# Wagner-Vengrenovich Distribution

Authors: Bohdan V. Ivanskii; Anatolii V. Moskalyuk; Sergey V. Yarema; Igor I. Panko; Miroslav O. Stasyk
Journal: ISRN Nanomaterials
 (2013)

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## Abstract

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 growing mechanism. 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.


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## Body

## 1. 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 (W) distributions separately.The products of nanotechnologies [6–8] become new objects of applying the LSW theory. As sizes become 100 nm and less [9], the characteristics of both separate NCs and of the system as a whole change cardinally that provides practically useful properties. The OR is among probable factors destabilizing microstructure and properties of nanocluster systems. Semiconductor heterosystems with quantum dots (QDs) obtained by electron-beam epitaxy technique [10–16] in the Stranski-Krastanov regime [17] as well as arrays of QDs synthesized from liquid matter by chemical and other techniques [18–25] are examples of such nanocluster systems.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 [26–30].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].


## 2. 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]
(1)j=jd+ji,
where jd is the diffusion part of flow that equals the number of atoms reaching cluster surface in unit time through dislocation diffusion and ji 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,f(r,t), let us use the continuity equation
(2)∂f(r,t)∂t+∂∂r[f(r,t)·r˙]=0,
where r˙≡dr/dt is the rate of growth of cluster with radius r, which is determined from the following condition:
(3)ddt(43πr3)=jυm,
where j is determined by (1).In accordance with [27], the diffusion part of a flow equals
(4)jd=Dd2Zq〈C〉-Crr,
where Dd is the diffusion coefficient along dislocations, Z is the the number of dislocation lines crossing a cluster (Z=const), q is the cross-section of dislocation pipe, 〈C〉 is the mean concentration of diffusing atoms into solution, Cr=C∞exp(2συm/rkT)≈C∞(1+2συm/rkT) is that concentration of atoms of solved matter at the boundary of a cluster of radius r, C∞ is that equilibrium concentration for the specified temperature T, and υm is that volume of an atom of solved matter.Wagner’s (kinetic) part of a flow equals [3]:
(5)ji=4πr2β(〈C〉-Cr),
where β is the kinetic coefficient.Knowingjd and ji, one finds out the rate of cluster growth from (3). Consider the following:
(6)drdt=14πr2[Dd2Zq(〈C〉-Cr)r+4πr2β(〈C〉-Cr)],
where 〈C〉-Cr=(2συm/kT)(1/rk-1/r), rk is the critical radius that, within the LSW theory approximation, assumes to be equal to the mean radius, 〈r〉.To determine the analytical form of the size distribution function of clusters with peculiarities intrinsic in Wagner’s (W) distribution [3] and Vengrenovich’s (V) distribution [27], one must express the rate of growth (6) as the ratio of flows jd and ji(7)jdji=x1-x,
where x is the part of flow jd in the total flow j:x=jd/j and (1-x) is the part of flow ji in the total flow j:1-x=ji/j. Taking into account that (4) and (5) for flows are valid for NCs of arbitrary radius r, including NCs of the maximal size rg [4, 5], one can rewrite (6) in the form
(8)drdt=A*r(x1-x1u3+1)(rrk-1),
where A*=2συm2C∞β/kT, u=r/rg, or in the form
(9)drdt=B*r4(1+1-xxu3)(rrk-1),
where B*=(2συm2C∞ZqDd)/πkT.Equation (8) corresponds to the rate of growth of NCs that is determined by the kinetic flow ji, namely, by the rate of formation of chemical connections, with the share contribution (1-x) of the diffuse flow (x<1), and the rate of growth (9) is determined by the diffuse flow jd with the share contribution x of the kinetic flow (x>0).From (8) or (9), for the rate of growth, one can determine the ratio rg/rk that in terms of paper [1] corresponds to the locking point magnitude, u0  (u0=rg/rk). Knowing u0 enables to integrate (8) and (9) and to find out temporal dependences for rg and rk, 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 (rg/rk) (u0) can be found out from the dependence of the specific rate of growth r˙/r on r shown schematically in Figure 1. Derivative equals zero at the maximum of this function
(10)ddr(r˙r)|r=0.Figure 1
Schematic representation of the specific rate of growth(r˙/r) on r.From (10), one obtains
(11)rgrk=2+3x1+3x.Assumingr=rg(u=1) in (8) and replacing the ratio rg/rk by its magnitude from (11), one obtains by integration
(12)rg2=2A*(1-x)(1+3x)t,rk2=2A*(1+3x)(1-x)(2+3x)2t.Equation (12) determines the temporal dependences for the maximal radius of NCs, rg, and the critical one, rk, when the Wagner’s mechanism of growth predominates. The limits of changing parameter x can be determined from comparison of experimental and theoretical data, for example, for size distribution function, which equals in our case 0≤x≤0,3, approximately.Forx=0, growth of NCs is completely controlled by the rate of formation of chemical connections (the w-mechanism) [3]
(13)rg2=2A*t,rk2=12A*t,rgrk=2.Similarly, one obtains from (9)
(14)rg5=5B*x(1+3x)t,rk5=5B*(1+3x)4x(2+3x)5t.Equation (14) corresponds to the temporal dependences of changing rg and rk, when diffusion along dislocation pipes predominates in the growth of NCs. For that, the parameter x changes within interval 0,7≤x≤1, approximately.Forx=1, growth of NCs is completely controlled by the diffusion of solved atoms along dislocation pipes (the v-mechanism) [26, 27]:
(15)rg5=54B*t,rk5=(45)4B*t,rgrk=54.In accordance with [26], the total change of NC’s volume, ΔVΣ, consists from ΔVd, that is, the change caused by the dislocation flux, jd, and ΔVi caused by the kinetic flux, ji,
(16)ΔVΣ=ΔVd+ΔVi,
or
(17)43πrkΣ3=43πrkd3+43πrki3,
where for the sake of simplicity, time reading is made from zero: t=t0=0; rkΣ3(t0)=0; rkd3(t0)=0; rki3(t0)=0. Taking into account (12) and (14), one obtains from (17)
(18)rkΣ=[f1(x)t3/5+f2(x)t3/2]1/3,
where f1(x)=5B*(1+3x)4/x(2+3x)5, f2(x)=2A*(1+3x)/(1-x)(2+3x)2, and the limits of changing total radius of NCs x are 0,3≤x≤0,7, approximately.In accordance with [31], the size distribution function of NCs is found out as the product of two functions
(19)f(r,t)=φ(rg)g′(u),
where rg=ψ(t).By substitution (19) into the mass conservation law for disperse phase (NCs),
(20)M=43πρ∫0rgr3f(r,t)dr,
one obtains
(21)φ(rg)=Qrg4,
where Q=3M/4πρ∫01u3g′(u)du.Taking into account (21), one can write the size distribution function in the form
(22)f(r,t)=1rg4g(u),
where
(23)g(u)=Qg′(u),
and g(u) 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 determiningg′(u), let us use the continuity equation (2) substituting in this equation, instead of f(r,t) and r˙, their magnitudes from (19) and (8) (or (9)). By transition in (2), from differentiation on r and t to differentiation on u [31], variables are separated
(24)dg′(u)g′(u)=-4υgu2+υ-u(dυ/du)u3υg-υudu,
where one takes into account that υ=r˙r/A*, υg=r˙gr/A*, du/dr=1/rg, and du/drg=-u/rg.Finding outυ, υg, and dυ/du from (8), one can rewrite (24) in the form
(25)dg′(u)g′(u)=-4u5+u3(3x2-2x-1)+ux(9x+6)-12x2-4xu(1-u)2[u3+u2x(3x-1)+3ux2+x(3x+1)]×du,
by integration of which one obtains the WV size distribution function:
(26)g′(u)=uσ(1-u)-α(u+b)β(u2+cu+d)γexp(ω1-u)g′(u)×exp[ε-γcd-c2/4tan-1(u+(c/2)d-(c2/4))],
with the magnitudes of the power indices, σ,  α,  β,  γ, and coefficients γ,  ε,  c,  and  d expressed through the parameter x are given in the appendix.


## 3. Results and Discussion

The WV distribution is represented in Figure2. Dependences represented in Figure 2(a) are computed with interval Δx=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.Dependences corresponding to the WV distribution computed with intervalΔx=0,1 (a); the same dependences normalized by the corresponding maxima (b).
(a)
(b)The results of comparison of the experimentally obtained histograms with our computations, (23), are shown in Figure 3 for various magnitudes of the parameter x.  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 u=r/rg:
(27)g′(u)=u2(1-u)-11/3(u+2)-7/3exp(-11-u).Comparison of experimentally obtained histograms of Al3Sc NCs [32] with the WV distribution (solid curve) for exposition of alloys for temperature 350°C over a period: (a) 2; (b) 5; (c) 72; (d) 104; (e) 168 h.
(a)
(b)
(c)
(d)
(e)Experimentally obtained histograms shown in Figure3 correspond to the size distribution function of Al3Sc NCs in binary alloys Al–Sc(0,12 am%Sc). Exposition of alloys has been performed at temperature 350°C over a period: (a) 2; (b) 5; (c) 72; (d) 104; (e) 168 h [32]. Dashed curves correspond to the LS, W(x=0), and V(x=1) distributions. Solid curve corresponds to the WV distribution for relevant magnitudes of the parameter x. One can see that neither the W distribution (x=0) nor the V distribution (x=1) is well fitting the experimentally obtained histograms. The LS distribution (dashed curve) and the WV distribution for relevant magnitude of the parameter x (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 Figure4 correspond to size distribution of NCs Al3Sc1-xZrx 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.Comparison of experimentally obtained histograms of Al3Sc1-xZrx NCs [33] with the WV distribution (solid curve) for exposition of aluminium-based alloys containing 0,14 Sc and 0,012 Zr at % for temperature 375°C over a period: (a) 3; (b) 192; (c) 288 h.
(a)
(b)
(c)


## 4. 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, jd, but also kinetic (Wagner’s) one, ji, it has been shown, for alloys containing Al–Sc and Al(Sc,Zr), correspondingly, NCs Al3Sc and Al3(Sc1-xZrx), that the size distribution function is satisfactorily fitted by the introduced Wagner-Vengrenovich (WV) distribution here.



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*Source: 651576-2013-06-20.xml*