**4. Mass transfer between clusters in heterostructures. The generalized Chakraverty-Wagner distribution**

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 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*υ, then a

mean cluster size, *r* , changes in time as 1 <sup>3</sup> *t* , and the particle size distribution is governed

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 the kinetic coefficient β , then *r* changes as <sup>1</sup> <sup>2</sup> *t* , and the size distribution function 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 distribution (Vengrenovich *et al.,* 2007b).

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

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

One can see from Fig. 8 that cupola-like clusters are the part of a sphere of radius *RC* , with the boundary angle θ . That is why, the radius of base of island, *r* , length of its perimeter, *l* , its surface, *S* , and volume, *V* , can be expressed through *RC* : *r R*= *<sup>C</sup>* sinθ , 2 sin *<sup>C</sup> l R* = π θ , 4 2 2 3cos cos θθθ

$$S = 4\pi R\_{\text{C}}^2 a\_2(\theta), \quad V = \frac{4}{3}\pi R\_{\text{C}}^3 a\_1(\theta), \quad \text{where} \quad a\_1(\theta) = \frac{2 - 3\cos\theta + \cos^2\theta}{4}, \quad a\_2(\theta) = \frac{1 - \cos\theta}{2}$$

(Hirth, Pound, 1963).

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

$$\mathbf{C}\_r = \mathbf{C}\_\phi \exp\left(\Delta P \frac{\nu\_m}{kT}\right) \approx \mathbf{C}\_\phi \left(\mathbf{1} + \Delta P \frac{\nu\_m}{kT}\right) \tag{57}$$

where *C*<sup>∞</sup> - equilibrium concentration at temperature *T* , υ*<sup>m</sup>* - volume of adatoms, *k* - the 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 *d*V to the caused by it free energy of island surface:

$$
\Delta P d\mathbf{V} = \sigma dS \quad \text{or} \quad \Delta P = \sigma \frac{dS}{dV} = 2 \frac{\sigma}{R\_\odot} \frac{a\_2(\theta)}{a\_1(\theta)} \,\prime \tag{58}
$$

Mass Transfer Between Clusters Under Ostwald's Ripening 127

diffusion coefficient, *DS* . Adatoms reaching island perimeters through surface diffusion and overcoming the potential barrier at the interface 'island-substrate', occur at their surfaces. Redistribution of adatoms at cluster surface is made by capillary forces, *viz*. surface tension

In accordance with Wagner, the diffusion growth mechanism with maintenance of island

interface 'island-substrate' and occurring at their surface in unite time have the time to form chemical connections necessary for reproduction of island matter structure. If it is not so, then adatoms are accumulated near the interface 'island-substrate' with concentration *C* , that is equal to mean concentration of a solution, *C* . For that, the process of growth is no more controlled by the surface diffusion coefficient, *DS* , but rather by the kinetic coefficient

Following to Wagner (Wagner, 1961), the number of adatoms crossing the boundary 'island-

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

<sup>2</sup> ( ) <sup>2</sup> <sup>2</sup> <sup>2</sup> 4 , sin *<sup>r</sup> <sup>j</sup> r C* α θ

 π

 βθ

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

 βθ

, at the boundary 'island-substrate':

*R r*

=

substrate' and occurring at the island surface at unite of time is determined as:

π α θβ

and the number of atoms leaving it at unite of time equals:

*R r*

*dC dR* <sup>=</sup> ⎛ ⎞ ⎜ ⎟ ⎝ ⎠

Solution of Eq. (66) can be represented in the form:

where *Cr* is determined by Eq. (59).

concentration gradient,

symmetry, takes the form:

(Chakraverty, 1967)) .

1 2 <sup>2</sup> 4 4 sin *<sup>C</sup> jR Cr C*

π

so that the total flow of atoms involved into formation of chemical connection is:

1 2 <sup>2</sup> 4 sin *i r j j j r CC* α θ

2 *S S*

*dC j rD dR* π

⎛ ⎞ <sup>=</sup> ⎜ ⎟ ⎝ ⎠

It can be determined by solving the Fick equation that describes concentration of adatoms in the vicinity of isolated island. This equation, within the conditions of stationarity and radial

> <sup>1</sup> <sup>0</sup> *<sup>S</sup> d dC RD R dR dR*

where *R* is changed within the interval *r R lr* ≤ ≤ , *l* = 2; 3 (screening distance

( ) 1 2 ln *<sup>R</sup> CR C C r*

π

At the same time, the diffusion flow of adatoms, *Sj* , to (from) an island is determined by the

θ

α θ

 βθ<sup>=</sup> <sup>=</sup> , (62)

=−= <sup>−</sup> , (64)

<sup>=</sup> (63)

. (65)

⎛ ⎞ <sup>=</sup> ⎜ ⎟ ⎝ ⎠ , (66)

= + , (67)

is possible, if atoms crossing the

form, i.e. with maintenance of the boundary angle

forces.

β.

where σ- specific magnitude of surface energy.

Taking into account Eq. (58), Eq. (57) can be rewritten in the form:

$$\mathbf{C}\_{r} = \mathbf{C}\_{\sigma} \exp\left(\frac{2\sigma}{R\_{\odot}} \frac{\upsilon\_{m}}{kT} \frac{a\_{2}(\theta)}{a\_{1}(\theta)}\right) = \mathbf{C}\_{\sigma} \exp\left(\frac{2\sigma}{r} \frac{\upsilon\_{m} \sin\theta}{kT} \frac{a\_{2}(\theta)}{a\_{1}(\theta)}\right) \approx \mathbf{C}\_{\sigma} \left(1 + \frac{2\sigma}{r} \frac{\upsilon\_{m} \sin\theta}{kT} \frac{a\_{2}(\theta)}{a\_{1}(\theta)}\right). \tag{59}$$

Note, the Gibbs-Thomson formula in the form Eq. (59) has been written for the first time in paper (Vengrenovich *et. al.,* 2008а).

Fig. 8. An island (cluster) in the form of spherical segment as a part of the sphere of radius *RC*

Thus, the concentration of adatroms at the boundary 'cluter-substarte' along the line of separation (along the cluster diameter) is determined by the curvature radius of cluster base, *r* , as it is expected for a plane problem. As the cluster radius diminishes, as the concentration of adatoms at the interface with the cluster must grow. And *vice versa*, as cluster size grows as *Cr* diminishes. For that, some mean concentration, *C* , is set in at a substrate that is determined by the critical radius *kr* :

$$\{\mathcal{C}\} = \mathbb{C}\_{\boldsymbol{\omega}} \exp\left(\frac{2\sigma}{\eta\_{k}} \frac{\boldsymbol{\nu}\_{m} \sin\theta}{kT} \frac{a\_{2}(\theta)}{a\_{1}(\theta)}\right) \approx \mathbb{C}\_{\boldsymbol{\omega}} \left(1 + \frac{2\sigma}{r\_{k}} \frac{\boldsymbol{\nu}\_{m} \sin\theta}{kT} \frac{a\_{2}(\theta)}{a\_{1}(\theta)}\right) \tag{60}$$

The clusters for which *C C <sup>r</sup>* > will dissolve. The clusters for which *C C <sup>r</sup>* < will grow. So, the clusters of critical size *kr* are in equilibrium with a solution of adatoms and radius of such adatoms is determined by Eq. (60):

$$r\_k = \frac{\alpha}{\Delta} \cdot \tag{61}$$

$$\Delta \cdot \sigma \cdot \tag{62}$$

where oversaturation is Δ= − *C C*<sup>∞</sup> , and ( ) ( ) <sup>1</sup> <sup>2</sup> <sup>2</sup> sin *<sup>C</sup> <sup>m</sup> kT* σ υ α θ α θ α θ <sup>∞</sup> = .

For the diffusion mechanism of growth of cupola-like clusters, the mass transfer between them is realized through surface diffusion under conditions of self-consistent diffusion field (Sagalovich, Slyozov, 1987; Kukushkin, Osipov, 1998) that is characterized by the surface

2 () 2 () 2 sin sin

*R kT r kT r kT*

Note, the Gibbs-Thomson formula in the form Eq. (59) has been written for the first time in

*h r=RCsinθ* 

Fig. 8. An island (cluster) in the form of spherical segment as a part of the sphere of radius *RC* Thus, the concentration of adatroms at the boundary 'cluter-substarte' along the line of separation (along the cluster diameter) is determined by the curvature radius of cluster base, *r* , as it is expected for a plane problem. As the cluster radius diminishes, as the concentration of adatoms at the interface with the cluster must grow. And *vice versa*, as cluster size grows as *Cr* diminishes. For that, some mean concentration, *C* , is set in at a

2 () 2 sin sin

*r kT r kT*

⎝ ⎠ ⎝ ⎠

<sup>2</sup> <sup>2</sup> sin *<sup>C</sup> <sup>m</sup> kT* σ υ

<sup>∞</sup> = .

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

( ) *m m*

> *kr* α

α

*k k*

The clusters for which *C C <sup>r</sup>* > will dissolve. The clusters for which *C C <sup>r</sup>* < will grow. So, the clusters of critical size *kr* are in equilibrium with a solution of adatoms and radius of

= Δ

For the diffusion mechanism of growth of cupola-like clusters, the mass transfer between them is realized through surface diffusion under conditions of self-consistent diffusion field (Sagalovich, Slyozov, 1987; Kukushkin, Osipov, 1998) that is characterized by the surface

exp 1

 θ  αθ

α θ

*C C C*

υ

σ

where oversaturation is Δ= − *C C*<sup>∞</sup> , and ( )

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

2 2 2 1 11

 αθ

 α θ

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

 θ ( ) ( )

. (59)

 θ α θ

> α θ

σ υ

> ( ) ( )

(60)

 θ α θ

( ) <sup>1</sup>

α θ

α θ

 θ  α θ

, (61)

2 2 1 1

σ υ


*C*

paper (Vengrenovich *et. al.,* 2008а).

σ αθ

υ

> α θ

substrate that is determined by the critical radius *kr* :

such adatoms is determined by Eq. (60):

Taking into account Eq. (58), Eq. (57) can be rewritten in the form:

*C C C C*

*θ*

*RC* 

exp exp 1 ( ) ( ) *mm m <sup>r</sup>*

σ

υ

where σ diffusion coefficient, *DS* . Adatoms reaching island perimeters through surface diffusion and overcoming the potential barrier at the interface 'island-substrate', occur at their surfaces. Redistribution of adatoms at cluster surface is made by capillary forces, *viz*. surface tension forces.

In accordance with Wagner, the diffusion growth mechanism with maintenance of island form, i.e. with maintenance of the boundary angle θ is possible, if atoms crossing the interface 'island-substrate' and occurring at their surface in unite time have the time to form chemical connections necessary for reproduction of island matter structure. If it is not so, then adatoms are accumulated near the interface 'island-substrate' with concentration *C* , that is equal to mean concentration of a solution, *C* . For that, the process of growth is no more controlled by the surface diffusion coefficient, *DS* , but rather by the kinetic coefficient β.

Following to Wagner (Wagner, 1961), the number of adatoms crossing the boundary 'islandsubstrate' and occurring at the island surface at unite of time is determined as:

$$j\_1 = 4\pi R\_\odot^2 \alpha\_2(\theta) \beta \{\text{C}\} = 4\pi r^2 \frac{\alpha\_2(\theta)}{\sin^2 \theta} \beta \{\text{C}\}. \tag{62}$$

and the number of atoms leaving it at unite of time equals:

$$j\_2 = 4\pi r^2 \frac{a\_2(\theta)}{\sin^2 \theta} \beta \mathbf{C}\_{rr} \tag{63}$$

so that the total flow of atoms involved into formation of chemical connection is:

$$j\_i = j\_1 - j\_2 = 4\pi r^2 \frac{a\_2(\theta)}{\sin^2 \theta} \beta \left( \langle \mathbb{C} \rangle - \mathbb{C}\_r \right),\tag{64}$$

where *Cr* is determined by Eq. (59).

At the same time, the diffusion flow of adatoms, *Sj* , to (from) an island is determined by the concentration gradient, *R r dC dR* <sup>=</sup> ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ , at the boundary 'island-substrate':

$$j\_S = 2\pi r D\_S \left(\frac{d\mathbf{C}}{d\mathbf{R}}\right)\_{R=r}.\tag{65}$$

It can be determined by solving the Fick equation that describes concentration of adatoms in the vicinity of isolated island. This equation, within the conditions of stationarity and radial symmetry, takes the form:

$$\frac{1}{R}\frac{d}{dR}\left(R D\_S \frac{dC}{dR}\right) = 0 \,\,\,\,\,\tag{66}$$

where *R* is changed within the interval *r R lr* ≤ ≤ , *l* = 2; 3 (screening distance (Chakraverty, 1967)) .

Solution of Eq. (66) can be represented in the form:

$$\mathbf{C}\left(\mathbf{R}\right) = \mathbf{C}\_1 \ln \frac{\mathbf{R}}{r} + \mathbf{C}\_2 \,' \,. \tag{67}$$

Mass Transfer Between Clusters Under Ostwald's Ripening 129

1 *S i*

where *<sup>S</sup> xjj* = is the contribution of the flow *Sj* in the total flow *j* and, correspondingly, (1 ) *<sup>i</sup>* − = *x j j* , the rate of growth of islands, Eq. (77), under surface diffusion with the share contribution (1 ) *<sup>i</sup>* − = *x j j* of the part of flow controlled by the kinetic coefficient

2 4 2 \* 2

\* <sup>3</sup> 1 *k*

⎛ ⎞ <sup>=</sup> ⎜ ⎟ <sup>−</sup> ⎝ ⎠

contribution of a flow due to surface diffusion, ( *<sup>S</sup> xjj* = ), then the rate of growth, Eq. (79),

2 2 2 2 <sup>2</sup> \* <sup>2</sup>

\*

⎛ ⎞ <sup>=</sup> ⎜ ⎟ <sup>−</sup> ⎝ ⎠

that coincides with the equation for the rate of island growth controlled by the kinetic

**4.2 Temporal dependences for critical (** *kr* **) and maximal (** *gr* **) sizes of islands (clusters)**  For integrating Eqs. (79) and (81) to determine the temporal dependences of *kr* and *gr* , it is necessary to determine the magnitudes of the locking point, *u rr* <sup>0</sup> = *<sup>g</sup> <sup>k</sup>* . Its magnitude, in

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

accordance with paper (Vengrenovich, 1982), is found from the condition:

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

*dr B r dt r r*

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

*m g g*

*dr C x r r r Bx r dt kT rx r r x r r r*

*dr A r dt r r*

coinciding with the diffusion rate of growth of islands (Chakraverty, 1967; Eq. (17)).

If the rate of island growth is controlled by the kinetic coefficient

( ) ( )

1

θα θ

α θ

1

α θ

2 2 2

<sup>∞</sup> = .

θα θ

( )

(Chakraverty, 1967; Eq. (31)).

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

2 32 3 2

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

sin 1 1 <sup>1</sup> 11 11

*g g k k*

2 2

*k k*

sin <sup>1</sup> 11 11 1 1

> 1 *k*

> > 0

( )

( )

( )

For 1 *x* = , Eq. (79) takes the following simplified form:

θα θ

θα θ

1

2 1 sin 2 ln

*kT l*

α θ

2 ln *m S*

2 4 \* 2

<sup>∞</sup> = .

α θ 2

rewritten in the form:

σ υ

where ( )

*C Dm S <sup>A</sup>*

σ υβ

where ( )

For 0 *x* = , Eq. (81) is rewritten as:

sin *<sup>C</sup> <sup>m</sup> <sup>B</sup> kT* σ υβ

\* 2

σ υ

takes the form:

coefficient

β

where, for example:

*<sup>j</sup> <sup>x</sup>* <sup>=</sup> <sup>−</sup> , (78)

, (80)

β

, (82)

, (83)

with the share

, (81)

β, is

, (79)

*j x*

where the constants *C*1 and *C*2 are determined form the boundary conditions:

$$\mathbf{C}\{R\} = \mathbf{C}\_r \text{ , if } R = r \text{ ,} \tag{68}$$

$$\mathcal{C}\{\mathcal{R}\} = \{\mathcal{C}\}\text{, if }\;\mathcal{R} = \mathcal{l}r\;\;\tag{69}$$

from which one obtains:

$$\mathbf{C}\_1 = \frac{\{\mathbf{C}\} - \mathbf{C}\_r}{\ln l}, \; \mathbf{C}\_2 = \mathbf{C}\_r \; . \tag{70}$$

Thus, the solution of Eq. (67) takes the form:

$$C(R) = \frac{\{\mathbf{C}\} - \mathbf{C}\_r}{\ln l} \ln \frac{R}{r} + \mathbf{C}\_r \,. \tag{71}$$

Knowing *C R*( ) , one can determine *Sj* :

$$j\_S = \frac{2\pi D\_S}{\ln I} \left( \left< \mathbf{C} \right> - \mathbf{C}\_r \right). \tag{72}$$

At the equilibrium state:

$$j\_i = j\_S = j\ . \tag{73}$$

Thus, the flow *j* to (from) a cluster can be written as:

$$j = \frac{1}{2}(j\_i + j\_S) \,. \tag{74}$$

In general case, the flow *j* equals:

$$
\dot{j} = \dot{j}\_i + \dot{j}\_S \,. \tag{75}
$$

Thus, the problem of determination of the cluster size distribution function is reduced to accounting the ratio between the flows *ij* and *Sj* in the equation of cluster growth rate.

#### **4.1 Island growth rate**

The rate of growth of isolated island (cluster) is determined form the following condition:

$$\frac{d}{dt}\left(\frac{4}{3}\pi R\_C^3 \alpha\_1(\theta)\right) = \frac{d}{dt}\left(\frac{4}{3}\pi \frac{r^3}{\sin^3\theta} \alpha\_1(\theta)\right) = j\upsilon\_{m'} \tag{76}$$

where *j* is given by Eq. (75). Taking into account Eqs. (64) and (72), one finds from Eq. (76) the rate of cluster growth:

$$\frac{dr}{dt} = \frac{1}{4\pi r^2} \frac{\sin^3 \theta}{a\_1(\theta)} \nu\_m \left[ 4\pi r^2 \frac{a\_2(\theta)}{\sin^2(\theta)} \beta \left( \{\mathbf{C}\} - \mathbf{C}\_r \right) + \frac{2\pi D\_S}{\ln l} \left( \{\mathbf{C}\} - \mathbf{C}\_r \right) \right]. \tag{77}$$

Designating the ratio of flows as:

$$\frac{j\_S}{j\_i} = \frac{\infty}{1 - \infty} \text{.}\tag{78}$$

where *<sup>S</sup> xjj* = is the contribution of the flow *Sj* in the total flow *j* and, correspondingly, (1 ) *<sup>i</sup>* − = *x j j* , the rate of growth of islands, Eq. (77), under surface diffusion with the share contribution (1 ) *<sup>i</sup>* − = *x j j* of the part of flow controlled by the kinetic coefficient β , is rewritten in the form:

$$\frac{dr}{dt} = \frac{\sigma C\_w \nu\_m^2 D\_S}{2kT} \frac{\sin^4 \theta \alpha\_2(\theta)}{\alpha\_1^2(\theta) \ln l} \frac{1}{r^3} \left(\frac{1-x}{x} \frac{r^2}{r\_\mathcal{g}^2} + 1\right) \left(\frac{r}{r\_k} - 1\right) = \frac{A^\*}{r^3} \left(\frac{1-x}{x} \frac{r^2}{r\_\mathcal{g}^2} + 1\right) \left(\frac{r}{r\_k} - 1\right),\tag{79}$$

where ( ) ( ) 2 4 \* 2 2 1 sin 2 ln *C Dm S <sup>A</sup> kT l* σ υ θα θ α θ <sup>∞</sup> = .

128 Mass Transfer - Advanced Aspects

*CR C* ( ) = *<sup>r</sup>* , if *R r* = , (68)

*CR C* ( ) = , if *R lr* = , (69)

<sup>−</sup> <sup>=</sup> , *C C* <sup>2</sup> <sup>=</sup> *<sup>r</sup>* . (70)

<sup>−</sup> <sup>=</sup> <sup>+</sup> . (71)

= − . (72)

<sup>2</sup> *i S j jj* = + . (74)

*i S jj j* = + . (75)

 υ , (76)

. (77)

*r*

where the constants *C*1 and *C*2 are determined form the boundary conditions:

<sup>1</sup> ln *C Cr <sup>C</sup> l*

( ) ln ln *r*

ln *S S r <sup>D</sup> <sup>j</sup> C C l* π

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

*l r*

( ) <sup>2</sup>

*i S jjj* = = . (73)

( ) <sup>1</sup>

Thus, the problem of determination of the cluster size distribution function is reduced to accounting the ratio between the flows *ij* and *Sj* in the equation of cluster growth rate.

The rate of growth of isolated island (cluster) is determined form the following condition:

3

2 2

υπ

π

( )

θ

1

α θ

π

Designating the ratio of flows as:

3

4 4

αθ

( ) ( )

3 3 sin *C m d dr <sup>R</sup> <sup>j</sup> dt dt*

 π

1 1 3

where *j* is given by Eq. (75). Taking into account Eqs. (64) and (72), one finds from Eq. (76)

 β

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

( )

 θ

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

2 2

*dt r l*

1 sin <sup>2</sup> <sup>4</sup> 4 sin ln

α θ

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

3

 αθ

( ) ( ) ( )

π

θ

*<sup>S</sup> m rr*

from which one obtains:

At the equilibrium state:

**4.1 Island growth rate** 

the rate of cluster growth:

In general case, the flow *j* equals:

Thus, the solution of Eq. (67) takes the form:

Knowing *C R*( ) , one can determine *Sj* :

Thus, the flow *j* to (from) a cluster can be written as:

For 1 *x* = , Eq. (79) takes the following simplified form:

$$\frac{dr}{dt} = \frac{A^\*}{r^3} \left(\frac{r}{r\_k} - 1\right) \tag{80}$$

coinciding with the diffusion rate of growth of islands (Chakraverty, 1967; Eq. (17)). If the rate of island growth is controlled by the kinetic coefficient β with the share contribution of a flow due to surface diffusion, ( *<sup>S</sup> xjj* = ), then the rate of growth, Eq. (79), takes the form:

$$\frac{dr}{dt} = \frac{\sigma \mathbb{C}\_{\alpha} \nu\_{m}^{2} \theta}{kT} \frac{\sin^{2} \theta \alpha\_{2}^{2} (\theta)}{\alpha\_{1} (\theta)} \frac{1}{r} \left( \frac{\mathbf{x}}{1 - \mathbf{x}} \cdot \frac{r\_{\text{g}}^{2}}{r^{2}} + 1 \right) \left( \frac{r}{r\_{k}} - 1 \right) = \frac{B}{r} \left( \frac{\mathbf{x}}{1 - \mathbf{x}} \cdot \frac{r\_{\text{g}}^{2}}{r^{2}} + 1 \right) \left( \frac{r}{r\_{k}} - 1 \right), \tag{81}$$

where ( ) ( ) 2 2 2 \* 2 1 sin *<sup>C</sup> <sup>m</sup> <sup>B</sup> kT* σ υβ θα θ α θ <sup>∞</sup> = .

For 0 *x* = , Eq. (81) is rewritten as:

$$\frac{dr}{dt} = \frac{B^\*}{r} \left(\frac{r}{r\_k} - 1\right),\tag{82}$$

that coincides with the equation for the rate of island growth controlled by the kinetic coefficient β(Chakraverty, 1967; Eq. (31)).

#### **4.2 Temporal dependences for critical (** *kr* **) and maximal (** *gr* **) sizes of islands (clusters)**

For integrating Eqs. (79) and (81) to determine the temporal dependences of *kr* and *gr* , it is necessary to determine the magnitudes of the locking point, *u rr* <sup>0</sup> = *<sup>g</sup> <sup>k</sup>* . Its magnitude, in accordance with paper (Vengrenovich, 1982), is found from the condition:

$$\left. \frac{d}{dr} \left( \frac{\dot{r}}{r} \right) \right|\_{r=r\_g} = 0 \,, \tag{83}$$

where, for example:

or:

(Chakraverty, 1967):

**(clusters)** 

where:

In the same way, one obtains from Eq. (81):

is represented as the product *f* ( ) *rt r* , =

distribution of clusters, *<sup>g</sup>*

law of disperse phase volume:

Using Eq. (92), one finds:

Mass Transfer Between Clusters Under Ostwald's Ripening 131

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

For 1 *x* = , growth of islands is fully controlled by the surface diffusion coefficient

*x x* <sup>=</sup> − + , ( )

Eq. (90) describes of island growth under conditions controlled by the kinetic coefficient

**4.3 Generalized Chakraverty-Wagner distribution for the case of cupola-like islands** 

ϕ

1 3

ϕ

1 3

4 1 3 sin

<sup>Φ</sup> <sup>=</sup>

πα θ

*Q*

Taking into account Eq. (93), the function *f* (,) *r t* is rewritten as:

4 1 , <sup>3</sup> sin

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

As previously, the size distribution function of clusters (islands) within the interval 0 1 ≤ ≤ *x*

ϕ

( ) ( ) <sup>3</sup>

 *r f r t dr* θ

*gr*

0

( ) ( ) 1 3

( ) ( ) ( ) 4 4 1 1 , *g g f rt Qg u gu*

0

 *u g u du* θ

′ ∫

controlled by kinetics of crossing the interface 'island-substrate' (Wagner, 1961):

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

with the contribution *x* of surface diffusion. If 0 *x* = , then the growth process is fully

2

*k*

4 \*

*k*

4 \* 4

( )( )

*u rr* = . The function

πα θ

\* <sup>2</sup> <sup>2</sup> 1 21 *<sup>g</sup> <sup>B</sup> r t*

4

<sup>3</sup> *gr At* <sup>=</sup> , 4 \* <sup>27</sup>

( ) ( )

2 1

2 2

<sup>64</sup> *kr At* <sup>=</sup> , <sup>4</sup>

2

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

3

2 1

( )( )

1 22

2

*g k r*

\*

*B x r t x x* <sup>+</sup> <sup>=</sup> − +

3

4

. (88)

*<sup>r</sup>* <sup>=</sup> . (89)

*<sup>r</sup>* <sup>=</sup> . (91)

( *<sup>g</sup>* ) *g*′( ) *u* , where *g*′(*u*) is the relative size

= , (93)

*r r* =⋅ = ′ , (95)

. (94)

Φ = ∫ . (92)

(*rg* ) is determined from the conservation

. (90)

β,

$$\frac{\dot{r}}{r} = \frac{A}{r^4} \left( \frac{1-\chi}{\chi} \frac{r^2}{r\_\S} + 1 \right) \left( \frac{r}{r\_k} - 1 \right). \tag{84}$$

By differentiation, one finds:

$$
\mu\_0 = \frac{r\_g}{r\_k} = \frac{2\varkappa + 2}{2\varkappa + 1}.\tag{85}
$$

For 0 *x* = (the Wagner mechanism of growth), 2 *g k r r* = , and for 1 *x* = (the diffusion mechanism of growth), 4 3 *g k r r* = .

Using the magnitude *uo* of the locking point, Eq. (85), one can express the specific rate of growth *r r* , Eq. (84), through dimensionless variable, *<sup>g</sup> u rr* = , that enables its representation not schematically, but in the form of a graph for various magnitudes of the parameter *x* :

$$\boldsymbol{\omega}^{\prime} = \frac{r\_{\mathcal{S}}^{4}}{A^{\*}} \frac{dr}{dt} = \frac{1}{\boldsymbol{\mu}^{4}} \left( \frac{1-\chi}{\chi} \boldsymbol{u}^{2} + 1 \right) \left( \frac{2\chi+2}{2\chi+1} \boldsymbol{u} - 1 \right) \tag{86}$$

where the dimensionless specific rate of growth is 4 \* *gr dr A dt* υ′ = .

Fig. 9 shows the dependence of υ′ on *u* as a function of *x* . The role of the locking point consists in that within the LSW theory all solutions, including the size distribution function, are determined for magnitudes *uo* alone. It means physically that under the Ostwald's ripening process, the relation between critical and maximal sizes of clusters is always the same, i.e. being constant one.

Fig. 9. Dependence of the dimensionless growth rate, υ′ , on *u* for various magnitudes of *x* For determining *gr* and *kr* , we use Eqs. (79) and (81). Substituting in Eq. (79) *<sup>g</sup> r r* = and replacing the ratio *<sup>g</sup> <sup>k</sup> r r* by its magnitude (85), one obtains after integrating:

( ) \* <sup>4</sup> <sup>4</sup> 2 1 *<sup>g</sup> <sup>A</sup> r t x x* <sup>=</sup> <sup>+</sup> , (87)

or:

130 Mass Transfer - Advanced Aspects

<sup>1</sup> 1 1 *g k*

> 2 2 2 1

. (84)

. (85)

*u rr* = , that enables its

, (86)

\* 2 4 2

0

*u*

growth *r r* , Eq. (84), through dimensionless variable, *<sup>g</sup>*

\* 4

υ

4

υ

where the dimensionless specific rate of growth is


Fig. 9. Dependence of the dimensionless growth rate,

0

2

υ*'*

4

6

8

*x=0.1*

By differentiation, one finds:

parameter *x* :

mechanism of growth), 4 3 *g k r r* = .

Fig. 9 shows the dependence of

same, i.e. being constant one.

*r A xr r rx r r r* ⎛ ⎞ <sup>−</sup> ⎛ ⎞ <sup>=</sup> ⎜ ⎟ + − ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

> *g k*

*r x*

*r x* <sup>+</sup> = = <sup>+</sup>

For 0 *x* = (the Wagner mechanism of growth), 2 *g k r r* = , and for 1 *x* = (the diffusion

Using the magnitude *uo* of the locking point, Eq. (85), one can express the specific rate of

representation not schematically, but in the form of a graph for various magnitudes of the

*gr dr x x*

*A u dt x x*

2

⎛ ⎞⎛ ⎞ − + ′ <sup>=</sup> = +− ⎜ ⎟⎜ ⎟

consists in that within the LSW theory all solutions, including the size distribution function, are determined for magnitudes *uo* alone. It means physically that under the Ostwald's ripening process, the relation between critical and maximal sizes of clusters is always the

01234

*u*

υ

′ , on *u* for various magnitudes of *x*

*x x* <sup>=</sup> <sup>+</sup> , (87)

*x=1*

For determining *gr* and *kr* , we use Eqs. (79) and (81). Substituting in Eq. (79) *<sup>g</sup> r r* = and

\* <sup>4</sup> <sup>4</sup> 2 1 *<sup>g</sup> <sup>A</sup> r t*

( )

replacing the ratio *<sup>g</sup> <sup>k</sup> r r* by its magnitude (85), one obtains after integrating:

11 2 2 1 1

υ

*u u*

⎝ ⎠⎝ ⎠ +

2 1

4 \* *gr dr A dt*

′ = .

′ on *u* as a function of *x* . The role of the locking point

$$r\_k^4 = 4A^\* \frac{\left(2\chi + 1\right)^3}{\chi \left(2\chi + 2\right)^4} t \cdot \tag{88}$$

For 1 *x* = , growth of islands is fully controlled by the surface diffusion coefficient (Chakraverty, 1967):

$$r\_{\mathcal{S}}^4 = \frac{4}{3} A^\* t \quad , \quad r\_k^4 = \frac{27}{64} A^\* t \quad , \quad \frac{r\_{\mathcal{S}}}{r\_k} = \frac{4}{3} \, . \tag{89}$$

In the same way, one obtains from Eq. (81):

$$r\_{\mathcal{S}}^2 = 2 \frac{B^\*}{(1-\alpha)(2\alpha+1)} t \; \; r\_k^2 = 2 \frac{B^\* \left(2\alpha+1\right)}{(1-\alpha)(2\alpha+2)} t \; \; \, \tag{90}$$

Eq. (90) describes of island growth under conditions controlled by the kinetic coefficient β , with the contribution *x* of surface diffusion. If 0 *x* = , then the growth process is fully controlled by kinetics of crossing the interface 'island-substrate' (Wagner, 1961):

$$r\_{\underline{g}}^2 = 2B^\*t \; \; \; r\_k^2 = \frac{1}{2}B^\*t \; \; \; \; \frac{r\_{\underline{g}}}{r\_k} = 2 \; \; \; \tag{91}$$

#### **4.3 Generalized Chakraverty-Wagner distribution for the case of cupola-like islands (clusters)**

As previously, the size distribution function of clusters (islands) within the interval 0 1 ≤ ≤ *x* is represented as the product *f* ( ) *rt r* , = ϕ ( *<sup>g</sup>* ) *g*′( ) *u* , where *g*′(*u*) is the relative size distribution of clusters, *<sup>g</sup> u rr* = . The function ϕ (*rg* ) is determined from the conservation law of disperse phase volume:

$$\Phi = \frac{4}{3}\pi\alpha\_1(\theta)\frac{1}{\sin^3\theta}\int\_0^{r\_3} r^3 f(r, t) dr \,. \tag{92}$$

Using Eq. (92), one finds:

$$\varphi(r\_{\mathcal{S}}) = \frac{\mathcal{Q}}{r\_{\mathcal{S}}^4} \,' \,. \tag{93}$$

where:

$$Q = \frac{\Phi}{\frac{4}{3}\pi\alpha\_1(\theta)\frac{1}{\sin^3\theta}\Big|\Bigu\_0^1 g'(u)du\Big|}\Big.\tag{94}$$

Taking into account Eq. (93), the function *f* (,) *r t* is rewritten as:

$$f\left(r,t\right) = \frac{1}{r\_{\mathcal{S}}^4} \mathbf{Q} \cdot \mathbf{g}'\left(\mu\right) = \frac{1}{r\_{\mathcal{S}}^4} \mathbf{g}\left(\mu\right),\tag{95}$$

Mass Transfer Between Clusters Under Ostwald's Ripening 133

For 0 *x* = : 5 *B* = , 3 *C* = − , *D* = −2 , 0 *F* = , *A* = 1 , and Eq. (99) is transformed into the Wagner

( ) ( )

1 23

*u uu*

Taking into account the volume (mass) conservation law for island condensate, one can find

The dependences shown in Fig. 10 *а* correspond to the size distribution function computed using Eq. (96) for various magnitudes of *x*. The extreme curves for *x* = 0 and *x* = 1 determine the Chakraverty distribution and the Wagner distributions, respectively (Wagner, 1961; Chakraverty, 1967). All other curves, within interval 0 < *x* < 1, describe the size distribution of islands for simultaneous action of the Wagner and diffusion mechanisms of cluster

The same dependences normalized by their maxima are shown in Fig. 10 *b*. In such form, being normalized by unity along the coordinate axes, such dependences are easy-to-use for

For the computed family of distributions, see Eq. (96), the magnitude of the locking point changes in accordance with Eq. (85) within the interval 4/3 ≤ *u*0 ≤ 2. For *x* = 0.5, one obtains *u*0 = 3/2, what coincides with similar magnitude for the Lifshitz-Slyozov distribution. At the

( )

4 22 <sup>2</sup> <sup>3</sup> <sup>5</sup>

( ) ( )

It means that one can not judge on the type of distribution proceeding from the locking point magnitude *u*0. It must be considered only as evaluating parameter for choice of the theoretical curve from the family Eq. (96), for comparison with specific experimentally

Once more important property of the found distribution, Eq. (96), consists in that it can be used not only for comparison with experimentally obtained histograms in the form of distribution of particles of radii *r* (or diameters *d*), but also for description of the particle

> ( ) 1 cos 1 cos sin *<sup>C</sup> hR r*

θ

θ

<sup>−</sup> =− = , (104)

θ

1 0,5 1

*uu u*

− ++

1.2 exp 1.084435tan 1.032795 0.258199 exp <sup>1</sup>

<sup>−</sup> ⎛ ⎞ ⎡ ⎤ <sup>−</sup> + ⋅−⎜ ⎟ ⎣ ⎦ ⎝ ⎠ <sup>−</sup> <sup>=</sup>

same time, the curve Eq. (96) for *x* = 0.5 is not the Lifshitz-Slyozov distribution:

*u u*

height distribution, *h*. One can see in Fig. 8 that island height is equal to:

− ++

<sup>23</sup> <sup>19</sup> <sup>2</sup> <sup>12</sup> <sup>6</sup>

exp exp tan ( ) 2 1 62 2

1 11

3 1

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

*u*

*u* <sup>−</sup> ⎛ ⎞ ′ =− −⎜ ⎟ ⎝ ⎠ <sup>−</sup> . (101)

<sup>6</sup> *<sup>D</sup>* = − , *<sup>F</sup>* <sup>=</sup> <sup>−</sup><sup>4</sup> , and Eq. (99) corresponds to the

. (102)

*u*

. (103)

*u*

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

<sup>2</sup> *<sup>C</sup>* = − , 23

( ) ( )

the cluster's relative size distribution function *g*(*u*) from Eq. (96).

growth (the generalized Chakraverty-Wagner distribution.).

3 1

comparison with experimentally obtained histograms.

( )

*g u*

obtained histogram.

so that *g g rr hh u* = = .

distribution (Wagner, 1961):

For 1 *x* = : 36 *A* = , 19

**4.4 Discussion** 

<sup>6</sup> *<sup>B</sup>* <sup>=</sup> , 1

*u*

Chakraverty distribution (Chakraverty, 1967):

*g u*

where the relative size distribution function is:

$$\mathbf{g}\left(\boldsymbol{u}\right) = \mathbf{Q} \cdot \mathbf{g}'\left(\boldsymbol{u}\right). \tag{96}$$

To determine *g*′( ) *u* , we use the continuity equation (8), substituting in it, instead of *f* (*r t*, ) and *r* , their magnitudes from Eqs. (95) and (81) (or 79)). Under preceding from differentiation on *r* and *t* to differentiation on *u* , the variables are separated, and Eq. (8) takes the form:

$$\frac{d\mathbf{g'}\left(\boldsymbol{u}\right)}{d\boldsymbol{g'}\left(\boldsymbol{u}\right)} = -\frac{4\nu\_{\mathcal{S}} - \frac{1}{\nu}\frac{d\nu}{d\boldsymbol{u}} + \frac{\nu}{\boldsymbol{u}^{2}}}{\nu\nu\_{\mathcal{S}} - \frac{\nu}{\boldsymbol{u}}}d\boldsymbol{u}\,\boldsymbol{u} \tag{97}$$

where \* 2 <sup>122</sup> 1 1 1 21 *rr x x u B u x x* υ ⋅ + ⎛ ⎞⎛ ⎞ = = + <sup>−</sup> ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠ − + , ( )( ) <sup>1</sup> \* 1 1 21 *g g g u r dr B dt x x* υ υ <sup>=</sup> = = = <sup>−</sup> <sup>+</sup> , <sup>1</sup> *g du dr r* <sup>=</sup> ,

*g g du u dr r* = − .

Substituting υ , υ*<sup>g</sup>* and *d du* υin Eq. в (97), one obtains the expression:

$$\frac{d\mathbf{g'(u)}}{d\mathbf{g'(u)}} = -\frac{4u^4 - u^2\left(1 + \mathbf{x} - 2\mathbf{x}^2\right) + 4u\left(\mathbf{x}^2 + \mathbf{x}\right) - 3\mathbf{x}\left(2\mathbf{x} + 1\right)}{u\left(1 - u\right)^2\left(u^2 + 2u\mathbf{x}^2 + 2\mathbf{x}^2 + \mathbf{x}\right)} du\,\tag{98}$$

after integration of which we find *g*′(*u*) , i.e. the generalized Chakraverty-Wagner distribution for islands of cupola-like form:

$$g'(u) = \frac{u^3 \left(u^2 + 2ux^2 + 2x^2 + x\right)^{1/2}}{\left(1 - u\right)^6} \exp\left(\frac{F - Dx^2}{\sqrt{2x^2 + x - x^4}} \tan^{-1}(\frac{u + x^2}{\sqrt{2x^2 + x - x^4}})\right) \times \tag{99}$$

$$\times \exp\left(\frac{C}{1 - u}\right)$$

where:

$$\begin{cases} B = \frac{32\mathbf{x}^4 + 16\mathbf{x}^3 + 48\mathbf{x}^2 + 13\mathbf{x} + 5}{A}, \\ C = -\frac{12\mathbf{x}^2 + 3\mathbf{x} + 3}{A}, \\ D = -\frac{80\mathbf{x}^4 + 40\mathbf{x}^3 + 15\mathbf{x}^2 + \mathbf{x} + 2}{A}, \\ F = -\frac{32\mathbf{x}^6 + 16\mathbf{x}^5 + 54\mathbf{x}^4 + 34\mathbf{x}^3 + 8\mathbf{x}^2}{A}, \\ A = 16\mathbf{x}^4 + 8\mathbf{x}^3 + 9\mathbf{x}^2 + 2\mathbf{x} + 1. \end{cases} \tag{100}$$

For 0 *x* = : 5 *B* = , 3 *C* = − , *D* = −2 , 0 *F* = , *A* = 1 , and Eq. (99) is transformed into the Wagner distribution (Wagner, 1961):

$$\log'(\mu) = \mu \left(1 - \mu\right)^{-5} \exp\left(-\frac{3}{1 - \mu}\right) \cdot \tag{101}$$

For 1 *x* = : 36 *A* = , 19 <sup>6</sup> *<sup>B</sup>* <sup>=</sup> , 1 <sup>2</sup> *<sup>C</sup>* = − , 23 <sup>6</sup> *<sup>D</sup>* = − , *<sup>F</sup>* <sup>=</sup> <sup>−</sup><sup>4</sup> , and Eq. (99) corresponds to the Chakraverty distribution (Chakraverty, 1967):

$$g'(u) = \frac{u^3 \exp\left(-\frac{1}{2\left(1-u\right)}\right) \exp\left(-\frac{1}{6\sqrt{2}} \tan^{-1}(\frac{u+1}{\sqrt{2}})\right)}{\left(1-u\right)^{19} \Big{{}^{\zeta}\left(u^2+2u+3\right)}^{\zeta\_0}}.\tag{102}$$

Taking into account the volume (mass) conservation law for island condensate, one can find the cluster's relative size distribution function *g*(*u*) from Eq. (96).

#### **4.4 Discussion**

132 Mass Transfer - Advanced Aspects

To determine *g*′( ) *u* , we use the continuity equation (8), substituting in it, instead of *f* (*r t*, ) and *r* , their magnitudes from Eqs. (95) and (81) (or 79)). Under preceding from differentiation on *r* and *t* to differentiation on *u* , the variables are separated, and Eq. (8)

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

υ

− + ′ = − ′ <sup>−</sup>

42 2 2

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

*g u u u u ux x x* ′ − +− + + − + = − ′ − + ++

*D*

*dg u u du <sup>u</sup> du*

*g*

υ

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

υ υ

*u*

*g u*

in Eq. в (97), one obtains the expression:

( ) ( )

1 22

after integration of which we find *g*′(*u*) , i.e. the generalized Chakraverty-Wagner

32 2 2 <sup>2</sup> 2 2

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

*u u ux x x F Dx u x*

1 2 2

*C u*

6 5 4 32

32 16 54 34 8 ,

exp <sup>1</sup>

32 16 48 13 5 ,

432

*xxxx <sup>B</sup> A*

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

432

80 40 15 2 ,

*A x x x xx <sup>F</sup> A*

432

⎪⎪ + + ++ <sup>⎨</sup> = −

*Axxxx*

<sup>⎪</sup> = + + ++ <sup>⎪</sup>

16 8 9 2 1.

<sup>⎪</sup> ++++ <sup>⎪</sup> = −

2

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

<sup>⎪</sup> + + <sup>⎪</sup> = −

12 3 3 ,

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

⎛ ⎞ <sup>×</sup> ⎜ ⎟ ⎝ ⎠ −

2 2 22 4 1 2 4 32 1

( ) ( ) ( )

*u x xx x xx*

υ

*d*

υ

2

*g g*

*B dt x x*

1 24 24

exp tan ( )

<sup>=</sup> = = = <sup>−</sup> <sup>+</sup> , <sup>1</sup>

*r dr*

υ

( ) ( )

<sup>122</sup> 1 1 1 21

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

*rr x x*

*B u x x*

*<sup>g</sup>* and *d du* υ

> ( ) ( )

distribution for islands of cupola-like form:

( ) ( ) ( )

2 2

⎪

⎪

⎪

⎪

⎪⎩

*B*

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

*u*

*g*(*u Q* ) = ⋅ *g*′(*u*) . (96)

, (97)

*g*

*du dr r* <sup>=</sup> ,

, (98)

, (99)

(100)

1 1 21

*du*

where the relative size distribution function is:

takes the form:

υ

*g g du u dr r* = − .

Substituting

where \* 2

υ , υ

*g u*

where:

The dependences shown in Fig. 10 *а* correspond to the size distribution function computed using Eq. (96) for various magnitudes of *x*. The extreme curves for *x* = 0 and *x* = 1 determine the Chakraverty distribution and the Wagner distributions, respectively (Wagner, 1961; Chakraverty, 1967). All other curves, within interval 0 < *x* < 1, describe the size distribution of islands for simultaneous action of the Wagner and diffusion mechanisms of cluster growth (the generalized Chakraverty-Wagner distribution.).

The same dependences normalized by their maxima are shown in Fig. 10 *b*. In such form, being normalized by unity along the coordinate axes, such dependences are easy-to-use for comparison with experimentally obtained histograms.

For the computed family of distributions, see Eq. (96), the magnitude of the locking point changes in accordance with Eq. (85) within the interval 4/3 ≤ *u*0 ≤ 2. For *x* = 0.5, one obtains *u*0 = 3/2, what coincides with similar magnitude for the Lifshitz-Slyozov distribution. At the same time, the curve Eq. (96) for *x* = 0.5 is not the Lifshitz-Slyozov distribution:

$$\log(u) = \frac{u^3 \exp\left[-1.084435 \tan^{-1}\left(1.032795u + 0.258199\right)\right] \cdot \exp\left(-\frac{1.2}{1-u}\right)}{\left(1-u\right)^{\frac{22}{5}} \left(u^2 + 0.5u + 1\right)^{\frac{4}{3}}} \cdot \tag{103}$$

It means that one can not judge on the type of distribution proceeding from the locking point magnitude *u*0. It must be considered only as evaluating parameter for choice of the theoretical curve from the family Eq. (96), for comparison with specific experimentally obtained histogram.

Once more important property of the found distribution, Eq. (96), consists in that it can be used not only for comparison with experimentally obtained histograms in the form of distribution of particles of radii *r* (or diameters *d*), but also for description of the particle height distribution, *h*. One can see in Fig. 8 that island height is equal to:

$$h = R\_{\mathbb{C}} \left( 1 - \cos \theta \right) = r \frac{1 - \cos \theta}{\sin \theta} \,\,\,\,\,\,\tag{104}$$

so that *g g rr hh u* = = .

Mass Transfer Between Clusters Under Ostwald's Ripening 135

Fig. 11 shows comparison of experimental histograms of nanodots *Mn* at substrate *Si* on diameters, *d* , (*а* and *b*) and on heights, *h* , (*c* and *d*), obtained by the molecular beam epitaxy technique at various temperatures, *viz.* at room temperature (fragments *а* and *с*) and at 180С (fragments *b* and *d)*, as well as for various thickness of molecular layers of *Mn* (ML) (De-yong Wang *et. al*., 2006), with theoretical dependence Eq. (96). It is seen from comparison that for experimental distributions on diameters, the contributions of the each of two mechanisms of growth, i.e. Wagner and diffusion ones, are approximately the same, cf.

At the same time, the diffusion mechanism occurs to be predominant for the height distribution functions, cf. Fig. 11 *c*, 0.8 *x* = , and Fig. 11 *d*, 0.9 *x* = . It means that as nanodots of *Mn* grow, increasing of height leaves behind increasing lateral size *d* , so that *h d* > 1 . Probably, this circumstance just explains of the form of nanodots of *Mn* obtained by the

Fig. 12 shows the results of comparison of the theoretical dependence, cf. Eq. (96), with experimentally obtained histograms of particles of gold obtained at temperature 525°С at silicon substrate ( *Au Si* / 111 ( )) - (Fig. 12 *а*), and later, after 180-min isothermal exposure - (Fig. 12 *b*) (Werner *et al*., 2006). Judging by the magnitude of *x* , particle growth is initially

changes, and after three-hour exposure it becomes predominantly diffusion one (Fig. 12 *b*).

0.0

0.2

0.4

0.6

*g(u)/gmax*

Fig. 12. Comparison of the dependence Eq. (96) with experimentally obtained histograms for particles of *Au* obtained by the molecular beam epitaxy technique for temperature 525°С (*а*)

The results of comparison of the theoretical dependence computed for 0.3 *x* = with the experimental histogram for nanoclusters of *Ag* obtained by the molecular beam epitaxy technique at room temperature at substrate *TiO*<sup>2</sup> (110) (Xiaofeng Lai *et al*., 1999), cf. Fig. 13,

Thus, the considered examples of comparison of computed and experimentally obtained data leads to the conclusion on the possibility to implement simultaneous action of both mechanisms of growth, i.e. Wagner and diffusion ones. What is more, the situation when both mechanisms of growth co-exist and act in parallel is, to all appearance, more general than separate manifestations of one of two mechanisms considered early by Wagner and

0.8

*/rk =1.417*

1.0

(Fig. 12 *а*). But later, the mechanism of growth

0.0 0.2 0.4 0.6 0.8 1.0

*u*

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

*x=0.7*

β

Fig. 11 *а*, 0.4 *x* = , and Fig. 11 *b*, 0.5 *x* = .

authors of paper (De-yong Wang *et al*., 2006).

0.0 0.2 0.4 0.6 0.8 1.0

*u*

also argue in favour of the proposed mechanism of growth.

and 180 min later, after isothermal exposure (*b*)

*<sup>a</sup> <sup>r</sup>*

*x=0.4*

controlled by the kinetic coefficient

0.0

Chakraverty.

0.2

0.4

0.6

*g(u)/gmax*

0.8

*g /rk =1.555*

1.0

Fig. 10. The generalized Chakraverty-Wagner distribution: *а –* dependences computed for various magnitudes of *x* following Eq. (96); *b –* the same dependences normalized by their maxima

Fig. 11. Comparison of the dependence represented by Eq. (96) with experimentally obtained histograms on diameter, *d* , and height, *h* , of nanodots of *Mn* at various temperatures and thickness of monolayer of *Mn* : *а -* room temperature, *Mn ML* 0.21 , 1.555 *g k r r* = ; *b* - temperature 180°С, 1.5 *g k r r* = ; *c -* room temperature, *Mn ML* 0.21 , 1.384 *g k h h* = ; *d* - temperature 180°С, 1.357 *g k h h* =

0.0

*<sup>u</sup>* 0.0 0.2 0.4 0.6 0.8 1.0

*/hk =1.357*

*/rk =1.5*

0.0

0.0

0.2

0.4

0.6

*g(u)/gmax*

Fig. 11. Comparison of the dependence represented by Eq. (96) with experimentally obtained histograms on diameter, *d* , and height, *h* , of nanodots of *Mn* at various temperatures and

thickness of monolayer of *Mn* : *а -* room temperature, *Mn ML* 0.21 , 1.555 *g k r r* = ; *b* - temperature 180°С, 1.5 *g k r r* = ; *c -* room temperature, *Mn ML* 0.21 , 1.384 *g k h h* = ;

0.8

1.0

0.2

0.4

0.6

*g(u)/gmax*

0.8

1.0

0.2

0.4

0.6

*g(u)/gmax*

Fig. 10. The generalized Chakraverty-Wagner distribution: *а –* dependences computed for various magnitudes of *x* following Eq. (96); *b –* the same dependences normalized by their

0.8

0.0 0.2 0.4 0.6 0.8 1.0

*u*

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

*u*

*<sup>d</sup> hg*

*x=0.9*

0.0 0.2 0.4 0.6 0.8 1.0

*u*

*x=0.5*

1.0 *<sup>b</sup>*

*x=0 x=1*

*a x=1*

*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*

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

*<sup>c</sup> hg*

*x=0.8*

0.0 0.2 0.4 0.6 0.8 1.0

*u*

*d* - temperature 180°С, 1.357 *g k h h* =

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

*x=0.4*

*x=0.1*

0.0

maxima

0.0

0.0

0.2

0.4

0.6

*g(u)/gmax*

0.8

1.0

0.2

0.4

0.6

*g(u)/gmax*

0.8

*/rk =1.555*

*/hk =1.384*

1.0

0.2

0.4

0.6

*g(u)*

0.8

1.0

Fig. 11 shows comparison of experimental histograms of nanodots *Mn* at substrate *Si* on diameters, *d* , (*а* and *b*) and on heights, *h* , (*c* and *d*), obtained by the molecular beam epitaxy technique at various temperatures, *viz.* at room temperature (fragments *а* and *с*) and at 180С (fragments *b* and *d)*, as well as for various thickness of molecular layers of *Mn* (ML) (De-yong Wang *et. al*., 2006), with theoretical dependence Eq. (96). It is seen from comparison that for experimental distributions on diameters, the contributions of the each of two mechanisms of growth, i.e. Wagner and diffusion ones, are approximately the same, cf. Fig. 11 *а*, 0.4 *x* = , and Fig. 11 *b*, 0.5 *x* = .

At the same time, the diffusion mechanism occurs to be predominant for the height distribution functions, cf. Fig. 11 *c*, 0.8 *x* = , and Fig. 11 *d*, 0.9 *x* = . It means that as nanodots of *Mn* grow, increasing of height leaves behind increasing lateral size *d* , so that *h d* > 1 . Probably, this circumstance just explains of the form of nanodots of *Mn* obtained by the authors of paper (De-yong Wang *et al*., 2006).

Fig. 12 shows the results of comparison of the theoretical dependence, cf. Eq. (96), with experimentally obtained histograms of particles of gold obtained at temperature 525°С at silicon substrate ( *Au Si* / 111 ( )) - (Fig. 12 *а*), and later, after 180-min isothermal exposure - (Fig. 12 *b*) (Werner *et al*., 2006). Judging by the magnitude of *x* , particle growth is initially controlled by the kinetic coefficient β (Fig. 12 *а*). But later, the mechanism of growth changes, and after three-hour exposure it becomes predominantly diffusion one (Fig. 12 *b*).

Fig. 12. Comparison of the dependence Eq. (96) with experimentally obtained histograms for particles of *Au* obtained by the molecular beam epitaxy technique for temperature 525°С (*а*) and 180 min later, after isothermal exposure (*b*)

The results of comparison of the theoretical dependence computed for 0.3 *x* = with the experimental histogram for nanoclusters of *Ag* obtained by the molecular beam epitaxy technique at room temperature at substrate *TiO*<sup>2</sup> (110) (Xiaofeng Lai *et al*., 1999), cf. Fig. 13, also argue in favour of the proposed mechanism of growth.

Thus, the considered examples of comparison of computed and experimentally obtained data leads to the conclusion on the possibility to implement simultaneous action of both mechanisms of growth, i.e. Wagner and diffusion ones. What is more, the situation when both mechanisms of growth co-exist and act in parallel is, to all appearance, more general than separate manifestations of one of two mechanisms considered early by Wagner and Chakraverty.

Mass Transfer Between Clusters Under Ostwald's Ripening 137

islands corresponds to heterostructure with more stable form of hut-clusters (Safonov &

The rate of change of volume of cluster with constant height *h* (Fig. 14) is determined by the

<sup>2</sup> ( ) *<sup>m</sup> <sup>d</sup> r h <sup>j</sup> dt* π=

1 2 *<sup>m</sup> dr <sup>j</sup> dt rh*

π

where *Sj* – the part of flow caused by surface diffusion, and *ij* – the part of flow of adatoms, which due to overcoming the potential barrier at the interface 'cluster-substrate' fall at cluster surface and, then, take part in formation of chemical connections (the Wagner

> 2 () *S S Rr dC j rD dR* <sup>=</sup> π

interface 'cluster-substrate', which can be represented in the form (Chakraverty, 1967;

<sup>1</sup> ( ) ln

where *l* determines the distance from an island, ( ) *R lr* = , at which a mean concentration of adatoms at a substrate, *C* , is set around separate cluster of radius *r* ( *l* = 2, 3 ). Taking into

> <sup>2</sup> ( ) ln *S S r <sup>D</sup> j CC <sup>l</sup>* π

Concentration of adatoms at the cluster base, *Cr* , is determined by the Gibbs-Thomson

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

where *C*∞ – the equilibrium concentration at temperature *T*, Δ*P* – the Laplacian pressure caused by island surface curvature, *k* – the Boltzmann constant. Pressure Δ*P* , in

∞ ∞ ⎛ ⎞ = Δ ≈ +Δ ⎜ ⎟ ⎝ ⎠

*R r dC C C dR l r* <sup>=</sup>

*dC*

*r*

*<sup>m</sup>* – adatom volume. From Eq. (105) one obtains:

Following to (Vengrenovich *et al*., 2008а), the flow *j* consists of two parts:

υ

υ

, (105)

<sup>=</sup> . (106)

*S i jj j* = + , (107)

<sup>=</sup> , (108)

<sup>−</sup> <sup>=</sup> <sup>⋅</sup> , (109)

= − . (110)

υ

*dR* <sup>=</sup> – concentration gradient at the

, (111)

Trushin, 2007).

mechanism of growth).

equation:

By definition, the diffusion part of a flow equals:

where *DS* – the surface diffusion coefficient, ( )*R r*

Vengrenovich 1980a, 1980b; Vengrenovich *et al*., 2008а):

account Eq. (109), one can rewrite Eq. (108) in the form:

accordance with (Vengrenovich et al., 2008а), equals:

where υ

flow *j* of adatoms to (from) a cluster:

Fig. 13. Comparison of the dependence Eq. (96) with experimental histogram of nanoclusters *Ag* obtained by the molecular beam epitaxy technique at substrate *TiO*<sup>2</sup> (110) at room temperature
