Δ*G*\* means a lower activation energy to be overcome in nucleation; that is, nucleation takes place more easily. The magnitude of the effect can be appreciated by considering values of *f*(θ). *f*(θ) is the ratio of the volume of the heterogeneous nucleus (the cap) to the volume of the sphere with the same radius of curvature (Ragone, 1994).

Crystallization Kinetics of Amorphous Materials 135

It is apparent that any factors that reduce the volume of the nucleus reduce the critical Gibbs free energy of formation of that nucleus, making nucleation more probable (Ragone, 1994). The activation energy barrier against heterogeneous nucleation (Δ*G*\*het) is smaller than Δ*G*\*hom by the shape factor *f*(θ). In addition, the critical nucleus radius (*r*\*) is unaffected by the mould wall and only depends on the undercooling. This result was to be expected since equilibrium across the curved interface is unaffected by the presence of the mould wall.

The effect of undercooling on Δ*G*\*het and Δ*G*\*hom is shown schematically in Fig. 7a. If there

∆*G*het \*

*kT* <sup>൰</sup> (26)

het becomes

het should not be very different from the critical

=*n* 1exp ൬-

Fig. 7. (a) Variation of Δ*G*\* with undercooling (Δ*T*) for homogeneous and heterogeneous nucleation, (b) The corresponding nucleation rates assuming the same critical value of Δ*G*\*

value for homogeneous nucleation. It will mainly depend on the magnitude of *n*1, in Equation 26. It can be seen from Fig. 7b that heterogeneous nucleation will be possible at

Therefore, heterogeneous nucleation should become feasible when Δ*G*\*

much lower undercoolings than are necessary for homogeneous nucleation.

are *n* atoms in contact with the mould wall the number of nuclei should be given by:

where *V\** is the volume of the critical nucleus.

*n*\*

(Porter & Easterling, 1992)

sufficiently small. The critical value for Δ*G*\*

Fig. 6, a graph of Δ*G* as a function of radius of curvature of the nucleus, shows the effect of wetting on the critical Gibbs free energy to be overcome for the nucleus to form.

Fig. 6. Plot of Δ*G* versus *r* for homogeneous nucleation and an example of heterogeneous nucleation (Ragone, 1994)

The critical Gibbs free energy for nucleation depends on the nucleus volume. This can be demonstrated by considering a nucleus having the shape of spherical cap with radius of curvature *r*. The Gibbs free energy of the nucleus depends on the interfacial energy and the volumetric Gibbs free energy change as follows:

$$
\Delta G\_{\rm r} = \alpha r^2 \chi + \beta r^3 \Delta G\_{\rm V} \tag{19}
$$

The parameters α and β are determined by the particular geometry of the nucleus. The surface energy term, γ, is an average surface energy for the nucleus determined according to the geometrical factors.

The volume of the nucleus is β*r*3. To determine *r*\*,

$$\left(\frac{\partial \Lambda G\_{\rm r}}{\partial \mathbf{r}}\right)\_{\rm T} = 0\tag{20}$$

$$2\alpha \mathbf{y}\mathbf{y}^\* + 3\beta \mathbf{y}^{\*2} \Delta \mathbf{C}\_\mathbf{V} = \mathbf{0} \tag{21}$$

$$r^\* = \frac{2a}{3\beta\Lambda G\_V} \text{ y or } \mathbf{u} = \mathbf{\bar{\phantom{x}}} \frac{3\beta\Lambda G\_V}{2\gamma} r^\* \tag{22}$$

Substituting in Equation 19, we have

$$
\Delta G\_r^\* = -\frac{3\hbar \Lambda G\_r r^\*}{2} \left(r^\*\right)^2 + \Delta G\_V \beta \left(r^\*\right)^3 \tag{23}
$$

$$
\Delta G\_{\rm r}^{\*} = -\frac{1}{2} \beta (r^{\*})^{3} \Delta G\_{\rm V} \tag{24}
$$

$$
\Delta G\_{\rm f}^{\*} = \frac{1}{2} \dot{V}^{\*} \Delta G\_{\rm V} \tag{25}
$$

Fig. 6, a graph of Δ*G* as a function of radius of curvature of the nucleus, shows the effect of

Fig. 6. Plot of Δ*G* versus *r* for homogeneous nucleation and an example of heterogeneous

The critical Gibbs free energy for nucleation depends on the nucleus volume. This can be demonstrated by considering a nucleus having the shape of spherical cap with radius of curvature *r*. The Gibbs free energy of the nucleus depends on the interfacial energy and the

 ∆*G*r=α*r*2γ+ β*r*3∆*G*v (19) The parameters α and β are determined by the particular geometry of the nucleus. The surface energy term, γ, is an average surface energy for the nucleus determined according to

> ∂∆Gr <sup>∂</sup><sup>r</sup> � T

> > γ or α = -

3β∆*G*<sup>V</sup> 2γ *r*\*

+ ∆*G*Vβ(*r*\*

) 3

3β∆*G*<sup>V</sup>

3β∆*G*v*r*\* 2 (*r*\* ) 2

> \* = - 1 <sup>2</sup> <sup>β</sup>(*r*\* ) 3

> > \* = - <sup>1</sup> 2 *V*\*

\* = -

= 0 (20)

(22)

(23)

∆*G*V (24)

∆*G*V (25)

+ 3β*r*\*2∆*G*<sup>V</sup> = 0 (21)

nucleation (Ragone, 1994)

the geometrical factors.

volumetric Gibbs free energy change as follows:

The volume of the nucleus is β*r*3. To determine *r*\*,

�

2αγ*r*\*

*r*\* = - <sup>2</sup><sup>α</sup>

∆*G*<sup>r</sup>

∆*G*<sup>r</sup>

Substituting in Equation 19, we have

∆*G*<sup>r</sup>

wetting on the critical Gibbs free energy to be overcome for the nucleus to form.

where *V\** is the volume of the critical nucleus.

It is apparent that any factors that reduce the volume of the nucleus reduce the critical Gibbs free energy of formation of that nucleus, making nucleation more probable (Ragone, 1994).

The activation energy barrier against heterogeneous nucleation (Δ*G*\*het) is smaller than Δ*G*\*hom by the shape factor *f*(θ). In addition, the critical nucleus radius (*r*\*) is unaffected by the mould wall and only depends on the undercooling. This result was to be expected since equilibrium across the curved interface is unaffected by the presence of the mould wall.

The effect of undercooling on Δ*G*\*het and Δ*G*\*hom is shown schematically in Fig. 7a. If there are *n* atoms in contact with the mould wall the number of nuclei should be given by:

\*

Fig. 7. (a) Variation of Δ*G*\* with undercooling (Δ*T*) for homogeneous and heterogeneous nucleation, (b) The corresponding nucleation rates assuming the same critical value of Δ*G*\* (Porter & Easterling, 1992)

Therefore, heterogeneous nucleation should become feasible when Δ*G*\* het becomes sufficiently small. The critical value for Δ*G*\* het should not be very different from the critical value for homogeneous nucleation. It will mainly depend on the magnitude of *n*1, in Equation 26. It can be seen from Fig. 7b that heterogeneous nucleation will be possible at much lower undercoolings than are necessary for homogeneous nucleation.

Crystallization Kinetics of Amorphous Materials 137

Supersaturation is the difference between the actual concentration and the solubility concentration at a given temperature which is the driving force for all solution crystallization processes. The figure below (Fig. 9) illustrates the concept of supersaturation and introduces the metastable zone width (MSZW), the kinetic boundary at which

Fig. 9. The excess free energy of solid clusters for homogeneous and heterogeneous

Supersaturation is critical because it is the driving force for crystal nucleation and growth. The relationship between supersaturation and nucleation and growth is defined by the

where, *G* is growth rate, *k*g is growth constant, *g* is growth order, *B* is nucleation rate, *k*b is

At low supersaturation, crystals can grow faster than they nucleate resulting in a larger crystal size distribution. However, at higher supersaturation, crystal nucleation dominates crystal growth, ultimately resulting in smaller crystals. This diagram, relating supersaturation to nucleation, growth and crystal size clearly illustrates how controlling supersaturation is vitally important when it comes to creating crystals of the desired size

The rate of crystal nucleation in glasses reaches its maximum at a temperature somewhat higher than the glass transition temperature and then decreases rapidly with increasing temperature, while the rate of crystal growth reaches its maximum at a temperature much

(30)

(31)

**2.4 Supersaturation** 

following equations.

crystallization occurs (Porter & Easterling, 1992).

nucleation. Note *r*\* is independent of nucleation site

*G* = *k*g∆*C*<sup>g</sup>

*B* = *k*b∆*C*<sup>b</sup>

and specification (see Fig. 10) (Porter & Easterling, 1992).

nucleation constant, b is nucleation order, Δ*C* is supersaturation.

#### **2.3 Growth**

Nucleation is the birth of new crystal nuclei either spontaneously from solution or in the presence of existing crystals. Crystal growth is the increase in size of crystals as solute is deposited from solution. These often competing mechanisms ultimately determine the final crystal size distribution. The rate of growth of a transformation product is determined by the driving force for the transformation and the frequency with which molecules successfully make the transition from the reactant phase to the product phase. To use solidification as an example, the driving force is the negative of the Δ*G* of solidification:

$$-\Delta G \equiv \frac{L}{T\_{\text{m}}} \left(T\_{\text{m}} \text{-- } T\right) \tag{27}$$

The jump frequency across the liquid-solid interface has a temperature dependence of the form:

$$f = f\_0 \exp\left(-\frac{\Delta G\_M}{kT}\right) \tag{28}$$

where Δ*GM* is the activation energy for movement across the liquid-solid interface. The product of the two is:

$$\text{solidification rate} = f\_0 \exp\left(-\frac{\Lambda G\_\text{M}}{kT}\right) \left(\frac{L}{T\_\text{m}}\right) \left(T\_\text{m} - T\right) \tag{29}$$

As the temperature decreases, the driving force increases but the jump frequency decreases. These two opposing dependencies can produce a maximum in the rate of growth as a function of temperature, as illustrated in Fig. 8.

Fig. 8. Solidification rate as a function of temperature (Ragone, 1994)

The temperature dependencies of both nucleation of a new phase and its rate of growth result in a strong temperature dependence of transformation rate.

#### **2.4 Supersaturation**

136 Advances in Crystallization Processes

Nucleation is the birth of new crystal nuclei either spontaneously from solution or in the presence of existing crystals. Crystal growth is the increase in size of crystals as solute is deposited from solution. These often competing mechanisms ultimately determine the final crystal size distribution. The rate of growth of a transformation product is determined by the driving force for the transformation and the frequency with which molecules successfully make the transition from the reactant phase to the product phase. To use solidification as an example, the driving force is the negative of the Δ*G* of solidification:

*T*<sup>m</sup>

0

solidification rate = *f*

Fig. 8. Solidification rate as a function of temperature (Ragone, 1994)

result in a strong temperature dependence of transformation rate.

The temperature dependencies of both nucleation of a new phase and its rate of growth

The jump frequency across the liquid-solid interface has a temperature dependence of the

exp �- ∆*G*<sup>M</sup>

exp �- ∆*G*<sup>M</sup>

*kT* � � *<sup>L</sup> T*<sup>m</sup>

where Δ*GM* is the activation energy for movement across the liquid-solid interface. The

0

As the temperature decreases, the driving force increases but the jump frequency decreases. These two opposing dependencies can produce a maximum in the rate of growth as a

(*T*m- *T*) (27)

*kT* � (28)

� (*T*<sup>m</sup> – *T*) (29)

**2.3 Growth** 

form:

product of the two is:


*f* = *f*

function of temperature, as illustrated in Fig. 8.

Supersaturation is the difference between the actual concentration and the solubility concentration at a given temperature which is the driving force for all solution crystallization processes. The figure below (Fig. 9) illustrates the concept of supersaturation and introduces the metastable zone width (MSZW), the kinetic boundary at which crystallization occurs (Porter & Easterling, 1992).

Fig. 9. The excess free energy of solid clusters for homogeneous and heterogeneous nucleation. Note *r*\* is independent of nucleation site

Supersaturation is critical because it is the driving force for crystal nucleation and growth. The relationship between supersaturation and nucleation and growth is defined by the following equations.

$$G = k\_{\mathcal{K}} \Delta C^{\mathcal{G}} \tag{30}$$

$$B = k\_b \Delta \mathbf{C}^b \tag{31}$$

where, *G* is growth rate, *k*g is growth constant, *g* is growth order, *B* is nucleation rate, *k*b is nucleation constant, b is nucleation order, Δ*C* is supersaturation.

At low supersaturation, crystals can grow faster than they nucleate resulting in a larger crystal size distribution. However, at higher supersaturation, crystal nucleation dominates crystal growth, ultimately resulting in smaller crystals. This diagram, relating supersaturation to nucleation, growth and crystal size clearly illustrates how controlling supersaturation is vitally important when it comes to creating crystals of the desired size and specification (see Fig. 10) (Porter & Easterling, 1992).

The rate of crystal nucleation in glasses reaches its maximum at a temperature somewhat higher than the glass transition temperature and then decreases rapidly with increasing temperature, while the rate of crystal growth reaches its maximum at a temperature much

Crystallization Kinetics of Amorphous Materials 139

a non-equilibrium structure, the driving force for which is brought about by cooling from one temperature to another. Consider the initial phase to be α and the resulting phase to be

α → β

Assume that the transformation from α to β is controlled by nucleation and growth, that is,

If we consider *N* is the nucleation rate per unit volume and *G* is the growth rate in one direction = *dr*/*dt* (assuming spherical form of β). Consider the time line from zero to a time, *t* (see Fig. 11*)*. We will consider another measure of time (τ), which starts when a nucleus is

 *NV*α *d*τ (34)

Assuming that the particles nucleated in this time *d*τ grow as spheres, the radius of the

 *r* = *G*(*t* – τ) (36)

(*t* - τ) 3

*N*(*V* - *V*<sup>β</sup>

Early in the transformations, when *V*β is small, *V*β can be considered negligible with respect

<sup>3</sup> <sup>π</sup>G3 N <sup>t</sup> <sup>0</sup> V(t - τ)

4

)(*t* - τ) 3

3

 *V* = *V*α + *V*β (32)

*<sup>V</sup>* (33)

<sup>τ</sup> dt (35)

(*NV*α)*d*τ (37)

dτ (38)

dτ (39)

The total volume of the sample is the sum of the volumes of α and β:

the nucleation of phase β within α and then the rate of growth of β.

formed. The number of nuclei formed in differential time *d*τ is equal to

particles formed during *d*τ, after they have grown to time *t*, is:

The volume of the particle nucleated during *d*τ at time *t* is:

*dV*<sup>β</sup>

dV <sup>V</sup> <sup>β</sup> <sup>β</sup>

Fig. 11. Definition of *t* and τ for derivation of solid state transformation equations

 dr r G = <sup>0</sup> t

= 4 <sup>3</sup> <sup>π</sup>*G*<sup>3</sup>

*dV*<sup>β</sup> = 4 <sup>3</sup> <sup>π</sup>*G*<sup>3</sup>

to *V*. In this case, the fraction transformed may be calculated as follows:

= <sup>0</sup>

*<sup>x</sup>*<sup>=</sup> *<sup>V</sup>*<sup>β</sup>

The fraction transformed can be represented in the literature as *x*, α or F:

β; the reaction is written:

(Ragone, 1994)

higher than the temperature at which the nucleation rate is highest. Therefore, when a glass is heated at a constant rate, crystal nuclei are formed at lower temperature and grow in size at higher temperatures without any increase in number (Matusita & Sakka, 1981).

Fig. 10. The relationship between supersaturation and nucleation and growth
