Effect of diffusion on nucleation of 2D and 3D nanoclusters in supersaturated solutions

Physica A 387 (2008) 2419–2426 www.elsevier.com/locate/physa

Effect of diffusion on nucleation of 2D and 3D nanoclusters in supersaturated solutions D.N. Korolev, M.V. Sorokin ∗ , A.E. Volkov Russian Research Centre ‘Kurchatov Institute’, Kurchatov Sq.1, Moscow, 123182, Russia Received 7 November 2006; received in revised form 6 December 2007 Available online 9 January 2008

Abstract The effect of diffusion on the steady-state nucleation of 2D and 3D nanoclusters is described analytically. Proposed approach takes self-consistently into account coupling between the kinetics of monomers near the cluster boundary, their diffusion and annihilation at other clusters. It has been shown that due to this coupling the nucleation barriers can considerably differ from those predicted by the thermodynamic approach. c 2008 Elsevier B.V. All rights reserved.

PACS: 64.60.Qb; 68.55.Ac; 68.43.Hn; 68.43.Jk Keywords: Nucleation; Nucleation barrier; Nanocluster; Diffusion

1. Introduction Decomposition of a solid solution supersaturated with impurity atoms results in creation of an ensemble of nanoclusters. Analytical approaches describing decomposition of supersaturated solutions are based on the adiabatic principle [1]. This principle declares that diffusion currents of monomers in the vicinity of clusters adjust themselves to the actual cluster sizes. In this case the probabilities of absorption/desorption of atoms on the cluster interfaces depend only on these sizes and clustering is a Markovian stochastic process completely described by these probabilities [2]. The thermodynamic approach assumes that forming clusters does not disturb the supersaturated solution [3] and the absorption probability depends on the average monomer concentration in the system. We will consider a more general case, when the absorption/desorption probabilities depend on both: (i) the kinetics of impurity atoms in the vicinity of the cluster interface including their interaction with this interface, and (ii) the diffusion current of these atoms from the interior to the cluster so that the difference between the absorption and desorption rates of monomers coincides with the value of this current at the cluster interface. The diffusion current results from the spatially non-uniform impurity concentration induced by a forming cluster. Having in mind the adiabatic principle the current can be determined from the solution of the steady-state diffusion problem in the vicinity of the selected cluster. The average concentration of monomers in the system and ∗ Corresponding author. Tel.: +7 4951969178.

E-mail address: [email protected] (M.V. Sorokin). c 2008 Elsevier B.V. All rights reserved. 0378-4371/$ - see front matter doi:10.1016/j.physa.2008.01.019

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their absorption by other clusters form the boundary conditions for this problem. Thus, the absorption/desorption mechanisms couple the kinetics of impurity atoms at the cluster interface with impurity diffusion which depends on monomer annihilation at other clusters. We demonstrate in this paper that such coupling can result in the forms of parameters governing clusterization which deviate from those predicted by the thermodynamic approach [2,4–6]. For illustration of this effect we choose the steady-state nucleation mode [6–8] when the nucleation barrier determines the steady-state nucleation rate of clusters. 2. Governing equations We assume the cluster ensembles consisting of either monolayer disks having radii R = (nω/πccl )1/2 in the case of 2D geometry or spheres with R = (3nω/4πccl )1/3 in the 3D case. Here n is a number of impurity atoms constituting a cluster, ω is the space occupied by one monomer (area in 2D case or volume in 3D); ccl is the atomic concentration of monomers in clusters. In order to elucidate the effect of monomer diffusion on the cluster kinetics we assume in this paper the spatially homogeneous cluster nucleation in the system. In order to describe the steady-state nucleation and the constant mean monomer concentration in the solution we assume that an amount of monomers accumulated in clusters remains negligible in comparison to that contained in the solution (dilute ensemble). We suggest also a low monomer supersaturation when the size of the critical nucleus considerably exceeds the interatomic distance. This permits us to use the continuous approximation for the cluster sizes in the vicinity of the critical cluster nucleus. And finally, no lattice misfits (no elastic effects) between a cluster and the matrix are considered. Nucleation in such dilute ensemble of clusters can be described in terms of “one cluster” distribution function f 1 (n, t) so that f 1 (n, t)dn is the densityRof clusters with sizes n . . . n + dn at time t [2]. This distribution function is normalized to the density ρ of clusters: f 1 (n, t)dn = ρ. Evolution of this distribution function in the vicinity of the critical size is described by the Fokker–Planck equation (FPE) [2]: ∂ f 1 /∂t = −∂ I /∂n.

(1)

We choose the Itoh form for FPE when the cluster current I (n; t) along the “size” axis n is defined as I (n; t) = A f 1 − ∂(B f 1 )/∂n. The kinetic coefficients A and B are, respectively, the “hydrodynamic” rate and the “diffusion” coefficient of clusters in the size space. These coefficients completely determine the evolution of the cluster ensemble. They depend on the probabilities of absorption P(n) and desorption Q(n) of a monomer at the cluster interface per a unit time. A(n) = P(n) − Q(n),

B(n) = [P(n) + Q(n)]/2.

(2)

In particular, A and B determine the form of the steady state cluster current into the growth region of the size axis Is (the steady-state nucleation rate) [2]: q Is = f 0 (n min )B(n min )(2π )−1/2 |d 2 ϕ/dn 2 |n=n c × exp(−ϕc ). (3) Here f 0 is the distribution function, which maintains the detailed balance in the cluster ensemble on the size axis: I0 ( f 0 ) = A f 0 − ∂(B f 0 )/∂n = 0. n c is the size of the critical cluster which is in equilibrium with the supersaturated solution, ϕc = ϕ(n c ) is the nucleation (critical) barrier and the function ϕ(n) is determined by the kinetics coefficients: Z n A(n 0 ) 0 ϕ(n) = − dn . (4) 0 n min B(n ) 3. Absorption and desorption rates The desorption probability Q depends on the number of monomers at the boundary layer having the thickness l, close to the diffusion jump length λ; the barriers for monomer evaporation, and the rate of monomer diffusion from the cluster [9] (see Fig. 1):     Ψ + δ Fs ν Φm + ε L d lξ ccl exp − exp − . (5) Q= ω T 2d T

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Fig. 1. The Gibbs energy of the system containing the cluster of the size of R and an impurity atom near the cluster interface.

Here d = 2, 3 indicate the geometry of the problem; L 2 = 2π R, L 3 = 4π R 2 ; ξ ≤ 1 takes into account the interface roughness; ν is the attempt frequency of atomic jumps, 1/(2d) takes into account geometry of the problem; Ψ - is the Gibbs energy of dissolution of a monomer from a flat interface; Fs = γ L d is the Gibbs energy of the curved cluster interface and δ Fs = −∂ Fs /∂n is its change due to evaporation of a monomer; Φm is the barrier for monomer diffusion and T is the temperature measured in the energy units. The interface tension γ is assumed as constant. An additional barrier ε in the interface vicinity [9] can originate from possible distortion of monomer jump barrier at the interface, or be due to interface roughness, or to peculiarities of monomer diffusion related to collective effects near the cluster interface. The equilibrium atomic concentration of monomers at the curved cluster interface Ceq (R) is evaluated from the equality of the chemical potentials of monomers at the interface and outside it. Taking into account that δ Fs = −γ ω/ccl R in 2D geometry and δ Fs = −2γ ω/ccl R in 3D case, we obtain: 0 Ceq (R) = ccl exp[−(Ψ − γ ω/ccl R)/T ] = Ceq exp(γ ω/ccl RT )

(d = 2)

0 Ceq (R) = ccl exp [−(Ψ − 2γ ω/ccl R)/T ] = Ceq exp(2γ ω/ccl RT )

(d = 3).

(6)

0 = c exp(−Ψ /T ) is the equilibrium concentration of monomers near the flat interface. Taking into account Here Ceq cl Eq. (6) and the usual form of the monomer diffusion coefficient D = (λ2 ν/2d) exp(−Φm /T ) we can finally rewrite the desorption rate (5) as:

Q = L d lξ ω−1 λ−2 D exp(−ε/T )Ceq (R).

(7)

The absorption probability (absorption rate) P(n) depends on the number of monomers in the outside matrix layer nearest to the cluster interface and the barrier Φm for the diffusion jump of a monomer between this layer and the interface (see Fig. 1):   L d lξ ν Φm + ε P(n) = C(E rs ) exp − = L d lξ ω−1 λ−2 D exp(−ε/T )C(E rs ). (8) ω 2d T Here C(E r ) is the local atomic concentration of monomers, points rEs belong to the cluster interface, and C(E rs ) is the concentration in the outside layer nearest to the interface. In Eqs. (7) and (8) we assume λ  R in agreement with the continuous approximation about the cluster sizes. 4. Diffusion in the cluster vicinity Due to the adiabatic principle [1] the diffusion currents of monomers in the vicinities of clusters adjust themselves to the actual cluster sizes and positions [9]. In this case the concentration of impurity atoms C(E r ) in between clusters, whose number positions and sizes are considered as fixed, is described by: ∇ 2 C(E r |Ξ ) = 0.

(9)

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Here Ξ = (E r 1 , R1 ; rE2 , R2 ; . . . ; rE N , R N ) indicates positions rEi and radii Ri of all N clusters constituting the ensemble. The boundary conditions for Eq. (9) are formed by: (i) absorption and desorption of monomers on the boundaries of all clusters and (ii) coincidence of the monomer concentration far from the cluster interfaces with the mean concentration C0 : I ( Ej(E rsk |Ξ ), eEk ) dlk = Q(n k ) − P(n k ); C(E r |Ξ ) = C0 , (|E r − rEk |  Rk ). (10) lk

Here Ej = −ω−1 D∇C is the diffusion current of impurity atoms, eEk is the unit vector normal to the kth cluster interface at points rEsk and integration is performed over all these points. Setting the parameters of the selected cluster (k = m) as rEm = 0 and Rm = R we can reduce the diffusion problem (9) and (10) by averaging over parameters of all clusters except the selected one and transferring the boundary conditions at the edges of residual clusters into Eq. (9) (for details see Ref. [9]): Z ∞ −1 2 J2 (E r |0, R; rE, R 0 ) p2 (E r , R 0 |0, R) dR 0 = 0 ω D∇ C1 (E r |0, R) − N (11) 0 C1 |r →∞ = C0 , L d Rω−1 D(Ee, ∇C1 )|r =R = J1 |r =R = P1 − Q. The second term in Eq. (11) is the loss intensity, which describes the loss of monomers at all remaining clusters of the ensemble besides the selected one. Subscript “2” indicates that during the averaging the positions of already two clusters are fixed: the selected one having the position and dimension (0, R) and the “probe” one having parameters (E r , R 0 ). p2 is the averaged conditional probability to find a cluster with parameters (E r , R 0 ), when parameters of the selected cluster are fixed (0, R). In the dilute cluster ensemble the conditional distribution function p2 can be approximated by the “one-cluster” distribution function [10]: p2 (E r , R 0 |0, R) = f 1 (E r , R 0 )/ρ

or

f 2 (0, R; rE, R 0 ) = f 1 (0, R) f 1 (E r , R 0 )/ρ

(12)

R0)

J2 (E r |0, R; rE, is the diffusion current of monomers which is determined by the difference of absorption and desorption rates at the interface of the probe cluster (E r , R 0 ): J2 = P2 − Q,

P2 = 2π R 0lξ ω−1 λ−2 D exp(−ε/T )C2 (rE0 s |0, R; rE, R 0 ).

(13)

The second concentration moment C2 (E r |0, R; rE, R 0 ) depends on the parameters of the selected and probe clusters [11]. It should be evaluated from the third concentration moment through the equations similar to Eq. (11) and so on. Thus, the diffusion equations for the concentration moments form the infinite coupled hierarchy originating from coupled hierarchy of the partial distribution functions [11]. Usually, this infinite hierarchy is truncated by the simplest fist-order additive or multiplicative approximations when the second concentration moment is replaced by the sum or product of the first moments [11]. In this paper we used the self-consistent approximation [9] of the second concentration moment: C2 (rE0 s |0, R; rE, R 0 ) = C1 (rE0 s |0, R) − α(R 0 )[C1 (rE0 s |0, R) − Ceq (R 0 )].

(14)

Substituting Eqs. (13) and (14) into (11) and assuming a spatially uniform system when f 1 (E r , R 0 ) = f 1 (R 0 )/S 0 0 (d = 2) and f 1 (E r , R ) = f 1 (R )/V (d = 3), we reduce Eq. (11) to the following: D(∇ 2 − k 2 )C1 + G eq = 0. Here k −1 = ls is the sink strength and ls is the screening length [12].  ε Z ∞ lξ k 2 = 2 exp − L d (R 0 )[1 − α(R 0 )] f 1 (R 0 ) dR 0 . T λ 0

(15)

(16)

This term in Eq. (15) provides the existence of a finite solution of the steady-state 2D diffusion problem in the vicinity of the selected island (0, R). The last term in Eq. (15) describes the averaged thermal monomer emission from clusters:  ε Z ∞ lξ G eq = 2 exp − L d (R 0 )[1 − α(R 0 )]Ceq (R 0 ) f 1 (R 0 ) dR 0 . (17) T λ 0

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Solving diffusion problem (15) with boundary conditions (11) we obtain the concentration of monomers at the boundary of the fixed cluster: C1 |r =R = C0 −

u [C0 − Ceq (R)] 1+u

(18)

u(R) = λ−2lξ exp(−ε/T )k −1 K 0 (k R)K 1−1 (k R), u(R) = λ

−2

lξ exp(−ε/T )R(1 + k R)

−1

,

(d = 2)

(d = 3).

(19)

Here K 0 , K 1 are the modified Bessel functions of the zeroth and the first order respectively. Averaging Eq. (14) over the parameters of the selected cluster (0, R) we obtain the value of the first moment at the boundary of the probe cluster (E r , R 0 ): C1 (rE0 s |E r , R 0 ) = C0 − α(R 0 )[C0 − Ceq (R 0 )].

(20)

Taking into account the equivalence of all clusters of the ensemble and comparing (18) and (20) we determine the relationship between the functions α(R) and u(α, R): α(R) = u(α, R)[1 + u(α, R)]−1

(21)

Eq. (21) self-consistently closes our hypothesis (14) about truncating of the hierarchy of the concentration moments. The parameter u(R) includes the combination of the effective length of the diffusion jumps of monomers at the interface of the cluster λ∗ = (λ/ξ ) exp(ε/T ) with the effective length leff (R) = ls K 0 (R/ls )/K 1 (R/ls ) (d = 2) and leff (R) = R/(1+ R/ls ) (d = 3) which can be treated as a typical distance from a cluster of size R, where the diffusion of impurity atoms is sensitive to the presence of the cluster. Comparing λ∗ with leff (R), two limiting shapes of the radial dependence of the monomer concentration (18) in the vicinity of the selected cluster can be distinguished: i. The concentration radial “step” occurs at the interface when (leff (R)  λ∗ , u(R)  1): C (r ) (R) ≈ C0 .

(22)

Such cluster does not disturb the monomer solution and the kinetics of monomers is determined by the rate of their capture and release by the cluster interface (reaction controlled case, rcc). The nucleation barrier for such clusters is not sensitive to the diffusion of monomers and coincides with that predicted by the thermodynamic approach. ¯ The well-known form of the sink strengths can be obtained in this case: k 2 = lξ exp(−ε/T )λ−2 L d ( R)ρ, where R ¯ R = R f 1 (R) dR is the mean radius of clusters. ii. In the opposite case (leff (R)  λ∗ , u(R)  1) a spatially non-uniform profile of the monomer concentration in the vicinity of the a cluster (diffusion “clouds”) occurs: C (d) (R) ≈ Ceq (R) 6= C0 .

(23)

Perturbation of the monomer concentration by such a cluster results in diffusion currents. This current controls the absorption rate of monomers (diffusion controlled case, dcc). Because diffusion is not an equilibrium process the kinetics of such clusters cannot be described under the thermodynamic assumptions. 5. The critical size and nucleation barrier In order to obtain the ratio of the kinetic coefficients A(n) and B(n), we substitute Eq. (18) into Eqs. (2)–(7). In the important size region close to the critical size, where |C0 − Ceq (R)|  C0 , we obtain: A ∼ C0 − Ceq (R) 1 . = B C0 1 + u(R)

(24)

Being in equilibrium with the supersaturated monomer solution the critical cluster does not disturb the concentration of monomers: Ceq (Rc ) = C0 . This leads to vanishing “hydrodynamic” rate at the critical size A(n c ) = 0 and gives

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the radius of the critical cluster: Rc = (γ ω/ccl )[Ψ − T ln(ccl /C0 )]−1 , Rc = (2γ ω/ccl )[Ψ − T ln(ccl /C0 )]

−1

(d = 2) ,

(25)

(d = 3).

Substituting Eq. (24) into Eq. (4) we obtain the nucleation barrier (n min = 0):   Z R 2π γ Rc dR 1− ϕc = , (d = 2) T R 1 + u(R) c 0  Z Rc  8π γ Rc R dR ϕc = 1− , (d = 3). T Rc 1 + u(R) 0

(26)

Eq. (26) demonstrates that in the general case the form of the nucleation barrier cannot be reduced to that predicted by the thermodynamic approach from the minimum work ∆Rmin (R, T ) of cluster creation ϕcT = π γ Rc /T, (d = 2)

and

ϕcT = 4π γ Rc2 /T,

(d = 3).

(27)

6. Effect of diffusion on the nucleation barrier Taking into account that Eq. (26) is integrated in the interval up to Rc , we can conclude that the thermodynamic form of the nucleation barrier cannot be applied when u(Rc ) = leff (Rc )/λ∗  1, because a part of the subcritical cluster sizes in Eq. (26) falls into the dcc domain where the diffusion effect is considerable. In the opposite case, when u(Rc )  1, the thermodynamic nucleation barrier can be applied. The parameter u(Rc ) combines three length parameters: (i) the critical size Rc , (ii) the screening length ls , which is determined by the summary efficiency of monomer absorption by all clusters of the ensemble, and (iii) the effective diffusion length λ∗ . It is important that the parameters Rc and ls cannot be reduced to those of some selected group of clusters. They reflect the general thermal and kinetic properties of the system as whole, including the supersaturated solution of monomers and all clusters of the ensemble. The third parameter λ∗ also has a general meaning and describes conditions of monomer absorption at cluster interfaces for the particular system. At the weak screening and/or the largest supersaturations, when Rc /ls  1, the parameter u(Rc ) is reduced to u c = Rc /λ∗ . At the strong screening and/or the smallest supersaturations, when Rc /ls  1, the parameter u(Rc ) is reduced to u s = ls /λ∗ . The nucleation barrier (26) can be expressed as a function of these dimensionless parameters u c and u s : Z uc (1 − x/u c )dx T 2 , (d = 2) ϕc = ϕc u c 0 1 + u s K 0 (x/u s )/K 1 (x/u s ) (28) Z uc 2 (1 − x/u c )dx ϕc = ϕcT , (d = 3). u c 0 (1 + xu s /(x + u s )) Eqs. (28) demonstrate considerable deviations of the barrier ϕc from its thermodynamic limit ϕcT . It is shown in Fig. 2 that the ratio ϕc /ϕcT varies by up to an order of magnitude in the parameter plane (u c , u s ). This plane can be divided into three domains corresponding to different behaviour of the nucleation barrier (see. Fig. 3). In the first domain, located in the close vicinity of axes 0u c and 0u s , the values of parameters u c and u s are small (u c  1, u s  1). In this region disturbance of the monomer solution by nucleating clusters is minor, leading to negligible deviations of the nucleation barrier ϕc from its thermodynamic limit ϕcT . The reasons, why subcritical clusters do not disturb the solution, are different for the different regions of the first domain. In the area along the axis 0u s , where u c  1, small critical nucleus or high barrier ε provide the thermodynamic barrier ϕcT . In the region along the axis 0u c , where u s  1, an extremely strong screening neutralizes the abilities of clusters to disturb the solution. It is interesting that in the second case non-equilibrium monomer diffusion provides the thermodynamic form of the nucleation barrier. However, it is hard to imagine a system having the required parameter values. In the second domain subcritical clusters do disturb the solution. As a result, the ratio ϕc /ϕcT decreases considerably, demonstrating a pronounced dependence on the screening length and critical radius. In contrast to the

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Fig. 2. Deviation of the nucleation barrier ϕc from its thermodynamic form ϕcT as the function of the dimensionless parameters u c = Rc /λ∗ and u s = ls /λ∗ (d = 3).

Fig. 3. The counter plot of the deviation of the ratio ϕc /ϕcT from the unity (d = 2).

thermodynamic limit, the parameters of cluster ensemble appear in the nucleation barrier via its dependence on the screening length. In the third domain the ratio ϕc /ϕcT is an order of magnitude smaller than the unity. Considerable perturbations of the solution by subcritical clusters falling into this domain provide large deviations of the nucleation barrier from its thermodynamic limit. 7. Conclusion Due to diffusion, the cluster vicinity can be enriched or depleted by monomers in comparison with average supersaturation level. It is demonstrated that this effect changes the probability of monomer absorption and therefore the energetic of cluster formation. The thermodynamic form of the nucleation barrier is obtained as the limit of the more general form. In both systems of 2D nanoislands and 3D nanoclusters the screening length appears in the generalized form of the nucleation barrier. This length depends on the density of clusters and the distribution of their sizes. Thus, even at the early nucleation stage the information about the cluster ensemble should be taken into account.

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Acknowledgements Financial support from the Russian Fund for Basic Research (grants 05-02-16994 and 06-08-81030) and INTAS (05-111-5118) is gratefully acknowledged. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

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