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Effect of solute–solute and solute–solvent interactions on the kinetics of nucleation in liquids
Journal of Colloid and Interface Science 342 (2010) 528–532
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Journal of Colloid and Interface Science www.elsevier.com/locate/jcis
Effect of solute–solute and solute–solvent interactions on the kinetics of nucleation in liquids Eli Ruckenstein *, Gersh O. Berim Department of Chemical and Biological Engineering, State University of New York at Buffalo, Buffalo, NY 14260, USA
a r t i c l e
i n f o
Article history: Received 26 September 2009 Accepted 18 October 2009 Available online 24 October 2009 Keywords: Nucleation rate Mean passage time Solute–solute interaction Solute–solvent interaction
a b s t r a c t One of the assumptions of the theory of nucleation developed by Ruckenstein et al. [1,2] is that the main contribution to the nucleation rate of a solid phase from a solution comes from the interaction of a solute molecule with those in a cluster (nucleus) of the solid phase. This assumption is avoided in this paper by including the interactions of the solute molecule with those outside the cluster and with the molecules of the solvent. For each of the above interactions the rate of nucleation changes when compared to the original theory by several orders of magnitudes when calculated at a fixed density of the solute, but changes less than one order of magnitude when calculated as a function of supersaturation. Such changes are usually small compared with the absolute magnitude of the nucleation rate. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction In papers [1,2] a kinetic theory of nucleation in liquids was developed in which the rate of dissociation of nuclei was calculated on the basis of a mean first passage time analysis. The theory did not employ the macroscopic concept of surface tension and was based on molecular interactions only. To simplify the problem, only the interactions between the molecules of solute and those of a cluster of the new solid phase were taken into account. The interactions between the solute molecules themselves as well as the interactions of these molecules with those of the solvent were taken into account only indirectly by modifying the energy parameter of the solute–cluster potential. In the present paper the theory is improved by including those interactions. 2. Background 2.1. Interaction potentials The considered system consists of a cluster of the new phase which is assumed to be an amorphous, uniform, spherical solid of density qc and radius R surrounded by a liquid which consists of molecules of solvent of density qw and molecules of solute of density qs . Note that the latter densities are connected by the relation
qw v 2 þ qs v s ¼ 1
where v 2 and v s are the volumes of a molecule of solvent and a molecule of solute, respectively. The dissociation of molecules from the cluster is supposed to occur only from the surface layer of thickness g, whereas the molecules beneath this layer do not dissolve. The interaction potential between two molecules of solute has the form
( /1 ðr 12 Þ ¼
/ðrÞ ¼
0021-9797/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2009.10.039
1;
6 g
r 12
; r 12 P g;
ð2Þ
r 12 < g
where r12 ¼ jr r0 j denotes the distance between the centers of the interacting molecules, g being the diameter of the hard core repulsion, and 1 the energy parameter of the London-van der Waals attraction. In Ref. [2], the effect of the presence of the surrounding liquid medium was accounted for by the appropriate estimation of the effective interaction constant 1 . In the present paper we introduce explicitly a new potential, /2 ðr 12 Þ, which accounts for the interactions between a molecule of solute and the molecules of solvent, which will be taken to be of the same form as the potential /1 ðr12 Þ but with another energy parameter 2 –1 , instead of 1 (see Eq. (2)). Even though the real solute–solvent interaction potential might have a different form, this approximation allows one to simplify the calculations. The interaction between a molecule of solute with the solute molecules located outside the cluster are also taken into account. The net potential exerted on a single molecule of solute is given by the expression
ð1Þ
* Corresponding author. Fax: +1 716 645 3822. E-mail addresses: [email protected], [email protected] (E. Ruckenstein).
1
Z
dr0 /1 ðjr r0 jÞqc ðr0 Þ þ Vc Z dr0 /1 ðjr r0 jÞqs ðr0 Þ þ V0
Z V0
dr0 /2 ðjr r0 jÞqw ðr0 Þ ð3Þ
E. Ruckenstein, G.O. Berim / Journal of Colloid and Interface Science 342 (2010) 528–532
where r provides the location of the solute molecule, V c and V 0 are the volumes of the cluster and of the remaining of the system, respectively, and the coordinates are taken with respect to the center of the cluster. The three integrals in Eq. (3) represent the interaction potentials of a solute molecule with the cluster, with the molecules of the solvent outside the cluster, and with the solute molecules located outside the cluster, respectively. (It is assumed that the molecules of the solvent do not penetrate into the cluster.) Only the first of the three potentials in Eq. (3) was taken into account in Refs. [1,2]. To evaluate the second integral of Eq. (3) it is convenient to write it in the form
Z V0
0
0
0
dr /2 ðjr r jÞqw ðr Þ ¼
Z ZV Vc
0
0
0
dr /2 ðjr r jÞqw ðr Þ dr0 /2 ðjr r0 jÞqw ðr0 Þ
ð4Þ
where V is the volume of the entire system, which is considered to be infinite. For uniform densities of the cluster, solute, and solvent ðqc ðrÞ ¼ qc ; qs ðrÞ ¼ qs ; qw ðrÞ ¼ qw Þ the case considered below, the first right hand side integral of Eq. (4) is a constant which can be omitted. Consequently
Z V0
dr0 /2 ðjr r0 jÞqw ðr0 Þ ¼ qw
Z
dr0 /2 ðjr r0 jÞ
ð5Þ
Vc
i.e. the potential of interaction of a solute molecule with the entire volume of the solvent is given by that for the interaction of this molecule with the solvent which occupies the volume V c of the cluster, taken with opposite sign. The third integral in Eq. (3) can be rewritten in a similar way as
Z V0
dr0 /1 ðjr r0 jÞqs ðr0 Þ ¼ qs
Z
dr0 /1 ðjr r0 jÞ:
ð6Þ
The explicit analytical expression for the net potential /ðrÞ depends on whether the center of the selected solute molecule is located inside the surface layer ðR 6 r 6 R þ gÞ of the cluster or outside of it ðr P R þ gÞ. Introducing the dimensionless potential UðrÞ ¼ /ðrÞ=kB T where kB is the Boltzmann constant one obtains
8 h i p le 8 þ 3 r 1 R2 þ 6 g rþ3R g3 ; R 6 r 6 R þ g; > < 12 1 3 g r r2 rðrþRÞ UðrÞ ¼ 3 > : 4p le1 g 3 R ; r >Rþg 3 rR rþR ð7Þ where
being dependent on the cluster size. The change of fi as function of time is described by the equation
dfi ¼ ðIiþ1 Ii Þ ¼ bi1 fi1 ðai þ bi Þfi þ aiþ1 fiþ1 : dt
ð10Þ
Assuming a smooth dependence of fi on i, Eq. (10) can be transformed into the following form [2]
@f ðg; tÞ @Iðg; tÞ 1 @ 2 @ ðb þ aÞf ðg; tÞ ðb aÞf ðg; tÞ; ¼ ¼ @t @g 2 @g 2 @g
ð11Þ
where the continuum variable g replaces the discrete variable i. The rate of nucleation, I, is provided by the stationary value of the flux
Iðg; tÞ ¼
1 @ ½ðb þ aÞf ðg; tÞ þ ðb aÞf ðg; tÞ 2 @g
ð12Þ
of clusters along the coordinate g [2,5]. To find the stationary solution of Eq. (12), hence the solution for Iðg; tÞ I ¼ const and f ðg; tÞ f ðgÞ, boundary conditions for f ðgÞ at g ! 1 (large clusters) and g ! 1 (smallest clusters, monomers) should be satisfied. Regarding the first boundary condition, it is assumed that clusters with very large sizes are absent in the system. Consequently, f ðgÞ ! 0 as g ! 1. Regarding the small clusters (monomers), it is assumed that nucleation does not cause their substantial depletion, hence, that their number density remains the initial one. Consequently, f ðgÞ ! qS as g ! 1 (in calculations, the limit 1 can be replaced by zero). Straightforward calculations provide the following expression for the rate of nucleation
I ¼ R1 0
Vc
529
1 b 2 1
qs
exp½2wðgÞdg
ð13Þ
where b1 is the condensation rate calculated for a cluster consisting of one molecule of the solute (see Section 2.4 for details) and
wðgÞ ¼
Z 0
g
ba dg: bþa
ð14Þ
In Eq. (14), the rates of dissociation ðaÞ and condensation ðbÞ, while functions of the discrete number i of the molecules in the cluster, are considered as continuous functions of i, or, equivalently, of the cluster radius R. Eqs. (13) and (14) will be used below to calculate the nucleation rate. 2.3. Dissociation rate
e1 ¼ 1 =kB T,
l ¼ lc þ d g3 =v 2 ls 1 d
v2 v1
ð8Þ
lc ¼ qc g3 ; d ¼ 2 =1 ; ls ¼ qs g3 . The general form of the net interaction potential, Eq. (7), coincides with that of the potential provided by Eq. (3) of Ref. [2] which was calculated by taking into account the interaction of a solute molecule with the cluster only. In the latter case, the parameter l in Eq. (7) is equal with the density of the molecules in the cluster, whereas in the present case l is given by Eq. (8).
The similarity between the interaction potential, Eq. (7), and that used in Ref. [2] provides the possibility to use the results of Ref. [2] for calculating the rate of dissociation a of the molecules from the external layer of the cluster. Dissociation is assumed to be due to Brownian motion of the surface molecules that is the result of collisions of those molecules with others. In Ref. [2] it was suggested to describe the Brownian motion using the Smoluchowski equation under an external field [6,7] which in the case of spherical symmetry has the form
2.2. Nucleation rate
@nðr; tÞ @ @ ¼ Dr2 r2 eUðrÞ eUðrÞ nðr; tÞ @t @r @r
To calculate the nucleation rate let us consider the populations fi of clusters of various sizes consisting of i molecules ði ¼ 1; 2; . . .Þ [3,4]. The flux of clusters passing from the populations fi1 to fi is given by the equation
where nðr; tÞ is the number density of diffusing molecules; D is the diffusion coefficient. The transition probability pðr; tjr0 Þ that a molecule initially at the distance r 0 will be located after time t at the point r satisfies the backward Smoluchowski equation [7–9]
Ii ¼ bi1 fi1 ai fi
@pðr; tjr 0 Þ @ @ Uðr 0 Þ r 20 eUðr0 Þ pðr; tjr 0 Þ ¼ Dr2 0 e @t @r 0 @r 0
ð9Þ
where fi is the number of clusters consisting of i molecules, ai and bi are the rates of dissociation and condensation of molecules from and on the surface of a cluster from population fi , respectively, both
ð15Þ
ð16Þ
The probability that a molecule initially at point r0 will leave the surface layer of the cluster (dissociate) during a time between 0
530
E. Ruckenstein, G.O. Berim / Journal of Colloid and Interface Science 342 (2010) 528–532
and t is 1 Q ðtjr0 Þ, where Q ðtjr 0 Þ is the so called survival probability
Q ðtjr0 Þ ¼
Z
r 2 pðr; tjr0 Þdr
Z
1
0
@Qðtjr 0 Þ t dt ¼ @t
Z
Q ðtjr 0 Þdt:
ð18Þ
0
@ @ r 20 eUðr0 Þ sðr0 Þ ¼ 1: @r 0 @r0
Uðr 0 Þ Dr 2 0 e
ð19Þ
The boundary conditions for sðr 0 Þ follow from the boundary conditions imposed on the function pðr; tjr 0 Þ. Considering the inner wall of the surface layer as a reflecting boundary on obtains
¼ 0:
ð20Þ
r¼R
The molecules reaching the outer boundary of the surface layer separate from the cluster and diffuse further in the outer space. The diffusive flux under an external field can be obtained from Eq. (15) and is given by the equation
jðr; tÞ ¼ DeUðrÞ
@ UðrÞ e nðr; tÞ @r
ð21Þ
Assuming quasi steady state and integrating with the boundary condition nðrÞ ! 0 as r ! 1 the following expression for the diffusive flux is obtained
r
DeUðrÞ r2 nðrÞ x2 eUðxÞ dx
ð22Þ
jðR þ gÞ ¼ k0 nðR þ gÞ
ð23Þ
where
DeUðRþgÞ R1 ðR þ gÞ Rþg x2 eUðxÞ dx
ð24Þ
2
Z
Rþg
r0
dy peq ðyÞ
Z R
y
dxpeq ðxÞ þ ½k0 peq ðR þ gÞ1 :
ds ¼ k0 D1 sðR þ gÞ: dr r¼Rþg
Rþg
r0
dy peq ðyÞ
Z R
y
1
dxpeq ðxÞ þ ½k0 peq ðR þ gÞ
ð26Þ
where peq is the equilibrium probability distribution given by
peq ðxÞ ¼ x2 eUðxÞ =Z R Rþg
Ns
ð29Þ
where N s is the number of surface molecules which is given by the expression
Ns ¼ 4plc ðR=gÞ2 ð1 þ g=R þ g2 =3R2 Þ:
ð30Þ
The rate of dissociation depends on the form of the intermolecular potential and on the size of the cluster. 2.4. Condensation rate The expression for the condensation rate b was obtained from the stationary solution of the diffusion equation in an external field, Eq. (15), and has the form
b ¼ c4pDRqs
ð31Þ R1
2 UðrÞ
where c ¼ R Rþg r e dr. In the limit R ! 1, the factor c tends to unity. The size Rc of the critical nucleus can be calculated from the balance between dissociation and condensation rates ða ¼ bÞ provided by Eqs. (29) and (31). In the limit of large clusters, Rc is provided by equation [2] 4pqc Rc D 1þ g=Rc þ 13 ðg=Rc Þ2 eA=3 ðeA 1Þ=Aþ Rgc 3ð1ð1þAÞeA Þ=A2 e4A=3 þeA=3 12 eA þ 2ðsinhAAÞ ð1eA ÞA2 1
¼ 4pqs Rc D where
A ¼ lp=2kT:
ð33Þ
The important consequence from this equation is the expression for the saturation number density ns of the solute which can be obtained from Eq. (32) in the limit of large clusters ðg=R ! 0Þ
qA 4A=3 e : 1 eA
ð34Þ
When solving Eq. (34), the density of the solute, qs , which is present in the parameter l (see Eq. (8)) should be replaced by ns .
ð25Þ
Using the boundary conditions Eqs. (20) and (25) the following solution of Eq. (19) is obtained
Z
The rate of dissociation of the surface molecules is provided by the equation [2]
ns ¼
As a consequence of Eq. (23), the following boundary condition for the mean passage time can be written
1 sðr0 Þ ¼ D
R
dr 0 peq ðr 0 Þ
ð32Þ
The latter equation at r ¼ R þ g becomes the radiation boundary condition
k0 ¼
Rþg
ð28Þ
a¼
1
Integrating Eq. (16) over r and t over their entire ranges and using the boundary conditions Q ð0jr 0 Þ ¼ 1 and Q ðtjr 0 Þ ! 0 as t ! 1 for any r 0 one arrives at the following equation for sðr0 Þ [2,7–10]
jðrÞ ¼ R 1
Z
ð17Þ
The probability density for the dissociation time is given by @Q =@t and the mean passage time can be calculated with the formula [2]
ds dr
1 D
Rþg
R
sðr0 Þ ¼
hsi ¼
ð27Þ
and Z ¼ R x2 eUðxÞ . Let us assume that the relaxation time to achieve equilibrium is much shorter than the dissociation time. Then the average mean passage time hsi can be obtained from Eq. (26) after integration over the initial distribution of r0 , which is assumed to be the equiR Rþg librium one, i.e. hsi ¼ R sðr 0 Þpeq ðr0 Þdr 0 . As a result, the average mean passage time is given by
3. Numerical estimation of effect of the solute–solute and solute–solvent interactions on the kinetics of nucleation To clarify the importance of the interactions of a solute molecule with other solute molecules located outside the cluster and with the molecules of solvent, the nucleation rate I was calculated separately for three different cases. The first one (i) coincides with the case considered in Ref. [2], where only the interaction of the solute molecule with those in the cluster was taken into account. In the second case (ii) in addition to that interaction the interaction of the solute molecule with the solute molecules outside the cluster is included. In the third case (iii) the interaction of the solute molecule with the molecules of the solvent is added to the two interactions mentioned above. To distinguish between these three cases, the subscripts 0, 1, and 2, respectively, will be used for the supersaturation s, critical radius Rc , nucleation rate I, and saturated density of the solute ns . By comparing the results for cases (ii) and
531
E. Ruckenstein, G.O. Berim / Journal of Colloid and Interface Science 342 (2010) 528–532 Table 1 Comparison between cases (i) and (ii) at e1 = 2 and e1 = 4 and various solute number densities ls.
ls
s0
s1
2I0/qsb1
2I1/qsb1
I1/I0
Rc0/g
Rc1/g
1.025 1.170 1.464
4.30 103 2.52 102 1.05 101
3.9 104 5.8 103 5.3 102
9.1 102 2.3 101 5.0 101
12.5 6.1 2.5
86 12.1 3.9
1.021 1.151 1.279 2.560 12.80
1.3 1013 5.3 108 1.8 106 3.1 103 2.10 101
6.1 1014 4.6 108 1.6 106 3.0 103 2.05 101
4.7 101 8.7 101 8.9 101 9.7 101 9.8 101
290 44 23.5 5.1 1.25
310 45 23.8 5.2 1.28
e1 = 2,ms1 = 0.03415, ms0 = 0.03038 3.5 102 4.0 102 5.0 102
1.15 1.32 1.64
e1= 4, ms1 = 3.909 104,ms0 = 3.897 104 3.99 104 4.50 104 5.00 104 1.00 104 5.00 104
1.0238 1.155 1.282 2.570 12.80
Fig. 1. Dimensionless saturation density d ¼ 2 =1 at 1 =kB T ¼ 2.
ls ¼ ns g3 as function of the parameter
(iii) with those for case (i) one can estimate the contribution of the additional interactions to the nucleation rate. In all calculations, the parameter lc was taken 1.2 as was selected in Ref. [2]. The energy parameter 1 of the solute–solute interaction was selected to be e1 ¼ 2 and e1 ¼ 4 where e1 ¼ 1 =kB T and the parameters v 2 and v s were selected equal to g3 . Because the results for case (i) can be obtained using the corresponding equations of Ref. [2], let us analyze first the role of the interaction of a solute molecule with other solute molecules located outside the cluster (case (ii)) by neglecting the interaction of the molecule with those of the solvent. To analyze this case one should take d ¼ 0 in the expression of l, the latter parameter thus becoming l ¼ lc ls . As an immediate consequence of the change in the value of the parameter l one can note the change of the value of ns of the saturation density calculated with Eq. (34). For example, at e1 ¼ 2; ms0 ¼ 0:03038 ðms ¼ ns g3 Þ for case (i) and ms1 ¼ 0:03415 if the solute–solute interaction is taken into ac-
count (case (ii)). This affects the supersaturations s qs =ns ¼ ls =ms which are different for the same solute density qs . The same observations can be made regarding the critical cluster size Rc and the rate of nucleation I. In Table 1, the nucleation rates I0 and I1 are listed for several values of ls along with the supersaturations s0 and s1 and the critical cluster radii Rc0 and Rc1 . This table shows that for a given ls the solute–solute interactions decrease the nucleation rate, and that the ratio I1 =I0 decreases with decreasing ls . For ls 6 ms1 ¼ 0:03415, this ratio becomes zero. The latter result occurs because for ls < ms1 the solution is undersaturated ðs1 < 1Þ in case (ii) but oversaturated ðs0 > 1Þ in case (i). One should note that the changes in the nucleation rate due to the solute–solute interactions are small (less than one order of magnitude, which for homogeneous nucleation constitutes a small number) for almost all values of ls . At the lower temperature ðe1 ¼ 4Þ; ms0 ’ ms1 ðms0 ¼ 3:897 104 ; ms1 ¼ 3:909 104 Þ and the difference between s0 and s1 for a given ls is negligible. In this case, Table 1 shows that I0 ’ I1 . The small contribution of solute–solute interactions to the nucleation rate can be explained by the relatively small density of the solute molecules outside the cluster and, consequently, the small change in the total potential energy of a solute molecule due to this interaction. Similar to the solute–solute interactions, the solute–solvent ones also affect (at a selected ls ) the calculated values of the saturation density ns and, as a consequence, the supersaturation s, critical cluster size R, and the rate of nucleation I. The dependence of ms2 on d ¼ 2 =1 is shown in Fig. 1 for e1 ¼ 2 and 0 6 d 6 0:5. The quantity ms increases with increasing d and rapidly acquires the solid-like magnitude 0:2. For this reason our considerations are restricted to the range of small values of d ðd 6 0:1Þ. In Table 2 the results for case (iii) are presented along with those for case (i), marked with subscript 0. The data in this table are calculated for e1 ¼ 2 and d ¼ 0:03 and d ¼ 0:1. In both cases the concentration ls is taken larger than ms2 . Note that for
Table 2 Comparison between cases (i) and (iii) at e1 = 2 and various solute number densities ls.
ls
s0
s2
2I0/qsb1
2I2/qsb1
I2/I0
Rc0/g
Rc1/g
1.0078 1.021 1.047 1.178 1.309 1.57
1.94 102 2.18 102 2.52 102 6.13 102 1.05 101 2.06 101
1.3 106 4.4 105 2.96 104 8.4 103 2.8 102 9.2 102
6.7 105 2.0 103 1.2 102 1.4 101 2.7 101 4.5 101
7.0 6.9 6.1 3.5 2.5 1.0
255 99 43 11.5 5.5 2.6
1.0028 1.0069 1.076 1.208
1.04 101 1.05 101 1.54 101 2.06 101
8.8 107 1.02 105 5.23 103 2.14 102
8.5 106 9.7 105 3.4 102 1.0 101
2.51 2.50 1.8 1.0
820 262 15.5 7.6
e2/e1= 0.03, ms2 = 0.03820, ms0 = 0.03038 3.85 102 3.90 102 4.00 102 4.50 102 5.00 102 6.00 102
1.267 1.283 1.32 1.481 1.64 1.97
e2/e1= 0.1, ms2 = 0.04966, ms0 = 0.03038 4.98 102 5.00 102 5.50 102 6.60 102
1.639 1.645 1.810 1.970
532
E. Ruckenstein, G.O. Berim / Journal of Colloid and Interface Science 342 (2010) 528–532
Table 3 Comparison between cases (i) and (iii) at e1 = 4 and various solute number densities ls.
ls
s0 4
2I0/qsb1
2I2/qsb1
I2/I0
Rc0/g
Rc1/g
1.020 1.122 1.427 1.613 4.080 12.20
1.77 106 2.28 105 2.66 104 8.02 104 3.79 102 2.68 101
1.07 1013 2.10 108 2.25 105 1.36 104 2.40 102 2.2 101
6.0 108 9.2 104 8.5 102 1.7 101 6.3 101 8.2 101
24 18 9.9 5.7 2.6 1.1
350 65 17 11 3.5 1.5
1.011 1.035 1.083 1.805 3.610 6.020 8.430
1.04 103 1.31 103 1.73 103 1.65 102 9.18 102 2.10 101 3.24 101
7.80 1014 2.97 1011 5.10 109 7.80 104 2.64 102 9.72 102 1.79 101
7.5 1011 2.3 108 2.9 106 4.7 102 2.9 101 4.6 101 5.5 101
7.1 6.8 6.0 3.5 1.8 1.3 1.0
450 160 65 8.0 3.2 2.1 1.6
s2 4
e2/e1= 0.03, ms2 = 4.9 10 ,ms0 = 3.897 10 5.0 5.5 7.0 8.0 2.0 6.0
104 104 104 104 103 103
1.282 1.411 1.790 2.05 5.13 15.4
e2/e1= 0.1, ms2 = 8.3 104,ms0 = 3.897 104 8.4 104 8.6 104 9.0 104 1.5 103 3.0 103 5.0 103 7.0 103
2.16 2.20 2.31 3.85 7.70 12.8 18.0
responding to d ¼ 0), and (iii) (two lines for d ¼ 0:03 and d ¼ 0:1). The largest difference between various nucleation rates at the same supersaturation does not exceed one order of magnitude for all considered supersaturations. 4. Conclusion
Fig. 2. Nucleation rate I calculated at 1 =kB T ¼ 2 as function of the supersaturation s for cases (i)–(iii). Case (i) considered in Ref. [2] is presented by the solid line. Case (ii) is represented by the curve for d ¼ 0. Case (iii) is represented by curves computed for d ¼ 0:03 and d ¼ 0:1.
In the present paper a modification of the Ruckenstein, Narasimhan, and Nowakowski (Refs. [1,2]) nucleation theory is developed which, in addition to the solute–cluster interactions, accounts for the solute–solute and solute–solvent interactions. It is shown that at the same supersaturation these additional interactions do not affect significantly the rate of nucleation and hence, the approximation made in Refs. [1,2] by neglecting them is valid in most cases. References
ls 6 ms2 , the nucleation rate I2 is equal to zero because s2 is smaller than one, and Rc2 ¼ 1. In all cases, I2 I0 and the ratio I2 =I0 decreases with decreasing ls as in case (ii), varying from I2 =I0 101 for larger ls to 105 for d ¼ 0:03, or to 106 for d ¼ 0:1 for ls close to ms2 . In Table 3 similar results are presented for e1 ¼ 4. In this case, I2 =I0 varies from I2 =I0 101 for large ls to 108 for d ¼ 0:03, or to 1011 for d ¼ 0:1. In Fig. 2, the nucleation rate is presented at e1 ¼ 2 as a function of the supersaturation s for the cases: (i) (solid line), (ii) (line cor-
[1] G. Narsimhan, E. Ruckenstein, J. Colloid Interface Sci. 128 (1989) 549. [2] E. Ruckenstein, B. Nowakowski, J. Colloid Interface Sci. 137 (1990) 583. [3] F. Abraham, Homogeneous Nucleation Theory, Academic Press, New York, 1974. [4] F.C. Goodrich, Proc. R. Soc. London A 277 (1964) 167. [5] E. Ruckenstein, Y.S. Djikaev, Adv. Colloid Interface Sci. 118 (2005) 51. [6] S. Chandrasekhar, Rev. Mod. Phys. 15 (1949) 1. [7] C.W. Gardiner, Handbook of Stochastic Methods, Springer, New York/Berlin, 1983. [8] N. Agmon, J. Chem. Phys. 81 (1984) 3644. [9] G.H. Weiss, J. Stat. Phys. 42 (1986) 1. [10] J.M. Deutch, J. Chem. Phys. 73 (1980) 4700.
Contents lists available at ScienceDirect
Journal of Colloid and Interface Science www.elsevier.com/locate/jcis
Effect of solute–solute and solute–solvent interactions on the kinetics of nucleation in liquids Eli Ruckenstein *, Gersh O. Berim Department of Chemical and Biological Engineering, State University of New York at Buffalo, Buffalo, NY 14260, USA
a r t i c l e
i n f o
Article history: Received 26 September 2009 Accepted 18 October 2009 Available online 24 October 2009 Keywords: Nucleation rate Mean passage time Solute–solute interaction Solute–solvent interaction
a b s t r a c t One of the assumptions of the theory of nucleation developed by Ruckenstein et al. [1,2] is that the main contribution to the nucleation rate of a solid phase from a solution comes from the interaction of a solute molecule with those in a cluster (nucleus) of the solid phase. This assumption is avoided in this paper by including the interactions of the solute molecule with those outside the cluster and with the molecules of the solvent. For each of the above interactions the rate of nucleation changes when compared to the original theory by several orders of magnitudes when calculated at a fixed density of the solute, but changes less than one order of magnitude when calculated as a function of supersaturation. Such changes are usually small compared with the absolute magnitude of the nucleation rate. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction In papers [1,2] a kinetic theory of nucleation in liquids was developed in which the rate of dissociation of nuclei was calculated on the basis of a mean first passage time analysis. The theory did not employ the macroscopic concept of surface tension and was based on molecular interactions only. To simplify the problem, only the interactions between the molecules of solute and those of a cluster of the new solid phase were taken into account. The interactions between the solute molecules themselves as well as the interactions of these molecules with those of the solvent were taken into account only indirectly by modifying the energy parameter of the solute–cluster potential. In the present paper the theory is improved by including those interactions. 2. Background 2.1. Interaction potentials The considered system consists of a cluster of the new phase which is assumed to be an amorphous, uniform, spherical solid of density qc and radius R surrounded by a liquid which consists of molecules of solvent of density qw and molecules of solute of density qs . Note that the latter densities are connected by the relation
qw v 2 þ qs v s ¼ 1
where v 2 and v s are the volumes of a molecule of solvent and a molecule of solute, respectively. The dissociation of molecules from the cluster is supposed to occur only from the surface layer of thickness g, whereas the molecules beneath this layer do not dissolve. The interaction potential between two molecules of solute has the form
( /1 ðr 12 Þ ¼
/ðrÞ ¼
0021-9797/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2009.10.039
1;
6 g
r 12
; r 12 P g;
ð2Þ
r 12 < g
where r12 ¼ jr r0 j denotes the distance between the centers of the interacting molecules, g being the diameter of the hard core repulsion, and 1 the energy parameter of the London-van der Waals attraction. In Ref. [2], the effect of the presence of the surrounding liquid medium was accounted for by the appropriate estimation of the effective interaction constant 1 . In the present paper we introduce explicitly a new potential, /2 ðr 12 Þ, which accounts for the interactions between a molecule of solute and the molecules of solvent, which will be taken to be of the same form as the potential /1 ðr12 Þ but with another energy parameter 2 –1 , instead of 1 (see Eq. (2)). Even though the real solute–solvent interaction potential might have a different form, this approximation allows one to simplify the calculations. The interaction between a molecule of solute with the solute molecules located outside the cluster are also taken into account. The net potential exerted on a single molecule of solute is given by the expression
ð1Þ
* Corresponding author. Fax: +1 716 645 3822. E-mail addresses: [email protected], [email protected] (E. Ruckenstein).
1
Z
dr0 /1 ðjr r0 jÞqc ðr0 Þ þ Vc Z dr0 /1 ðjr r0 jÞqs ðr0 Þ þ V0
Z V0
dr0 /2 ðjr r0 jÞqw ðr0 Þ ð3Þ
E. Ruckenstein, G.O. Berim / Journal of Colloid and Interface Science 342 (2010) 528–532
where r provides the location of the solute molecule, V c and V 0 are the volumes of the cluster and of the remaining of the system, respectively, and the coordinates are taken with respect to the center of the cluster. The three integrals in Eq. (3) represent the interaction potentials of a solute molecule with the cluster, with the molecules of the solvent outside the cluster, and with the solute molecules located outside the cluster, respectively. (It is assumed that the molecules of the solvent do not penetrate into the cluster.) Only the first of the three potentials in Eq. (3) was taken into account in Refs. [1,2]. To evaluate the second integral of Eq. (3) it is convenient to write it in the form
Z V0
0
0
0
dr /2 ðjr r jÞqw ðr Þ ¼
Z ZV Vc
0
0
0
dr /2 ðjr r jÞqw ðr Þ dr0 /2 ðjr r0 jÞqw ðr0 Þ
ð4Þ
where V is the volume of the entire system, which is considered to be infinite. For uniform densities of the cluster, solute, and solvent ðqc ðrÞ ¼ qc ; qs ðrÞ ¼ qs ; qw ðrÞ ¼ qw Þ the case considered below, the first right hand side integral of Eq. (4) is a constant which can be omitted. Consequently
Z V0
dr0 /2 ðjr r0 jÞqw ðr0 Þ ¼ qw
Z
dr0 /2 ðjr r0 jÞ
ð5Þ
Vc
i.e. the potential of interaction of a solute molecule with the entire volume of the solvent is given by that for the interaction of this molecule with the solvent which occupies the volume V c of the cluster, taken with opposite sign. The third integral in Eq. (3) can be rewritten in a similar way as
Z V0
dr0 /1 ðjr r0 jÞqs ðr0 Þ ¼ qs
Z
dr0 /1 ðjr r0 jÞ:
ð6Þ
The explicit analytical expression for the net potential /ðrÞ depends on whether the center of the selected solute molecule is located inside the surface layer ðR 6 r 6 R þ gÞ of the cluster or outside of it ðr P R þ gÞ. Introducing the dimensionless potential UðrÞ ¼ /ðrÞ=kB T where kB is the Boltzmann constant one obtains
8 h i p le 8 þ 3 r 1 R2 þ 6 g rþ3R g3 ; R 6 r 6 R þ g; > < 12 1 3 g r r2 rðrþRÞ UðrÞ ¼ 3 > : 4p le1 g 3 R ; r >Rþg 3 rR rþR ð7Þ where
being dependent on the cluster size. The change of fi as function of time is described by the equation
dfi ¼ ðIiþ1 Ii Þ ¼ bi1 fi1 ðai þ bi Þfi þ aiþ1 fiþ1 : dt
ð10Þ
Assuming a smooth dependence of fi on i, Eq. (10) can be transformed into the following form [2]
@f ðg; tÞ @Iðg; tÞ 1 @ 2 @ ðb þ aÞf ðg; tÞ ðb aÞf ðg; tÞ; ¼ ¼ @t @g 2 @g 2 @g
ð11Þ
where the continuum variable g replaces the discrete variable i. The rate of nucleation, I, is provided by the stationary value of the flux
Iðg; tÞ ¼
1 @ ½ðb þ aÞf ðg; tÞ þ ðb aÞf ðg; tÞ 2 @g
ð12Þ
of clusters along the coordinate g [2,5]. To find the stationary solution of Eq. (12), hence the solution for Iðg; tÞ I ¼ const and f ðg; tÞ f ðgÞ, boundary conditions for f ðgÞ at g ! 1 (large clusters) and g ! 1 (smallest clusters, monomers) should be satisfied. Regarding the first boundary condition, it is assumed that clusters with very large sizes are absent in the system. Consequently, f ðgÞ ! 0 as g ! 1. Regarding the small clusters (monomers), it is assumed that nucleation does not cause their substantial depletion, hence, that their number density remains the initial one. Consequently, f ðgÞ ! qS as g ! 1 (in calculations, the limit 1 can be replaced by zero). Straightforward calculations provide the following expression for the rate of nucleation
I ¼ R1 0
Vc
529
1 b 2 1
qs
exp½2wðgÞdg
ð13Þ
where b1 is the condensation rate calculated for a cluster consisting of one molecule of the solute (see Section 2.4 for details) and
wðgÞ ¼
Z 0
g
ba dg: bþa
ð14Þ
In Eq. (14), the rates of dissociation ðaÞ and condensation ðbÞ, while functions of the discrete number i of the molecules in the cluster, are considered as continuous functions of i, or, equivalently, of the cluster radius R. Eqs. (13) and (14) will be used below to calculate the nucleation rate. 2.3. Dissociation rate
e1 ¼ 1 =kB T,
l ¼ lc þ d g3 =v 2 ls 1 d
v2 v1
ð8Þ
lc ¼ qc g3 ; d ¼ 2 =1 ; ls ¼ qs g3 . The general form of the net interaction potential, Eq. (7), coincides with that of the potential provided by Eq. (3) of Ref. [2] which was calculated by taking into account the interaction of a solute molecule with the cluster only. In the latter case, the parameter l in Eq. (7) is equal with the density of the molecules in the cluster, whereas in the present case l is given by Eq. (8).
The similarity between the interaction potential, Eq. (7), and that used in Ref. [2] provides the possibility to use the results of Ref. [2] for calculating the rate of dissociation a of the molecules from the external layer of the cluster. Dissociation is assumed to be due to Brownian motion of the surface molecules that is the result of collisions of those molecules with others. In Ref. [2] it was suggested to describe the Brownian motion using the Smoluchowski equation under an external field [6,7] which in the case of spherical symmetry has the form
2.2. Nucleation rate
@nðr; tÞ @ @ ¼ Dr2 r2 eUðrÞ eUðrÞ nðr; tÞ @t @r @r
To calculate the nucleation rate let us consider the populations fi of clusters of various sizes consisting of i molecules ði ¼ 1; 2; . . .Þ [3,4]. The flux of clusters passing from the populations fi1 to fi is given by the equation
where nðr; tÞ is the number density of diffusing molecules; D is the diffusion coefficient. The transition probability pðr; tjr0 Þ that a molecule initially at the distance r 0 will be located after time t at the point r satisfies the backward Smoluchowski equation [7–9]
Ii ¼ bi1 fi1 ai fi
@pðr; tjr 0 Þ @ @ Uðr 0 Þ r 20 eUðr0 Þ pðr; tjr 0 Þ ¼ Dr2 0 e @t @r 0 @r 0
ð9Þ
where fi is the number of clusters consisting of i molecules, ai and bi are the rates of dissociation and condensation of molecules from and on the surface of a cluster from population fi , respectively, both
ð15Þ
ð16Þ
The probability that a molecule initially at point r0 will leave the surface layer of the cluster (dissociate) during a time between 0
530
E. Ruckenstein, G.O. Berim / Journal of Colloid and Interface Science 342 (2010) 528–532
and t is 1 Q ðtjr0 Þ, where Q ðtjr 0 Þ is the so called survival probability
Q ðtjr0 Þ ¼
Z
r 2 pðr; tjr0 Þdr
Z
1
0
@Qðtjr 0 Þ t dt ¼ @t
Z
Q ðtjr 0 Þdt:
ð18Þ
0
@ @ r 20 eUðr0 Þ sðr0 Þ ¼ 1: @r 0 @r0
Uðr 0 Þ Dr 2 0 e
ð19Þ
The boundary conditions for sðr 0 Þ follow from the boundary conditions imposed on the function pðr; tjr 0 Þ. Considering the inner wall of the surface layer as a reflecting boundary on obtains
¼ 0:
ð20Þ
r¼R
The molecules reaching the outer boundary of the surface layer separate from the cluster and diffuse further in the outer space. The diffusive flux under an external field can be obtained from Eq. (15) and is given by the equation
jðr; tÞ ¼ DeUðrÞ
@ UðrÞ e nðr; tÞ @r
ð21Þ
Assuming quasi steady state and integrating with the boundary condition nðrÞ ! 0 as r ! 1 the following expression for the diffusive flux is obtained
r
DeUðrÞ r2 nðrÞ x2 eUðxÞ dx
ð22Þ
jðR þ gÞ ¼ k0 nðR þ gÞ
ð23Þ
where
DeUðRþgÞ R1 ðR þ gÞ Rþg x2 eUðxÞ dx
ð24Þ
2
Z
Rþg
r0
dy peq ðyÞ
Z R
y
dxpeq ðxÞ þ ½k0 peq ðR þ gÞ1 :
ds ¼ k0 D1 sðR þ gÞ: dr r¼Rþg
Rþg
r0
dy peq ðyÞ
Z R
y
1
dxpeq ðxÞ þ ½k0 peq ðR þ gÞ
ð26Þ
where peq is the equilibrium probability distribution given by
peq ðxÞ ¼ x2 eUðxÞ =Z R Rþg
Ns
ð29Þ
where N s is the number of surface molecules which is given by the expression
Ns ¼ 4plc ðR=gÞ2 ð1 þ g=R þ g2 =3R2 Þ:
ð30Þ
The rate of dissociation depends on the form of the intermolecular potential and on the size of the cluster. 2.4. Condensation rate The expression for the condensation rate b was obtained from the stationary solution of the diffusion equation in an external field, Eq. (15), and has the form
b ¼ c4pDRqs
ð31Þ R1
2 UðrÞ
where c ¼ R Rþg r e dr. In the limit R ! 1, the factor c tends to unity. The size Rc of the critical nucleus can be calculated from the balance between dissociation and condensation rates ða ¼ bÞ provided by Eqs. (29) and (31). In the limit of large clusters, Rc is provided by equation [2] 4pqc Rc D 1þ g=Rc þ 13 ðg=Rc Þ2 eA=3 ðeA 1Þ=Aþ Rgc 3ð1ð1þAÞeA Þ=A2 e4A=3 þeA=3 12 eA þ 2ðsinhAAÞ ð1eA ÞA2 1
¼ 4pqs Rc D where
A ¼ lp=2kT:
ð33Þ
The important consequence from this equation is the expression for the saturation number density ns of the solute which can be obtained from Eq. (32) in the limit of large clusters ðg=R ! 0Þ
qA 4A=3 e : 1 eA
ð34Þ
When solving Eq. (34), the density of the solute, qs , which is present in the parameter l (see Eq. (8)) should be replaced by ns .
ð25Þ
Using the boundary conditions Eqs. (20) and (25) the following solution of Eq. (19) is obtained
Z
The rate of dissociation of the surface molecules is provided by the equation [2]
ns ¼
As a consequence of Eq. (23), the following boundary condition for the mean passage time can be written
1 sðr0 Þ ¼ D
R
dr 0 peq ðr 0 Þ
ð32Þ
The latter equation at r ¼ R þ g becomes the radiation boundary condition
k0 ¼
Rþg
ð28Þ
a¼
1
Integrating Eq. (16) over r and t over their entire ranges and using the boundary conditions Q ð0jr 0 Þ ¼ 1 and Q ðtjr 0 Þ ! 0 as t ! 1 for any r 0 one arrives at the following equation for sðr0 Þ [2,7–10]
jðrÞ ¼ R 1
Z
ð17Þ
The probability density for the dissociation time is given by @Q =@t and the mean passage time can be calculated with the formula [2]
ds dr
1 D
Rþg
R
sðr0 Þ ¼
hsi ¼
ð27Þ
and Z ¼ R x2 eUðxÞ . Let us assume that the relaxation time to achieve equilibrium is much shorter than the dissociation time. Then the average mean passage time hsi can be obtained from Eq. (26) after integration over the initial distribution of r0 , which is assumed to be the equiR Rþg librium one, i.e. hsi ¼ R sðr 0 Þpeq ðr0 Þdr 0 . As a result, the average mean passage time is given by
3. Numerical estimation of effect of the solute–solute and solute–solvent interactions on the kinetics of nucleation To clarify the importance of the interactions of a solute molecule with other solute molecules located outside the cluster and with the molecules of solvent, the nucleation rate I was calculated separately for three different cases. The first one (i) coincides with the case considered in Ref. [2], where only the interaction of the solute molecule with those in the cluster was taken into account. In the second case (ii) in addition to that interaction the interaction of the solute molecule with the solute molecules outside the cluster is included. In the third case (iii) the interaction of the solute molecule with the molecules of the solvent is added to the two interactions mentioned above. To distinguish between these three cases, the subscripts 0, 1, and 2, respectively, will be used for the supersaturation s, critical radius Rc , nucleation rate I, and saturated density of the solute ns . By comparing the results for cases (ii) and
531
E. Ruckenstein, G.O. Berim / Journal of Colloid and Interface Science 342 (2010) 528–532 Table 1 Comparison between cases (i) and (ii) at e1 = 2 and e1 = 4 and various solute number densities ls.
ls
s0
s1
2I0/qsb1
2I1/qsb1
I1/I0
Rc0/g
Rc1/g
1.025 1.170 1.464
4.30 103 2.52 102 1.05 101
3.9 104 5.8 103 5.3 102
9.1 102 2.3 101 5.0 101
12.5 6.1 2.5
86 12.1 3.9
1.021 1.151 1.279 2.560 12.80
1.3 1013 5.3 108 1.8 106 3.1 103 2.10 101
6.1 1014 4.6 108 1.6 106 3.0 103 2.05 101
4.7 101 8.7 101 8.9 101 9.7 101 9.8 101
290 44 23.5 5.1 1.25
310 45 23.8 5.2 1.28
e1 = 2,ms1 = 0.03415, ms0 = 0.03038 3.5 102 4.0 102 5.0 102
1.15 1.32 1.64
e1= 4, ms1 = 3.909 104,ms0 = 3.897 104 3.99 104 4.50 104 5.00 104 1.00 104 5.00 104
1.0238 1.155 1.282 2.570 12.80
Fig. 1. Dimensionless saturation density d ¼ 2 =1 at 1 =kB T ¼ 2.
ls ¼ ns g3 as function of the parameter
(iii) with those for case (i) one can estimate the contribution of the additional interactions to the nucleation rate. In all calculations, the parameter lc was taken 1.2 as was selected in Ref. [2]. The energy parameter 1 of the solute–solute interaction was selected to be e1 ¼ 2 and e1 ¼ 4 where e1 ¼ 1 =kB T and the parameters v 2 and v s were selected equal to g3 . Because the results for case (i) can be obtained using the corresponding equations of Ref. [2], let us analyze first the role of the interaction of a solute molecule with other solute molecules located outside the cluster (case (ii)) by neglecting the interaction of the molecule with those of the solvent. To analyze this case one should take d ¼ 0 in the expression of l, the latter parameter thus becoming l ¼ lc ls . As an immediate consequence of the change in the value of the parameter l one can note the change of the value of ns of the saturation density calculated with Eq. (34). For example, at e1 ¼ 2; ms0 ¼ 0:03038 ðms ¼ ns g3 Þ for case (i) and ms1 ¼ 0:03415 if the solute–solute interaction is taken into ac-
count (case (ii)). This affects the supersaturations s qs =ns ¼ ls =ms which are different for the same solute density qs . The same observations can be made regarding the critical cluster size Rc and the rate of nucleation I. In Table 1, the nucleation rates I0 and I1 are listed for several values of ls along with the supersaturations s0 and s1 and the critical cluster radii Rc0 and Rc1 . This table shows that for a given ls the solute–solute interactions decrease the nucleation rate, and that the ratio I1 =I0 decreases with decreasing ls . For ls 6 ms1 ¼ 0:03415, this ratio becomes zero. The latter result occurs because for ls < ms1 the solution is undersaturated ðs1 < 1Þ in case (ii) but oversaturated ðs0 > 1Þ in case (i). One should note that the changes in the nucleation rate due to the solute–solute interactions are small (less than one order of magnitude, which for homogeneous nucleation constitutes a small number) for almost all values of ls . At the lower temperature ðe1 ¼ 4Þ; ms0 ’ ms1 ðms0 ¼ 3:897 104 ; ms1 ¼ 3:909 104 Þ and the difference between s0 and s1 for a given ls is negligible. In this case, Table 1 shows that I0 ’ I1 . The small contribution of solute–solute interactions to the nucleation rate can be explained by the relatively small density of the solute molecules outside the cluster and, consequently, the small change in the total potential energy of a solute molecule due to this interaction. Similar to the solute–solute interactions, the solute–solvent ones also affect (at a selected ls ) the calculated values of the saturation density ns and, as a consequence, the supersaturation s, critical cluster size R, and the rate of nucleation I. The dependence of ms2 on d ¼ 2 =1 is shown in Fig. 1 for e1 ¼ 2 and 0 6 d 6 0:5. The quantity ms increases with increasing d and rapidly acquires the solid-like magnitude 0:2. For this reason our considerations are restricted to the range of small values of d ðd 6 0:1Þ. In Table 2 the results for case (iii) are presented along with those for case (i), marked with subscript 0. The data in this table are calculated for e1 ¼ 2 and d ¼ 0:03 and d ¼ 0:1. In both cases the concentration ls is taken larger than ms2 . Note that for
Table 2 Comparison between cases (i) and (iii) at e1 = 2 and various solute number densities ls.
ls
s0
s2
2I0/qsb1
2I2/qsb1
I2/I0
Rc0/g
Rc1/g
1.0078 1.021 1.047 1.178 1.309 1.57
1.94 102 2.18 102 2.52 102 6.13 102 1.05 101 2.06 101
1.3 106 4.4 105 2.96 104 8.4 103 2.8 102 9.2 102
6.7 105 2.0 103 1.2 102 1.4 101 2.7 101 4.5 101
7.0 6.9 6.1 3.5 2.5 1.0
255 99 43 11.5 5.5 2.6
1.0028 1.0069 1.076 1.208
1.04 101 1.05 101 1.54 101 2.06 101
8.8 107 1.02 105 5.23 103 2.14 102
8.5 106 9.7 105 3.4 102 1.0 101
2.51 2.50 1.8 1.0
820 262 15.5 7.6
e2/e1= 0.03, ms2 = 0.03820, ms0 = 0.03038 3.85 102 3.90 102 4.00 102 4.50 102 5.00 102 6.00 102
1.267 1.283 1.32 1.481 1.64 1.97
e2/e1= 0.1, ms2 = 0.04966, ms0 = 0.03038 4.98 102 5.00 102 5.50 102 6.60 102
1.639 1.645 1.810 1.970
532
E. Ruckenstein, G.O. Berim / Journal of Colloid and Interface Science 342 (2010) 528–532
Table 3 Comparison between cases (i) and (iii) at e1 = 4 and various solute number densities ls.
ls
s0 4
2I0/qsb1
2I2/qsb1
I2/I0
Rc0/g
Rc1/g
1.020 1.122 1.427 1.613 4.080 12.20
1.77 106 2.28 105 2.66 104 8.02 104 3.79 102 2.68 101
1.07 1013 2.10 108 2.25 105 1.36 104 2.40 102 2.2 101
6.0 108 9.2 104 8.5 102 1.7 101 6.3 101 8.2 101
24 18 9.9 5.7 2.6 1.1
350 65 17 11 3.5 1.5
1.011 1.035 1.083 1.805 3.610 6.020 8.430
1.04 103 1.31 103 1.73 103 1.65 102 9.18 102 2.10 101 3.24 101
7.80 1014 2.97 1011 5.10 109 7.80 104 2.64 102 9.72 102 1.79 101
7.5 1011 2.3 108 2.9 106 4.7 102 2.9 101 4.6 101 5.5 101
7.1 6.8 6.0 3.5 1.8 1.3 1.0
450 160 65 8.0 3.2 2.1 1.6
s2 4
e2/e1= 0.03, ms2 = 4.9 10 ,ms0 = 3.897 10 5.0 5.5 7.0 8.0 2.0 6.0
104 104 104 104 103 103
1.282 1.411 1.790 2.05 5.13 15.4
e2/e1= 0.1, ms2 = 8.3 104,ms0 = 3.897 104 8.4 104 8.6 104 9.0 104 1.5 103 3.0 103 5.0 103 7.0 103
2.16 2.20 2.31 3.85 7.70 12.8 18.0
responding to d ¼ 0), and (iii) (two lines for d ¼ 0:03 and d ¼ 0:1). The largest difference between various nucleation rates at the same supersaturation does not exceed one order of magnitude for all considered supersaturations. 4. Conclusion
Fig. 2. Nucleation rate I calculated at 1 =kB T ¼ 2 as function of the supersaturation s for cases (i)–(iii). Case (i) considered in Ref. [2] is presented by the solid line. Case (ii) is represented by the curve for d ¼ 0. Case (iii) is represented by curves computed for d ¼ 0:03 and d ¼ 0:1.
In the present paper a modification of the Ruckenstein, Narasimhan, and Nowakowski (Refs. [1,2]) nucleation theory is developed which, in addition to the solute–cluster interactions, accounts for the solute–solute and solute–solvent interactions. It is shown that at the same supersaturation these additional interactions do not affect significantly the rate of nucleation and hence, the approximation made in Refs. [1,2] by neglecting them is valid in most cases. References
ls 6 ms2 , the nucleation rate I2 is equal to zero because s2 is smaller than one, and Rc2 ¼ 1. In all cases, I2 I0 and the ratio I2 =I0 decreases with decreasing ls as in case (ii), varying from I2 =I0 101 for larger ls to 105 for d ¼ 0:03, or to 106 for d ¼ 0:1 for ls close to ms2 . In Table 3 similar results are presented for e1 ¼ 4. In this case, I2 =I0 varies from I2 =I0 101 for large ls to 108 for d ¼ 0:03, or to 1011 for d ¼ 0:1. In Fig. 2, the nucleation rate is presented at e1 ¼ 2 as a function of the supersaturation s for the cases: (i) (solid line), (ii) (line cor-
[1] G. Narsimhan, E. Ruckenstein, J. Colloid Interface Sci. 128 (1989) 549. [2] E. Ruckenstein, B. Nowakowski, J. Colloid Interface Sci. 137 (1990) 583. [3] F. Abraham, Homogeneous Nucleation Theory, Academic Press, New York, 1974. [4] F.C. Goodrich, Proc. R. Soc. London A 277 (1964) 167. [5] E. Ruckenstein, Y.S. Djikaev, Adv. Colloid Interface Sci. 118 (2005) 51. [6] S. Chandrasekhar, Rev. Mod. Phys. 15 (1949) 1. [7] C.W. Gardiner, Handbook of Stochastic Methods, Springer, New York/Berlin, 1983. [8] N. Agmon, J. Chem. Phys. 81 (1984) 3644. [9] G.H. Weiss, J. Stat. Phys. 42 (1986) 1. [10] J.M. Deutch, J. Chem. Phys. 73 (1980) 4700.