**2.2 Temporal dependences of** *gr* **and** *kr*

One of the main parameters of the LSW theory is the ratio *<sup>g</sup> <sup>k</sup> r r* (in terms of the papers (Lifshitz and Slyozov, 1958, 1961), locking point *u*<sup>0</sup> ), whose magnitude together with the equation for the rate of growth (14) or (15) provides integration of Eq. (8) after separation of variables and determination of the analytical form of the size distribution function. This ratio can be determined from the dependence of the specific rate of growth *r r* on *r* , that is schematically shown in Fig. 1, where *r* is determined by Eq. (14) or (15) (Vengrenovich, 1982).

Fig. 1. Schematic dependence of the specific rate of growth *<sup>r</sup> r* on *r*

At the point where the rate of growth on the unite length of cluster radius reaches its maximal magnitude, derivation equals zero:

$$\left. \frac{d}{dr} \left( \frac{\dot{r}}{r} \right) \right|\_{r=r\_{\rm g}} = 0 \,, \tag{16}$$

From the physical point of view, it means that the maximal size of *gr* is reached for the particle, for which the rate of growth of the unit of length of its radius is maximal. Thus, one obtains from Eq. (16):

$$\frac{r\_{\mathcal{S}}}{r\_k} = \frac{2+\infty}{1+\infty}.\tag{17}$$

Assuming in Eq. (14) *<sup>g</sup> r r* = and replacing the ratio *<sup>g</sup> k r r* by its magnitude from Eq. (17), one obtains by integration:

$$r\_{\mathcal{g}}^3 = A^\* \frac{t}{\varkappa(1+\varkappa)} \, ^\prime \tag{18}$$

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

112 Mass Transfer - Advanced Aspects

One of the main parameters of the LSW theory is the ratio *<sup>g</sup> <sup>k</sup> r r* (in terms of the papers (Lifshitz and Slyozov, 1958, 1961), locking point *u*<sup>0</sup> ), whose magnitude together with the equation for the rate of growth (14) or (15) provides integration of Eq. (8) after separation of variables and determination of the analytical form of the size distribution function. This ratio can be determined from the dependence of the specific rate of growth *r r* on *r* , that is schematically shown in Fig. 1, where *r* is determined by Eq. (14) or (15) (Vengrenovich,

> *d r dr r r r* ⎛ ⎞

<sup>0</sup> *<sup>g</sup>*

*r* on *r*

⎜ ⎟ <sup>=</sup> ⎝ ⎠ <sup>=</sup>

At the point where the rate of growth on the unite length of cluster radius reaches its

*<sup>g</sup> r r*

From the physical point of view, it means that the maximal size of *gr* is reached for the particle, for which the rate of growth of the unit of length of its radius is maximal. Thus, one

> 2 1

*r x r x* + = +

*g k*

3 \* 1 *<sup>g</sup> <sup>t</sup> r A x x* <sup>=</sup> <sup>+</sup>

*d r dr r* <sup>=</sup> ⎛ ⎞ <sup>=</sup> ⎜ ⎟ ⎝ ⎠

0

*k r r*

( )

*r*

. (16)

. (17)

by its magnitude from Eq. (17), one

, (18)

**2.2 Temporal dependences of** *gr* **and** *kr*

*r r* 

<sup>0</sup>*rk* 

Fig. 1. Schematic dependence of the specific rate of growth *<sup>r</sup>*

Assuming in Eq. (14) *<sup>g</sup> r r* = and replacing the ratio *<sup>g</sup>*

maximal magnitude, derivation equals zero:

obtains from Eq. (16):

obtains by integration:

1982).

$$r\_k^3 = A^\* \frac{\left(1 + \chi\right)^2}{\chi \left(2 + \chi\right)^3} t \cdot \tag{19}$$

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

$$r\_{\mathcal{S}}^{\mathcal{S}} = \frac{1}{2} A^\* t \; \; \; r\_k^{\mathcal{S}} = \frac{4}{27} A^\* t \; \; \; \; \frac{r\_{\mathcal{S}}}{r\_k} = \frac{3}{2} \, \, \, \, \tag{20}$$

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

$$r\_{\mathcal{S}}^2 = B^\* \frac{t}{1 - \mathbf{x}^2} \,' \tag{21}$$

where <sup>2</sup> \* <sup>2</sup> *mC <sup>B</sup> kT* συ β<sup>∞</sup> = , or:

$$r\_k^2 = B^\* \frac{1+\chi}{\left(1-\chi\right)\left(2+\chi\right)^2} t \,. \tag{22}$$

Eqs. (21) and (22) describe changing in time cluster sizes, when growth of them is controlled by the kinetic coefficient β , with the share contribution *x* of matrix diffusion. If 0 *x* = , then the process of growth is fully controlled by kinetics of transition through the interface 'cluster-matrix':

$$r\_{\mathcal{g}}^2 = B^\* t \; \; \; r\_k^2 = \frac{1}{4} B^\* t \; \; \; \; \frac{r\_{\mathcal{g}}}{r\_k} = 2 \; \; \; \tag{23}$$

#### **2.3 Size distribution function**

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

$$f\left(r,\mathbf{t}\right) = \varphi\left(r\_{\mathcal{S}}\right)\mathbf{g'}\left(\boldsymbol{\mu}\right)\,\tag{24}$$

where *g*′(*u*) is the relative size distribution function of clusters, *g r u <sup>r</sup>* <sup>=</sup> .

To determine the function ϕ (*rg* ) , let us apply the conservation law for mass of disperse phase:

$$M = \frac{4}{3}\pi\rho\int\_0^{r\_\ddagger} r^3 f(r, t) dr \,, \tag{25}$$

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

Mass Transfer Between Clusters Under Ostwald's Ripening 115

2 4 12 10 5 3 3 3 , ,

*xxxx D A <sup>x</sup> <sup>x</sup> <sup>x</sup> <sup>x</sup>*

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

() ( ) <sup>5</sup> <sup>3</sup> 1 exp <sup>1</sup> *gu u u*

Within the interval 0 1 ≤ *x* ≤ , the size distribution function is represented by the generalized LSW function. However, for graphic representation of the size distribution function one must compute following Eq. (28), where the conservation law for mass (volume) of a film is

To obtain the distributions represented by Eqs. (34) and (35) in the form derived by Lifshitz and Slyozov (Lifshitz and Slyozov, 1958, 1961) and by Wagner (Wagner, 1961), one must go

> *k r r* ρ= :

*u*

0.0

0.2

0.4

0.6

 *g(u)/gmax*

Fig. 2. The curves computed following Eq. (28): а – depending on *x*; b – normalized by

Fig. 2,a illustrates the curves corresponding to the distribution Eq. (28) computed for various magnitudes of the parameter *x* with interval 0.1 Δ*x* = . Inset shows the Wagner function ( ) *x* = 0 , which is hardly to be shown in the main graph in its scale. One can see gradual

0.8

1.0

For 0 *x* = , 5 *B* = , 3 *C* = − , *D* = −1 Eq. (32) corresponds to the Wagner distribution:

<sup>⎪</sup> + + ++ <sup>=</sup> − = <sup>+</sup> <sup>+</sup> <sup>+</sup> <sup>+</sup> ⎪⎩

For 1 *x* = , *B* = 11 3 , 1 *C* = − , *D* = −7 3 Eq. (32) corresponds to the Lifshitz-Slyozov

4 8 6 21 , 2 3 2 1.

*A A*

432

− − ⎛ ⎞ ′ =− + −⎜ ⎟ ⎝ ⎠ <sup>−</sup> . (34)

*u* <sup>−</sup> ⎛ ⎞ ′ =− −⎜ ⎟ ⎝ ⎠ <sup>−</sup> . (35)

*u*

0

= = = , where *u*<sup>0</sup> − the locking

*x=0 x=1*

*b*

0.0 0.2 0.4 0.6 0.8 1.0

*u*

ρ

*k g kg r r r*

*r rr u*

(33)

43 2 2

*xx x x xx B C*

<sup>⎧</sup> + + + + ++ <sup>⎪</sup> <sup>=</sup> <sup>=</sup> <sup>−</sup> <sup>⎪</sup>

432

⎨

distribution:

taken into account.

from the variable

*u*

**2.4 Discussion** 

0.00000

maximal magnitudes

0.00005

0.00010

*g(u)*

0.00015

0.00020

*g k r*

8 10-8 6 10-8 4 10-8

point ( <sup>0</sup>

*g r u*

0.0 0.2 0.4 0.6 0.8 1.0

2 10-8 *x=0.9*

*<sup>r</sup>* <sup>=</sup> ), and *kr* <sup>−</sup> the critical radius.

0.0 0.2 0.4 0.6 0.8 1.0

*u*

*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=1 a x=0*

*<sup>r</sup>* <sup>=</sup> to the variable

*A*

$$
\varphi \left( r\_{\mathcal{S}} \right) = \frac{\mathcal{Q}}{r\_{\mathcal{S}}^4} r\_{\mathcal{S}} \tag{26}
$$

where ( ) 1 3 0 4 3 *<sup>M</sup> <sup>Q</sup>* πρ *u g u du* = ′ ∫

Substituting Eq. (26) in Eq. (24), one obtains:

.

$$f\left(r,t\right) = \frac{\mathbb{Q}}{r\_{\mathcal{g}}^3} \mathbf{g}'\left(\mu\right) = \frac{\mathbf{g}\left(\mu\right)}{r\_{\mathcal{g}}^3},\tag{27}$$

where:

$$\log(\mu) = Q \cdot \mathcal{g}'(\mu) \,. \tag{28}$$

The relative size distribution function *g*′(*u*) is determined from the continuity equation. For that, one substitutes in Eq. (8) the magnitude *f* (*r t*, ) from Eq. (24) and takes into account Eq. (26), as well as the magnitude of *r* from Eqs. (14) or (15). After the mentioned substitution and transition in Eq. (8) from differentiation on *r* and *t* to differentiation on

$$\mu = \frac{r}{r\_{\text{g}}} \quad (\frac{\partial}{\partial r} = \frac{\partial}{\partial u}\frac{du}{dr}) \text{ where } \frac{du}{dr} = \frac{1}{r\_{\text{g}}};\\\frac{\partial}{\partial t} = \frac{\partial}{\partial u}\frac{du}{dr\_{\text{g}}}\frac{dr\_{\text{g}}}{dt}, \text{ where } \frac{\partial u}{\partial r\_{\text{g}}} = -\frac{u}{r\_{\text{g}}}\text{)}\text{, the variables are: }$$

separated, and Eq. (8) takes the form:

$$\frac{d\lg'(u)}{\lg'(u)} = -\frac{4\nu\_{\mathcal{S}} - \frac{1}{u^2}\frac{d\nu}{du} + 2\frac{\nu}{u^3}}{u\nu\_{\mathcal{S}} - \frac{\nu}{u^2}}du\,u\,d\nu\_{\mathcal{S}}\tag{29}$$

where it is taken into account that:

$$\boldsymbol{\nu} = \frac{r^2 \dot{r}}{\boldsymbol{B}^\*} = \left( 1 + \frac{1-\chi}{\chi} \boldsymbol{\mu} \right) \left( \frac{2+\chi}{1+\chi} \boldsymbol{\mu} - 1 \right), \ \boldsymbol{\nu}\_g = \frac{r\_g^2}{\boldsymbol{B}^\*} \frac{d r\_g}{dt} = \boldsymbol{\nu} \Bigg|\_{\boldsymbol{\mu} = 1} = \frac{1}{\chi \left( 1+\chi \right)} \tag{30}$$

Substituting the magnitudes υ , υ*g* and *<sup>d</sup> du* υ into Eq. (29), after straightforward transformations one obtains:

$$\frac{d\mathbf{g'(u)}}{d\mathbf{g'(u)}} = -\frac{4\boldsymbol{\mu}^3 + \boldsymbol{\mu}\left(2\boldsymbol{\omega}^2 + 2\boldsymbol{\omega} - 1\right) - 2\left(1 + \mathbf{x}\right)\mathbf{x}}{\boldsymbol{\mu}\left(1 - \boldsymbol{\mu}\right)^2 \left(\boldsymbol{\mu} + \boldsymbol{\omega} + \boldsymbol{\omega}^2\right)} d\boldsymbol{u} \,\tag{31}$$

Integration of Eq. (31) provides obtaining the analytical form of the generalized LSW distribution, which has been for the first time obtained by us (Vengrenovich *et al*., 2007b) :

$$\log'(\mu) = \mu^2 \left(1 - \mu\right)^{-B} \left(\mu + \mathbf{x}^2 + \mathbf{x}\right)^D \exp\left[\frac{\mathbf{C}}{1 - \mu}\right],\tag{32}$$

where

$$\begin{cases} B = \frac{2\mathbf{x}^4 + 4\mathbf{x}^3 + 12\mathbf{x}^2 + 10\mathbf{x} + 5}{A}, & \mathcal{C} = -\frac{3\mathbf{x}^2 + 3\mathbf{x} + 3}{A}, \\ D = -\frac{4\mathbf{x}^4 + 8\mathbf{x}^3 + 6\mathbf{x}^2 + 2\mathbf{x} + 1}{A}, & A = \mathbf{x}^4 + 2\mathbf{x}^3 + 3\mathbf{x}^2 + 2\mathbf{x} + 1. \end{cases} \tag{33}$$

For 1 *x* = , *B* = 11 3 , 1 *C* = − , *D* = −7 3 Eq. (32) corresponds to the Lifshitz-Slyozov distribution:

$$\log'(u) = u^2 \left(1 - u\right)^{-1} \bigvee\_{3}^{\prime} \left(u + 2\right)^{-\frac{7}{3}} \exp\left(-\frac{1}{1 - u}\right). \tag{34}$$

For 0 *x* = , 5 *B* = , 3 *C* = − , *D* = −1 Eq. (32) corresponds to the Wagner distribution:

$$\log'(\mu) = \mu \left(1 - \mu\right)^{-5} \exp\left(-\frac{3}{1 - \mu}\right). \tag{35}$$

Within the interval 0 1 ≤ *x* ≤ , the size distribution function is represented by the generalized LSW function. However, for graphic representation of the size distribution function one must compute following Eq. (28), where the conservation law for mass (volume) of a film is taken into account.

To obtain the distributions represented by Eqs. (34) and (35) in the form derived by Lifshitz and Slyozov (Lifshitz and Slyozov, 1958, 1961) and by Wagner (Wagner, 1961), one must go from the variable *g r u <sup>r</sup>* <sup>=</sup> to the variable *k r r* ρ = : 0 *k g kg r r r u r rr u* ρ = = = , where *u*<sup>0</sup> − the locking *r*

point ( <sup>0</sup> *g k u <sup>r</sup>* <sup>=</sup> ), and *kr* <sup>−</sup> the critical radius.

#### **2.4 Discussion**

114 Mass Transfer - Advanced Aspects

( ) *<sup>g</sup>* <sup>4</sup> *g <sup>Q</sup> <sup>r</sup> r*

( ) ( ) ( ) 3 3 , *g g*

The relative size distribution function *g*′(*u*) is determined from the continuity equation. For that, one substitutes in Eq. (8) the magnitude *f* (*r t*, ) from Eq. (24) and takes into account Eq. (26), as well as the magnitude of *r* from Eqs. (14) or (15). After the mentioned substitution and transition in Eq. (8) from differentiation on *r* and *t* to differentiation on

> ; , *<sup>g</sup> g g du du dr dr r t u dr dt* ∂ ∂ = = ∂ ∂

> > 2 3

υ

<sup>1</sup> 4 2 *<sup>g</sup>*

*dg u u u du du*

υ

− + ′ = − ′ <sup>−</sup>

*g* and *<sup>d</sup>*

3 2

*g u u u uxx* ′ + + −− + = − ′ − ++

*gu u u u x x*

1 *dg u u u x x xx*

*d*

υ

*g*

, ( )

υ

*du* υ

4 2 2 1 21

Integration of Eq. (31) provides obtaining the analytical form of the generalized LSW distribution, which has been for the first time obtained by us (Vengrenovich *et al*., 2007b) :

() ( ) 2 2 1 ( ) exp <sup>1</sup>

<sup>−</sup> <sup>⎡</sup> <sup>⎤</sup> ′ = − ++ <sup>⎢</sup> <sup>⎥</sup> <sup>⎣</sup> <sup>−</sup> <sup>⎦</sup>

( ) ( )

2 2

*<sup>B</sup> <sup>D</sup> C*

( ) ( )

*g*

υ

2

2 \*

*g g*

*r dr*

υ

*u*

*<sup>Q</sup> g u frt gu r r*

= , (26)

= = ′ , (27)

*g*(*u Q* ) = ⋅ *g*′(*u*) . (28)

where ),

*g g u u r r* <sup>∂</sup> = − <sup>∂</sup>

the variables are

. (30)

. (31)

, (32)

, (29)

1

into Eq. (29), after straightforward

1 1

*B dt x x u*

*du*

*u*

 υ= = = <sup>=</sup> <sup>+</sup>

ϕ

where

where:

*g r u*

where

*<sup>r</sup>* <sup>=</sup> ( , *du r u dr* ∂ ∂ <sup>=</sup> ∂ ∂

separated, and Eq. (8) takes the form:

where it is taken into account that:

υ

Substituting the magnitudes

transformations one obtains:

2 \* ( )

Substituting Eq. (26) in Eq. (24), one obtains:

.

where <sup>1</sup>

( ) ( )

1 2 1 1 1

> υ , υ

*u u*

*rr x x*

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

*B x x*

( ) ( )

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

*u g u du*

′ ∫

1 3 0

4 3

=

*<sup>M</sup> <sup>Q</sup>*

πρ

Fig. 2. The curves computed following Eq. (28): а – depending on *x*; b – normalized by maximal magnitudes

Fig. 2,a illustrates the curves corresponding to the distribution Eq. (28) computed for various magnitudes of the parameter *x* with interval 0.1 Δ*x* = . Inset shows the Wagner function ( ) *x* = 0 , which is hardly to be shown in the main graph in its scale. One can see gradual

Mass Transfer Between Clusters Under Ostwald's Ripening 117

However, the distribution Eq. (28) computed for two mechanisms of mass transfer controlled by the volume diffusion coefficient and kinetics of transition of solved atoms through the interface 'cluster-martix', i.e. by the kinetic coefficient *β*, has been firstly obtained in analytic form by us. As the rate of forming the chemical connection is higher, as more simply solved atoms overcome potential barrier at the interface 'cluster-martix'. In this case, the rate of cluster growth is in less degree controlled by kinetics at the interface and in more degree by the diffusion processes of mass transfer. For that, the contribution of diffusion flow *jv* in general flow of matter *j* to (from) a particle increases, and the size distribution

function becomes more and more close to the Lifshitz-Slyozov distribution, Eq. (34).

between the flows (*x* ⋅ 100%) and find, in this way, what mechanism is predominant.

parameter for the choice of theoretical curve and comparison with desired histogram. It follows from Fig. 3 that increasing of the exposure time for temperature 300ºС up to 350 hours results in changing the mechanism of particle growth from one limited predominantly by diffusion processes of mass transfer, cf. fragments a – 0.8 *x* = ; b – 0.9 *x* = , to one controlled predominantly by kinetics at the interface 'cluster-matrix', cf. fragment с – *x* = 0.2 . Increasing the exposure temperature to 400ºС leads to particle growth under conditions controlled predominantly by kinetics at the interface, cf. fragments d – 0.3 *x* = ; е

Besides, knowing *x* , one can find the ratio *<sup>g</sup>*

0.0 0.2 0.4 0.6 0.8 1.0

*u*

– 0.2 *x* = .

coefficient,

0.0

*et al*., 2005)

0.2

0.4

0.6

*g(u)/gmax*

0.8

1.0

2005).

β

*508K, 1 min*

*r g* /*r k =1.83*

Fig. 3 illustrates the results of comparison of the theoretical dependence, Eq. (28), with the experimentally obtained histograms of nano-scale particles *Al Sc* 3 in alloys *Al Sc* − (Marquis and Seidman, 2001) corresponding to temperature 300ºС and exposure times *а* – 6, *в –* 72, *с –* 350 hours; to temperature 400ºС and exposure times *d* – 1, *е* – 5 hours. Using the magnitudes of *x* from the results of comparison, one can determine percentage ratio

> *k r r*

The possibility for implementation of the considered mechanism of particle growth controlled simultaneously both by the volume diffusion coefficient, *Dv* , and by the kinetic

aluminium obtained under crystallization of amorphous alloy *Al Ni Y Co* 85 8 5 2 (Nitsche *et al.,*

*a*

Fig. 4. Comparison with experimental histograms for nano-crystals *Al* , obtained under annealing of amorphous alloy *Al Ni Y Co* 85 8 5 2 (508°К) during: а – 1 min; b – 2.5 min (Nitsche

*x=0.2*

, is also proved by the experimentally obtained histograms for nano-crystals of

0.0

0.2

0.4

0.6

*g(u)/gmax*

0.8

*r g* /*r k =1.91*

*508K, 2.5 min*

1.0

that then may be used as the evaluation

0.0 0.2 0.4 0.6 0.8 1.0

*u*

*b*

*x=0.1*

transition from the Lifshitz-Slyozov distribution, Eq. (34) ( 1) *x* = , to the Wagner distribution, Eq. (35). The same curves normalized by their maxima are shown in Fig. 2, b. In this form, these curves are suitable for comparison with the corresponding normalized experimentally obtained histograms.

Note, that computation of the theoretical curve under simultaneous (combined) action of two mass transfer mechanisms, *viz.* volume diffusion and chemical reaction at the interface 'extraction-matrix' has been performed earlier by using numerical techniques (Sagalovich, Slyozov, 1987).

Fig. 3. Comparison of the dependence (28) with the experimentally obtained histograms of nano-scale particles *Al Sc* 3 in alloys *Al Sc* − (Marquis and Seidman, 2001) for various temperatures and exposure times shown in fragments a, b, c, d, and e

transition from the Lifshitz-Slyozov distribution, Eq. (34) ( 1) *x* = , to the Wagner distribution, Eq. (35). The same curves normalized by their maxima are shown in Fig. 2, b. In this form, these curves are suitable for comparison with the corresponding normalized experimentally

Note, that computation of the theoretical curve under simultaneous (combined) action of two mass transfer mechanisms, *viz.* volume diffusion and chemical reaction at the interface 'extraction-matrix' has been performed earlier by using numerical techniques (Sagalovich,

0.0

0.0

0.2

0.4

0.6

*g(u)/gmax*

0.0 0.2 0.4 0.6 0.8 1.0

Fig. 3. Comparison of the dependence (28) with the experimentally obtained histograms of nano-scale particles *Al Sc* 3 in alloys *Al Sc* − (Marquis and Seidman, 2001) for various

*u*

0.8

*400o*

*e x=0.2*

*r g* /*r k =1.77*

*C, 1 hour*

1.0

0.2

0.4

0.6

*g(u)/gmax*

*c*

*a x=0.8*

0.8

*300o*

*r g* /*r k =1.53*

*C, 72 hours*

1.0

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

*u*

*b x=0.9*

*d x=0.3*

obtained histograms.

*300o*

*r g* /*r k =1.55*

*C, 6 hours*

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

*g(u)/gmax*

0.8

*400O*

temperatures and exposure times shown in fragments a, b, c, d, and e

*r g* /*r k =1.83*

*C, 5 hours*

1.0

*u*

*x=0.2 300O*

0.0 0.2 0.4 0.6 0.8 1.0

*u*

*C, 350 hours*

Slyozov, 1987).

0.0

0.2

0.4

0.6

*g(u)/gmax*

0.8

*r g* /*r k =1.83*

1.0

0.0

0.2

0.4

0.6

*g(u)/gmax*

0.8

1.0

However, the distribution Eq. (28) computed for two mechanisms of mass transfer controlled by the volume diffusion coefficient and kinetics of transition of solved atoms through the interface 'cluster-martix', i.e. by the kinetic coefficient *β*, has been firstly obtained in analytic form by us. As the rate of forming the chemical connection is higher, as more simply solved atoms overcome potential barrier at the interface 'cluster-martix'. In this case, the rate of cluster growth is in less degree controlled by kinetics at the interface and in more degree by the diffusion processes of mass transfer. For that, the contribution of diffusion flow *jv* in general flow of matter *j* to (from) a particle increases, and the size distribution function becomes more and more close to the Lifshitz-Slyozov distribution, Eq. (34).

Fig. 3 illustrates the results of comparison of the theoretical dependence, Eq. (28), with the experimentally obtained histograms of nano-scale particles *Al Sc* 3 in alloys *Al Sc* − (Marquis and Seidman, 2001) corresponding to temperature 300ºС and exposure times *а* – 6, *в –* 72, *с –* 350 hours; to temperature 400ºС and exposure times *d* – 1, *е* – 5 hours. Using the magnitudes of *x* from the results of comparison, one can determine percentage ratio between the flows (*x* ⋅ 100%) and find, in this way, what mechanism is predominant.

Besides, knowing *x* , one can find the ratio *<sup>g</sup> k r r* that then may be used as the evaluation

parameter for the choice of theoretical curve and comparison with desired histogram.

It follows from Fig. 3 that increasing of the exposure time for temperature 300ºС up to 350 hours results in changing the mechanism of particle growth from one limited predominantly by diffusion processes of mass transfer, cf. fragments a – 0.8 *x* = ; b – 0.9 *x* = , to one controlled predominantly by kinetics at the interface 'cluster-matrix', cf. fragment с – *x* = 0.2 . Increasing the exposure temperature to 400ºС leads to particle growth under conditions controlled predominantly by kinetics at the interface, cf. fragments d – 0.3 *x* = ; е – 0.2 *x* = .

The possibility for implementation of the considered mechanism of particle growth controlled simultaneously both by the volume diffusion coefficient, *Dv* , and by the kinetic coefficient, β , is also proved by the experimentally obtained histograms for nano-crystals of aluminium obtained under crystallization of amorphous alloy *Al Ni Y Co* 85 8 5 2 (Nitsche *et al.,* 2005).

Fig. 4. Comparison with experimental histograms for nano-crystals *Al* , obtained under annealing of amorphous alloy *Al Ni Y Co* 85 8 5 2 (508°К) during: а – 1 min; b – 2.5 min (Nitsche *et al*., 2005)

Mass Transfer Between Clusters Under Ostwald's Ripening 119

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

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

2 1 4 *<sup>m</sup> dr jv dt* π

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

= =

*R r R r*

⎛⎞ ⎛⎞ = ⋅ <sup>+</sup> ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ (40)

π

0 2

<sup>1</sup> <sup>2</sup> 2 41

π

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

*<sup>m</sup> d v dr v C Z q <sup>r</sup> D qD r dt r R <sup>r</sup> <sup>r</sup>*

Designating, as previously, the shares *vj* and *dj* in the general flow *j* as *x* and (1 − *x*) ,

2 3/2 3

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

1 1 1 1 \*

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

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

2 \* 1 *m d*

3 2 2 3

*<sup>m</sup> <sup>g</sup> dr v C Dv x r r dt R x r r r*

*dr v C Zq D xr r dt rR r x r*

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

*Z q dC dC jD q D r r dR dR*

Substituting Eq. (40) in Eq. (39) and taking into account that

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

4 2

respectrively, one can represent the rate of growth, Eq. (41), in the form

π

σ

0 5 2 3 <sup>1</sup> 1 1

2 1/2

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

1/2

π

where we take into account that, in a flow *dj* , there is

, where

π σ

4 \*

σ

σ

function of the ratio of flows *dj* and *vj* .

**sizes of clusters** 

2002).

σ

or:

\* 2 <sup>2</sup> <sup>1</sup> <sup>1</sup> *vm <sup>r</sup> <sup>С</sup> R r r*

∞ ⎛ ⎞ = ⋅− ⎜ ⎟ <sup>Τ</sup> ⎝ ⎠

κ

inequality (36), respectively:

*d v jj j* = + . (38)

*<sup>r</sup>* <sup>=</sup> , (39)

1/2 0 2 *Z q <sup>Z</sup>* π

is the surface energy, *C*∞ is the equilibrium concentration

π

*g*

κ

κ

κ

*<sup>r</sup>* <sup>=</sup> (Vengrenovich *et al*.,

*R r*

*dC dR* <sup>=</sup> ⎛ ⎞ ⎜ ⎟ ⎝ ⎠

. (41)

, (42)

. (43)

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* = (fragment b).

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 separately by Lifshitz and Slyozov, and Wagner.
