**3. Cluster growth under dislocation-matrix diffusion. Size distribution function**

The Ostwald's ripening of disperse phases in metallic alloys at the final stage of forming their structure reflecting the late stage of the development of nucleation centers of a new phase in time, when oversaturation between them decreases and their diffusion fields overlap.

In respect to metallic alloys strengthened by disperse extractions of the second phase, the Ostwald ripening is one of the causes of loss of strength of them. As large particles grow and small particles disappear (due to dissolution), distance between particles increases resulting in decreasing of tension necessary for pushing the dislocations between particles and, correspondingly, to decreasing of the creep strength.

For the dislocation mechanism of growth of particles that are coherent with a matrix, the flow along dislocations, *dj* , much exceeds the flow of matrix diffusion, *vj* :

$$D\_d Zq \left(\frac{d\mathbf{C}}{dR}\right)\_{R=r} \gg D\_v 4\pi r^2 \left(\frac{d\mathbf{C}}{dR}\right)\_{R=r} \tag{36}$$

where , *Dd Dv* – the coefficients of dislocation and matrix diffusion, respectively, *Z* – the number of dislocation lines that are fixed or crossing a particle of radius *r*, *q* – the square of dislocation pipe cross-section, *dC dR* <sup>=</sup> ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ – gradient of concentration at the boundary of a

*R r*

particle. Taking into account that for disturbed coherence (as a consequence of relaxation of elastic tensions (Kondratyev & Utyugow, 1987)) *Z* is not constant (*Z const* ≠ ) being changed in inverse proportion to the particle radius, inequality (36) determines limitations on particle sizes for which the pipe mechanism of diffusion is yet possible (Vengrenovich *et al*., 2002):

$$r \ll \sqrt[3]{\frac{D\_d Z\_0 q^{3/2}}{4\pi^2 D\_v}},\tag{37}$$

where *Z*0 is the initial number of dislocations fixed at particle surface. If the condition (37) is violated, it means that one can not neglect the component *vj* caused by matrix diffusion in full flow of matter *j* to (from) a particle. In this case, particle growth takes place under diffusion of mixed type (dislocation-matrix one), when one can not neglect any of two components, *dj* or *vj* , in the resulting flow

$$j = j\_d + j\_v \,. \tag{38}$$

Below we represent the results of investigation of peculiarities of the Ostwald ripening of clusters under dislocation-matrix diffusion and, in part, computation of the size distribution function and temporal dependences for mean (critical) and maximal particle sizes as a function of the ratio of flows *dj* and *vj* .

#### **3.1 The rate of growth and temporal dependences for the mean (critical) and maximal sizes of clusters**

As in previous case, the rate of growth is determined from Eq. (9):

$$\frac{dr}{dt} = \frac{1}{4\pi r^2} jv\_{m'} \tag{39}$$

where *j* is given by Eq. (38), and *dj* and *vj* take the magnitudes of left and right parts of inequality (36), respectively:

$$j = D\_d \cdot 2 \frac{Z\_0 q^{1/2}}{2 \pi r} q \left(\frac{d\mathcal{C}}{dR}\right)\_{R=r} + D\_v 4 \pi r^2 \left(\frac{d\mathcal{C}}{dR}\right)\_{R=r} \tag{40}$$

where we take into account that, in a flow *dj* , there is 1/2 0 2 *Z q <sup>Z</sup>* π*<sup>r</sup>* <sup>=</sup> (Vengrenovich *et al*., 2002).

Substituting Eq. (40) in Eq. (39) and taking into account that *R r dC dR* <sup>=</sup> ⎛ ⎞ ⎜ ⎟ ⎝ ⎠

\* 2 <sup>2</sup> <sup>1</sup> <sup>1</sup> *vm <sup>r</sup> <sup>С</sup> R r r*κ σ ∞ ⎛ ⎞ = ⋅− ⎜ ⎟ <sup>Τ</sup> ⎝ ⎠ , where σis the surface energy, *C*∞ is the equilibrium concentration

of solid solution, \* *R* is the gas constant, and Τ is a temperature, one obtains:

$$\frac{dr}{dt} = \frac{1}{4\pi r^4} \frac{2\sigma v\_m^2 \mathbb{C}\_\alpha}{R^\ast \mathbb{T}} \left( D\_d \mathfrak{L} \cdot \frac{Z\_0 q^{1/2}}{2\pi r} \cdot q + D\_v \, 4\pi r^2 \right) \left( \frac{r}{r\_\kappa} - 1 \right). \tag{41}$$

Designating, as previously, the shares *vj* and *dj* in the general flow *j* as *x* and (1 − *x*) , respectrively, one can represent the rate of growth, Eq. (41), in the form

$$\frac{dr}{dt} = \frac{1}{r^5} \frac{\sigma \upsilon\_m^2 C\_\infty Z\_0 q^{3/2} D\_d}{2\pi^2 R^4 \,\mathrm{T}} \left( 1 + \frac{\mathrm{x}}{1 - \mathrm{x}} \frac{r^3}{r\_\infty^3} \right) \left( \frac{r}{r\_\kappa} - 1 \right),\tag{42}$$

or:

118 Mass Transfer - Advanced Aspects

Fig. 4 shows comparison of the experimental histograms obtained under crystallization of amorphous alloy for temperature 508°K during 1 min and 2.5 min (fragments a and b, respectively) with the theoretical dependence, Eq. (28). One can see that theoretical dependences well fit the experimental histograms for 0.2 *x* = (fragment а) and 0.1 *x* =

Thus, the considered examples of comparison with the experimental data prove the conclusion that the distribution Eq. (28) is quite eligible for description of experimentally obtained histograms, if particle growth in the process of the Ostwald ripening is controlled simultaneously by two mechanisms of mass transfer, which earlier were considered

The Ostwald's ripening of disperse phases in metallic alloys at the final stage of forming their structure reflecting the late stage of the development of nucleation centers of a new phase in time, when oversaturation between them decreases and their diffusion fields

In respect to metallic alloys strengthened by disperse extractions of the second phase, the Ostwald ripening is one of the causes of loss of strength of them. As large particles grow and small particles disappear (due to dissolution), distance between particles increases resulting in decreasing of tension necessary for pushing the dislocations between particles

For the dislocation mechanism of growth of particles that are coherent with a matrix, the

⎛⎞ ⎛⎞ >> ⎜⎟ ⎜⎟

where , *Dd Dv* – the coefficients of dislocation and matrix diffusion, respectively, *Z* – the number of dislocation lines that are fixed or crossing a particle of radius *r*, *q* – the square

particle. Taking into account that for disturbed coherence (as a consequence of relaxation of elastic tensions (Kondratyev & Utyugow, 1987)) *Z* is not constant (*Z const* ≠ ) being changed in inverse proportion to the particle radius, inequality (36) determines limitations on particle sizes for which the pipe mechanism of diffusion is yet possible (Vengrenovich *et* 

> <sup>0</sup> <sup>3</sup> <sup>2</sup> , <sup>4</sup> *d*

where *Z*0 is the initial number of dislocations fixed at particle surface. If the condition (37) is violated, it means that one can not neglect the component *vj* caused by matrix diffusion in full flow of matter *j* to (from) a particle. In this case, particle growth takes place under diffusion of mixed type (dislocation-matrix one), when one can not neglect any of two

*DZq <sup>r</sup>* π*D*

*R r R r*

= =

3/2

*v*

⎝⎠ ⎝⎠ , (36)

⎝ ⎠ – gradient of concentration at the boundary of a

<< (37)

flow along dislocations, *dj* , much exceeds the flow of matrix diffusion, *vj* :

*dC dR* <sup>=</sup> ⎛ ⎞ ⎜ ⎟

<sup>2</sup> 4 *d v*

*R r*

*dC dC D Zq D r dR dR* π

**3. Cluster growth under dislocation-matrix diffusion. Size distribution** 

(fragment b).

**function** 

overlap.

*al*., 2002):

separately by Lifshitz and Slyozov, and Wagner.

and, correspondingly, to decreasing of the creep strength.

of dislocation pipe cross-section,

components, *dj* or *vj* , in the resulting flow

$$\frac{dr}{dt} = \frac{1}{r^2} \frac{\sigma v\_m^2 C\_\infty D v}{R^\* \, ^\circ \text{T}} \left( \frac{1 - \chi}{\chi} \frac{r\_g^3}{r^3} + 1 \right) \left( \frac{r}{r\_\kappa} - 1 \right). \tag{43}$$

Eq. (42) describes the rate of particle growth for predominant contribution in the general flow of the diffusion matter along dislocations, with the share contribution *x* of matrix

Mass Transfer Between Clusters Under Ostwald's Ripening 121

( ) ( )

*r В t* κ

()( ) <sup>2</sup> 11/3 7/3 <sup>1</sup> () 1 2 exp . <sup>1</sup> *gu u u u*

− − ⎛ ⎞ ′ =− + ⎜ ⎟ ⎝ ⎠ <sup>−</sup>

The size distribution function of clusters within the interval 0 1 ≤ *x* ≤ is represented, as previously, in the form Eq. (24) (Vengrenovich, 1982), where *g u*′( ) - relative size distribution

– particle density.

If one replaces in the continuity equation (8) *f* (,) *r t* and *r* by their magnitudes from Eqs. (24) and (42) (or Eq. (43)) and differentiates *u* instead of on *r* and *t* , then variables in Eq.

> <sup>1</sup> 4 2 ( ) , ( ) *g*

υ

*dg u u u du du*

υ

+ − ′ = − ′ <sup>−</sup>

*g u <sup>u</sup>*

3 \* , *rr B* υ

<sup>=</sup> <sup>3</sup>

*g*

υ

υ

*g*

*r r B*

and decomposing in denominator the second-order polynomial into prime factors, one gets

\* , *g g*

υ

3 2

υ

*d*

 υ

*g*

1 1 63 1 1 5 3

*x x*

*x x u*

*dr r* <sup>=</sup> and

2

*du*

*u*

<sup>=</sup> <sup>1</sup> ,

<sup>=</sup> <sup>3</sup> ,

*gr r*κ

*<sup>r</sup>* <sup>=</sup> From the mass conservation law and disperse phase, Eq. (25), one

2

*u*

*g*(*u Q* ) = ⋅ *g*′(*u*) , (52)

(53)

υ<sup>=</sup> <sup>−</sup>

1 5 3 *<sup>g</sup> x x*

*g g du u dr r* <sup>=</sup> <sup>−</sup> .

*u*

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

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

3 \* <sup>3</sup> , <sup>2</sup> *gr Bt* <sup>=</sup> 3 \* <sup>4</sup> , <sup>9</sup>

and the size distribution function is described by the Lifshitz-Slyozov function, Eq. (34):

κ

where \* <sup>2</sup>

\* *m v vCD <sup>B</sup> R* σ<sup>∞</sup> <sup>=</sup> <sup>Τ</sup> .

Another limiting case corresponds to 1 *x* = :

**3.2 Size distribution function of clusters** 

*g r u*

> 3 0

′ ∫

πρ

where we take into account that

the following form of Eq. (53):

<sup>Μ</sup> <sup>=</sup>

, <sup>4</sup> ( ) <sup>3</sup>

*u g u du*

*r* , Eq. (26), and, correspondingly,

 and ρ

Substituting in Eq. (53) the magnitudes 3

function, and .

finds ( ) *<sup>g</sup>* ϕ

where 1

(8) are separated:

*Q*

1/3 \* <sup>2</sup> 3 6 53 , 6 3 *B x r t x x*

(50)

= (51)

diffusion; and Eq. (43) describes the rate of growth under matrix diffusion, with the share contribution (1 − *x*) along dislocations.

Eqs. (42) or (43) provide determining the locking point 0 *g k r u <sup>r</sup>* <sup>=</sup> , and one finds out from the continuity equation (8), after separation of variables, the specific size distribution function, *f* (*u*) , where *g r u <sup>r</sup>* <sup>=</sup> . The ratio *gr r*κ , in accordance with (Vengrenovich, 1982), equals:

$$\frac{r\_{\mathcal{S}}}{r\_{\kappa}} = \frac{6 - 3\varkappa}{5 - 3\varkappa}. \tag{44}$$

If we let *<sup>g</sup> r r* <sup>=</sup> in Eq. (42), and the ratio *gr r*κ is replaced by its magnitude from Eq. (44), then after integration one obtains the temporal dependence for maximal

$$r\_{\mathcal{S}} = \left(\frac{6A^\*}{(5-3x)(1-x)}t\right)^{1/6},\tag{45}$$

and critical

$$r\_{\kappa} = \left(\frac{6A^\*\left(5-3\mathfrak{x}\right)^5}{\left(6-3\mathfrak{x}\right)^6\left(1-\mathfrak{x}\right)}t\right)^{1/6},\tag{46}$$

 particle sizes, where <sup>2</sup> 3 2 \* <sup>0</sup> \* 2 *m d v C Zq D <sup>A</sup> R T* σ<sup>∞</sup> = .

Eqs. (45) and (46) describe changing in time the sizes of particles under dislocation-matrix diffusion for predominant contribution of matter diffusion along dislocations. For 0 *x* = , that corresponds to the first limiting case, particle growth is limited by diffusion along dislocation:

$$r\_{\mathcal{S}}^6 = \frac{6}{5} A^\* t, \quad r\_{\kappa}^6 = \left(\frac{5}{6}\right)^5 A^\* t, \quad \frac{r\_{\mathcal{S}}}{r\_k} = \frac{6}{5} \cdot \tag{47}$$

For that ( 0 *x* = ), the specific size distribution function has a form (Vengrenovich *et al*., 2002):

$$g'(u) = \frac{u^5 \exp\left(-\frac{0.2}{\left(1 - u\right)}\right) \exp\left(-0.0287 \tan^{-1}(\frac{2u + a}{\sqrt{4b - a^2}})\right) \exp\left(-0.1127 \tan^{-1}(\frac{2u + c}{\sqrt{4d - c^2}})\right)}{\left(1 - u\right)^a \left(u^2 + au + b\right)^\beta \left(u^2 + cu + d\right)^\gamma}, \text{(48)}$$

where *a* ≅ 2.576, 2.394, *b* ≅ 0.576, *c* ≅ − 0.088, *d* ≅ 41 /15, α ≅ 1.562, β ≅ 1.572. γ ≅ Integrating for the same conditions Eq. (43), one obtains:

$$r\_g = \left(\frac{6B}{\varkappa \left(5-3\varkappa\right)} t\right)^{1/3} \tag{49}$$

$$r\_{\kappa^\*} = \left(\frac{6B^\*\left(5-3\chi\right)^2}{\varkappa\left(6-3\chi\right)^3}t\right)^{1/3},\tag{50}$$

where \* <sup>2</sup> \* *m v vCD <sup>B</sup> R* σ<sup>∞</sup> <sup>=</sup> <sup>Τ</sup> .

120 Mass Transfer - Advanced Aspects

diffusion; and Eq. (43) describes the rate of growth under matrix diffusion, with the share

continuity equation (8), after separation of variables, the specific size distribution function,

6 3 . 5 3 *gr x r x* κ

*r*κ

( )( ) 1/6 \* <sup>6</sup> , 53 1 *<sup>g</sup> <sup>A</sup> r t*

*x x* ⎛ ⎞ <sup>=</sup> ⎜ ⎟ − − ⎝ ⎠

> ( ) ( )( )

*x x*

6 6 53 , 63 1 *A x r t*

Eqs. (45) and (46) describe changing in time the sizes of particles under dislocation-matrix diffusion for predominant contribution of matter diffusion along dislocations. For 0 *x* = , that corresponds to the first limiting case, particle growth is limited by diffusion along

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

For that ( 0 *x* = ), the specific size distribution function has a form (Vengrenovich *et al*., 2002):

0.2 <sup>2</sup> <sup>2</sup> exp exp 0.0287 tan ( ) exp 0.1127 tan ( ) <sup>1</sup> 4 4 ( )

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

( ) 1/3 \* <sup>6</sup> , 5 3 *<sup>g</sup> <sup>B</sup> r t x x* ⎛ ⎞ <sup>=</sup> ⎜ ⎟ <sup>−</sup> ⎝ ⎠

5 11

( ) ( )( )

− ++ ++

2 2

*u u au b u cu d* β

α

*r At r At*

κ

5 6 \*6 \* 65 6 , , 56 5

*g*

*k r*

*r*

*u b a d c*

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

1/6 \* <sup>5</sup>

*g k r*

<sup>−</sup> <sup>=</sup> <sup>−</sup> (44)

is replaced by its magnitude from Eq. (44), then

*<sup>r</sup>* <sup>=</sup> , and one finds out from the

(45)

(46)

. (47)

, (48)

2 2

 ≅ 1.572. γ≅

(49)

*u a u c*

 γ

 ≅ 1.562, β

*u*

, in accordance with (Vengrenovich, 1982), equals:

contribution (1 − *x*) along dislocations.

*g r u*

If we let *<sup>g</sup> r r* <sup>=</sup> in Eq. (42), and the ratio *gr*

*<sup>r</sup>* <sup>=</sup> . The ratio *gr*

*f* (*u*) , where

and critical

dislocation:

*g u*

particle sizes, where

*u*

Eqs. (42) or (43) provide determining the locking point 0

*r*κ

after integration one obtains the temporal dependence for maximal

κ

<sup>∞</sup> = .

<sup>2</sup> 3 2 \* <sup>0</sup> \* 2 *m d v C Zq D <sup>A</sup> R T*

*g*

1

where *a* ≅ 2.576, 2.394, *b* ≅ 0.576, *c* ≅ − 0.088, *d* ≅ 41 /15,

Integrating for the same conditions Eq. (43), one obtains:

α

( )

σ

Another limiting case corresponds to 1 *x* = :

$$r\_{\mathcal{S}}^{\;3} = \frac{3}{2} B^\* t, \quad r\_{\kappa^\*}^{\;3} = \frac{4}{9} B^\* t, \quad \frac{r\_{\mathcal{S}}}{r\_{\kappa}} = \frac{3}{2}. \tag{51}$$

and the size distribution function is described by the Lifshitz-Slyozov function, Eq. (34):

$$\lg'(\mu) = \mu^2 \left(1 - \mu\right)^{-11/3} \left(\mu + 2\right)^{-7/3} \exp\left(\frac{1}{1 - \mu}\right).$$

#### **3.2 Size distribution function of clusters**

The size distribution function of clusters within the interval 0 1 ≤ *x* ≤ is represented, as previously, in the form Eq. (24) (Vengrenovich, 1982), where *g u*′( ) - relative size distribution function, and . *g r u <sup>r</sup>* <sup>=</sup> From the mass conservation law and disperse phase, Eq. (25), one finds ( ) *<sup>g</sup>* ϕ*r* , Eq. (26), and, correspondingly,

$$\mathbf{g}\left(\boldsymbol{u}\right) = \mathbf{Q} \cdot \mathbf{g}'\left(\boldsymbol{u}\right),\tag{52}$$

where 1 3 0 , <sup>4</sup> ( ) <sup>3</sup> *Q* πρ *u g u du* <sup>Μ</sup> <sup>=</sup> ′ ∫ and ρ– particle density.

If one replaces in the continuity equation (8) *f* (,) *r t* and *r* by their magnitudes from Eqs. (24) and (42) (or Eq. (43)) and differentiates *u* instead of on *r* and *t* , then variables in Eq. (8) are separated:

$$\frac{d\mathbf{g'(u)}}{\mathbf{g'(u)}} = -\frac{4\nu\_{\mathcal{S}} + 2\frac{\nu}{\mu^3} - \frac{1}{\mu^2}\frac{d\nu}{du}}{\mu\nu\_{\mathcal{S}} - \frac{\nu}{\mu^2}}d\mu\_{\nu} \tag{53}$$

where we take into account that 3 \* , *rr B* υ <sup>=</sup> <sup>3</sup> \* , *g g g r r B* υ <sup>=</sup> <sup>1</sup> , *g du dr r* <sup>=</sup> and *g g du u dr r* <sup>=</sup> <sup>−</sup> . Substituting in Eq. (53) the magnitudes 3 1 1 63 1 1 5 3 *x x u x x u* υ ⎛ ⎞⎛ ⎞ − − <sup>=</sup> + − ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠ <sup>−</sup> and ( ) 1 5 3 *<sup>g</sup> x x* υ<sup>=</sup> <sup>−</sup>

and decomposing in denominator the second-order polynomial into prime factors, one gets the following form of Eq. (53):

Mass Transfer Between Clusters Under Ostwald's Ripening 123

histogram than the curve in Fig. 6,*а*. It means that formation of quantum dots of *C Sd* in process of the Ostwald's ripening obtained by chemical evaporation is realized through mixed diffusion, with 70% share of matrix ( 0.7 *x* = ) and 30% dislocation ( 0.3 *x* = ) diffusion. For that, it is of importance that temporal growth of nanocrystals of *C Sd* obeys the cubic

*r* ~ *t* , cf. Eq. (50). It shows that, in first, that the size distribution is formed in process of the Ostwald's ripening, and, secondly, that growth of *C Sd* nanocrystals is limited, mainly, by matrix diffusion with the mentioned above share contribution of dislocation

0.0

0.0

0.2

0.4

0.6

*g(u)/gmax*

0.8

1.0

0.2

0.4

0.6

0.2 0.4 0.6 0.8 1.0

0.8 0.9 1.0 0.0

*x=0*

*x=1*

*g(u)/gmax*

Fig. 5. Size distribution functions, Eq. (52), for various magnitudes of *x* - *а*; the same

*a*

Fig. 6. Comparison with experimentally obtained histogram for nanocrystals of *C Sd*

(Katsikas *et. al*., 1990) with the theoretical dependence: *а* – the Lifshitz-Slyozov distribution,

Let us note that there is the set of quantum dots in semiconductor compounds II-IV obtained by chemical evaporation techniques and having sizes from 1 to 5 nm (Gaponenko, 1996), for which the size distribution function occurs be narrower than one for the Lifshitz-Slyozov

Similarly to as crystalline gratings of numerous matters are controlled by simultaneous (combined) action of various connection types, the cluster growth goes on under mixed

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

*x=1 x=0*

*u*

0.0 0.2 0.4 0.6 0.8 1.0

*x=0.7*

*u*

*b*

*b*

law, <sup>3</sup>

diffusion.

0.0

0.0

distribution.

0.2

0.4

0.6

*g(u)/gmax*

0.8

1.0

0.2

0.4

0.6

*g(u)*

0.8

2,5 10-10 2,0 10-10 1,5 10-10 1,0 10-10 5,0 10-11

0.0 0.2 0.4 0.6 0.8 1.0

*u*

0.0 0.2 0.4 0.6 0.8 1.0

*x=1*

*u*

Eq. (34), *b* – distribution corresponding to Eq. (55) for 0.7 *x* =

*x=0.9 x=0.8 x=0.7 x=0.6 x=0.5 x=0.4 x=0.3 x=0.2 x=0.1 x=0*

distributions normalized by their maxima - *b*

1.0 *a*

0.0 0.2 0.4 0.6 0.8 1.0

*x=1*

$$\begin{split} \frac{d\mathbf{g'(u)}}{\mathbf{g'(u)}} &= -\frac{4u^6 + u^4 \left(6\mathbf{x} - 3\mathbf{x}^2\right) - 2u^3 \left(5\mathbf{x} - 3\mathbf{x}^2\right) + 4u\left(3\mathbf{x}^2 - 9\mathbf{x} + 6\right) - 5\left(3\mathbf{x}^2 - 8\mathbf{x} + 5\right)}{u\left(1 - u\right)^2 \left(u^2 + au + d\right) \left(u^2 + bu + p\right)} du \\ &= A\frac{du}{u} + B\frac{du}{1 - u} + C\frac{du}{\left(1 - u\right)^2} + \left(Du + E\right)\frac{du}{u^2 + au + d} + \left(Fu + G\right)\frac{du}{u^2 + bu + p} .\end{split} \tag{54}$$

where 2.575 *a* = ; 0.575 *b* = − ; 2.398 *d* = ; *p* = 2.089 .

Integrating Eq. (54), one obtains the analytical form of the relative size distribution function for arbitrary 0 1 ≤ ≤ *x* :

$$g'(u) = \frac{u^A \left(u^2 + au + d\right)^D \left(u^2 + bu + p\right)^F}{\left(1 - u\right)^B} \exp\left(\frac{C}{1 - u}\right) \exp\left(\frac{E - \frac{Da}{2}}{\sqrt{d - \frac{a^2}{4}}} \tan^{-1}(\frac{u + \frac{a}{2}}{\sqrt{d - \frac{a^2}{4}}})\right) \tag{55}$$

$$\times \exp\left(\frac{G - \frac{Fb}{2}}{\sqrt{p - \frac{b^2}{4}}} \tan^{-1}(\frac{u + \frac{b}{2}}{\sqrt{p - \frac{b^2}{4}}})\right)$$

where the coefficients *ABC DEFG* ,,, ,,, are found out by matrix solving (Gauss method) the system of seven equations obtained by integrating Eq. (54) ( 5; 2.731; *A B* = = *C* = −0.2; *DEF* =− =− =− 3.117; 4.037; 3.142; *G* = 0.747) .

#### **3.3 Discussion**

Fig. 5 *a* shows the dependences corresponding to the size distribution function, Eq. (52), computed for various magnitudes of *x* . It is hardly to represent such dependence for 1 *x* = (the Lifshitz-Slyozov distribution) at the same scale; that is why this case is illustrated in other scale at inset.

It is clearly seen that the maxima of curves reached at point *u*′ diminish, as *x* grows, taking the maximal magnitude for the curve 1 *x* = . Magnitude *u*′ itself is determined for the specified *x* from the following equation:

$$4\mu^6 + \mu^4 \left(6\mathbf{x} - 3\mathbf{x}^2\right) - 2\mu^3 \left(5\mathbf{x} - 3\mathbf{x}^2\right) + 4\mu \left(3\mathbf{x}^2 - 9\mathbf{x} + 6\right) - 5 \left(3\mathbf{x}^2 - 8\mathbf{x} + 5\right)\Big|\_{\mu=\mu'} = 0\tag{56}$$

One can see from Fig. 5 *b*, showing the same dependences normalized by their maxima, that as *x* grows, as magnitudes *u*′ are shifted to the left (diminish), cf. the inset.

Fig. 6 shows the results of comparison of experimentally obtained histogram with the Liwfitz-Slyozov distribution – (*а*), and the distribution (52) for 0.7 *x* = – (*b*). It is regularly *a priory* assumed (Gaponenko, 1996) that the experimentally obtained histogram shown in Fig. 6 and taken from the paper (Katsikas et al., 1990) that corresponds to the size distribution of nanoclusters of *C Sd* is described by the Lifshitz-Slyozov distribution. However, as one can see from Fig. 6,*b*, the dependence computed by us is narrower, being better fitting a

64 2 3 2 2 2

4 6 3 2 5 3 4 3 9 6 53 8 5 ( )

*du du du du du A B C Du E Fu G u u u uaud u bu p*

= + + + + ++ <sup>−</sup> − + + + <sup>+</sup>

Integrating Eq. (54), one obtains the analytical form of the relative size distribution function

2 2 ( ) exp exp tan ( )

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

*u u au d u bu p E u <sup>C</sup>*

1 1

*D F*

1 2 2

−

− − ⎝ ⎠

*b b p p*

4 4

where the coefficients *ABC DEFG* ,,, ,,, are found out by matrix solving (Gauss method) the system of seven equations obtained by integrating Eq. (54) ( 5; 2.731; *A B* = = *C* = −0.2;

Fig. 5 *a* shows the dependences corresponding to the size distribution function, Eq. (52), computed for various magnitudes of *x* . It is hardly to represent such dependence for 1 *x* = (the Lifshitz-Slyozov distribution) at the same scale; that is why this case is illustrated in

It is clearly seen that the maxima of curves reached at point *u*′ diminish, as *x* grows, taking the maximal magnitude for the curve 1 *x* = . Magnitude *u*′ itself is determined for the

One can see from Fig. 5 *b*, showing the same dependences normalized by their maxima, that

Fig. 6 shows the results of comparison of experimentally obtained histogram with the Liwfitz-Slyozov distribution – (*а*), and the distribution (52) for 0.7 *x* = – (*b*). It is regularly *a priory* assumed (Gaponenko, 1996) that the experimentally obtained histogram shown in Fig. 6 and taken from the paper (Katsikas et al., 1990) that corresponds to the size distribution of nanoclusters of *C Sd* is described by the Lifshitz-Slyozov distribution. However, as one can see from Fig. 6,*b*, the dependence computed by us is narrower, being better fitting a

as *x* grows, as magnitudes *u*′ are shifted to the left (diminish), cf. the inset.

( ) ( ) ( ) ( ) 64 2 3 2 2 <sup>2</sup> 4 6 3 2 5 3 4 3 9 6 53 8 5 0 *u u x x u x x ux x x x* + − − − + − +− − + = *u u*<sup>=</sup> ′ . (56)

2 2 exp tan ( )

*Fb <sup>b</sup> G u*

⎛ ⎞ ⎜ ⎟ − +

*g u u u u au d u bu p*

*u u x x u x x ux x x x dg u du*

( ) 1

for arbitrary 0 1 ≤ ≤ *x* :

*g u*

**3.3 Discussion** 

other scale at inset.

*A*

where 2.575 *a* = ; 0.575 *b* = − ; 2.398 *d* = ; *p* = 2.089 .

( ) ( ) ( )

×

*DEF* =− =− =− 3.117; 4.037; 3.142; *G* = 0.747) .

specified *x* from the following equation:

2 2 2 2

*B*

( ) ( ) ( ) ( )

22 2

*<sup>u</sup> <sup>u</sup> a a d d*

(54)

, (55)

1 2 2

−

*Da a*

⎛ ⎞

4 4

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

( ) ( )( )

2 2 2

′ + − − − + − +− − + = − <sup>=</sup> ′ − ++ ++

histogram than the curve in Fig. 6,*а*. It means that formation of quantum dots of *C Sd* in process of the Ostwald's ripening obtained by chemical evaporation is realized through mixed diffusion, with 70% share of matrix ( 0.7 *x* = ) and 30% dislocation ( 0.3 *x* = ) diffusion. For that, it is of importance that temporal growth of nanocrystals of *C Sd* obeys the cubic law, <sup>3</sup> *r* ~ *t* , cf. Eq. (50). It shows that, in first, that the size distribution is formed in process of the Ostwald's ripening, and, secondly, that growth of *C Sd* nanocrystals is limited, mainly, by matrix diffusion with the mentioned above share contribution of dislocation diffusion.

Fig. 5. Size distribution functions, Eq. (52), for various magnitudes of *x* - *а*; the same distributions normalized by their maxima - *b*

Fig. 6. Comparison with experimentally obtained histogram for nanocrystals of *C Sd* (Katsikas *et. al*., 1990) with the theoretical dependence: *а* – the Lifshitz-Slyozov distribution, Eq. (34), *b* – distribution corresponding to Eq. (55) for 0.7 *x* =

Let us note that there is the set of quantum dots in semiconductor compounds II-IV obtained by chemical evaporation techniques and having sizes from 1 to 5 nm (Gaponenko, 1996), for which the size distribution function occurs be narrower than one for the Lifshitz-Slyozov distribution.

Similarly to as crystalline gratings of numerous matters are controlled by simultaneous (combined) action of various connection types, the cluster growth goes on under mixed

Mass Transfer Between Clusters Under Ostwald's Ripening 125

The structure and phase dispersion (the particle size distribution function) at the late stages of decay of oversaturated solid solution, i.e. under the stage of the Ostwald's ripening, are

> υ, then a

<sup>3</sup> *t* , and the particle size distribution is governed

<sup>2</sup> *t* , and the size distribution function

θ

α θ<sup>−</sup> <sup>=</sup>

, (57)

*<sup>m</sup>* - volume of adatoms, *k* - the

2

− + <sup>=</sup> , 2 ( ) 1 cos

θ  , 2 sin *<sup>C</sup> l R* = π

θ,

θ

2

**4. Mass transfer between clusters in heterostructures. The generalized** 

determined by the mechanisms of mass transfer between the structure components. If the particle growth is limited by the coefficient of volume or matrix diffusion *D*

, then *r* changes as <sup>1</sup>

1

by the Lifshitz-Slyozov distribution function (Lifshits, Slyozov, 1958, 1961). But if the cluster growth is controlled by the processes at the boundary 'particle-matrix', being governed by

corresponds to the Wagner distribution (Wagner, 1961). In the case of simultaneous action of two mechanisms of growth, dispersion of extractions is described by the generalized LSW

Generalization of the LSW theory for surface disperse systems, in part, for island films, is of especial interest. This generalization becomes urgent now in connection with development of nanotechnologies and forming nanostructures (Alekhin, 2004; Alfimov *et al*., 2004; Andrievskii, 2002; Dunaevskii *et al*., 2003; Dmitriev, Reutov, 2002; Roko, 2002; Gerasimenko, 2002). In part, semiconductor heterostructures with quantum dots obtained under the Stranskii-Kastranov self-organizing process find out numerous practical applications (Bartelt, Evans, 1992; Bartelt *et al.,* 1996; Goldfarb *et al.,* 1997a, 1997b; Joyce *et al.,* 1998; Kamins *et al.,* 1999; Pchelyakov *et al.,* 2000; Ledentsov *et al.,* 1998; Vengrenovich *et al.,* 2001b,

Chakraverty (Chakraverty, 1967) for the first time applied the LSW theory to describe evolution of structure of discrete films containing of separate islands (clusters) of the form of spherical segments, cf. Fig. 8. Within the Chakraverty model, a film consists of separate cupola-like islands, which are homogeneously (in statistics sense) distributed into

One can see from Fig. 8 that cupola-like clusters are the part of a sphere of radius *RC* , with

1

α θ

Concentration of adatoms at the cluster base, *Cr* , is given by the Gibbs-Thomson formula:

exp <sup>1</sup> *m m CC P C P <sup>r</sup> kT kT* υ

Boltzmann's constant, Δ*P* - the Laplasian pressure caused by island surface's curvature. It can be determined by equaling the work necessary for diminishing of an island volume by

> σ

∞ ∞ ⎛ ⎞⎛ ⎞ = Δ ≈ +Δ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

V or 2

*dS Pd dS P*

. That is why, the radius of base of island, *r* , length of its perimeter,

2 3cos cos 4 θ

> υ

υ

V *<sup>C</sup>*

σ α θ

*d R*

( ) 2 1

α θ

Δ = Δ= = , (58)

oversaturated 'sea' (solution) of atoms absorbed by a substrate, so-called adatoms.

, where ( )

*l* , its surface, *S* , and volume, *V* , can be expressed through *RC* : *r R*= *<sup>C</sup>* sin

**Chakraverty-Wagner distribution** 

mean cluster size, *r* , changes in time as

distribution (Vengrenovich *et al.,* 2007b).

θ

, ( ) <sup>3</sup>

4 3 *V R* = π *C*α θ

1

where *C*<sup>∞</sup> - equilibrium concentration at temperature *T* ,

( )

σ

*d*V to the caused by it free energy of island surface:

β

the kinetic coefficient

2005, 2006a, 2006b, 2007a).

the boundary angle

<sup>2</sup> 4 *S R* = π *C*α θ

( ) <sup>2</sup>

(Hirth, Pound, 1963).

diffusion, where only one of the types of diffusion can be predominant (matrix, surface, dislocation at the grain boundaries, etc.).

Fig. 7. Example of accidental concurrence of experimentally obtained histogram with the theoretical dependence, Eq. (55), for heterogeneous nucleation of aluminum nanoclusters (Aronin *et al.,* 2001), much earlier than the stage of the Ostwald's ripening comes: *а – x* = 0 *; b* – 0.4 *x* =

Note, the idea of combined action of several mechanisms of diffusion mass transfer has been formulated in several earlier papers (Slyzzov *et al.*, 1978; Sagalovich and Slyozov, 1987). However, the size distribution function for particles coherently connected with the matrix Eq. (55) for combined action of two mechanisms of mass transfer, i.e. diffusion along dislocations and matrix diffusion, has been firstly found by us (Vengrenovich *et al.,* 2007a).

Let us emphasize once more point connected with the study of particle growth under the Ostwald's ripening. It occurs that comparison of the experimentally obtained histograms with the theoretically found dependences does not provide an unambiguous answer the question: What is the mechanism of particle growth? and: is the stage of the Ostwald's ripening occurred? To elucidate these questions, the temporal dependences for a mean particle size, *r* , are needed.

For example, Fig. 7 shows comparison of experimental histograms for nanoclusters of aluminum obtained by annealing of amorphous alloy *Al Ni Yb* 86 11 3 (Aronin *et al.,* 2001) with theoretical dependence, Eq. (55), for (*а*) – 0 *x* = , and (*b*) – 0.4 *x* = . Satisfactory concurrence, however, is accidental. As it is shown in paper (Aronin *et al.,* 2001), the LWS theory is not applicable to this case. Growth of aluminum nanocrystals obeys parabolic dependence *r* ~ 1 2 *t* , rather than to the dependence *r* ~ 1 6 *t* . Histograms in Fig. 7 correspond to heterogeneous nucleation of aluminium clusters that precedes the Ostwald's ripening, which follows much later.

Thus, for estimation of a share (percentage) of the each component, *dj* and *vj* , in the diffusion flow, one must compare both experimentally obtained histograms with the theoretical dependences and temporal dependences for mean (critical) particle sizes. In the case of metallic alloys strengthened by disperse particles, it enables establishing the mechanism of particle's enlargement, while for quasi-zero-dimension semiconductor structures it makes possible to study, under the Ostwald's ripening, nanoclusters (quantum dots) obtaining by chemical evaporation techniques.

diffusion, where only one of the types of diffusion can be predominant (matrix, surface,

0.0

0.2

0.4

0.6

*g(u)/gmax*

Fig. 7. Example of accidental concurrence of experimentally obtained histogram with the theoretical dependence, Eq. (55), for heterogeneous nucleation of aluminum nanoclusters (Aronin *et al.,* 2001), much earlier than the stage of the Ostwald's ripening comes: *а – x* = 0 *;* 

Note, the idea of combined action of several mechanisms of diffusion mass transfer has been formulated in several earlier papers (Slyzzov *et al.*, 1978; Sagalovich and Slyozov, 1987). However, the size distribution function for particles coherently connected with the matrix Eq. (55) for combined action of two mechanisms of mass transfer, i.e. diffusion along dislocations and matrix diffusion, has been firstly found by us (Vengrenovich *et al.,* 2007a). Let us emphasize once more point connected with the study of particle growth under the Ostwald's ripening. It occurs that comparison of the experimentally obtained histograms with the theoretically found dependences does not provide an unambiguous answer the question: What is the mechanism of particle growth? and: is the stage of the Ostwald's ripening occurred? To elucidate these questions, the temporal dependences for a mean

For example, Fig. 7 shows comparison of experimental histograms for nanoclusters of aluminum obtained by annealing of amorphous alloy *Al Ni Yb* 86 11 3 (Aronin *et al.,* 2001) with theoretical dependence, Eq. (55), for (*а*) – 0 *x* = , and (*b*) – 0.4 *x* = . Satisfactory concurrence, however, is accidental. As it is shown in paper (Aronin *et al.,* 2001), the LWS theory is not applicable to this case. Growth of aluminum nanocrystals obeys parabolic dependence

heterogeneous nucleation of aluminium clusters that precedes the Ostwald's ripening,

Thus, for estimation of a share (percentage) of the each component, *dj* and *vj* , in the diffusion flow, one must compare both experimentally obtained histograms with the theoretical dependences and temporal dependences for mean (critical) particle sizes. In the case of metallic alloys strengthened by disperse particles, it enables establishing the mechanism of particle's enlargement, while for quasi-zero-dimension semiconductor structures it makes possible to study, under the Ostwald's ripening, nanoclusters (quantum

0.8

*b*

1.0

0.0 0.2 0.4 0.6 0.8 1.0

*x=0.4*

*u*

*t* . Histograms in Fig. 7 correspond to

dislocation at the grain boundaries, etc.).

0.0 0.2 0.4 0.6 0.8 1.0

*x=0*

*t* , rather than to the dependence *r* ~ 1 6

dots) obtaining by chemical evaporation techniques.

*u*

0.0

*b* – 0.4 *x* =

*r* ~ 1 2

particle size, *r* , are needed.

which follows much later.

0.2

0.4

0.6

*g(u)/gmax*

0.8

*a*

1.0
