**1. Introduction**

26 Will-be-set-by-IN-TECH

106 Mass Transfer - Advanced Aspects

[28] Nield, D.A. (1968). Onset of thermohaline convection in a porous medium. *Water*

[29] Nield, D.A. & Bejan, A. (1992). *Convection in Porous Media*, Springer-Verlag, New York. [30] Onsager, L. (1931). Reciprocal Relations in Irreversible Processes. *Phys. Rev*, 37, pp. 405 -

[31] Partha, M. K. (2009). Suction/injection effects on thermophoresis particle deposition in a non-Darcy porous meduim under the influence of Soret, Dufour effects. *International*

[33] Pop, I. & Na, T. Y. (1994). Natural convection of a Darcian fluid about a cone. *International*

[34] Pop, I. & Na, T. Y. (1995). Natural convection over a frustum of a wavy cone in a porous

[35] Stern, M. E. (1960). The 'salt fountain' and thermohaline convection. *Tellus*, 12, pp. 172 -

[36] Stern, M. E. (1969). Collective instability of salt fingers. *Journal of Fluid Mechanics*, 35, pp.

[37] Sunil; Sharma, A. & Sharma, R. C. (2006). Effect of dust particles on ferrofluid heated and

[38] Yih, K. A. (1999). Coupled heat and mass transfer by free convection over a truncated cone in porous media: VWT/ VWC or VHF/VMF. *Acta Mechanica*, 137, pp. 83-97.

*Resources Research*, 5, pp. 553 - 560.

*Journal of heat and mass transfer*, 52, pp. 1971-1979.

[32] Pop, I. & Ingham, D. B. (2001). *Convective Heat Transfer*, Elsevier.

*Communications in Heat and Mass Transfer*, 12, pp. 891 - 899.

soluted from below. *Int. J. Therm. Sci*. 45, pp. 347 - 458.

medium. *Mechanics Research Communications*, 22, pp. 181 - 190.

426.

175.

209 - 218.

Decay of oversaturated solid solutions with forming a new phase includes three stages, *viz.* nucleation of centers (clusters, nucleation centers, extractions), independent growth of them and, at last, development of these centers interconnecting to each other. This last stage, socalled late stage of decay of oversaturated solid solution has been firstly revealed by Ostwald (Ostwald, 1900). Its peculiarity consists in the following. Diffusion mass transfer of a matter from clusters with larger magnitudes of surface curvature to ones with smaller magnitudes of surface curvature (owing to the Gibbs-Thomson effect) results in dissolving and disappearing small clusters that causes permanent growth of the mean size of extractions. In accordance with papers (Sagalovich, Slyozov, 1987; Kukushkin, Osipov, 1998), interaction between clusters is realized through the 'generalized self-consistent diffusion field'. This process, when large clusters grow for account of small ones is referred to as the Ostwald's ripening. Investigation of the Ostwald's ripening resulted in determination of the form of the size distribution function in respect of the mass transfer mechanisms. The first detailed theory of the Ostwald's ripening for the diffusion mass transfer mechanism has been developed by Lifshitz and Slyozov (Lifshitz and Slyozov, 1958, 1961). Under diffusion mass transfer mechanism, atoms of a solved matter reaching clusters by diffusion are then entirely absorbed by them, so that cluster growth is controlled by matrix diffusion and, in part, by the volume diffusion coefficient, *Dv* . In paper (Wagner, 1961), Wagner has firstly showed that it is possible, if the atoms crossing the interface 'cluster-matrix' and falling at a cluster surface in unit of time have a time to form chemical connections necessary for reproduction of cluster matter structure. If it is not so, solved atoms are accumulated near the interface 'cluster-matrix' with concentration *C* that is equal to the mean concentration of a solution, *C* . For that, growing process is not controlled by the volume diffusion coefficient, *Dv* , but rather by kinetic coefficient, β . Thus, in his paper published three years later than the papers by Lifshitz and Slyozov, Wagner considered other mechanism of cluster growth controlled by the rate of formation of chemical connection at cluster surface. The quoted papers (Lifshitz, Slyozov, 1958, 1961; Wagner, 1961) form the base of the theory of the Ostwald's ripening that is conventionally referred to as the Lifshitz-Slyozov-Wagner (LSW) theory. Within the framework of this theory, several

Mass Transfer Between Clusters Under Ostwald's Ripening 109

boundary of a cluster of radius *r* , *C*∞ – equilibrium concentration for specified temperature

The flows 1*j* (Eq. 1) (to the cluster) and 2*j* (Eq. 2) (from the cluster) are caused by thermal motion of atoms. 1*j* in Eq. 1 is proportional to the mean concentration of the solution, *С* .

Both in 1*j* (Eq. 1) and in 2*j* (Eq. 2), the kinetic diffusion coefficient equals the flow density for the unit concentration. Thus, taking into account the nature of flows, the kinetic diffusion

character of the processes occurring both at the cluster surfaces and at their interfaces with a matrix. On this reason, one can not write the flow *ij* through the concentration gradient at

( ) 2 2 4 4 *<sup>r</sup>*

<sup>⋅</sup>*<sup>r</sup>* has a dimension of the diffusion coefficient; however, such diffusion coefficient, \* *D r* <sup>=</sup>

( ) <sup>1</sup>

where *vj* – the number of atoms reaching a cluster surface in unite of time through

For determining the size distribution function of particles, *f* (*r t*, ) , one must know the rate

*dt* <sup>=</sup> , that is connected with the size distribution function of the

 πβ

*C C j r C C rr*

<sup>2</sup>*j* in Eq. 2 is proportional to concentration *С<sup>r</sup>* , that is set at the cluster boundary in

⎛ ⎞⎛ ⎞ = ≈+ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠ – concentration of atoms of solved matter at the

exp *<sup>m</sup> С<sup>r</sup> C*

∞

*<sup>m</sup>* – volume of an atom of solved matter, and *k* – the

*kTr* συ

, determining the flow *ij* is caused by non-equilibrium

⎛ ⎞ <sup>=</sup> ⎜ ⎟ ⎝ ⎠ ).

*r*

<sup>−</sup> = −= , (4)

*i v jj j* = = , (5)

<sup>2</sup> *i v j jj* = + , (6)

*i v jj j* = + . (7)

συ

υ

where 2 2

Introducing the kinetic coefficient,

*T* , σ

where

β

In equilibrium state one has:

**2.1 The rate of cluster growth** 

*r*

of particle's growth, *dr*

continuity equation:

β

diffusion.

Boltzmann constant.

exp 1 *m m C C <sup>r</sup> C*

συ

∞ ∞

– interface surface energy,

*kTr rkT*

accordance with the Gibbs-Thomson formula: ( <sup>2</sup>

coefficients are regarded to be equal to each other in 1*j* and in 2*j* .

β

the interface. Formally, it can be represented through concentration gradient:

, has no physical sense. That is why, one proceeds to the kinetics.

*i r*

that is why the flow *j* of atoms to (from) a cluster can be determined as

The flow *j* in Eq. (7) provides determination of the rate of cluster growth.

In general case, the flow *j* of atoms to (from) a cluster will be

πβ

other problems connected with the Ostwald's ripening for diffusion at grain boundaries (Slyozov, 1967; Kirchner, 1971), for surface diffusion (Chakraverty, 1967; Vengrenovich, 1977), for diffusion along dislocation pipes (Ardell, 1972; Kreye, 1970; Vengrenovich, 1975, 1982; Vengrenovich *et al.*, 2001a, 2002) etc. have been solved later. A new phase extracted during decay of oversaturated solid solution as specific matrices of particles (clusters) is the strengthening phase. Its extractions act as a stopper for traveling dislocations. Elastic strength fields arising around clusters and interacting with matrix dislocations, depending on their energy, can be fixed at cluster surfaces or cut of them. Cutting the extracted particles (clusters) by dislocations or fixing of them at particle surfaces leads to the pipe mechanism of diffusion along dislocations with diffusion coefficient *Dd* (Vengrenovich, 1980a, 1980b, 1983; Vengrenovich *et al*., 1998).

For some time past, the LSW theory is successfully used for analysis of evolution of island structure resulting from self-organization in semiconductor heterosystems (Bartelt *et al.,* 1992, 1996; Goldfarb *et al.,* 1997a, 1997b; Joyce *et al.,* 1998; Kamins *et al.,* 1999; Vengrenovich *et al.,* 2001b, 2005, 2010; Pchelyakov *et al.,* 2000; Ledentsov *et al.,* 1998; Xiaosheng Fang *et al.,*  2011). It is also used for dscription of dissipative structures in non-equilibrium semiconductor systems (Gudyma, Vengrenovich, 2001c; Vengrenovich *et al.,* 2001d).

Mass transfer between clusters under the Ostwald's ripening depends on the kind of diffusion than, in its turn, determines the rate of growth of clusters and the size distribution function of them. As it has been noted above, the size distribution function of clusters for matrix diffusion mechanism has been for the first time obtained by Lifshitz and Slyozov within the framework of hydrodynamic approximation. So, this distribution is referred to as the Lifshitz-Slyozov distribution.

This chapter is devoted to the computing of the size distribution function of clusters under mass transfer corresponding to simultaneous (combined) action of various diffusion mechanisms. Topicality of this study follows from the fact that often in practice (due to various reasons) mass transfer between clusters is controlled in parallel, to say, by the kinetic diffusion coefficient, β , and by the matrix diffusion coefficient *Dv* , or, alternatively, by the coefficients *Dv* and *Dd* , simultaneously, ect. All following computations are carried out within the Lifshitz-Slyozov hydrodynamic approximation using the approach developed earlier by one of the authors of this chapter (Vengrenovich, 1982).

### **2. Cluster growth under diffusion and Wagner mechanisms of mass transfer. Generalized Lifshitz-Slyozov-Wagner distribution**

Following to Wagner, the number of atoms crossing the interface 'cluster-matrix' and getting to the cluster surface in unite of time, 1*j* , is

$$j\_1 = 4\pi r^2 \beta \left< \mathbb{C} \right>\,,\tag{1}$$

and the number of atoms leaving it in unite of time is

$$j\_2 = 4\pi r^2 \mathcal{B} \mathbb{C}\_r \, , \tag{2}$$

so that the resulting flux of atoms involving into formation of chemical connections is

$$j\_i = j\_1 - j\_2 = 4\pi r^2 \beta \left( \left< \mathbb{C} \right> - \mathbb{C}\_r \right),\tag{3}$$

other problems connected with the Ostwald's ripening for diffusion at grain boundaries (Slyozov, 1967; Kirchner, 1971), for surface diffusion (Chakraverty, 1967; Vengrenovich, 1977), for diffusion along dislocation pipes (Ardell, 1972; Kreye, 1970; Vengrenovich, 1975, 1982; Vengrenovich *et al.*, 2001a, 2002) etc. have been solved later. A new phase extracted during decay of oversaturated solid solution as specific matrices of particles (clusters) is the strengthening phase. Its extractions act as a stopper for traveling dislocations. Elastic strength fields arising around clusters and interacting with matrix dislocations, depending on their energy, can be fixed at cluster surfaces or cut of them. Cutting the extracted particles (clusters) by dislocations or fixing of them at particle surfaces leads to the pipe mechanism of diffusion along dislocations with diffusion coefficient *Dd* (Vengrenovich,

For some time past, the LSW theory is successfully used for analysis of evolution of island structure resulting from self-organization in semiconductor heterosystems (Bartelt *et al.,* 1992, 1996; Goldfarb *et al.,* 1997a, 1997b; Joyce *et al.,* 1998; Kamins *et al.,* 1999; Vengrenovich *et al.,* 2001b, 2005, 2010; Pchelyakov *et al.,* 2000; Ledentsov *et al.,* 1998; Xiaosheng Fang *et al.,*  2011). It is also used for dscription of dissipative structures in non-equilibrium semiconductor

Mass transfer between clusters under the Ostwald's ripening depends on the kind of diffusion than, in its turn, determines the rate of growth of clusters and the size distribution function of them. As it has been noted above, the size distribution function of clusters for matrix diffusion mechanism has been for the first time obtained by Lifshitz and Slyozov within the framework of hydrodynamic approximation. So, this distribution is referred to as

This chapter is devoted to the computing of the size distribution function of clusters under mass transfer corresponding to simultaneous (combined) action of various diffusion mechanisms. Topicality of this study follows from the fact that often in practice (due to various reasons) mass transfer between clusters is controlled in parallel, to say, by the

by the coefficients *Dv* and *Dd* , simultaneously, ect. All following computations are carried out within the Lifshitz-Slyozov hydrodynamic approximation using the approach developed

**2. Cluster growth under diffusion and Wagner mechanisms of mass transfer.** 

Following to Wagner, the number of atoms crossing the interface 'cluster-matrix' and

2 <sup>1</sup>*j rC* = 4π β

2 <sup>2</sup> 4 *<sup>r</sup> j* = π*r C*β

( ) <sup>2</sup>

so that the resulting flux of atoms involving into formation of chemical connections is

1 2 4 *i r jjj* =−= − π β

, and by the matrix diffusion coefficient *Dv* , or, alternatively,

, (1)

, (2)

*r CC* , (3)

systems (Gudyma, Vengrenovich, 2001c; Vengrenovich *et al.,* 2001d).

β

**Generalized Lifshitz-Slyozov-Wagner distribution** 

getting to the cluster surface in unite of time, 1*j* , is

and the number of atoms leaving it in unite of time is

earlier by one of the authors of this chapter (Vengrenovich, 1982).

1980a, 1980b, 1983; Vengrenovich *et al*., 1998).

the Lifshitz-Slyozov distribution.

kinetic diffusion coefficient,

where 2 2 exp 1 *m m C C <sup>r</sup> C kTr rkT* συ συ ∞ ∞ ⎛ ⎞⎛ ⎞ = ≈+ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠ – concentration of atoms of solved matter at the

boundary of a cluster of radius *r* , *C*∞ – equilibrium concentration for specified temperature *T* , σ – interface surface energy, υ*<sup>m</sup>* – volume of an atom of solved matter, and *k* – the Boltzmann constant.

The flows 1*j* (Eq. 1) (to the cluster) and 2*j* (Eq. 2) (from the cluster) are caused by thermal motion of atoms. 1*j* in Eq. 1 is proportional to the mean concentration of the solution, *С* . <sup>2</sup>*j* in Eq. 2 is proportional to concentration *С<sup>r</sup>* , that is set at the cluster boundary in

accordance with the Gibbs-Thomson formula: ( <sup>2</sup> exp *<sup>m</sup> С<sup>r</sup> C kTr* συ ∞ ⎛ ⎞ <sup>=</sup> ⎜ ⎟ ⎝ ⎠ ).

Both in 1*j* (Eq. 1) and in 2*j* (Eq. 2), the kinetic diffusion coefficient equals the flow density for the unit concentration. Thus, taking into account the nature of flows, the kinetic diffusion coefficients are regarded to be equal to each other in 1*j* and in 2*j* .

Introducing the kinetic coefficient, β , determining the flow *ij* is caused by non-equilibrium character of the processes occurring both at the cluster surfaces and at their interfaces with a matrix. On this reason, one can not write the flow *ij* through the concentration gradient at the interface. Formally, it can be represented through concentration gradient:

$$j\_i = 4\pi r^2 \beta \left( \left< \mathbf{C} \right> - \mathbf{C}\_r \right) = 4\pi r^2 \beta r \frac{\left< \mathbf{C} \right> - \mathbf{C}\_r}{r} \,, \tag{4}$$

where β <sup>⋅</sup>*<sup>r</sup>* has a dimension of the diffusion coefficient; however, such diffusion coefficient, \* *D r* <sup>=</sup> β , has no physical sense. That is why, one proceeds to the kinetics. In equilibrium state one has:

*i v jj j* = = , (5)

that is why the flow *j* of atoms to (from) a cluster can be determined as

$$j = \frac{1}{2}(j\_i + j\_v) \, , \tag{6}$$

where *vj* – the number of atoms reaching a cluster surface in unite of time through diffusion.

In general case, the flow *j* of atoms to (from) a cluster will be

$$j = j\_i + j\_v \,. \tag{7}$$

The flow *j* in Eq. (7) provides determination of the rate of cluster growth.

#### **2.1 The rate of cluster growth**

For determining the size distribution function of particles, *f* (*r t*, ) , one must know the rate

of particle's growth, *dr r dt* <sup>=</sup> , that is connected with the size distribution function of the continuity equation:

Mass Transfer Between Clusters Under Ostwald's Ripening 111

<sup>∞</sup> ⎛ ⎞ <sup>−</sup> ⎛ ⎞ <sup>=</sup> ⎜ ⎟ + − ⎜ ⎟

where *kr* is the critical radius. Within the LSW theory, *kr* coincides with a mean size of

Eq. (14) corresponds to the rate of cluster growth through matrix diffusion with the share

(14) coincides with the rate of growth Eq. (2.15) from the review paper (Sagalovich, Slyozov,

1 *m g*

⎛ ⎞⎛ ⎞ <sup>=</sup> ⎜ ⎟ + − ⎜ ⎟ <sup>−</sup> ⎝ ⎠⎝ ⎠

Eq. (15) determines the rate of cluster growth under conditions controlled by the kinetic

is fully determined by the kinetic coefficient, and our Eq. (15) coincides with Eq. (2.15) form

the rate of growth that are the combinations of the Wagner and conventional diffusion mechanisms of cluster's enlargement, one assumes that no any term in the general flow *j* , Eq. (7), can be neglected. It means that the flows *vj* and *ij* must be commensurable. However, the intrigue consists in that formation of chemical connections is electron process, while the classical diffusion is the atomic activation process with considerably different temporal scale. Thus, the question arises: what are the conditions for two qualitatively

question on the ratio of flows *vj* and *ij* is reduced, in fact, to the ratio of the relaxation

the proposed mechanism of cluster growth. To obtain answer on this question is, in general,

electron process of formation of chemical connections is activation one, and if the activation

In paper (Wagner, 1961), the solution is obtained for the limiting cases: 0 *x* = , *diffus*.

diffusion mechanism of growth). Note, Wagner (Wagner, 1961) does not discuss the

In the case under consideration here, when the solution is found for arbitrary magnitude of *x* within the interval 0 1 < *x* < , relaxation times must be comparable to each other at least for the systems whose histograms are represented by the computed curves. We provide this

τ

 and *diffus*. τ

, (the Wagner mechanism of growth), and, 1 *x* = , *chem con* . .

*dr C x r r dt kT r x r r*

*Rk* α <sup>⎞</sup> Δ = <sup>⎟</sup> ⎠

<sup>2</sup> <sup>1</sup> 1 1

with the share contribution *x* of matrix diffusion. If 0 *x* = , then growth

β,

1 1 1 1 *m v*

*g k*

, (14)

β

for 3 *n* = , where *D D* <sup>2</sup> = *<sup>v</sup>* ,

2

, become comparable to each other? Thus the

τ

, one obtains:

π*r* β

*k*

1 *<sup>C</sup> <sup>m</sup> <sup>a</sup> kT* ∞συ

, and, as a result, to the question on the possibility to implement

 and *diffus*. τ

. For 1 *x* = , Eq.

2 *<sup>C</sup> <sup>m</sup> <sup>a</sup> kT* ∞συ= .

. (15)

= . In Eqs. (14) or (15) for

are commensurable, if the

 << *difus*. τ

τ

, (the

2

⎝ ⎠⎝ ⎠

2

*dr C D xr r dt kT x r r r*

2

contribution (1 ) − *x* of the part of flow controlled by the kinetic coefficient

, where

∞συ β

σ υ

<sup>1</sup> <sup>1</sup> *n n <sup>n</sup> <sup>k</sup>*

(Sagalovich, Slyozov, 1987) for 2 *n* = , where, *D*<sup>1</sup> =

τ

To all appearance, the relaxation times *chem con* . .

energies for both processes (electron and diffusion) are comparable.

Repeating this procedure and taking out of the brackets <sup>2</sup> 4

*dR D a R dt R R* − − − <sup>⎛</sup> ⎛ ⎞ <sup>⎜</sup> <sup>=</sup> ⎜ ⎟ <sup>−</sup> <sup>⎜</sup> ⎝ ⎝ ⎠

particles, *kr r* = .

coefficient

times *chem con* . . τ

too hardly.

<< *chem con* . . τ

relaxation times.

comparison below.

1987), *viz*. 1 1

β

different relaxation times, *chem con* . .

 and *diffus*. τ

$$\frac{\partial f(r,t)}{\partial t} + \frac{\partial}{\partial r} (f(r,t)\dot{r}) = 0 \,. \tag{8}$$

The rate of cluster growth is determined from a condition:

4 <sup>3</sup> 3 *<sup>m</sup> d <sup>r</sup> <sup>j</sup> dt* π υ ⎛ ⎞ <sup>=</sup> ⎜ ⎟ ⎝ ⎠ , (9)

where *j* is determined by Eq. (7). There is the diffusion part of a flow:

$$j\_v = 4\pi r^2 D\_v \left(\frac{d\mathbf{C}}{dr}\right)\_{R=r} = 4\pi r^2 D\_v \frac{\{\mathbf{C}\} - \mathbf{C}\_r}{r} \,. \tag{10}$$

Taking into account Eqs. (3) and (10), one finds from Eq. (9):

$$\frac{dr}{dt} = \frac{\left(\left\{\mathbf{C}\right\} - \mathbf{C}\_r\right)\nu\_m}{4\pi r^2} \left(4\pi r^2 \beta + 4\pi r^2 \mathbf{D}\_v \frac{1}{r}\right). \tag{11}$$

Let us denote the shares *vj* and *ij* in general flow *j* as *x* and (1 ) − *x* , respectively:

$$\mathbf{x} = \frac{\dot{j}\_v}{\dot{j}} \text{ , } \mathbf{1} - \mathbf{x} = \frac{\dot{j}\_i}{\dot{j}} \text{ , } \frac{\dot{j}\_v}{\dot{j}\_i} = \frac{\mathbf{x}}{1 - \mathbf{x}} \text{ } \tag{12}$$

To represent the rate of growth (11) through the share flows *ij* and *vj* , let us take out of the brackets the second term, <sup>2</sup> <sup>1</sup> <sup>4</sup> *<sup>v</sup> r D r* π , and multiply nominator and denominator of the firs term by ( ) <sup>2</sup> *<sup>g</sup> CCr* − *<sup>r</sup> <sup>g</sup>* , where *<sup>g</sup> Cr* is the concentration at the boundary with a cluster of maximal size *gr* :

$$\frac{dr}{dt} = \left( \{\mathbf{C}\} - \mathbf{C}\_r \right) \frac{D\_v \nu\_m}{r} \left( \frac{4\pi r\_s^2 \beta \left( \{\mathbf{C}\} - \mathbf{C}\_{r\_s} \right)}{4\pi r\_s^2 D\_v} \frac{r}{r\_\beta} + 1 \right). \tag{13}$$

The ratio ( ) ( ) 2 2 4 4 *g g g r r g v g r CC C C r D r* π β π − − equals the ratio of the flows *<sup>i</sup> v j j* for a particle of the maximal

size, and, in accordance with Eq. (12), it can be replaced by <sup>1</sup> *<sup>x</sup> x* ⎛ ⎞ <sup>−</sup> ⎜ ⎟ ⎝ ⎠ , while there are not any limitations on particle size in Eq. (12). Besides, taking into account that <sup>2</sup> *<sup>m</sup>* 1 1 *<sup>r</sup> k <sup>C</sup> C C kT r r* ∞συ ⎛ ⎞ −= − ⎜ ⎟ ⎝ ⎠ , the rate of growth (13) can be rewritten in the following form:

( ) ( ) ( ) , , 0 *f rt f rtr t r*

4 <sup>3</sup> 3 *<sup>m</sup>*

*<sup>r</sup> <sup>j</sup> dt* π

*v v v R r*

*j rD r D*

2

*r m*

Let us denote the shares *vj* and *ij* in general flow *j* as *x* and (1 ) − *x* , respectively:

*<sup>j</sup>* <sup>=</sup> , 1 *ij <sup>x</sup>*

4

*vj <sup>x</sup>*

*r*

π

π

⎛ ⎞ <sup>=</sup> ⎜ ⎟

2 2 4 4 *<sup>r</sup>*

= ⎛ ⎞ <sup>−</sup> <sup>=</sup> <sup>=</sup> ⎜ ⎟

( ) 2 2

<sup>−</sup> ⎛ ⎞ <sup>=</sup> <sup>+</sup> ⎜ ⎟

*dt r r* υ

υ

*dC C C*

<sup>1</sup> 4 4

*r rD*

*j x*

 π

*v*

, and multiply nominator and denominator of the firs

*dr r*

πβ

*<sup>j</sup>* <sup>−</sup> <sup>=</sup> , 1 *v i*

*<sup>g</sup> CCr* − *<sup>r</sup> <sup>g</sup>* , where *<sup>g</sup> Cr* is the concentration at the boundary with a cluster of

( )

−

⎛ ⎞ ⎜ ⎟ −

*g*

*r*

⎝ ⎠

*g*

*g*

*g r*

*v j*

, the rate of growth (13) can be rewritten in the following form:

*x* ⎛ ⎞ <sup>−</sup> ⎜ ⎟

1

. (13)

*j* for a particle of the maximal

⎝ ⎠ , while there are not any

To represent the rate of growth (11) through the share flows *ij* and *vj* , let us take out of the

( ) ( )

*<sup>D</sup> r CC dr <sup>r</sup> C C dt r C C r*

υ

4

equals the ratio of the flows *<sup>i</sup>*

limitations on particle size in Eq. (12). Besides, taking into account that

π

4

= − +

2

2

*r D*

*g v*

*g r v m <sup>r</sup>*

π β

 π

<sup>+</sup> <sup>=</sup> ∂ ∂ . (8)

⎝ ⎠ , (9)

⎝ ⎠ . (10)

⎝ ⎠ . (11)

*<sup>j</sup> <sup>x</sup>* <sup>=</sup> <sup>−</sup> (12)

∂ ∂

*d*

where *j* is determined by Eq. (7). There is the diffusion part of a flow:

π

*dr C C*

Taking into account Eqs. (3) and (10), one finds from Eq. (9):

brackets the second term, <sup>2</sup> <sup>1</sup> <sup>4</sup> *<sup>v</sup> r D*

term by ( ) <sup>2</sup>

The ratio ( )

π β

4

4

*<sup>C</sup> C C*

π

2

*r D*

*g v*

<sup>2</sup> *<sup>m</sup>* 1 1 *<sup>r</sup>*

∞συ⎛ ⎞ −= − ⎜ ⎟

2

( )

−

*C C*

−

*r*

*k*

⎝ ⎠

*kT r r*

*g*

*g r*

*r CC*

*g*

*g*

size, and, in accordance with Eq. (12), it can be replaced by <sup>1</sup> *<sup>x</sup>*

*r*

maximal size *gr* :

The rate of cluster growth is determined from a condition:

$$\frac{dr}{dt} = \frac{\sigma \mathbf{C}\_{\infty} \boldsymbol{\nu}\_m^2 D\_v}{kT} \frac{1}{r^2} \left(\frac{1-\mathbf{x}}{\mathbf{x}} \frac{r}{r\_g} + 1\right) \left(\frac{r}{r\_k} - 1\right),\tag{14}$$

where *kr* is the critical radius. Within the LSW theory, *kr* coincides with a mean size of particles, *kr r* = .

Eq. (14) corresponds to the rate of cluster growth through matrix diffusion with the share contribution (1 ) − *x* of the part of flow controlled by the kinetic coefficient β . For 1 *x* = , Eq. (14) coincides with the rate of growth Eq. (2.15) from the review paper (Sagalovich, Slyozov,

$$(1987)\_{\prime} \,\mathrm{v} \,\mathrm{i}. \left( \frac{d\mathbb{R}}{dt} = \frac{D\_{n-1}a\_{n-1}}{R^{n-1}} \left( \frac{\mathbb{R}}{R\_k} - 1 \right), \text{where } \Lambda = \frac{a}{R\_k} \right) \text{ for } n = 3 \text{ , where } D\_2 = D\_{v\prime} \,\, a\_2 = \frac{C\_w \sigma \nu\_m^2}{kT}.$$

Repeating this procedure and taking out of the brackets <sup>2</sup> 4π*r* β, one obtains:

$$\frac{dr}{dt} = \frac{\mathbb{C}\_{\infty} \sigma \upsilon\_m^2 \mathcal{B}}{kT} \frac{1}{r} \left( \frac{\infty}{1 - \infty} \frac{r\_{\mathcal{S}}}{r} + 1 \right) \left( \frac{r}{r\_k} - 1 \right). \tag{15}$$

Eq. (15) determines the rate of cluster growth under conditions controlled by the kinetic coefficient β with the share contribution *x* of matrix diffusion. If 0 *x* = , then growth is fully determined by the kinetic coefficient, and our Eq. (15) coincides with Eq. (2.15) form

(Sagalovich, Slyozov, 1987) for 2 *n* = , where, *D*<sup>1</sup> = β , 2 1 *<sup>C</sup> <sup>m</sup> <sup>a</sup> kT* ∞συ= . In Eqs. (14) or (15) for

the rate of growth that are the combinations of the Wagner and conventional diffusion mechanisms of cluster's enlargement, one assumes that no any term in the general flow *j* , Eq. (7), can be neglected. It means that the flows *vj* and *ij* must be commensurable. However, the intrigue consists in that formation of chemical connections is electron process, while the classical diffusion is the atomic activation process with considerably different temporal scale. Thus, the question arises: what are the conditions for two qualitatively different relaxation times, *chem con* . . τ and *diffus*. τ , become comparable to each other? Thus the question on the ratio of flows *vj* and *ij* is reduced, in fact, to the ratio of the relaxation times *chem con* . . τ and *diffus*. τ , and, as a result, to the question on the possibility to implement the proposed mechanism of cluster growth. To obtain answer on this question is, in general, too hardly.

To all appearance, the relaxation times *chem con* . . τ and *diffus*. τ are commensurable, if the electron process of formation of chemical connections is activation one, and if the activation energies for both processes (electron and diffusion) are comparable.

In paper (Wagner, 1961), the solution is obtained for the limiting cases: 0 *x* = , *diffus*. τ << *chem con* . . τ , (the Wagner mechanism of growth), and, 1 *x* = , *chem con* . . τ << *difus*. τ , (the diffusion mechanism of growth). Note, Wagner (Wagner, 1961) does not discuss the relaxation times.

In the case under consideration here, when the solution is found for arbitrary magnitude of *x* within the interval 0 1 < *x* < , relaxation times must be comparable to each other at least for the systems whose histograms are represented by the computed curves. We provide this comparison below.

Mass Transfer Between Clusters Under Ostwald's Ripening 113

( ) ( )

*x rA t x x* <sup>+</sup> <sup>=</sup> <sup>+</sup>

1 2

3 \*

<sup>2</sup> *gr At* <sup>=</sup> , 3 \* <sup>4</sup>

2 \* <sup>2</sup> 1 *<sup>g</sup> <sup>t</sup> r B*

2 \*

*k*

2 \*

Following to paper (Vengrenovich, 1982), *f* (*r t*, ) is found as the product:

*M* =

where *g*′(*u*) is the relative size distribution function of clusters,

ϕ

*gr Bt* <sup>=</sup> , 2 \* <sup>1</sup>

*f* (*rt r* , ) = ϕ

( )( )

*x x*

<sup>4</sup> *kr Bt* <sup>=</sup> , 2 *<sup>g</sup> k r*

( ) <sup>3</sup>

0 <sup>4</sup> , <sup>3</sup> *gr*

πρ

1 1 2

*<sup>x</sup> r B <sup>t</sup>*

Eqs. (21) and (22) describe changing in time cluster sizes, when growth of them is controlled

the process of growth is fully controlled by kinetics of transition through the interface

The size distribution function, *f* (*r t*, ) , and the rate of growth, *r* , are connected by the continuity equation (8). Knowing *r* (Eqs. (14) or (15)), one can find *f* (*r t*, ) from Eq. (8).

<sup>+</sup> <sup>=</sup> − +

2

, with the share contribution *x* of matrix diffusion. If 0 *x* = , then

*k*

For 1 *x* = particle growth is full controlled by the volume diffusion coefficient:

3 \* 1

2

<sup>27</sup> *kr At* <sup>=</sup> , <sup>3</sup>

2

*g k r*

3

. (19)

*<sup>r</sup>* <sup>=</sup> . (20)

. (22)

*<sup>r</sup>* <sup>=</sup> . (23)

( *<sup>g</sup>* ) *g*′(*u*) , (24)

*g r u <sup>r</sup>* <sup>=</sup> .

*<sup>r</sup> <sup>f</sup> r t dr* ∫ , (25)

(*rg* ) , let us apply the conservation law for mass of disperse

*<sup>x</sup>* <sup>=</sup> <sup>−</sup> , (21)

where,

where

<sup>2</sup> \* <sup>3</sup> *m v C D <sup>A</sup> kT* συ<sup>∞</sup> = , or:

By analogy, one obtains from Eq. (15):

 β<sup>∞</sup> = , or:

β

<sup>2</sup> \* <sup>2</sup> *mC <sup>B</sup> kT* συ

by the kinetic coefficient

**2.3 Size distribution function** 

To determine the function

by substituting in it *f* (*r t*, ) from Eq. (24):

phase:

'cluster-matrix':
