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Critical Comparison of Droplet Models in Homogeneous Nucleation Theory
JOURNAL OF COLLOID AND INTERFACE SCIENCE ARTICLE NO.
201, 194–199 (1998)
CS975379
Critical Comparison of Droplet Models in Homogeneous Nucleation Theory Richard B. McClurg 1 and Richard C. Flagan 2 Spalding Laboratory (210-41), Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125 Received September 3, 1997; accepted December 8, 1997
We review the droplet models commonly used in calculations of the rate of homogeneous nucleation from the vapor. A statistical mechanical framework is used to form a common basis for comparison of the models. The asymptotic models (Capillarity, Tolman, and higher order models) and the semiphenomenological models introduced by A. Dillmann and G. E. A. Meier (1991) ( J. Chem. Phys. 94, 3872) are shown to be consistent with the statistical mechanical approach. The Lothe/Pound (J. Lothe and G. M. Pound (1962), J. Chem. Phys. 36, 2080) formalism overcounts the degrees of freedom for a cluster and is therefore inconsistent. The self-consistency correction is shown to be an arbitrary method for handling a truncation error. q 1998 Academic Press Key Words: homogeneous nucleation; droplet models.
single monomers or evaporate by elimination of single monomers. Under this assumption, the flux (Jn ) of clusters through a size (n) is Jn Å anbAnCn 0 En/1Cn /1 ,
[1]
where an is the accommodation coefficient (commonly assumed to be unity), b is the flux of monomers through a unit area, An is the surface area of the cluster, Cn is the number concentration, and En is the frequency of monomer evaporation from an n-mer. To determine En , we follow the approach of Katz (7). Applying detailed balancing at full thermodynamic equilibrium yields
I. INTRODUCTION
eq eq eq eq J eq n Å 0 Å anb AnC n 0 E n/1 C n/1 .
There are currently a wide variety of droplet models used to calculate homogeneous nucleation rates (1–5). This paper compares these models in the framework of the statistical mechanics of an ideal gas mixture of polyatomic gases. The common comparison reveals the assumptions behind each model and suggests that some have a more solid physical basis than others. The outline of the balance of this paper is as follows. In Section II, we briefly review classical nucleation theory to motivate Section III, in which we compute the partition function of a cluster as an asymptotic series in the cluster size, n. This series is the common framework for comparing the various classes of cluster models in Sections IV, V, and VI. Finally, we draw several conclusions in Section VII.
[2]
Assuming that the evaporation rate is independent of the saturation ratio, i.e., En/1 Å E eq n/1 , leads to the following estimate for En/1 : eq En/1 Å anb eq AnC eq n /C n/1 .
[3]
Substituting Eq. [3] into Eq. [1],
Jn Å anbAnC eq n
F
Cn b eqCn/1 0 C eq bC eq n n/1
G
,
[4]
multiplying and dividing by ( b eq / b ) n , and rearranging gives II. CLASSICAL NUCLEATION THEORY
S D b eq b
n
S D
n/ 1
Cn Jn Å eq eq eq n anbAnC n ( b / b ) Cn
1 Current address: Department of Chemical Engineering, University of Minnesota, Minneapolis, MN 55455. 2 To whom correspondence should be addressed.
For steady-state nucleation, the net flux through each size is equal. Therefore, J1 Å J2 Å rrr Å Jb Å J. Summing Eq. [5] over n yields
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0
Cn/1 C eq n/1
b eq b
According to classical theory (6, 7), homogeneous nucleation is a Markov process. Clusters grow by addition of
.
[5]
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DROPLET MODELS IN HOMOGENEOUS NUCLEATION THEORY b
J ∑ nÅ1
1 anbAnC ( b / b eq ) n eq n
Å
C1 C eq 1
S D b eq b
0
Cb/1 C eq b/1
S D b eq b
This is the law of mass-action. In its more familiar form, the cluster concentrations are related through a free energy of formation ( DG):
b/ 1
.
[6] C eq n Å exp( 0 DG/kT ) C eq 1
From the kinetic theory of gases, b Å P/(2pmkT ) 1 / 2
[7]
b / b eq Å P/P vap Å C1 /C eq 1 Å S,
Y∑ `
n 01 [ an AnC eq . n S ]
[9]
nÅ1
Equation [9] differs by a factor of 1/S from the original expression developed by Volmer and Weber (8), Becker and Do¨ring (9), and Zeldovich (10). The 1/S factor was originally proposed by Courtney (11). It is now commonly accepted that Eq. [9] is the correct form (6). The various formalisms for calculating the homogeneous nucleation rate (J) share Eq. [9] but differ in their methods for estimating the equilibrium cluster distribution (C eq n ). III. STATISTICAL MECHANICS
One rigorous way to determine the equilibrium cluster distribution (C eq n ) is through statistical mechanics. We begin with the partition function for an ideal gas mixture. The canonical partition function for an ideal gas mixture of clusters is (12) `
Qmix Å
∏
q inn /(in )!,
[10]
nÅ1
where qn is the partition function for an n-mer and in is the number of n-mers in the mixture. This leads directly to the chemical potential for each component, mn Å 0kT
S
Ì ln(Qmix ) Ìin
D
Å 0kT ln(qn /in ).
[11]
T ,V ,ia xn
Thus, at full thermodynamic equilibrium (n m Å mn ), the ratio of equilibrium cluster concentrations can be written as C eq in (qn /i 1 ) n Å Å . eq C1 i 1 (q1 /i 1 ) n
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SD q1 i1
0 kT ln
SD qn i1
Å 0n m / Gn
[14]
[8]
where P is the monomer pressure, m is the monomer mass, k is Boltzmann’s constant, and S is the saturation ratio. Finally, using Eq. [8] and allowing b to become large gives the desired form for the nucleation rate, JÅb
DG Å nkT ln
[13]
[12]
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Note that the free energy of formation of a cluster of size one from one monomer is precisely zero. This is the self-consistency requirement as defined by Girshick and Chiu (5). Molecular dynamics (MD) and Monte Carlo (MC) simulations can be used to calculate the free energy of clusters (Gn ) (6). The computational cost of creating a table of free energy versus n and T makes this approach impractical for routine use, however. Density functional methods have also been used (6). Although these methods are less computationally intensive, it is not clear if the assumed monotonic density profile is appropriate for clusters of a few atoms or molecules, particularly below the melting temperature. The balance of this report is dedicated to another approach, droplet models, in which one uses properties of the vapor and/ or bulk liquid to estimate cluster free energies (Gn ). We motivate the functional form for those models using the statistical mechanics of an ideal polyatomic gas. The partition function for a cluster of n monomers (qn ) can be separated into terms for translation (qtr ), rotation (qrot ), libration of monomers about their equilibrium positions (qlib ), vibration within the monomers (qvib ), degeneracy (dj ), and the Boltzmann factor: qn Å ∑ dj qtr qrotqlibqvib exp( 0 Ej /RT ).
[15]
j
The degeneracy and Boltzmann factor account for the contributions from multiple isomers (13, 14) that are important above the melting temperature of the cluster. In writing Eq. [15], we have assumed that the cluster rotation, monomer libration, and internal vibrations are decoupled. This is justified since the characteristic frequencies for these modes are very different. (The frequencies are roughly 0.1, 10, and 10 3 cm01 , respectively.) For an ideal polyatomic gas, the translational degrees of freedom (qtr ) factor out exactly. This is a direct consequence of conservation of momentum. Some methods (e.g., MC simulation) are easier to perform if the center of mass is allowed to fluctuate (15), and such fluctuations can lead to modifications to the nucleation rate expression, Eq. [9] (16). These issues are due to unphysical fluctuations in the simulation method and are not fundamental to nucleation theory.
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McCLURG AND FLAGAN imax
TABLE 1 Material Property Scalings
E zp Å ∑
iÅ1
Property
Symbol
Leading power
Expansion
Mass Moments of inertia Minimum potential energy Zero point energy Einstein temperature
M IAIBIC Emin E zp UE
1 5/3 1 1 1
mn (iain5/30i/3 (iain10i/3 (iain10i/3 (iain10i/3
Assuming that the cluster rotates rigidly and vibrates harmonically, the partition function becomes 2pnmkT h2
j
1
p 1/2 sj
imax
1
iÅ1
0 RT ln
j
J
3/2
(IA IB IC ) 1 / 2
0hni 2kT 0hni 1 0 exp kT
exp
exp( 0 Ej /RT ),
[16]
where m is the mass of the monomer, s is the rotation symmetry number, {IA , IB , IC } are the principal moments of inertia, V is the volume per cluster, k is the Boltzmann constant, R is the gas constant, and h is Planck’s constant. We assume that various isomers differ mainly in their degeneracies, binding energies, and symmetry numbers. This is reasonable since the cluster mass (nm) for each isomer is conserved and the product of the principal moments of inertia (IA IB IC ) is nearly constant for roughly spherical isomers. We further assume that the vibrational frequency distribution { ni } is barely changed by the small number of defects defining the differences between isomers (17). In particular, we have assumed that the internal vibrations (qvib ) are unchanged by the condensation process and they will be ignored hereafter. This is equivalent to treating the molecules as rigid bodies. The limits on i are 1 to 3n 0 6 for atomic condensation and 1 to 6n 0 6 for molecular condensation. Also, Eq. [16] assumes fixed volume, restricting these results to the low pressure limit. To simplify Eq. [16], we introduce the Einstein temperature
S∏
imax
UE Å
iÅ1
hni /k
D
1 / imax
[17]
and the average zero-point energy per mode
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[18]
8p 2kT h2
2pnmkT h2
3/2
3/2
(IA IB IC ) 1 / 2
1 1 0 exp( 0 UE /kT )
0 RTimax ln
3/2
8p 2kT h2
imax .
HS DJ H S D J H J H∑ S D S DJ
Gn Å E min / imax E zp 0 RT ln V 0 RT ln p 1 / 2
HS DJ HS DS D S D ∏ S D
Y
Then, from Eqs. [14] and [16], the free energy can be written as
Note. See Eq. [20].
qn Å ∑ dj V
hni 2
dj Ej 0 E min exp 0 sj RT
,
[19]
where E min denotes the global minimum potential energy of the cluster. Thus, Eq. [19 ] shows that the free energy is a sum of contributions from the binding energy, zeropoint energy, translation, rotation, vibration, and multiple configurations. In order to express the free energy as a function of cluster size (n), we first express the material properties in Eq. [19] as a series in n 1 / 3 `
Y/n p Å ∑ ai /n i / 3 .
[20]
iÅ0
Here, Y stands for any of the material properties in Eq. [19] and ai is an expansion coefficient. The 13 power is expected for three-dimensional clusters. The highest powers (p) for each property (Y ) are determined by scaling analysis (18). See Table 1. Gathering terms of the same order in n yields `
Gn Å a1n / a2n 2 / 3 / a3n 1 / 3 / a4 ln(n) / ∑ ai n ( 50 i ) / 3 . iÅ5
[21] It has previously been recognized that many droplet models are in the form of the first five terms in Eq. [21] (19), but we believe this is the first derivation of the infinite expansion. The a1 term has contributions from 3n vibrational degrees of freedom for atoms, or 6n degrees of freedom for molecules. The proper values (imax Å 3n 0 6 for atoms or 6n 0 6 for molecules) is recovered by the a5 term. Translations and rotations of the cluster as a whole lead to contributions to the a4 and a5 terms. These assignments of the parentage of the various a values are important for understanding several of the cluster models discussed below.
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DROPLET MODELS IN HOMOGENEOUS NUCLEATION THEORY
TABLE 2 Cluster Models Model
a1n
/
a2n2/3
/
a3n1/3
/
a4 ln(n)
/
a5
Capillarity Tolman Lothe/Pound SCC Dillmann/Meier
mn mn mn mn f`n
/ / / / /
kTUn2/3 kTUn2/3 kTUn2/3 kTUn2/3 s` s1n2/3
0 / / /
kTUBn1/3 0 0 a1s` s1n1/3
0 / /
4kT ln(n) 0 tkT ln(n)
0 /
kTU a2s `s1 0kT ln(q˜0)
Note. Zeros denote terms which have been implicitly set to zero in the models. In both cases, lower order terms have been included before higher order terms. Thus, those models are not asymptotically correct. See Eqs. [24], [28], and [32]–[34].
S
IV. ASYMPTOTIC MODELS
G nd Å mn / sAn 1 0
A. Capillarity Approximation The oldest cluster model is the capillarity approximation and is attributed to Gibbs (1). He proposed that the free energy of a droplet is characterized by volume and surface terms, G
cap n
Å mn / sAn .
U Å A1s /kT,
[23]
Eq. [22] becomes [24]
Here, U is a nondimensional surface tension which is not to be confused with the Einstein temperature ( UE ), and A1 is the apparent surface area of a monomer which is generally approximated using the bulk molar volume ( £ ) A1 Å (36p ) 1 / 3(£ /Na)2 / 3 .
[25]
B. Tolman Correction A paper by Tolman (2), published posthumously in 1949, extended Eq. [22] to include the first size corrections to the surface tension. He introduced what is now called the Tolman length ( d ), which is the radial distance between the Gibbs surface of tension and the equimolar dividing surface. It leads to the cluster model
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[26]
B Å d /r1 Å
S D 4p 3£
1/3
[27]
Eq. [26] becomes G nd Å mn / kTUn 2 / 3 0 kTUBn 1 / 3 .
[28]
Therefore, Eq. [26] is equivalent to using the first three terms of Eq. [21]. Several methods have been developed to estimate d (14, 22–24). Equation [26] has been applied to homogeneous nucleation rate calculations (21, 25). V. SEMIPHENOMENOLOGICAL DROPLET MODELS
G cap Å mn / kTUn 2 / 3 . n
AID
D
Once again, it is useful to introduce notation that makes the scaling more transparent (21). Using Eq. [23] and the definition
[22]
In Eq. [22], m is the bulk chemical potential, s is the surface free energy, and An is the surface area of an n-mer. Since the surface area scales as n 2 /3 , Eq. [22] is equivalent to using the first two terms of Eq. [21]. For use in nucleation calculations, it is useful to introduce nomenclature that makes this scaling more transparent (20). Using the definition
2d / O(r 02 ) r
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Using an approach similar to that in Section III, but using different notation, Fisher (4) derived the droplet model F Fisher Å nf` / ss1n 2 / 3 / tkT ln(n) 0 kT ln(qI 0 ). n
[29]
Except for obvious differences in notation, this is the same as the first, second, fourth, and fifth terms in Eq. [21]. The microscopic surface tension ( s ) is an unspecified quantity in this model. Dillmann and Meier (26) (DM) later extended the Fisher model by specifying a functional form for s s Å s`kn k n Å 1 / a 1n
[30] 01 / 3
/ a2n
2/3
.
[31]
Using this assumed functional form (Eq. [30]) in Fisher’s
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McCLURG AND FLAGAN
model (Eq. [29]) yields a functional form equivalent to the first five terms in Eq. [21] F nDM Å nf` / s`s1n 2 / 3 / a1s`s1n 1 / 3 / tkT ln(n) / a2s`s1 0 kT ln(qI 0 ).
[32]
DM determined a1 and a2 by comparing the power series for the vapor density as a function of vapor pressure implied by Eq. [32] with the virial equation of state. The predicted rates were in good agreement with experimental values for several substances. Later, it was noted that the DM theory has an inconsistency in its treatment of the chemical potential (27–29). The corrected version of the DM theory does not agree as well with experimental results. There is continued effort to find new ways to determine the parameters in Eq. [32] which lead to better rate predictions (30, 31). VI. OTHER MODELS
A. Lothe and Pound Correction Lothe and Pound (3) noted that the capillarity approximation (Eq. [22]) does not contain any explicit contributions from translation or rotation of the cluster. They noted that the highest order terms contributed by these degrees of freedom are O(ln n) and they simply added this term to the capillarity model G nLP Å mn / kTUn 2 / 3 0 4kT ln(n).
[33]
This is equivalent to setting the third term of Eq. [21] equal to zero. Unfortunately, this term is needed to have the proper number of vibrational degrees of freedom (3n 0 6 for atoms or 6n 0 6 for rigid molecules). Neglecting it is equivalent to double counting six degrees of freedom and is, therefore, inconsistent. B. Self-Consistent Cluster (SCC) Correction Girshick and Chiu (5) noted that the capillarity approximation (Eq. [22]) leads to a nonzero free energy of formation for a cluster of one monomer. They proposed subtracting a constant to recover self-consistency: G nSCC Å G cap 0 DG cap Å mn / sAn 0 sA1 . n 1
[34]
Their model is equivalent to setting the coefficients of the n 1 / 3 , the ln(n), and all of the n 0i / 3 (i ú 0) terms to zero. We have already noted in discussing the Lothe and Pound Correction that the first of these is needed to ensure that the model contains the proper number of degrees of freedom. The second neglected term contains the temperature-dependent contributions of the translations and rotations. Setting
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this term to zero is equivalent to assuming that the mass and moment of inertia of the cluster are zero. Finally, this solution is not in keeping with the asymptotic nature of the capillarity approximation. As noted above, the capillarity approximation is simply the first two terms in an infinite series of terms for the cluster free energy. The self-consistency problem is simply a truncation error. Subtracting a constant from an asymptotic series to make the shifted series give a correct result at a particular size does not have any theoretical basis. If this ad hoc adjustment leads to improved predictions for nucleation rates for some systems, then this should be regarded as fortuitous (6). VII. CONCLUSIONS
All of the cluster models reviewed in this paper can be viewed as special cases of the general expansion, Eq. [21]. The capillarity model, Eq. [22], the Tolman model, Eq. [26], and the Dillmann and Meier extension of the Fisher model, Eq. [32], are consistent with Eq. [21] in that they include less significant terms only after including all more significant terms. Therefore, these models can be expected to be asymptotically correct for large cluster size, n. The Lothe and Pound model, Eq. [33], and the SCC model, Eq. [34], include less significant terms before more significant terms and are therefore not asymptotically correct. The empirical success of these two models should be regarded as fortuitous, especially since both overcount the number of vibrational degrees of freedom.
REFERENCES 1. Gibbs, J. W., ‘‘The Scientific Papers of J. Willard Gibbs,’’ p. 105. Dover, New York, 1961. 2. Tolman, R. C., J. Chem. Phys. 17, 333 (1949). 3. Lothe, J., and Pound, G. M., J. Chem. Phys. 36, 2080 (1962). 4. Fisher, M. E., Physics 3, 255 (1967). 5. Girshick, S. L., and Chiu, C-P., J. Chem. Phys. 93, 1273 (1990). 6. Oxtoby, D. W., J. Phys. Condens. Matter 4, 7627 (1992). 7. Katz, J. L., Pure Appl. Chem. 64, 1661 (1992). 8. Volmer, M., and Weber, A., Z. Phys. Chem. (Leipzig) 119, 277 (1926). 9. Becker, R., and Do¨ring, W., Ann. Phys. (Leipzig) 24, 719 (1935). 10. Zeldovich, J., J. Exp. Theor. Phys. 12, 525 (1942). 11. Courtney, W. G., J. Chem. Phys. 35, 2249 (1961). 12. Hill, T. L., ‘‘Statistical Thermodynamics,’’ pp. 104, 161–167. Dover, New York, 1986. 13. Hoare, M. R., Adv. Chem. Phys. 40, 49 (1979), and references therein. 14. McClurg, R. B., Flagan, R. C., and Goddard, W. A., J. Chem. Phys. 105, 7648 (1996). 15. Reiss, H., Katz, J. L., and Cohen, E. R., J. Chem. Phys. 48, 5553 (1968). 16. Weakliem, C. L., and Reiss, H., J. Phys. Chem. 98, 6408 (1994). 17. Stillinger, F. H., and Weber, T. A., J. Chem. Phys. 81, 5095 (1984). 18. McClurg, R. B., Flagan, R. C., and Goddard, W. A., J. Chem. Phys. 102, 3322 (1995).
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DROPLET MODELS IN HOMOGENEOUS NUCLEATION THEORY 19. Kiang, C. S., Stauffer, D., Walker, G. H., Puri, O. P., Wise, Jr., J. D., and Patterson, E. M., J. Atmos. Sci. 28, 1222 (1971). 20. Warren, D. R., and Seinfeld, J. H., Aerosol. Sci. Technol. 3, 135 (1984). 21. Rao, N. P., and McMurry, P. H., Aerosol. Sci. Technol. 13, 183 (1990). 22. Kirkwood, J. G., and Buff, F. P., J. Chem. Phys. 17, 338 (1949). 23. Blokhuis, E. M., and Bedeaux, D., J. Chem. Phys. 97, 3576 (1992). 24. Kegel, W. K., J. Chem. Phys. 102, 1094 (1995). 25. Heerman, D. W., J. Stat. Phys. 29, 631 (1982). 26. Dillmann, A., and Meier, G. E. A., J. Chem. Phys. 94, 3872 (1991).
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27. Delale, C. F., and Meier, G. E. A., J. Chem. Phys. 98, 9850 (1993). 28. Ford, I. J., Laaksonen, A., and Kulmala, M., J. Chem. Phys. 99, 764 (1993). 29. Laaksonen, A., Ford, I. J., and Kulmala, M., Phys. Rev. E 49, 5517 (1994). 30. Kalikmanov, V. I., and van Dongen, M. E. H., Phys. Rev. E 47, 3532 (1993). 31. Kalikmanov, V. I., and van Dongen, M. E. H., J. Chem. Phys. 103, 4250 (1995).
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CS975379
Critical Comparison of Droplet Models in Homogeneous Nucleation Theory Richard B. McClurg 1 and Richard C. Flagan 2 Spalding Laboratory (210-41), Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125 Received September 3, 1997; accepted December 8, 1997
We review the droplet models commonly used in calculations of the rate of homogeneous nucleation from the vapor. A statistical mechanical framework is used to form a common basis for comparison of the models. The asymptotic models (Capillarity, Tolman, and higher order models) and the semiphenomenological models introduced by A. Dillmann and G. E. A. Meier (1991) ( J. Chem. Phys. 94, 3872) are shown to be consistent with the statistical mechanical approach. The Lothe/Pound (J. Lothe and G. M. Pound (1962), J. Chem. Phys. 36, 2080) formalism overcounts the degrees of freedom for a cluster and is therefore inconsistent. The self-consistency correction is shown to be an arbitrary method for handling a truncation error. q 1998 Academic Press Key Words: homogeneous nucleation; droplet models.
single monomers or evaporate by elimination of single monomers. Under this assumption, the flux (Jn ) of clusters through a size (n) is Jn Å anbAnCn 0 En/1Cn /1 ,
[1]
where an is the accommodation coefficient (commonly assumed to be unity), b is the flux of monomers through a unit area, An is the surface area of the cluster, Cn is the number concentration, and En is the frequency of monomer evaporation from an n-mer. To determine En , we follow the approach of Katz (7). Applying detailed balancing at full thermodynamic equilibrium yields
I. INTRODUCTION
eq eq eq eq J eq n Å 0 Å anb AnC n 0 E n/1 C n/1 .
There are currently a wide variety of droplet models used to calculate homogeneous nucleation rates (1–5). This paper compares these models in the framework of the statistical mechanics of an ideal gas mixture of polyatomic gases. The common comparison reveals the assumptions behind each model and suggests that some have a more solid physical basis than others. The outline of the balance of this paper is as follows. In Section II, we briefly review classical nucleation theory to motivate Section III, in which we compute the partition function of a cluster as an asymptotic series in the cluster size, n. This series is the common framework for comparing the various classes of cluster models in Sections IV, V, and VI. Finally, we draw several conclusions in Section VII.
[2]
Assuming that the evaporation rate is independent of the saturation ratio, i.e., En/1 Å E eq n/1 , leads to the following estimate for En/1 : eq En/1 Å anb eq AnC eq n /C n/1 .
[3]
Substituting Eq. [3] into Eq. [1],
Jn Å anbAnC eq n
F
Cn b eqCn/1 0 C eq bC eq n n/1
G
,
[4]
multiplying and dividing by ( b eq / b ) n , and rearranging gives II. CLASSICAL NUCLEATION THEORY
S D b eq b
n
S D
n/ 1
Cn Jn Å eq eq eq n anbAnC n ( b / b ) Cn
1 Current address: Department of Chemical Engineering, University of Minnesota, Minneapolis, MN 55455. 2 To whom correspondence should be addressed.
For steady-state nucleation, the net flux through each size is equal. Therefore, J1 Å J2 Å rrr Å Jb Å J. Summing Eq. [5] over n yields
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Cn/1 C eq n/1
b eq b
According to classical theory (6, 7), homogeneous nucleation is a Markov process. Clusters grow by addition of
.
[5]
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DROPLET MODELS IN HOMOGENEOUS NUCLEATION THEORY b
J ∑ nÅ1
1 anbAnC ( b / b eq ) n eq n
Å
C1 C eq 1
S D b eq b
0
Cb/1 C eq b/1
S D b eq b
This is the law of mass-action. In its more familiar form, the cluster concentrations are related through a free energy of formation ( DG):
b/ 1
.
[6] C eq n Å exp( 0 DG/kT ) C eq 1
From the kinetic theory of gases, b Å P/(2pmkT ) 1 / 2
[7]
b / b eq Å P/P vap Å C1 /C eq 1 Å S,
Y∑ `
n 01 [ an AnC eq . n S ]
[9]
nÅ1
Equation [9] differs by a factor of 1/S from the original expression developed by Volmer and Weber (8), Becker and Do¨ring (9), and Zeldovich (10). The 1/S factor was originally proposed by Courtney (11). It is now commonly accepted that Eq. [9] is the correct form (6). The various formalisms for calculating the homogeneous nucleation rate (J) share Eq. [9] but differ in their methods for estimating the equilibrium cluster distribution (C eq n ). III. STATISTICAL MECHANICS
One rigorous way to determine the equilibrium cluster distribution (C eq n ) is through statistical mechanics. We begin with the partition function for an ideal gas mixture. The canonical partition function for an ideal gas mixture of clusters is (12) `
Qmix Å
∏
q inn /(in )!,
[10]
nÅ1
where qn is the partition function for an n-mer and in is the number of n-mers in the mixture. This leads directly to the chemical potential for each component, mn Å 0kT
S
Ì ln(Qmix ) Ìin
D
Å 0kT ln(qn /in ).
[11]
T ,V ,ia xn
Thus, at full thermodynamic equilibrium (n m Å mn ), the ratio of equilibrium cluster concentrations can be written as C eq in (qn /i 1 ) n Å Å . eq C1 i 1 (q1 /i 1 ) n
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SD q1 i1
0 kT ln
SD qn i1
Å 0n m / Gn
[14]
[8]
where P is the monomer pressure, m is the monomer mass, k is Boltzmann’s constant, and S is the saturation ratio. Finally, using Eq. [8] and allowing b to become large gives the desired form for the nucleation rate, JÅb
DG Å nkT ln
[13]
[12]
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Note that the free energy of formation of a cluster of size one from one monomer is precisely zero. This is the self-consistency requirement as defined by Girshick and Chiu (5). Molecular dynamics (MD) and Monte Carlo (MC) simulations can be used to calculate the free energy of clusters (Gn ) (6). The computational cost of creating a table of free energy versus n and T makes this approach impractical for routine use, however. Density functional methods have also been used (6). Although these methods are less computationally intensive, it is not clear if the assumed monotonic density profile is appropriate for clusters of a few atoms or molecules, particularly below the melting temperature. The balance of this report is dedicated to another approach, droplet models, in which one uses properties of the vapor and/ or bulk liquid to estimate cluster free energies (Gn ). We motivate the functional form for those models using the statistical mechanics of an ideal polyatomic gas. The partition function for a cluster of n monomers (qn ) can be separated into terms for translation (qtr ), rotation (qrot ), libration of monomers about their equilibrium positions (qlib ), vibration within the monomers (qvib ), degeneracy (dj ), and the Boltzmann factor: qn Å ∑ dj qtr qrotqlibqvib exp( 0 Ej /RT ).
[15]
j
The degeneracy and Boltzmann factor account for the contributions from multiple isomers (13, 14) that are important above the melting temperature of the cluster. In writing Eq. [15], we have assumed that the cluster rotation, monomer libration, and internal vibrations are decoupled. This is justified since the characteristic frequencies for these modes are very different. (The frequencies are roughly 0.1, 10, and 10 3 cm01 , respectively.) For an ideal polyatomic gas, the translational degrees of freedom (qtr ) factor out exactly. This is a direct consequence of conservation of momentum. Some methods (e.g., MC simulation) are easier to perform if the center of mass is allowed to fluctuate (15), and such fluctuations can lead to modifications to the nucleation rate expression, Eq. [9] (16). These issues are due to unphysical fluctuations in the simulation method and are not fundamental to nucleation theory.
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McCLURG AND FLAGAN imax
TABLE 1 Material Property Scalings
E zp Å ∑
iÅ1
Property
Symbol
Leading power
Expansion
Mass Moments of inertia Minimum potential energy Zero point energy Einstein temperature
M IAIBIC Emin E zp UE
1 5/3 1 1 1
mn (iain5/30i/3 (iain10i/3 (iain10i/3 (iain10i/3
Assuming that the cluster rotates rigidly and vibrates harmonically, the partition function becomes 2pnmkT h2
j
1
p 1/2 sj
imax
1
iÅ1
0 RT ln
j
J
3/2
(IA IB IC ) 1 / 2
0hni 2kT 0hni 1 0 exp kT
exp
exp( 0 Ej /RT ),
[16]
where m is the mass of the monomer, s is the rotation symmetry number, {IA , IB , IC } are the principal moments of inertia, V is the volume per cluster, k is the Boltzmann constant, R is the gas constant, and h is Planck’s constant. We assume that various isomers differ mainly in their degeneracies, binding energies, and symmetry numbers. This is reasonable since the cluster mass (nm) for each isomer is conserved and the product of the principal moments of inertia (IA IB IC ) is nearly constant for roughly spherical isomers. We further assume that the vibrational frequency distribution { ni } is barely changed by the small number of defects defining the differences between isomers (17). In particular, we have assumed that the internal vibrations (qvib ) are unchanged by the condensation process and they will be ignored hereafter. This is equivalent to treating the molecules as rigid bodies. The limits on i are 1 to 3n 0 6 for atomic condensation and 1 to 6n 0 6 for molecular condensation. Also, Eq. [16] assumes fixed volume, restricting these results to the low pressure limit. To simplify Eq. [16], we introduce the Einstein temperature
S∏
imax
UE Å
iÅ1
hni /k
D
1 / imax
[17]
and the average zero-point energy per mode
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[18]
8p 2kT h2
2pnmkT h2
3/2
3/2
(IA IB IC ) 1 / 2
1 1 0 exp( 0 UE /kT )
0 RTimax ln
3/2
8p 2kT h2
imax .
HS DJ H S D J H J H∑ S D S DJ
Gn Å E min / imax E zp 0 RT ln V 0 RT ln p 1 / 2
HS DJ HS DS D S D ∏ S D
Y
Then, from Eqs. [14] and [16], the free energy can be written as
Note. See Eq. [20].
qn Å ∑ dj V
hni 2
dj Ej 0 E min exp 0 sj RT
,
[19]
where E min denotes the global minimum potential energy of the cluster. Thus, Eq. [19 ] shows that the free energy is a sum of contributions from the binding energy, zeropoint energy, translation, rotation, vibration, and multiple configurations. In order to express the free energy as a function of cluster size (n), we first express the material properties in Eq. [19] as a series in n 1 / 3 `
Y/n p Å ∑ ai /n i / 3 .
[20]
iÅ0
Here, Y stands for any of the material properties in Eq. [19] and ai is an expansion coefficient. The 13 power is expected for three-dimensional clusters. The highest powers (p) for each property (Y ) are determined by scaling analysis (18). See Table 1. Gathering terms of the same order in n yields `
Gn Å a1n / a2n 2 / 3 / a3n 1 / 3 / a4 ln(n) / ∑ ai n ( 50 i ) / 3 . iÅ5
[21] It has previously been recognized that many droplet models are in the form of the first five terms in Eq. [21] (19), but we believe this is the first derivation of the infinite expansion. The a1 term has contributions from 3n vibrational degrees of freedom for atoms, or 6n degrees of freedom for molecules. The proper values (imax Å 3n 0 6 for atoms or 6n 0 6 for molecules) is recovered by the a5 term. Translations and rotations of the cluster as a whole lead to contributions to the a4 and a5 terms. These assignments of the parentage of the various a values are important for understanding several of the cluster models discussed below.
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TABLE 2 Cluster Models Model
a1n
/
a2n2/3
/
a3n1/3
/
a4 ln(n)
/
a5
Capillarity Tolman Lothe/Pound SCC Dillmann/Meier
mn mn mn mn f`n
/ / / / /
kTUn2/3 kTUn2/3 kTUn2/3 kTUn2/3 s` s1n2/3
0 / / /
kTUBn1/3 0 0 a1s` s1n1/3
0 / /
4kT ln(n) 0 tkT ln(n)
0 /
kTU a2s `s1 0kT ln(q˜0)
Note. Zeros denote terms which have been implicitly set to zero in the models. In both cases, lower order terms have been included before higher order terms. Thus, those models are not asymptotically correct. See Eqs. [24], [28], and [32]–[34].
S
IV. ASYMPTOTIC MODELS
G nd Å mn / sAn 1 0
A. Capillarity Approximation The oldest cluster model is the capillarity approximation and is attributed to Gibbs (1). He proposed that the free energy of a droplet is characterized by volume and surface terms, G
cap n
Å mn / sAn .
U Å A1s /kT,
[23]
Eq. [22] becomes [24]
Here, U is a nondimensional surface tension which is not to be confused with the Einstein temperature ( UE ), and A1 is the apparent surface area of a monomer which is generally approximated using the bulk molar volume ( £ ) A1 Å (36p ) 1 / 3(£ /Na)2 / 3 .
[25]
B. Tolman Correction A paper by Tolman (2), published posthumously in 1949, extended Eq. [22] to include the first size corrections to the surface tension. He introduced what is now called the Tolman length ( d ), which is the radial distance between the Gibbs surface of tension and the equimolar dividing surface. It leads to the cluster model
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[26]
B Å d /r1 Å
S D 4p 3£
1/3
[27]
Eq. [26] becomes G nd Å mn / kTUn 2 / 3 0 kTUBn 1 / 3 .
[28]
Therefore, Eq. [26] is equivalent to using the first three terms of Eq. [21]. Several methods have been developed to estimate d (14, 22–24). Equation [26] has been applied to homogeneous nucleation rate calculations (21, 25). V. SEMIPHENOMENOLOGICAL DROPLET MODELS
G cap Å mn / kTUn 2 / 3 . n
AID
D
Once again, it is useful to introduce notation that makes the scaling more transparent (21). Using Eq. [23] and the definition
[22]
In Eq. [22], m is the bulk chemical potential, s is the surface free energy, and An is the surface area of an n-mer. Since the surface area scales as n 2 /3 , Eq. [22] is equivalent to using the first two terms of Eq. [21]. For use in nucleation calculations, it is useful to introduce nomenclature that makes this scaling more transparent (20). Using the definition
2d / O(r 02 ) r
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Using an approach similar to that in Section III, but using different notation, Fisher (4) derived the droplet model F Fisher Å nf` / ss1n 2 / 3 / tkT ln(n) 0 kT ln(qI 0 ). n
[29]
Except for obvious differences in notation, this is the same as the first, second, fourth, and fifth terms in Eq. [21]. The microscopic surface tension ( s ) is an unspecified quantity in this model. Dillmann and Meier (26) (DM) later extended the Fisher model by specifying a functional form for s s Å s`kn k n Å 1 / a 1n
[30] 01 / 3
/ a2n
2/3
.
[31]
Using this assumed functional form (Eq. [30]) in Fisher’s
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model (Eq. [29]) yields a functional form equivalent to the first five terms in Eq. [21] F nDM Å nf` / s`s1n 2 / 3 / a1s`s1n 1 / 3 / tkT ln(n) / a2s`s1 0 kT ln(qI 0 ).
[32]
DM determined a1 and a2 by comparing the power series for the vapor density as a function of vapor pressure implied by Eq. [32] with the virial equation of state. The predicted rates were in good agreement with experimental values for several substances. Later, it was noted that the DM theory has an inconsistency in its treatment of the chemical potential (27–29). The corrected version of the DM theory does not agree as well with experimental results. There is continued effort to find new ways to determine the parameters in Eq. [32] which lead to better rate predictions (30, 31). VI. OTHER MODELS
A. Lothe and Pound Correction Lothe and Pound (3) noted that the capillarity approximation (Eq. [22]) does not contain any explicit contributions from translation or rotation of the cluster. They noted that the highest order terms contributed by these degrees of freedom are O(ln n) and they simply added this term to the capillarity model G nLP Å mn / kTUn 2 / 3 0 4kT ln(n).
[33]
This is equivalent to setting the third term of Eq. [21] equal to zero. Unfortunately, this term is needed to have the proper number of vibrational degrees of freedom (3n 0 6 for atoms or 6n 0 6 for rigid molecules). Neglecting it is equivalent to double counting six degrees of freedom and is, therefore, inconsistent. B. Self-Consistent Cluster (SCC) Correction Girshick and Chiu (5) noted that the capillarity approximation (Eq. [22]) leads to a nonzero free energy of formation for a cluster of one monomer. They proposed subtracting a constant to recover self-consistency: G nSCC Å G cap 0 DG cap Å mn / sAn 0 sA1 . n 1
[34]
Their model is equivalent to setting the coefficients of the n 1 / 3 , the ln(n), and all of the n 0i / 3 (i ú 0) terms to zero. We have already noted in discussing the Lothe and Pound Correction that the first of these is needed to ensure that the model contains the proper number of degrees of freedom. The second neglected term contains the temperature-dependent contributions of the translations and rotations. Setting
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this term to zero is equivalent to assuming that the mass and moment of inertia of the cluster are zero. Finally, this solution is not in keeping with the asymptotic nature of the capillarity approximation. As noted above, the capillarity approximation is simply the first two terms in an infinite series of terms for the cluster free energy. The self-consistency problem is simply a truncation error. Subtracting a constant from an asymptotic series to make the shifted series give a correct result at a particular size does not have any theoretical basis. If this ad hoc adjustment leads to improved predictions for nucleation rates for some systems, then this should be regarded as fortuitous (6). VII. CONCLUSIONS
All of the cluster models reviewed in this paper can be viewed as special cases of the general expansion, Eq. [21]. The capillarity model, Eq. [22], the Tolman model, Eq. [26], and the Dillmann and Meier extension of the Fisher model, Eq. [32], are consistent with Eq. [21] in that they include less significant terms only after including all more significant terms. Therefore, these models can be expected to be asymptotically correct for large cluster size, n. The Lothe and Pound model, Eq. [33], and the SCC model, Eq. [34], include less significant terms before more significant terms and are therefore not asymptotically correct. The empirical success of these two models should be regarded as fortuitous, especially since both overcount the number of vibrational degrees of freedom.
REFERENCES 1. Gibbs, J. W., ‘‘The Scientific Papers of J. Willard Gibbs,’’ p. 105. Dover, New York, 1961. 2. Tolman, R. C., J. Chem. Phys. 17, 333 (1949). 3. Lothe, J., and Pound, G. M., J. Chem. Phys. 36, 2080 (1962). 4. Fisher, M. E., Physics 3, 255 (1967). 5. Girshick, S. L., and Chiu, C-P., J. Chem. Phys. 93, 1273 (1990). 6. Oxtoby, D. W., J. Phys. Condens. Matter 4, 7627 (1992). 7. Katz, J. L., Pure Appl. Chem. 64, 1661 (1992). 8. Volmer, M., and Weber, A., Z. Phys. Chem. (Leipzig) 119, 277 (1926). 9. Becker, R., and Do¨ring, W., Ann. Phys. (Leipzig) 24, 719 (1935). 10. Zeldovich, J., J. Exp. Theor. Phys. 12, 525 (1942). 11. Courtney, W. G., J. Chem. Phys. 35, 2249 (1961). 12. Hill, T. L., ‘‘Statistical Thermodynamics,’’ pp. 104, 161–167. Dover, New York, 1986. 13. Hoare, M. R., Adv. Chem. Phys. 40, 49 (1979), and references therein. 14. McClurg, R. B., Flagan, R. C., and Goddard, W. A., J. Chem. Phys. 105, 7648 (1996). 15. Reiss, H., Katz, J. L., and Cohen, E. R., J. Chem. Phys. 48, 5553 (1968). 16. Weakliem, C. L., and Reiss, H., J. Phys. Chem. 98, 6408 (1994). 17. Stillinger, F. H., and Weber, T. A., J. Chem. Phys. 81, 5095 (1984). 18. McClurg, R. B., Flagan, R. C., and Goddard, W. A., J. Chem. Phys. 102, 3322 (1995).
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DROPLET MODELS IN HOMOGENEOUS NUCLEATION THEORY 19. Kiang, C. S., Stauffer, D., Walker, G. H., Puri, O. P., Wise, Jr., J. D., and Patterson, E. M., J. Atmos. Sci. 28, 1222 (1971). 20. Warren, D. R., and Seinfeld, J. H., Aerosol. Sci. Technol. 3, 135 (1984). 21. Rao, N. P., and McMurry, P. H., Aerosol. Sci. Technol. 13, 183 (1990). 22. Kirkwood, J. G., and Buff, F. P., J. Chem. Phys. 17, 338 (1949). 23. Blokhuis, E. M., and Bedeaux, D., J. Chem. Phys. 97, 3576 (1992). 24. Kegel, W. K., J. Chem. Phys. 102, 1094 (1995). 25. Heerman, D. W., J. Stat. Phys. 29, 631 (1982). 26. Dillmann, A., and Meier, G. E. A., J. Chem. Phys. 94, 3872 (1991).
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27. Delale, C. F., and Meier, G. E. A., J. Chem. Phys. 98, 9850 (1993). 28. Ford, I. J., Laaksonen, A., and Kulmala, M., J. Chem. Phys. 99, 764 (1993). 29. Laaksonen, A., Ford, I. J., and Kulmala, M., Phys. Rev. E 49, 5517 (1994). 30. Kalikmanov, V. I., and van Dongen, M. E. H., Phys. Rev. E 47, 3532 (1993). 31. Kalikmanov, V. I., and van Dongen, M. E. H., J. Chem. Phys. 103, 4250 (1995).
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