Wagner-Vengrenovich Distribution

The process of coarsening of nanoclusters or nanocrystals (NCs) is investigated for the case when cluster growth (dissolution) is governed simultaneously by both diffusion along dislocation pipes and the rate of formation of chemical connections (chemical reaction) at cluster surface, namely,Wagner’s growingmechanism. For that, the total flow of atoms to (from) a cluster is represented by two parts, namely, diffusion part and the Wagner (kinetic) one. The dependence of the rate of growth of NC on the ratio of the parts of the total flow has been determined as well as the NC’s size distribution function referred to as the Wagner-Vengrenovich distribution. Computed distribution is compared with experimentally obtained histograms.


Introduction
Ostwald's ripening (OR) is the final stage of formation of a new phase as a result of phase transformation, such as decay of oversaturated solid solutions. Nanoclusters or nanocrystals (NCs) of new phase having different sizes interact through the Gibbs-Thomson effect that results in dissolution of small NC and growth of large ones. Diffusion growth of NC under matrix of volume diffusion (ls-mechanism) has been firstly studied by Lifshitz and Slyozov [1,2]. Wagner has showed later [3] that beside diffusion mechanism, another mechanism of NC growth is possible, which is governed by the rate of formation of chemical connections (chemical reaction) at NC surface (w-mechanism). The theory developed in the cited papers is referred to as the LSW theory. Practical verification of this theory shows that in many cases, it is proper for the description of experimental data on temporal behavior of the mean NC size and the NC size the distribution function, while in other cases the LSW theory must be refined.
In this connection, NC growth is considered in papers [4,5] as a result of combined action of two growing mechanisms, diffusion (ls) and Wagner's (w) ones. In the framework of the modified LSW theory and taking into account both mechanisms of growing (ls and w), we can obtain a size distribution in the form of the generalized Lifschitz-Slyozov-Wagner distribution [4]. This distribution can refer to a much wider range of the experimental histograms than each of the Lifschitz-Slyozov (LS) and Wagner's ( ) distributions separately.
As it has been mentioned above, the generalized LSW distribution has been obtained by combination of the diffusion (ls) and Wagner's (w) mechanisms of NC growth. However, in presence of a matrix of free dislocations interacting with elastic fields of NCs or in presence of NCs located at dislocation network, just dislocation diffusion (v-mechanism of NC growth) predominates rather than matrix one. Peculiarities of the Ostwald's ripening under diffusion along dislocation pipes have been considered supra [26][27][28][29][30].

ISRN Nanomaterials
The purpose of this paper consists in finding out the size distribution function and temporal dependences of NC mean size for the case when NC growth is governed simultaneously by both mechanisms, namely, dislocation diffusion (v) and the rate of formation of chemical connections (chemical reaction) at cluster surface (w). Computations are carried out within the framework of the modified LSW theory [4], using the technique introduced in paper [31].

Theoretical Part
As NC growth is governed by two mechanisms (v and w) simultaneously, the total flow of atoms to (from) NC consists of two parts [4] = + , where is the diffusion part of flow that equals the number of atoms reaching cluster surface in unit time through dislocation diffusion and is the Wagner's part of flow that equals the number of atoms involved in forming chemical connections at cluster surface in unit time.
To determine the size distribution function, ( , ), let us use the continuity equation wherė≡ / is the rate of growth of cluster with radius , which is determined from the following condition: where is determined by (1). In accordance with [27], the diffusion part of a flow equals where is the diffusion coefficient along dislocations, is the the number of dislocation lines crossing a cluster ( = const), is the cross-section of dislocation pipe, ⟨ ⟩ is the mean concentration of diffusing atoms into solution, is that concentration of atoms of solved matter at the boundary of a cluster of radius , ∞ is that equilibrium concentration for the specified temperature , and is that volume of an atom of solved matter.
Wagner's (kinetic) part of a flow equals [3]: where is the kinetic coefficient.
Knowing and , one finds out the rate of cluster growth from (3). Consider the following: where ⟨ ⟩ − = (2 / )(1/ − 1/ ), is the critical radius that, within the LSW theory approximation, assumes to be equal to the mean radius, ⟨ ⟩.
To determine the analytical form of the size distribution function of clusters with peculiarities intrinsic in Wagner's ( ) distribution [3] and Vengrenovich's ( ) distribution [27], one must express the rate of growth (6) as the ratio of flows and where is the part of flow in the total flow : = / and (1 − ) is the part of flow in the total flow : 1 − = / . Taking into account that (4) and (5) for flows are valid for NCs of arbitrary radius , including NCs of the maximal size [4,5], one can rewrite (6) in the form where * = 2 2 ∞ / , = / , or in the form where * = (2 2 ∞ )/ . Equation (8) corresponds to the rate of growth of NCs that is determined by the kinetic flow , namely, by the rate of formation of chemical connections, with the share contribution (1 − ) of the diffuse flow ( < 1), and the rate of growth (9) is determined by the diffuse flow with the share contribution of the kinetic flow ( > 0).
From (8) or (9), for the rate of growth, one can determine the ratio / that in terms of paper [1] corresponds to the locking point magnitude, 0 ( 0 = / ). Knowing 0 enables to integrate (8) and (9) and to find out temporal dependences for and , as well as to separate the variables in the continuity equation (2) and to find out the analytical form of the size distribution function. In accordance with [31], the ratio ( / ) ( 0 ) can be found out from the dependence of the specific rate of growth/ on shown schematically in Figure 1. Derivative equals zero at the maximum of this function From (10), one obtains Assuming = ( = 1) in (8) and replacing the ratio / by its magnitude from (11), one obtains by integration Equation (12) determines the temporal dependences for the maximal radius of NCs, , and the critical one, , when the Wagner's mechanism of growth predominates. The limits of changing parameter can be determined from comparison ISRN Nanomaterials of experimental and theoretical data, for example, for size distribution function, which equals in our case 0 ≤ ≤ 0, 3, approximately.
For = 0, growth of NCs is completely controlled by the rate of formation of chemical connections (the wmechanism) [3] Similarly, one obtains from (9) Equation (14) corresponds to the temporal dependences of changing and , when diffusion along dislocation pipes predominates in the growth of NCs. For that, the parameter changes within interval 0, 7 ≤ ≤ 1, approximately. For = 1, growth of NCs is completely controlled by the diffusion of solved atoms along dislocation pipes (the vmechanism) [26,27]: In accordance with [26], the total change of NC's volume, Δ Σ , consists from Δ , that is, the change caused by the dislocation flux, , and Δ caused by the kinetic flux, , where for the sake of simplicity, time reading is made from zero: = 0 = 0; 3 Σ ( 0 ) = 0; 3 ( 0 ) = 0; 3 ( 0 ) = 0. Taking into account (12) and (14), one obtains from (17) where 1 ( ) = 5 * (1 + 3 ) 4 / (2 + 3 ) 5 , 2 ( ) = 2 * (1 + 3 )/(1 − )(2 + 3 ) 2 , and the limits of changing total radius of NCs are 0, 3 ≤ ≤ 0, 7, approximately. In accordance with [31], the size distribution function of NCs is found out as the product of two functions where = ( ). By substitution (19) into the mass conservation law for disperse phase (NCs), one obtains where = 3 /4 ∫ Taking into account (21), one can write the size distribution function in the form where and ( ) is the required distribution function on relative sizes of NCs, called by us the Wagner-Vengrenovich (WV) distribution function, as the distributions corresponding to each of the mechanisms, w and v, separately have been early obtained by Wagner (W) [3] and Vengrenovich (V) [27]. For determining ( ), let us use the continuity equation (2) substituting in this equation, instead of ( , ) and, their magnitudes from (19) and (8) (or (9)). By transition in (2), from differentiation on and to differentiation on [31], variables are separated where one takes into account that =̇/ * , =̇/ * , / = 1/ , and / = − / . Finding out , , and / from (8), one can rewrite (24) in the form  by integration of which one obtains the WV size distribution function: with the magnitudes of the power indices, , , , , and coefficients , , , and expressed through the parameter are given in the appendix.

Results and Discussion
The WV distribution is represented in Figure 2. Dependences represented in Figure 2(a) are computed with interval Δ = 0, 1. The same dependences normalized by their maxima are represented in Figure 2(b). In the normalized form, such dependences are well behaved for comparison with the normalized experimentally obtained histograms.
The results of comparison of the experimentally obtained histograms with our computations, (23), are shown in Figure 3 for various magnitudes of the parameter . All experimentally obtained histograms and theoretically found dependences are normalized by their maxima. For convenience, the LS distribution is represented, accurate within a constant, as a function of variable = / : Experimentally obtained histograms shown in Figure  Solid curve corresponds to the WV distribution for relevant magnitudes of the parameter . One can see that neither the distribution ( = 0) nor the distribution ( = 1) is well fitting the experimentally obtained histograms. The LS distribution (dashed curve) and the WV distribution for relevant magnitude of the parameter (solid curve) fit experimentally obtained histograms much more reliably. Moreover, one can see from Figure 3 that the WV distribution (23) provides better fitting of experimental data than the LS distribution.
The same is observed for alloys Al(Sc,Zr). Histograms in Figure 4 correspond to size distribution of NCs Al 3 Sc 1− Zr in such alloys [33]. Alloys (Al-0, 14Sc-0, 012Zr)am% were exposed at temperature 375 ∘ C over a period: (a) 3; (b) 192; (c) 288 h. Similarly to the case of binary alloys, the WV distribution (23) satisfactorily fits experimentally obtained histograms. It means that the proposed mechanism of cluster coarsening governed by two mechanisms (w and ls ones) in parallel is realized under the OR. For that, interpretation of experimentally obtained histograms is made by applying the WV distribution.

Conclusions
Summarizing, it has been shown that the modified Lifshitz-Slyozov-Wagner (LSW) theory [4,5] can be applied to nanocluster systems. Theory of the Ostwald's ripening for alloys containing nanocrystalline phases must be assumed taking into account not only diffusion flow, , but also kinetic (Wagner's) one, , it has been shown, for alloys containing Al-Sc and Al(Sc, Zr), correspondingly, NCs Al 3 Sc and Al 3 (Sc 1− Zr ), that the size distribution function is satisfactorily fitted by the introduced Wagner-Vengrenovich (WV) distribution here.