**7. Conclusions**

We have developed the theory of the Ostwald's ripening, taking into account not only mass transfer between clusters due to diffusion (volume, surface, dislocation), but also the kinetics of mass transfer through the interface 'cluster-matrix' ('cluster-substrate') determining the formation of chemical connections at cluster surface (the Wagner mechanism of cluster growth).

Within the developed by us theory, diffusion and kinetics of mass transfer through the interface 'cluster-matrix' are taken into account as the corresponding flows, ( ) *V S j j* and *ij* , in the resulting flow to (from) a cluster: ( ) *VS i jjj j* = + . The contribution of the each mechanism of mass transfer in the resulting flow, *j* , is represented as the ratio of the partial

flows: *V S j j <sup>x</sup> j j* ⎛ ⎞ <sup>=</sup> ⎜ ⎟ ⎝ ⎠ and 1 *ij <sup>x</sup> <sup>j</sup>* <sup>−</sup> <sup>=</sup> . Taking into account both diffusion and kinetics of mass

transfer through interface of two structural components means that one can not neglect any of the components of flow *j* , both ( ) *V S j j* and *ij* . It corresponds to the model of cluster ripening, in accordance with which growth of them is governed by two mechanisms, i.e. by the Wagner and by the diffusion ones. Within the framework of this model, the size

distributions shown in Fig. 19 can be used also for comparison with experimentally obtained

One of such comparisons is illustrated in Fig. 20 (*а – x* = 0.8 , *b – x* = 0.9 ). Experimentally obtained histogram normalized by unity on axes *uhh* ( *<sup>g</sup>* ) and ( ) max *gu g* corresponds to

0.0

0.2

0.4

0.6

*g(u)/gmax*

Fig. 21. Comparison of the experimentally obtained histograms with theoretically computed

In Fig. 21, the experimentally obtained histogram normalized in the same manner as in previous case, corresponds to the height distribution of islands of germanium (Ge/Si (001)) for the quantity of fall out of germanium 5.5 monolayers ( 5.5 *Ge d ML* = ) (Vostokov *et al*., 2000). Theoretical curves have been computed for *а – x* = 0.2 , and *b* – 0.4 *x* = . One can see that as *x* increases, as discrepancy between the experimentally obtained histogram and

We have developed the theory of the Ostwald's ripening, taking into account not only mass transfer between clusters due to diffusion (volume, surface, dislocation), but also the kinetics of mass transfer through the interface 'cluster-matrix' ('cluster-substrate') determining the formation of chemical connections at cluster surface (the Wagner

Within the developed by us theory, diffusion and kinetics of mass transfer through the

in the resulting flow to (from) a cluster: ( ) *VS i jjj j* = + . The contribution of the each mechanism of mass transfer in the resulting flow, *j* , is represented as the ratio of the partial

transfer through interface of two structural components means that one can not neglect any

ripening, in accordance with which growth of them is governed by two mechanisms, i.e. by the Wagner and by the diffusion ones. Within the framework of this model, the size

*<sup>j</sup>* <sup>−</sup> <sup>=</sup> . Taking into account both diffusion and kinetics of mass

*j j* and *ij* . It corresponds to the model of cluster

interface 'cluster-matrix' are taken into account as the corresponding flows, ( ) *V S*

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

*x=0.4*

*u*

*b*

*j j* and *ij* ,

the height (on *h* ) distribution function in (Ge/ZnSe) (Neizvestnii *et al.,* 2001).

*a*

histograms, when the island height, *h* , is constant.

0.0 0.2 0.4 0.6 0.8 1.0

*x=0.2*

*u*

theoretically computed dependences increases also.

and 1 *ij <sup>x</sup>*

of the components of flow *j* , both ( ) *V S*

dependences (Vostokov *et al*., 2000) *x* = 0.2 (*а*), 0.4 *x* = (*b*)

0.0

**7. Conclusions** 

flows: *V S j j <sup>x</sup> j j* ⎛ ⎞ <sup>=</sup> ⎜ ⎟ ⎝ ⎠

mechanism of cluster growth).

0.2

0.4

0.6

*g(u)*/*gmax*

0.8

1.0

distribution function of clusters is described by the generalized Lifshitz-Slyozov-Wagner distribution (alloys, nanocomposites as nc*CdS* /polimer (Savchuk *et al*., 2010a, 2010b, 2010c) or the generalized Chakraverty-Wagner distribution (island films, heterostructures with quantum dots, etc.)

If in the resulting flow *j* the component *i V j j* << , it can lead to new mechanism of cluster growth under dislocation-matrix or dislocation-surface diffusion, for the each of which the specific size distribution function and the corresponding temporal dependences of *r* and *gr* are intrinsic.

Comparison of the theoretically computed size distribution functions with experimentally obtained histograms leads to the following two main conclusions.

1. The introduced model of cluster ripening under simultaneous (combined) action of both the diffuse mechanism and the Wagner one is proved experimentally. Other of the considered models is also finds out experimental proof, *viz.* the case when one neglects the Wagner mechanism of growth and cluster ripening results from mixed dislocation-matrix and dislocation-surface diffusion. Thus, it is the most likelihood that, in practice, cluster growth follows to not only one isolated of the considered early mechanisms of growth, i.e. the diffusion mechanism or the Wagner one, but rather to the mixed (combined) mechanism, when two mentioned limiting mechanisms act together.

It also follows from the results of comparison of the computed and experimental data, that cluster growth under mixed (combined) dislocation-matrix or dislocation-surface diffusion is most probable than cluster growth under any of two mentioned mechanisms, if isolated.

2. In connection with intense development of nanotechnologies and related techniques for generating of nanostructures, the problem arises: in what framework is the LSW theory applied to analysis of nanosystems containing nanoclusters. The final answer on this question is now absent. Also, the main question concerning stability of nanosystems in respect to the Ostwald's ripening leaves opened. Nevertheless, it follows from the represented by us results of comparison of theoretical and experimental data, that in many cases the experimentally obtained histograms built for nanoparticles (nanoclusters) by many authors for various nanosystems are quite satisfactory fitted by the computed by us theoretical distributions (the generalized Lifshitz-Slyozov-Wagner distribution, the generalized Chakraverty-Wagner distribution etc.). In means that the developed by us LSW theory can be, in principle, be used for analysis of phase and structural transformations in nanosystems with nanophases. Of course, derived by us approach requires further investigations, both theoretical and experimental.
