Density functional theory of nucleation: A ... - Semantic Scholar

Density functional theory of nucleation: A semiempirical approach Rebecca M. Nyquist,a) Vicente Talanquer, and David W. Oxtobyb) James Franck Institute, University of Chicago, Chicago, Illinois 60637

~Received 13 January 1995; accepted 12 April 1995! We present a semiempirical approach to the density functional theory of gas–liquid nucleation, in which the same experimental properties used in classical nucleation theory ~equilibrium vapor pressure, liquid density, and surface tension! are used to fit three adjustable parameters in the intermolecular potential. This approach allows direct comparison of nucleation rates with experimental data. Agreement with results on nonane from three different experimental groups is reasonable, although the comparison clearly reveals the scatter in those results and suggests that further experimental measurements by different groups on the same systems would be very valuable. For water and the n-alcohols, this version of density functional theory gives results quite close to classical nucleation theory, implying that it is not a good approximation to describe polar fluids by effective spherically symmetric potentials. © 1995 American Institute of Physics.

I. INTRODUCTION

Predictions of gas-liquid nucleation rates fall into two categories. On the one hand are the approaches that might be labeled ‘‘first-principles’’ in that they begin with a Hamiltonian ~an assumed intermolecular potential!. Included in this category are molecular dynamics simulations of actual cluster formation and growth1 and Monte Carlo techniques2 that predict the free energies of small clusters. Weakliem and Reiss3 have shown how computer simulations of clusters inside closed volumes can lead to predictions of nucleation rates. Also included in the first-principles category are the density functional approaches to nucleation that we and our collaborators have explored.4 –7 In them, an intermolecular potential is assumed ~it is often spherically symmetric but may also have angular dependences8!. From that potential, an approximate free-energy functional is constructed, using hard-sphere perturbation theory, and that functional is then used to calculate cluster free energies and from them nucleation rates. The second category of nucleation theories might be termed ‘‘semiempirical’’ in that they employ as input independent experimental data on the materials whose nucleation is being predicted. Classical nucleation theory9 fits into this category; it uses three pieces of experimental data ~the surface tension, the equilibrium vapor pressure of the liquid, and the density of the liquid! to predict the free energy of the critical nucleus and from it the rate of nucleation under arbitrary conditions of temperature and pressure. Other semiempirical theories include modifications of classical nucleation theory, such as the approach of Dillmann and Meier10 in which curvature dependence of the surface tension is introduced. Here, the experimental data used in the classical theory is still employed and, in addition, the model is fit to the experimentally measured critical point properties and to the experimental second virial coefficient. Until now, only the semiempirical theories have been used to make direct comparisons with experimental nuclea!

Present address: Department of Chemistry, University of California at Los Angeles, Los Angeles, California 90024. Author to whom correspondence should be addressed.

b!

ation rates. The first-principles approaches have been limited to examining trends. They have been successful in this; for example the temperature and supersaturation dependence of nucleation rates calculated for Lennard-Jones fluids5 resembles that found experimentally for materials such as nonane11–13 and the n-alcohols.14 However, there have not been any predictions of nucleation rates from first-principles approaches that could be directly compared with experiment. There are two reasons for this. First, the materials studied experimentally are relatively complex and may not be describable by simple spherical potentials; there are not accurate rate data for simple substances such as argon for which the intermolecular potentials are well known. Second, nucleation rates are exponentially sensitive to almost everything in the calculation; a small error in calculating the surface tension from first principles could translate into a huge error in the predicted nucleation rate. Our goal in this paper is to present a semiempirical version of density functional theory that can be directly compared to experimental rates on real materials. Our approach is to adjust the intermolecular potential to match certain key pieces of experimental data ~in particular, exactly the same three pieces of data required to use classical nucleation theory! and then to proceed with the calculation of nucleation rates within the context of density functional theory. Such a theory is guaranteed ~by construction! to give the correct macroscopic limit ~bulk thermodynamics and surface tension!, but it is more realistic than classical theory in its treatment of small-size corrections. Moreover, it has a more solid theoretical basis than other semiempirical extensions of classical theory in which arbitrary assumptions are made about the dependence of properties on cluster size. The organization of this paper is as follows. In Sec. II, we describe how the potential is parametrized and how the values of the parameters are determined by comparing the density functional calculation with experimental data. ~Note that we do not use any nucleation data in the fitting process, as some other semiempirical approaches do.15! In Sec. III, we compare predicted and observed nucleation rates for several substances, and show that the two agree within the accuracy of the experimental data. We argue there that further

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Nyquist, Talanquer, and Oxtoby: Density functional theory of nucleation

progress on nonpolar fluids cannot be made until there is a clearer experimental resolution of the discrepancy between rates measured by different groups. In Sec. IV, we point out the failure of our approach for strongly polar materials. We regard this failure in a positive light, however, as simply showing that polar liquids cannot be modeled with spherically symmetric potentials. Some extensions to treat such cases in the future are outlined.

accurate Carnahan–Starling form.16 The bulk properties of a homogeneous fluid depend only on this free energy evaluated for fixed density r: F @ r # /V5 f h ~ r ! 2 21 ar 2 . This depends on the two parameters d and a, but not on the range parameter ~l or s!. From it, the equilibrium coexistence curve can be calculated by setting the pressures p and chemical potentials m of liquid and vapor equal, where p5p h ~ r ! 2 21 ar 2 ,

II. SEMIEMPIRICAL DENSITY FUNCTIONAL THEORY

The density functional approaches that we have investigated all begin with an intermolecular potential that is then approximated through hard-sphere perturbation theory. In other words, the actual potential is replaced with a hardsphere repulsive term @with a temperature-dependent hardsphere diameter d (T)# and an attractive tail fat~r!. Because we intend to fit the same three pieces of bulk experimental data as classical nucleation theory ~surface tension, vapor pressure, and liquid density at each temperature!, we need a three-parameter potential. We take the three parameters to be the hard-sphere diameter d, the integrated strength a of the attractive potential

a 524 p

E

dr r 2 f at~ r !

and the range of the attractive potential. Most of the calculations described used a Yukawa potential:

f at~ r ! 52 a l 3 exp~ 2lr ! /4 p lr but we also examined the effect of a Lennard-Jones potential treated using Weeks–Chandler–Andersen perturbation theory16

H

2 e , r,2 1/6 s s 12 s6 f at~ r ! 5 4 e 12 2 6 , r r

S

D

r.2 1/6 s

.

In the latter case, we have

a5

32 p A2 es 3 . 9

E

dr f h @ r ~ r!#

1

1 2

EE

where p h and mh are the hard-sphere pressure and chemical potential. It is then easy to use experimental values for the equilibrium vapor pressure and for the liquid density to calculate the two parameters d and a at each temperature needed. The calculation of the third parameter, l or s, requires a little more effort. To do this, at each temperature we must calculate the equilibrium gas-liquid planar density profile r~z! for ld or d/s equal to 1 by solving the variational equation that results from density functional theory:

m h @ r ~ z !# 5 m 2

E

dr8 f at~ u r2r8u ! r ~ z 8 ! .

The numerical method of solving this equation is described in Refs. 4 and 5. From this, the surface tension can easily be calculated. It scales4 as 1/ld or as s/d, so it is straightforward to use experimental data for the surface tension of the substance studied to extract the third parameter, l or s. The three potential parameters that result from fitting to experimental data depend on temperature. If the shape of the potential and the free-energy calculation are reasonably accurate models for a particular substance, this temperature dependence should be relatively modest, but it is unavoidable. In the case of nonane, as the temperature is increased from 200 to 320 K ~approximately the range of accessible experimental nucleation data!, the hard-sphere diameter d decreases by 2%, the integrated strength a decreases by 14%, and the inverse range parameter l decreases by 10%. III. NUCLEATION RATES

Note that in this paper we do not use the mapping between d(T) and s described in Ref. 5 for the Lennard-Jones potential, but rather treat d and s ~along with a! as independently variable parameters at each temperature. The Helmholtz free energy is then taken to be the sum of a hard-sphere term ~treated as a purely local free energy! and a long-range attraction term ~treated using perturbation theory!. This gives4 F @ r ~ r!# 5

m 5 m h ~ r ! 2 ar ,

dr dr8 r ~ r! r ~ r8! f at~ u r2r8u ! ,

where f h ~r! is the Helmholtz free-energy density of a hardsphere fluid with density r, for which we use the highly

Once the potential parameters are set, density functional theory is used4 –7 to calculate the spherically averaged density profile r~r! and the excess free energy DV* of the critical nucleus at a series of temperatures and supersaturations, corresponding to those studied experimentally. The nucleation rate is then J5J 0 exp~ 2DV * /kT ! , where the preexponential J0 is taken from classical theory to have the value

S D

r 2v 2 g J 05 r1 pm

1/2

,

where rv and r1 are the vapor and liquid densities, g is the surface tension, and m is the molecular mass. We have shown17 that a full dynamical treatment of density functional theory gives results almost indistinguishable from this simple choice of preexponential.

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FIG. 1. Nucleation rates for nonane as a function of supersaturation for several temperatures as measured by Adams et al. ~triangles, Ref. 11! and Wagner and Strey ~squares, Ref. 12!, compared with the results of the present theory ~smooth and dashed lines!. The temperatures given correspond to the initial temperature of the chamber before expansion, not to the actual nucleation temperature.

Let us begin our comparison of experiment and density functional theory with nonane, a substance studied by three research groups using different experimental techniques. Figure 1 compares calculated results ~using the Yukawa potential! with the experiments of Adams et al.,11 who used an expansion cloud chamber, and those of Wagner and Strey,12 who used a double-piston expansion chamber. Figure 2 is a

FIG. 2. Nucleation rates for nonane as a function of supersaturation for several temperatures as measured by Hung et al. ~squares, Ref. 13!, compared with the results of the present theory ~smooth lines!. The temperatures given in this case are the actual nucleation temperatures.

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comparison with the results of Hung et al.,13 who used a diffusion cloud chamber. The three types of experiments probe ranges of nucleation rates from 1024 to 1010 cm23 s21. There are two important points to be made about the two sets of expansion chamber data in Fig. 1. The first is that the temperatures shown in Fig. 1 correspond to the temperature at the beginning of the expansion, not to the actual temperature at which nucleation occurs, which is calculated from the adiabatic equation of state of the gas mixture. A series of data points from a given initial temperature thus corresponds to a range of nucleation temperatures. The second point is that the nucleation data from Refs. 11 and 12 has been recalculated using the more accurate equilibrium vapor pressure curve selected in Ref. 13. It is difficult to derive a consistent picture of the success of this semiempirical density functional theory from Figs. 1 and 2. First, the dependence on supersaturation is correctly given, as shown by the match of experimental and theoretical slopes in all cases. However, this is also true of classical nucleation theory. Because the slopes are related to the number of molecules in the critical nucleus,18 this quantity is accurately predicted by theory. There is more discrepancy in the absolute values of the rates and their temperature dependence. The theory is closest to fitting the data of Adams et al.; although the predicted results lie consistently about two orders of magnitude below the measured ones, they systematically track the temperature dependence of those experiments. On the other hand, the predicted temperature dependence is stronger than that seen in the experiments of Hung et al. and of Wagner and Strey; they lie above the experimental data at high temperatures and below the data at low temperatures. Figure 3 summarizes the comparison between experiment and theory in a different way. The fact that the slopes of graphs of ln J against ln S are the same for experiment, density functional theory, and classical nucleation theory implies that the ratio Jexp/Jcl ~or Jdft/Jcl! depends on temperature but only weakly on supersaturation.13,14 Figure 3 compares the predicted and measured average values of this ratio at a series of temperatures. @The three different theoretical curves arise from the fact that over a large range of rates ~14 orders of magnitude! there is some curvature in graphs of ln J against ln S, so that the appropriate average value of this ratio changes.# As was already evident from our discussion of Figs. 1 and 2, the calculated temperature dependence of this ratio is consistent with that measured by Adams et al.,11 but not with the other two experiments. Figure 3 shows that the predicted results lie in the middle of the experimental data, but that the spread in the latter is so great that it is difficult to make a more quantitative assessment. Further progress will require a reconciliation of different experimental results. As pointed out by Hung et al.13 there is also some uncertainty in the lowtemperature extrapolation of measured surface tensions. We have tested the effect of the choice of attractive potential, by replacing the Yukawa with a Lennard-Jones potential, as described in Sec. II. Figure 4 shows the results for nonane, with only the data of Adams et al.11 included for comparison. In spite of the relatively large difference in the


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FIG. 3. The ratio of the nucleation rate ~either that measured experimentally or that predicted by the present theory! to the classical nucleation rate for nonane as a function of inverse temperature. The experimental results are those from Ref. 11 ~stars!, Ref. 12 ~crosses!, and Ref. 13 ~squares!. The corresponding three sets of theoretical predictions from density functional theory are given by the same symbols, with a line drawn through the data points. Each point shown is obtained by averaging over the series of measured supersaturations.

shapes of these two potentials, the predicted rates are virtually identical. ~Note that classical theory gives significantly poorer agreement with experiment.! This lack of sensitivity to the shape of the potential ~provided that the surface ten-

FIG. 4. A comparison of the Yukawa potential mode ~solid line! and the Lennard-Jones potential model ~dotted line! with the data for the rates of nucleation of nonane measured by Adams et al. ~Ref. 11!. The two theoretical curves almost overlap. Classical theory ~dashed–dotted line! gives significantly lower rates.

FIG. 5. The same comparison as that in Fig. 4 applied to toluene. The experimental data are from Ref. 19.

sion is used to account for the known sensitivity to its range! is encouraging for future applications of this semiempirical approach. Figure 5 compares the predicted nucleation rates for toluene with those measured by Schmitt et al.19 The picture is very much like that measured by the same group11 for nonane ~Fig. 4!. IV. POLAR FLUIDS

Although toluene has a small permanent dipole moment, and nonane has a small transient dipole moment depending on its fluctuating conformation, both can be thought of as essentially nonpolar substances. We next attempted to apply our semiempirical density functional theory to polar fluids whose nucleation rates had been measured: water20,21 and the n-alcohols.14 Here we found that it gave rates practially indistinguishable from classical nucleation theory. The reason is simple to understand. To fit the measured surface tensions of these polar fluids a rather large value of l is required. This means that the effective potential is short ranged and curvature corrections to the droplet free energy are greatly reduced; the classical theory of planar interfaces works in this case. Attempts to fit molecules that have intemolecular potentials that depend strongly on orientation ~such as water and the lower alcohols! by spherical potentials are thus misguided; they end up using an artificially short-ranged spherical potential to replace the actual orientational ordering at the liquid surface. Experiments by Wright et al.22 on still more strongly polar liquids such as acetonitrile have revealed very large deviations from classical nucleation theory. Although these experiments do not measure nucleation rates directly, the critical supersaturations are all much higher than those predicted by classical theory. Our first attempt to incorporate dipole–dipole interactions8 used a Stockmayer fluid model



~Lennard-Jones plus point dipole interactions!. The effects of the dipole moment on the nucleation were relatively modest in that case, in part because of the up-down symmetry of this simple model ~reversing the direction of the dipole moment does not change the free energy!. We are now23 using distributed charge models in which fused spheres ~of different sizes! carry positive and negative charges. The combination of dipole moments and shape asymmetries can have a much larger effect on cluster free energies. An extension in this direction will be needed to treat water and other polar fluids in a more realistic fashion. ACKNOWLEDGMENTS

This work was supported by the National Science Foundation ~Grant Nos. CHE 9123172 and 9422999!, by the Petroleum Research Fund of the American Chemical Society ~Grant No. 26950-AC9!, and by a Richter grant for undergraduate research ~to R. N.! from the University of Chicago. Support to V. T. from the Facultad de Quimica and DGAPA at UNAM is also gratefully acknowledged. We thank J. Schmitt, J. Katz, and R. Strey for helpful discussion and for providing experimental data. 1

2

M. Rao, B. J. Berne, and M. H. Kalos, J. Chem. Phys. 68, 1325 ~1978!; G. H. Peters and J. Eggebrecht, J. Phys. Chem. 95, 909 ~1991!. J. K. Lee, J. A. Barker, and F. F. Abraham, J. Chem. Phys. 58, 3166 ~1973!.

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C. Weakliem and H. Reiss, J. Chem. Phys. 99, 5374 ~1993!. D. Oxtoby and R. Evans, J. Chem. Phys. 89, 7521 ~1988!. 5 X. C. Zeng and D. W. Oxtoby, J. Chem. Phys. 94, 4472 ~1991!. 6 D. W. Oxtoby, in Fundamentals of Inhomogeneous Fluids, edited by D. Henderson ~Marcel-Dekker, New York, 1992!, p. 407. 7 D. W. Oxtoby, J. Phys. Condensed Matt. 4, 7627 ~1992!. 8 V. Talanquer and D. W. Oxtoby, J. Chem. Phys. 99, 4670 ~1993!. 9 For a general reference, see J. P. Hirth and G. M. Pound, Condensation and Evaporation: Nucleation and Growth Kinetics ~Pergamon, London, 1963!. 10 A. Dillmann and G. E. A. Meier, Chem. Phys. Lett. 160, 71 ~1989!; J. Chem. Phys. 94, 3872 ~1991!. See, however, I. J. Ford, A. Laaksonen, and M. Kulmala, ibid. 99, 764 ~1993!. 11 G. W. Adams, J. L. Schmitt, and R. A. Zalabsky, J. Chem. Phys. 81, 5074 ~1984!. 12 P. E. Wagner and R. Strey, J. Phys. Chem. 80, 5266 ~1984!. 13 C. Hung, M. J. Krasnopoler, and J. L. Katz, J. Chem. Phys. 90, 1856 ~1989!. 14 R. Strey, P. E. Wagner, and T. Schmeling, J. Chem. Phys. 84, 2325 ~1986!. 15 H. R. Kobraei and B. R. Anderson, J. Chem. Phys. 95, 8398 ~1991!. 16 See, J. P. Hansen and I. R. McDonald, Theory of Simple Liquids ~Academic, London, 1986!. 17 V. Talanquer and D. W. Oxtoby, J. Chem. Phys. 100, 5190 ~1994!. 18 D. Oxtoby and D. Kashchiev, J. Chem. Phys. 100, 7667 ~1994!. 19 J. L. Schmitt, R. A. Zalabsky, and G. W. Adams, J. Chem. Phys. 79, 4496 ~1983!. 20 R. C. Miller, R. J. Anderson, J. L. Kassner, Jr., and D. E. Hagen, J. Chem. Phys. 78, 3204 ~1983!. 21 Y. Viisanen, R. Strey, and H. Reiss, J. Chem. Phys. 99, 4680 ~1993!. 22 D. Wright, R. Caldwell, and M. S. El-Shall, Chem. Phys. Lett. 176, 46 ~1991!; D. Wright, R. Caldwell, C. Moxely, and M. S. El-Shall, J. Chem. Phys. 98, 3356 ~1993!. 23 V. Talanquer and D. W. Oxtoby, J. Chem. Phys. ~to be published!. 3 4
