J. Phys. Chem. 1995,99, 12335-12340
12335
Observation of Droplet Growth and Coalescence in Phase-Separating Lennard-Jones Fluids Giinther H. Peters*??and John Eggebrechts H.C. Brsted Institute, Chemistry Department 3, 2100 Copenhagen 0,Denmark, and Illinois Mathematics and Science Academy, Arora, Illinois 60601 Received: August 17, 1994; In Final Form: April 24, 1 9 9 9
Molecular dynamics computer simulations of the vaporAiquid phase transition of a Lennard-Jones fluid are presented. The simulations reveal that, in the late stage of the transition, clusters grow linearly with time. This linear growth regime can be described by an extension of the Lifshitz-Slyozov theory with an asymptotic droplet growth power law t 1 I 3 . This growth mode is interrupted by the coalescence of clusters, whose motions have Brownian character. The Brownian nature of the cluster trajectories is a consequence of internal degrees of freedom rather than of environmental noise. The diffusion coefficients of these clusters, as a function of cluster size, are found to be well represented by the kinetic theory of the ideal gas.
1. Introduction Aggregation phenomena are observed in a wide range of physical systems including crystallization; precipitation from solutions of colloids and macromolecules; aerosol coagulation; crack formation; sintering, segregation at grain boundaries; as well as boiling, melting, and condensation. The time evolution of these systems toward their equilibrium state subsequent to a temperature or field-induced quench into the phase coexistence region has been of great interest experimentally and theoretically.'-s Population balances have been widely used to describe the dynamic behavior of aerodisperse systems, where the aerosols are formed by dispersion or condensation. They have become tremendously important in technology, since they are encountered, e.g., by grinding solids, spray drying of viscous liquids and suspensions, using fluidized catalysts, or manufacturing powders. Here, small liquid or solid particles are transferred to the gas phase by air currents or vibrations. Condensation aerosols are formed when supersaturated vapors condense or as a result of gas-phase reactions leading to the formation of nonvolatile products. These aerosols play an important role in nature. For instance, the water cycle involves bulk condensation of vapor, to form clouds, and subsequent precipitation from them. Scientists in cloud physics have attempted to model rain drop formation, but the problem is complicated by the interaction of many processes. A realistic cloud model must consider the cloud dynamics. Environmental parameters such as gravitational and electrical fields, hydrodynamic drag, and fluctuating forces resulting from the thermal motion of the ambient fluid may play a substantial role in the formation of precipitation. As a result of coalescence of particles induced by their relative motion, the particle size distribution changes continually. Each of these processes has been investigated in varying detail, and depending on the concentration of the dispersed particles, different theories are involved in describing the kinetics of the system, ranging from the free molecule regime (Chapman-Enskog theory) to the continuum limit (Brownian coagulation). The theory of coagulation induced by Brownian motion has been developed by Smoluchow~ki,~ leading to an integrodifferential equation for the continuous model of coalescence and condensational growth (absorption of monomers). Many collision frequency factors
' H. C. 0rsted Institute.
Illinois Mathematics and Science Academy. @Abstractpublished in Advance ACS Absrracrs, July 1, 1995.
0022-3654I95l2099-12335$09.00/0
appear in the literature describing coagulation, e.g., under the influence of gravitational force, turbulent fluid field, charged aerosols, condensation, etc., and in general, these factors are complicated and the population balance can only be solved n ~ m e r i c a l l y . ' ~ -Using '~ this approach, the effect of monomer depletion by scavenging particles on the nucleation rate has been considered either by negle~ting'~.''or cluster scavenging. In the late stage of the phase separation, droplets of the minority phase in a supersaturated solution grow and shrink by Ostwald ripening or coarsening.2.20The dispersity of the second phase results in a large surface area, and the system has not reached its thermodynamic equilibrium. The system naturally escapes from this metastable state by minimizing the contribution of the surface free energy to the total free energy. Droplets with a lower interfacial curvature grow at the expense of domains with a higher curvature through a mass diffusion process. A mathematical formulation of the Ostwald ripening process was developed for crystallization from the melt by Lifshitz and Slyozov21 and for the aging of precipitates in electrolyte solutions by Wagner.22 Both groups associated in predicting a temporal power law of the droplet growth and a scaling behavior of the droplet size distribution in the long time limit. The mean field nature of their theory has prevented them from including a dependence on the volume fraction of droplets in their theories. Continued efforts have been made to deal with these problem^.*^-^^ Recently several a ~ t h o r s ~have ~-~~ developed statistical theories beyond the mean field LifshitzSlyozov theory, where droplet interactions in terms of medium polarization and direct correlation between any pairs of droplets are included. The properties of small droplets at thermodynamically stable states have recently been examined?O where thermodynamically stable droplets confined to a finite volume were considered. From the classical model, only the assumption that equilibrium thermodynamics can be applied to a stable, isolated droplet in terms of the conditions of mechanical, thermal, and chemical equilibrium was retained. This was accomplished through a solution of the Yvon-Bom-Green e q ~ a t i o n ~ 'for . ~ *the structure of an isolated droplet, subject to the constraint of mass conservation, with boundary values consistent with the conditions of thermodynamic stability. This approach was shown to very accurately predict the equilibrium structure of droplets of Lennard-Jones atoms by comparison with molecular dynamics computer ~ i m u l a t i o n .It~ ~was found that the representation of 0 1995 American Chemical Society
Peters and Eggebrecht
12336 J. Phys. Chem., Vol. 99, No. 32, 1995 the pressure within the drop as that of a homogeneous liquid at the density of the center of the drop is significantly in error for drops of less than 3000 atoms. This identifies the limit of applicability of bulk phase thermodynamics for this particular model. In particular, the droplet size dependence of the surface tension was evaluated for clusters of Lennard-Jones atoms at equilibrium states. It was shown that the identification of a critical droplet as a rate-limiting transition state is sensitive to the manner in which equilibrium thermodynamics is superimposed on the continuum of nonequilibrium droplet sizes. In this study, we have applied molecular dynamics computer simulations which provide a tool for the direct observation of cluster growth during the quench from a thermodynamically stable state into the two-phase region to investigate the dynamics of the vaporfliquid phase transitions of a Lennard-Jones fluid. The system may be interpreted as a condensation aerosol on a nanoscopic level. The computer simulations of condensation processes are an idealized case of cloud formation. Environmental parameters are excluded, and small clusters grow by condensation and binary coalescence. Our presentation is organized as follows. In the next section, the potential model and parameters used in the simulations are briefly described. In section 3, our results are presented, and finally, in section 4 the main findings of our investigation are summarized. Throughout the presentation, the numerical values are given in reduced units. Energy and length parameters are reduced by the Lennard-Jones parameters.
0
640+ 1 120
0
‘1
(l l.OO)
M
170t260
90+360
0
0
130
260 time/psec
520
390
Figure 1. Time evolution of the cluster size distribution of a N = 2744 Lennard-Jones fluid. Each set of numbers displayed at the trajectories refers to the cluster size (top) and time (bottom) involved in a binary coalescence event: Q = N/Vu3 = 0.05 and r* = kT/e =
0.76.
”1
640+1310
0
(5.25)
rlc
300+990
2 2. Computer Simulations
Molecular dynamics computer simulation of a vapor condensation process is free of uncertainties associated with impurities, surfaces, and the translation of an experimental observable in terms of the assumed rate-limiting mechanism. In a molecular dynamics simulation the time evolution of the process is completely determined by the integration of Newton’s equations. As yet, molecular dynamics methods are restricted to ensembles containing a fixed number of molecules and to systems which are too small to examine the effects of weak extemal fields, such as gravity. Therefore, in a simulated condensation the final equilibrium state consists of a single liquid droplet immersed in a coexisting vapor. A nonequilibrium path to the final stable droplet can be observed in complete detail with no approximation beyond the inaccuracies associated with the numerical integration. Results from these calculations are described in this section. The technique which we employ to study nucleation dynamics is to temperature quench a homogeneous fluid composed of Lennard-Jones atoms at a thermodynamically stable state to a point within the metastable region close to the spinodal curve. This final state point is defined by the reduced density, Q = NfVa3 = 0.05, and T* = kTk = 0.76, w h e r e E and 0 are t h e usual Lennard-Jones parameters. The potential is truncated at 3.5 0. Periodic images are used to mimic the thermodynamic limit. For comparison, the critical constants of the LennardJones fluid calculated from integral equations are Tz 1.3 and ec 3: 0.3.34 The triple point is in the vicinity of T: = 0.67 and et = 0.86 though considerable uncertainty remains regarding these values.35 The coexisting @v,-, @L.-, and spinodal densities at T* = 0.76 are 0.0060, 0.79, 0.072, and 0.62, re~pectively.~~ These are obtained from an equation of state fit to simulation data.37 The metastable state described here corresponds to a supersaturation, j?fpv,-,of 5.2. Earlier calculations of this type were described in ref 30 for two systems containing 1000 atoms in the central simulation
35>+820
“i 2 0
200
400
600
time/psec
Figure 2. Time evolution of the cluster size distribution of a N = 5324 Lennard-Jones fluid. Numbers displayed at the trajectories are explained in Figure 1: Q = N/Vu3 0.05 and F = kT/c = 0.76.
cell. Here we examine systems composed of 2744,5324, and 10648 atoms. Each system has been quenched to the same state point. The equations of motion are integrated with a fifth-order predictor-corrector algorithm. Velocities were scaled to maintain the desired global temperature. The stable state was equilibrated thoroughly preceding the quench, which was accomplished over a single time step. Following the quench the evolution of the cluster size distribution, NC1(r),was obtained. Clusters are defined in terms of connectivity matrices C,(t) =
1 if rij 5 d 0 otherwise
Each of these matrices was then reduced to a set of rows such that the intersection of any pair of rows in a reduced matrix is empty. Each of the rows in the reduced matrix represents a cluster disjoint from any other. The parameter d was taken to be 1.2 atomic diameters. A discussion of the sensitivity to this parameter is given in ref 30. Representative cluster size distributions are shown in Figures 1-3 for each system.
Phase-Separating Lennard-Jones Fluids
J. Phys. Chem., Vol. 99, No. 32, 1995 12337
0
0
%l 0,
.-
Fi
t 0
0
2.0
n
\
N ul"
. -
e -
-E
1.5
$ n
1.0
0
I\
1 :
0.5
0.0 90
0
370
185
555
time/psec
Figure 3. Time evolution of the cluster size distribution of a N = 10 648 Lennard-Jones fluid: 0 = N t V d = 0.05 and r" = k T k = 0.76.
3. Results and Discussion
Within the first 100 ps after the quench, several clusters form. The size of these clusters is comparable to the classical critical cluster size, which for the simulated state contains approximately 15 atoms8 These clusters are dispersed throughout the system, and each is contained within a region of lower density or depletion zone. The number of atoms in these clusters then increases linearly in time as they scavenge material from the surrounding depletion zone. At a fixed time at which several large clusters are present, this linear growth rate is found to be proportional to cluster size as Large clusters grow more rapidly than small clusters, and the growth mechanism is a ballistic capture which is proportional to surface area. This linear growth mode following a short-lived induction period has also been observed in lattice dynamics simulation^.^^-^^ Interrupting these linear expansions are coalescence events in which smaller clusters are absorbed by larger ones. In Figures 1 and 2 the duration of time between first encounter and coalescence is shown in parentheses for several binary cluster encounters and the number of atoms in each of these clusters is indicated. Upon encounter we have observed in several instances an oscillatory motion of the center of mass separations of the interacting clusters. While we note a possible trend toward increasing duration between encounter and coalescence as cluster size increases, the completion time required for this absorption in all instances is very short. As the vapor phase is depleted by the growing clusters, the system continuously enters a regime where the growth process is governed by the diffusion and coalescence of large clusters. The complex many-body dynamics which describe the condensation events are intractable. However, the identification of distinct kinetic processes allow simplification into two continuum approaches which we now consider. Both growth modes, cluster coalescence and linear growth, can be understood in terms of simple continuum models. In the following section, expressions are derived for the characteristic times of collisional encounters of clusters and relaxation of the collision products to a local equilibrium. Analysis of the trajectories of large clusters shows that the motion has distinctive Brownian character. The density of the fluid in which these large clusters are immersed is very nearly that of the macroscopic coexisting vapor density. Therefore, the mean free path of the background fluid atoms is large relative to the cluster radius, and the trajectories would, in the usual interpretation of the Knudsen number, be expected to be much
N;F.
690
1290
740
1890
2490
N,, Figure 4. Diffusion coefficient of Lennard-Jones clusters determined from the simulations as a function of cluster size (0). (-) is the diffusion coefficient calculated from the kinetic theory of transport phenomena of dilute gases, and (- - -) is the best fit to the form D,,= aNb9I8.
more linear. In this case the Brownian character is primarily a consequence of the dynamic processes of the clusters and not due to the background noise. A cluster, as defined above, is a connected set of atoms. The membership of this set is constantly changed as monomers and larger clusters are desorbed and then reabsorbed in a diffuse zone at the cluster surface. The velocity of the cluster center of mass is, therefore, a discontinuous function of time, which is the signature of Brownian motion. In addition, capture of monomers and clusters from the background gas occurs. If only a monomer deabsorptiodreabsorption process is considered, Binder and S t a ~ f f e r "showed ~ that the cluster diffusion coefficient should decay with increasing size as This monomer exchange between the cluster and the depletion zone surrounding the cluster is also present in our simulations and for large drops, as was assumed by Binder and Stauffer, occurs with a rate proportional to the cluster surface area.@ As will be seen in the following section the size dependence of cluster mobility is critical to the time dependence of the cluster growth rate, as predicted by scaling arguments. The size dependence of the cluster mobility can be determined directly from simulation from the slope of the mean-squared di~placemenl?~
c3.
13(t+At)
Dc, =
(
- 7(t)I2
At
(3)
for large At, where 7(r) is the center-of-mass position of the cluster at time t . In applying this expression to the evaluation of the cluster diffusion coefficient, we must recognize that since cluster size evolves with time, the size of the cluster considered must be identified with the mean size over the period At, during which linear growth occurs, and that the environment also varies with time. Therefore, the diffusion process is not stationary. However, we apply this expression only to large droplets, for which changes in size over the period of linear growth are small fractions of the total size (approximately 5%), and point out that the environment of these large droplets is nearly stationary in that the densities of the depletion zones which contain them only change slowly. Since the environment for smaller clusters changes more rapidly than that for bigger clusters, which are surrounded by larger depletion zones, the diffusion coefficients for NCl 5 200 are less accurate. The dependence of the cluster
Peters and Eggebrecht
12338 J. Phys. Chem., Vol. 99, No. 32, 1995 diffusion coefficients on size is shown in Figure 4 to closely follow the free molecule result
Re is the equimolecular dividing surface$6 m is the mass of a single atom, ji? = l/k& where kB is the Boltzmann constant, T is the absolute temperature, and eclis the density of clusters in the simulation cell. The second approximate expression follows from the assumption that clusters are dilute and that RJo 1. The asymptotic decay in eq 5 is as N-7/6. In Figure 4, eq 5 (-) has been plotted but cannot be discemed from eq 4 on this scale. Also shown in Figure 4 is the best fit to a form D,I = aN-" (- - -). For the latter we find Y = 9/~. However, due to the limited data we cannat distinguish this exponent from that given by eq 5. The constraint of mass conservation applied to a portion of the total volume containing a single cluster of mean radius, (R:(t)), can be used to approximately define the radius of this mean subvolume, ( R L ( ~ ) ) , ~ O . ~
Equation 6 agrees well with the solutions of the Yvon-BornGreen equation, even for very small droplek4 Since Re locates the minimum free energy of formation in a closed system, eq 6 constitutes a statement of the succession of mean subvolumes through which the system passes during the late stage of the transition. Clusters occupy a local stable subvolume and can only escape from the quasi-equilibrium state by coalescence. Here, we consider a model which neglects cluster size dispersion and long-ranged cluster-cluster interactions to derive an expression for the velocity at which the mean cluster radius descends through this succession of subvolumes. The time required to achieve a cluster of a specified size can then be obtained in closed form. Consider a set of 2" clusters, each of which has the mean radius (Re(to))and occupies a subvolume whose radial dimension is (RL(t0)). The first passage time through a separation of ~ ( R L- Re) of two of these clusters can be computed with diffusion coefficients given by eq 5 . At a time to Qtl, 2V2 collisions occur and the number of clusters is reduced to 2"-'. At the kth iteration of this process the first passage time is given by
where numerical constants have been evaluated. We note that the first passage time in eq 10 scales like (cZustersize)1'/6 and that the dependence on the fluid density occurs in Qt like 1/(Q - QV,,.)~/~. A critical evaluation of eqs 7-10 by comparison with simulation is not possible due to the limited size of systems which can be studied at the present time. The linear growth regime, as we now show, can be understood on the basis of a simple continuum model developed thirty years ago by Lifshitz and Slyozov.21 Their analysis was performed for crystallization from a melt and must be modified slightly to be applied to vapor condensation. The development of their approach has been reviewed many times, and we refer the reader to one of these.20 The statement of mass conservation on a spherical volume of radius RLcontaining N atoms can be written in terms of the local number density, e(r, t ) .
The time derivative is taken, and the density flux is replaced with the approximate diffusion equation, with monomer diffusion coefficient, D,
in which the droplet boundary is a density discontinuity. Assuming an ideal vapor in a linearized form of the GibbsThomson equation, one obtains
where 0 = g(r) is the metastable fluid density. Here (Re(t)) is the mean cluster radius. The analytic solution of this differential equation was obtained by Lifshitz and Slyozov in the long time limit. The result is that the mean cluster size is a linear function of time
+
where t* = t(E/ma2)112, T* = e/& and s = [ eL.JeV,m
-
se- 1
'1
where S, = @@v,mis a measure of supersaturation. The total time required to advance this process to a mean cluster size of Re.AT) containing 2" atoms is found to be z=
Csti i
(9)
Although this solution of eq 13 has been considered by the linearization of the Gibbs-Thomson equation for the effect of curvature on vapor density used in eq 13 is unusual and requires further comments. As always the chemical potentials of the liquid, L, and vapor, V, phases are integrated along an expanding radius of curvature (all quantities are reduced by the appropriate Lennard-Jones length (a)and energy ( E ) parameters)
Assuming a liquid density which is independent of curvature and an ideal vapor density, this becomes
where p ~ ( 0is) the normal component of the pressure tensor at the center of the liquid droplet. We replace the Laplace
J. Phys. Chem., Vol. 99, No. 32, 1995 12339
Phase-Separating Lennard-Jones Fluids
\ \
\
..
A factor of 2 has been introduced, since both clusters may translate. If it is assumed that the clusters are separated by a distance comparable to twice their radii, then
Since droplet volume is conserved in a coalescence event, the change in cluster radius of the coalescence product is proportional to the sizes of the original droplets, and if it is assumed that the local equilibrium state negligibly affects the partitioning between the liquid and vapor phase, then
--6Re
0
0.
I
I
1
1
dr
or letting Re(t
Re dr
+ df) = Re(t) i- dRe,
Substituting from eq 22 gives for small dt
Subsequent concern regarding the neglect of curvature on the surface tension of a molecularly small droplet is avoided. We now expand the In term in a Taylor series, retain the first two terms, and obtain
which rearranges to the familiar form on assumption of an ideal vapor
The factor @v*-in the numerator of eq 19 is absent in the usual linearization given by eq 20 and, as shown in Figure 5 , considerably improves the agreement with the vapor densities obtained from the integral equation analysis of ref 30.
The Lifshitz-Slyozov theory, eq 14, can be tested by comparison with the simulation results of the preceding section. The mean growth rate in the linear regime, taken over all system sizes, is found to be 0.30 f 0.11, The growth rate predicted by eq 14 for the state point of the simulations is 0.28. This is remarkable agreement for an asymptotic result from a continuum theory incorporating all of the approximation leading to eq 14. However, altemative routes also lead to a prediction of linear growth of cluster size, Ncl.4i.48*53 First we consider an approach based on a simple argument for the scaling of growth dominated by coalescence. Consider two clusters of equivalent size, Re, separated by a distance ~ ( R L Re). The expectation of the time to coalescence of these clusters, dr, which we will refer to as the first passage time in later considerations, is simply expressed from the usual definition of the diffusion coefficient, Dc1.45
Assuming that the cluster diffusion coefficient scales with size like DCl= DIR; 43 and integrating yields the following expression. Ry2’(r
+ dr) - R y 2 ) ( r )= Z(Y + 2)Ddr
(26)
A decay of the cluster mobility like l/Re leads to linear growth in Ncl. However, Figure 4 indicates a much stronger decay, at least for larger clusters where the quality of our estimates of the cluster diffusion coefficient is good. However, for small droplets, where the surface tension driven mechanism contained in the theory of Lifshitz and Slyozov is incompatible with our earlier observation of very strong dependence of surface tension on cluster size, we cannot reject the possibility that the important mechanism is that proposed by Binder and S t a ~ f f e r .The ~ ~ free energy density expansion developed by Langer et a1.50-53also leads to a linear growth rate when hydrodynamics effects are taken into a c c o ~ n t . ~However, ~ . ~ ~ this gradient expansion of the free energy is valid only near the critical point, where departures from linear growth are encountered in light-scattering e~periments.5~,~’ 4. Conclusions
Cluster growth and coalescence have been investigated on the basis of a self-consistent evaluation of the properties of small droplets and direct observations of the transition intermediates during molecular dynamics computer simulations. Each of the simulated systems displays, independent of system size, the same characteristic time evolution. During a very short induction period small clusters are formed throughout the system. This is followed by a linear growth in time for these cluster volumes as they scavenge the surrounding fluid. These linear growth processes are interrupted by coalescence events in which smaller clusters are absorbed by larger ones. The time evolution of the growth mode is found to be qualitatively described by a combination of a diffusion equation model given by Lifshitz and Slyozov2’ and a treatment of the collisional encounters of large clusters as Brownian motion. The Brownian .nature of
12340 J. Phys. Chem., Vol. 99, No. 32, 1995
the cluster trajectories is shown to be a consequence of intemal degrees of freedom rather than of environmental noise. The diffusion coefficients of these clusters, as a function of cluster size, are found to be well represented by the kinetic theory of ideal gases. It is then possible to construct a simple model of the growth process in which an array of clusters of the mean cluster size, each confined to a fixed subvolume, advance through a sequence of coalescence events. In the time interval between these events the clusters undergo linear growth from a depleting vapor phase. As the vapor phase is depleted by the growing clusters, the system continuously enters a regime where the growth process is govemed by the diffusion and coalescence of large clusters. In this late stage of the growth, the inverse collision frequency for these droplets is much larger than the relaxation time of the vapor phase surrounding these drops, and the transition can be treated as a cascade through finite subvolumes defined by local minima in a free energy surface. Such a cascade process can be very simply modeled by initiating the process with 2" clusters of size proportional to (&(0)3) whose centers of mass are separated by a distance (RdO)). Coalescence can be treated by binary collision, and the time required to achieve a particular cluster size can be estimated. Such an approach is clearly only a first step in the application of the vast literature on aggregation processes which can be brought to bear on the problem of cluster growth during vaporlliquid phase transitions, as well as transitions of a more general nature.
References and Notes (1) Bray, A. J. Phase Transitions and Relaxation in Systems with Competing Energy Scales; NATO Advanced Study Institute Series: Geilo, Norway, 1993. (2) Gunton, J. D.; San Miguel, M.; Sahni, P. S. In Phase transitions and Critical Phenomena; Domb, C., Lebowitz, J. L., Eds.; Academic Press: New York, 1983. (3) Morgan, N. Y.; Seul, M. J. Phys. 1995, 99, 2088. (4) Binder, K. Physica 1986, 140A, 35. ( 5 ) Gibbs, J. W. The Scient& Papers of J. W.Gibbs; Dover: New York, 1961. (6) Becker, R.; Diiring, W. Ann. Phys. (Leipzig) 1935, 24, 719. (7) Frenkel, J. Kinetic Theory of Liquids; Dover: New York, 1946. (8) Abraham, F. F. Homogeneous nucleation theory; Academic Press Inc.: New York, 1974. (9) Smoluchowski, M. V. Z. Phys. Chem. 1918, 92, 129. (10) Hidy, G. M.; Brock, J. R. International Rev. Aer. Physical Chemistry; Pergamon Press: New York, 1970 Vols. 1-3. (1 1) Gelbard, F.; Seinfeld, J. H. J . Colloid Interface 1979, 68, 363. (12) Hidy, G.M.; Brock, J. R. J. Colloid Sci. 1965, 20, 477. (13) Hidy, G. M. J . Colloid Sci. 1965, 20, 123. (14) Pich, J.; Friedlander, S. K.; Lai, F. S. Aerosol Sci. 1970, I , 115. (15) Lai, F. S.; Friedlander, S. K.; Pich, J.; Pich, G. M. J. Colloid Interface Sci. 1972, 39, 395. (i6) Pesthy, A.; Flagan, R. C.; Seinfeld, J. H. Colloid Interface Sci. 1983, 92, 525. (17) Stem. J. E.: Wu. J.-J.: Flagan. R. C.: Seinfeld. J. H. J . Colloid Interface 1986, 110, 533. (18) Shi, G.; Seinfeld, J. H. J. Chem. Phys. 1990, 92, 687.
-
Peters and Eggebrecht (19) Vega, E.; Peters, L. K. Aerosol Technol. 1983, 2, 513. (20) Vorhees, P. W. J . Stat. Phys. 1985, 38, 231. (21) Lifshitz, I. M.; Slyozov, V. V. J . Phys. Chem. Solids 1961, 19, 35. (22) Wagner, C. Z. Elektrochem. 1961, 65, 581. (23) Weins, J. J.; Cahn, J. W. In Sintering and Related Phenomena; Kuczynski, G. C., Eds.; Plenum: New York, 1973. (24) Davies, C. K. L.; Nash, P.; Stevans, R. N. Acta Metall. 1980, 28, 179. (25) Marqusee, J. A.; Ross, J. J . Chem. Phys. 1984, 80, 536. (26) Marder, M. Phys. Rev. Lett. 1985, 55, 2953. (27) Marder, M. Phys. Rev. A 1987, 36, 858. (28) Tokuyama, M.; Kawasaki, K. Physica A 1984, 123, 386. (29) Tokuyama, M.; Kawasaki, K.; Enomoto, Y. Physica A 1986, 134, 323. (30) Peters, G. H.; Eggebrecht, J. J. Phys. Chem. 1991, 95, 909. (31) Yvon, J. Acta Sci. lnd. 1935, 203. (32) Green, H. S. The Molecular Theory of Fluids; Interscience: New York, 1952. (33) Thompson, S. M.; Gubbins, K. E.; Walton, J. P. R. B.; Rowlins, J. J. Chem. Phys. 1984, 81, 530. (34) Levesque, D. Physica 1966, 32, 1985. (35) Hansen, J. P.; Donald, I. R. Theory of Simple Liquids; Academic Press: New York, 1976. (36) Nicolas, J. J.; Gubbins, K. E.; Streett, W. B.; Tildesley, D. J. Mol. Phys. 1979, 37, 1429. (37) It came to our attention that an improved equation of state for Lennard-Jones fluid has been published: Johnson, J. K.; Zollweg, J. A.; Gubbins, K. E. Mol. Phys. 1993, 78, 591. At the temperature and density applied in this study, vapor-liquid equilibrium data calculated from the new equation of state are comparable with the data from ref 36. (38) Penrose, 0.;Lebowitz, J. L.; M m o , J.; Kalos, M. H.; Sur, A. J . Stat. Phys. 1978, 243. (39) Penrose, 0.; Lebowitz, J. L. Studies Mechanics; Lebowitz, J. L., Montroll, E. W., Eds.; North Holland: Amsterdam, 1979; Vol. 7. (40) Marro, J.; Bortz, A. B.; Kalos, M. H.; Lebowitz, J. L. Phys. Rev. B 1975, 12, 2000. (41) M m o , J.; Lebowitz, J. L.; Kalos, M. H. Phys. Rev. Lett. 1979, 43, 282. (42) Marro, J.; Valles, J. L. Phys. Lett. A 1983, 95, 443. (43) Binder, K.; Stauffer, D. Adv. Phys. 1976, 25, 343. (44)Peters, G. H. Ph.D. Dissertation, Iowa State University, Ames, IA, 1991. (45) Einstein, A. Investigations on the theory of the Brownian movement; Fiirth, R., Ed.; Dover Publications, Inc.: New York, 1956. (46) The equimolar surface is a particular choice of the Gibbs dividing surface, where the contribution of the interfacial layer to the actual number of molecules vanishes. (47) Desai, R. C.; Koch, S. W.; Abraham, F. F. Physica 1983, 118A, 36. (48) Furukawa, H.; Binder, K. Phys. Rev. 1982, A26, 556. (49) Siggia, E. Phys. Rev. A 1979, 20, 595. (50) Langer, J. S.; Schwartz, A. J. Phys. Rev. A 1980, 21, 948. (51) Langer, J. S.; Turski, L. A. Phys. Rev. A 1973, 8, 3230. (52) Langer, J. S.; Turski, L. A. Phys. Rev. A 1980, 22, 2189. (53) Langer, J. S.; Bar-on, M.; Miller, H. D. Phys. Rev. A 1975, 11, 417.
(54) Otha, T.; Kawasaki, K. Prog. Theor. Phys. 1977, 58, 467. (55) Ohta, T.; Jasnow, D.; Kawasaki, K. Phys. Rev. Lett. 1982,49, 1223. (56) Goldburg, W. I.; Shaw, C. H.; Huang, J. S.; Pilant, M. S. J . Chem. Phys. 1978, 68, 484. (57) Goldburg, W. I.; Schwartz, A. J.; Kim, M. W. Prog. Theor. Phys. Suppl. 1918, 64, 477. JP942 195 1
