Molecular dynamics of homogeneous nucleation in the vapor...

Molecular dynamics of homogeneous nucleation in the vapor phase. I.Lennard-Jones fluid

Kenji Yasuokaa) and Mitsuhiro Matsumotob)

Department of Applied Physics, School of Engineering, Nagoya University, Nagoya 464-01, Japan

�Received 4 April 1997; accepted 10 August 1998�

Molecular dynamics computer simulation was carried out to investigate the dynamics of vaporphase homogeneous nucleation at the triple point temperature under supersaturation ratio 6.8 for aLennard-Jones fluid. To control the system temperature, the 5000 target particles were mixed with5000 soft-core carrier gas particles. The observed nucleation rate is seven orders of magnitude largerthan prediction of a classical nucleation theory. The kinetically defined critical nucleus size, atwhich the growth and decay rates are balanced, is 30–40, as large as the thermodynamically definedvalue of 25.4 estimated with the classical theory. From the cluster size distribution in the steadystate region, the free energy of cluster formation is estimated, which diminishes the differencebetween the theoretical prediction and the simulational result concerning the nucleation rate.© 1998 American Institute of Physics. �S0021-9606�98�50643-5�

I. INTRODUCTION

Nucleation processes are widely observed and have im-portant roles in many fields of science and technology. Het-erogeneous nucleation, in which already-existent nuclei startthe process, is much more frequently seen, but understandinghomogeneous nucleation is important as the fundamentals ofphase change.

There is a long history of experimental and theoreticalstudies of homogeneous nucleation processes.1,5 However,partly due to various experimental difficulties, it still requiressome caution to quantitatively compare experimental resultsand theoretical predictions. In this paper, we examine thehomogeneous nucleation from supersaturated vapor to liquiddroplets using molecular simulation technique. The mainpurpose of this study is to evaluate the nucleation rate, makecomparison with theoretical prediction, and investigate thedynamics of fluid phase change at a molecular level.

Traditional experiments of vapor homogeneous nucle-ation have concentrated on measuring the critical supersatu-ration ratio,1 which is an observation limit of the nucleationprocess. However, to evaluate the nucleation rate, it is nec-essary to measure the density of nuclei generated per unittime, which was extremely difficult. Over the past few de-cades, a considerable number of studies have been made onthe nucleation rate because developments of experimentaltechniques made possible the observation of small clusters.The size of the critical nucleus can be estimated from theslope of the isothermal nucleation rate versus the supersatu-ration ratio, independently of the model calculation.2,3 Thismethod, the nucleation theorem, was generalized later.4

Starting from the work of Volmer, Becker and Doring,and Zeldovich, the standard classical nucleation theory5 is

based on equilibrium statistical mechanics and unimolecularreaction kinetics. Let us briefly describe the theory here. Thecluster formation free energy �G is expressed as a sum ofbulk and surface contributions as

�G��A��gV . �1�

The first term is the surface contribution, where � is thesurface tension �surface excess free energy� and A the sur-face area of the cluster, while the second term is the freeenergy difference between bulk phases, where �g is the dif-ference per unit volume and V volume of the cluster. Whenthe cluster is assumed to be of a spherical shape, A�4�r2

and V�(4/3)�r3, where r is the radius of the cluster. Wealso make an assumption of the vapor as an ideal gas, whichleads to the expression of �g�� lkBT ln(�/�e), where �,� l , and �e are the number density of supersaturated vapor,bulk liquid, and the saturated vapor, respectively. Here, kB isthe Boltzmann constant, and T is the system temperature.

Apparently, �G is dependent on r, or the cluster size�number of molecules consisting of the cluster� n defined asn�� lV . �G takes a maximum value �G* at n�n*, whichcorresponds to the thermodynamically defined criticalnucleus. From the above expression of �G , we obtain

n*�32�

3

�3

�kBT ln S �3�� l�2 �2�

and

�G*�16�

3

�3

�kBT� l ln S �2 , �3�

where S��/�e is the supersaturation ratio. The nucleationrate Jcl , defined as the number of nucleus of size n* per unitvolume per unit time, is expressed5 in terms of �G* as

Jcl�Z�n*�c�n*�. �4�

The Zeldovich factor Z is given as

a�Current address: Department of Mechanical Engineering, Keio University,Yokohama 223-8522, Japan.

b�Current address: Department of Engineering Physics and Mechanics,Kyoto University, Kyoto 606-8501, Japan.

JOURNAL OF CHEMICAL PHYSICS VOLUME 109, NUMBER 19 15 NOVEMBER 1998

84510021-9606/98/109(19)/8451/12/$15.00 © 1998 American Institute of Physics

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Z�� 1

2�kBT� 2�G�n �

n2 �n�n*

� 1/2

�5�

and the forward rate as

�n �� kBT

2�m�A�n �, �6�

where m is the molecular mass. The concentration c(n) ofclusters of size n is related to the free energy difference as

c�n �� exp��G�n �

kBT � . �7�

Combining Eqs. �4�–�7�, we obtain the usual expression

Jcl�� l�1� 2�

�m�2 exp� �

�G*

kBT � . �8�

This theory assumes the applicability of equilibriumthermodynamics to clusters, and also uses the steady stateapproximation. Thus, in the case of very large supersatura-tion ratio where the critical nucleus is very small and thenucleation rate is very large, the appropriateness of thistheory can be questionable. For example, surface tension ofsmall nuclei could be very different from those of bulkphases, or the temperature of clusters could be varying withtime due to the latent heat release during the nucleation.

To investigate these points, molecular dynamics �MD�and Monte Carlo simulations have been applied recently.Thompson et al.6 studied the structural and thermodynamicproperties of isolated small droplets with an MD technique.More recently, Zhukhovitskii7 also adopted an MD techniqueto investigate the growth-decay behavior of single cluster insupersaturated vapor and found that the size of the criticalnucleus is larger than that predicted by the classical nucle-ation theory. However, this result is highly questionable be-cause experimental values of argon surface tension, satura-tion vapor pressure, and liquid density are used in this paperto estimate the size of critical nucleus with classical nucle-ation theory. Peters and Eggebrecht adopted a straightfor-ward simulation with simple velocity scaling.8 They did nothave the detail analysis of the nucleation rate and the size ofcritical nucleus except the analysis in growth stage.

In this study, we use a fairly large system consisting of5000 vapor molecules with carrier gas, and carried out anMD simulation to directly observe the nucleation underrather large supersaturation ratio. From this simulation, weobserve various dynamic behavior of cluster formation, fromwhich the nucleation rate is clearly defined and evaluated.

II. SIMULATION METHOD

If we use a microcanonical ensemble MD method, thesystem temperature will increase during the nucleation pro-cess due to the condensation heat, which prevents the systemfrom further condensation. A simple remedy for this is tocontrol the system temperature either by simple velocityscaling or by using more elegant thermostat techniques. Withthese techniques, however, the extra heat is removed frominside the condensed clusters as well as from their surfaces,which is very unnatural.

To observe a nucleation process in a more natural way,we adopt a mixture system of target particles and carrier gasparticles in a manner identical with that of manyexperiments.1 In the simulation, only the carrier gas is con-nected to a hypothetical heat bath using the Nose–Hoovermethod9 so that we control the system temperature, avoidingunnatural energy exchanges.

We use a rather large system �5000 target particles�5000 carrier gas particles� to obtain a good statistics. Theperiodic boundary conditions are assumed for all three di-mensions, and the leap-frog algorithm is adopted for numeri-cal integration of the classical equations of motion.

The target–target interaction potential U tt(r) is aLennard-Jones �LJ� type,

U tt�r ��4��/r �12��/r �6� , �9�

where r is the distance between centers of mass of mol-ecules. The potential between target–carrier or carrier–carrier is a soft-core type,

Ucc�r ��U tc�r ��4��/r �12. �10�

The potential parameters � and � and the particle mass m areset to be equal for target and carrier particles.

In the following part, we choose �, �, and m as the unitsof length, energy, and mass, respectively. Thus, the unit timeis ��m�2/�; in the case of argon, m is 6.63�10�26 kg, �3.405 Å, � 119.8 K, leading to �2.15 ps.

The cell size is 60�60�60 in the dimensionless unit.The cutoff radius is 4.5, and no tail correction is made forenergy calculation because of the spatial inhomogeneity.

The phase diagram of the Lennard-Jones system and thesimulation condition are shown in Fig. 1. Using the equationof state for Lennard-Jones system,10 the binodal �liquid–vapor coexistence� line is estimated with Maxwell’s theoremof equal areas, and the spinodal line �boundary of thermody-namically unstable region� using the condition that partialderivative of the pressure with the volume becomes zero. Inthe MD simulation, the whole system was initially equili-brated at state A (T�1.25) for 60 , and was quenched tostate B �T�0.67, near the triple point temperature� by asimple velocity scaling.

FIG. 1. Phase diagram of Lennard-Jones fluid �Ref. 10�. The solid linerepresents the binodal �liquid-vapor coexistence� line, and the broken lineshows the spinodal line �boundary of thermodynamically unstable region�.The critical temperature is Tc�1.35 and the triple point Tt�0.68�0.02�Ref. 11�. In the simulation, the system was equilibrated at state A, and wasquenched to state B.

8452 J. Chem. Phys., Vol. 109, No. 19, 15 November 1998 K. Yasuoka and M. Matsumoto


Using the density of saturated vapor �1.61�10�3 in di-mensionless unit12�, the supersaturation ratio at state A isevaluated to be �14.4. Then, the size of the thermodynami-cally defined critical nucleus n* is estimated with the clas-sical nucleation theory �Eq. �2��. Using the surface tension offlat surface12 ��1.03, n*�9.6 is obtained; the effect of sur-face curvature on the surface tension will be discussed later.However, we need to reestimate the supersaturation ratio af-ter the system became in the steady state region because themonomer density decreases with time.

The total run after the quench is 1800 �3.87 ns forargon parameters�. A large time step of 0.010 is sufficientin the earlier stage �0–450 �. As the nucleation proceeds,there appears density fluctuations and the energy conserva-tion becomes worse. Thus, after 450 , we adopt a smallertime step of 0.005 .

III. SIMULATION RESULTS

A. Thermodynamic properties

First, let us examine the thermodynamic properties ofthe whole system. Figure 2 shows the time development ofkinetic and potential energies of each species. As the con-densation heat is taken out by the carrier gas only throughthe cluster surfaces, the mean kinetic energy of target par-ticles increases with time, while that of carrier gas particlesis almost constant because of the temperature control. Thedifference increases with time and thus the way of tempera-ture control may affect the nucleation dynamics. The poten-tial energy of target particles decreases with time, which sug-gests that nucleation process takes place. The bottom figure

shows the condensation heat �Q , or the decrease of the totalenergy, which is taken through the heat bath connected to thecarrier gas. After time �600, �Q increases linearly withtime, which means that condensation takes place at almostconstant rate.

Molecular configurations are shown in Fig. 3, whereonly target particles are drawn. The start configuration �justafter the quench� seems almost uniform. At time�600, clus-ters of various sizes appear, some of which keep growing.Finally, at time�1800, five large clusters survive, the largestof which consists of about 900 molecules.

B. Cluster analyses

In this section, we inquire into cluster properties so thatwe can examine the assumptions used in the classical nucle-ation theory.

We define a ‘‘cluster’’ as a group of mutually connectedmolecules. There is some arbitrariness concerning the ‘‘con-nection.’’ We try three different definitions.

First definition of the connection is based on an instan-taneous molecular configuration; two molecules are con-nected if their center-of-mass distance r0 is less than�1.5��, the value of which roughly corresponds to the firstminimum of the radial distribution function of LJ liquid nearthe triple point.13 To see the r0 dependence of cluster prop-erties, we also use another value of r0�2.0. However, theclusters with these definitions based on an instantaneous con-figuration may include molecular pairs of mere collision.Thus, we try third definition including a time criterion inwhich the pair is connected when the r0 distance is keptcontinuously less than 1.5 for more than t0�3( ); the valueof time criterion corresponds to the time required for a mol-ecule of typical velocity to cross the boundary of r0�1.5around another molecule, rebound from it, and go away re-crossing the boundary.

In Fig. 4 we plot time development of the mean clustersize using each definition of connection. The mean clustersize based on the third definition �i.e., with time criterion� ismuch smaller than the results of other definitions; molecularmotions near cluster surface are so active that applying thetime criterion causes very large underestimation of the clus-ter size, which may overlook relevant growth-decay dynam-ics of clusters. On the other hand, the results with r0�1.5and with r0�2.0 are very similar. Therefore, we adopt thefirst definition �r0�1.5 and t0�0� in the following analyses.

Now let us look into the thermodynamic properties. InFig. 5, the average potential energy (Uclus) and the tempera-ture are plotted against the cluster size. Cluster temperature�lower figure� in general is higher than that of heat bath (T�0.67), which suggests that the latent heat released duringthe condensation of molecules is not instantly removed bythe carrier gas. The effect of carrier gas on the nucleationprocess was discussed before by Barrett et al.14 using a car-rier gas-cluster collision model. They concluded that thecluster temperature increases with the cluster size growth;however, the dependence of nucleation rate on carrier gasdensity is predicted to be rather weak. As to this point, weare carrying out a series of similar MD simulations with

FIG. 2. Time development of kinetic energies per molecule �top� and po-tential energies per molecule �middle� for each species, and condensationheat �bottom�. After time �600, the condensation heat increases linearlywith time.

8453J. Chem. Phys., Vol. 109, No. 19, 15 November 1998 K. Yasuoka and M. Matsumoto


different carrier gas density, the results of which will bepublished separately.

The potential energy �upper figure� is not proportional tothe cluster size because of the surface effect. To estimate thesurface excess energy, the potential energy Ul of a hypotheti-cal cluster consisting of bulk liquid molecules is calculatedas ul�n , where ul is the potential energy per molecule ofbulk LJ liquid �ul��5.4 at T�0.67 �80 K for argon�, and�5.3 at T�0.83 �100 K�12� and n is the cluster size.

The surface energy Us is calculated as the difference ofUclus and Ul , and shown in Fig. 6 as a function of the clustersize. In the calculation, we use ul��5.4 at T�0.67 as thebulk liquid energy; however, the temperature dependence oful is rather small as shown in Fig. 5. For the clusters of n�50, Us is proportional to n2/3, which means that the surfaceexcess energy �surface energy per unit area� is constant. For

smaller clusters, Us deviates from the n2/3 function, and ap-proaches n� with ��0.8. This deviation is due to the largesurface curvature, or the decrease of neighbor molecules, forsmaller clusters.

Large fluctuations and poor samplings bring a technicaldifficulty in evaluation of surface tension for these smallclusters, so we do not calculate the surface tension using oursystem. Instead, let us use the results of Thompson et al.6 forlarger clusters. They calculated the surface tension �s withrespect to the surface of tension �the radius Rs� for the clus-ters of size 54–2048 at reduced temperature T�0.71, andfound that �s /Rs �corresponding to the pressure drop due tothe Laplace equation� is fairly constant for the smaller drops�Fig. 19 in Ref. 6�. They also estimated the difference �between Rs and the Gibbs’ dividing surface Rd . The surfaceexcess free energy f s can be evaluated from these data as15

FIG. 3. Four sequential snapshots of the MD simulation. Only target particles are drawn. Note that the periodic boundary conditions are used for all threedimensions.



f s��s� Rs2

3Rd2 �

2Rd

3Rs� . �11�

In Fig. 6, the surface free energy Fs� f s�(surface area) isplotted, where the surface area is estimated with the bulkliquid density12 0.827 at T�0.67. Since �s /Rs is almost con-stant, the resulting Fs is proportional to n.

The surface entropy Ss is obtained through the relationFs�Us�TSs , where Fs is estimated by extrapolation of thedata in Ref. 6, though the temperature of their system �0.71�is slightly different from that of ours. Since Us is muchlarger than Fs , the behavior of Ss is similar to that of Us ; inparticular, Ss becomes proportional to n2/3 for clusters largerthan 50. Thus, properties of the cluster surface are greatlydifferent from that of flat surface for clusters smaller than 50.

Now, let us look into the structure of clusters. Takingclusters of size 20 for an example, we show the density,temperature, and potential energy profiles in Fig. 7; the num-ber of samples is 2233. The size is so small that the densityeven at the center is less than the bulk liquid density. Theposition of the Gibbs dividing surface is estimated to be 1.79

using the bulk values, but the tail of the profile reaches as faras r�3, which suggests the structure becomes relaxed nearthe surface. The loose structure is also apparent in the poten-tial profile because the value of bulk liquid is �5.4 at T�0.67. Although the heat exchange takes place only on thecluster surface, the temperature is almost constant inside thecluster, which indicates the heat transfer in the cluster israther fast.

C. Nucleation rate

The nucleation rate J is defined as the number of nuclei�larger than the critical nucleus� generated per unit volumeper unit time. In this section, growth-decay dynamics of clus-ters is investigated in order to estimate J.

FIG. 4. Time development of the mean cluster size. Clusters are identifiedbased on three different definitions of connection with distance criterion r0

and time criterion t0 . Circles: r0�1.5 and t0�0. Crosses: r0�2.0 and t0

�0. Triangles: r0�1.5 and t0�3.

FIG. 5. Cluster size dependence of potential energy per cluster �upper� andtemperature �lower�. The heat bath temperature is T�0.67. Two lines in theupper figure are the potential energy of a hypothetical cluster consisting ofbulk liquid molecules at T�0.67 �solid� and T�0.83 �dashed�.

FIG. 6. Surface energy Us , the surface free energy Fs , and the surfaceentropy Ss are plotted against the cluster size n. Fs is estimated from thedata in Ref. 6. The dotted lines are power functions of n with an exponent of2/3. The dashed line is an extrapolation of Fs and proportional to n.

FIG. 7. Example of cluster profiles; the size is n�20. The density �top�,temperature �middle� and potential energy per molecule �bottom� are plottedagainst the distance r from the center of mass of a cluster.



First, the time development of cluster size distribution isshown in Fig. 8. In an earlier stage �before time 600�, manysmall clusters are repeatedly formed and broken by molecu-lar collisions, among which several clusters larger than somesize survive; this is named the nucleation process stage, orStage I, in this paper. The surviving large clusters keepgrowing after time 600, but still later, we see the decrease ofnumber of large clusters probably due to a mechanism simi-lar to the Ostwald ripening. This later stage �time �600� isnamed the cluster growth process stage, or Stage II. In thisstage, the growth rate is almost constant as shown in Fig. 2where condensation heat �Q increases linearly with time.

The change of the numbers of monomers and clustersare shown in Fig. 9. Using the average of the monomer den-sity �top figure� between time 300 and 600, we can reesti-mate the supersaturation ratio and the size of critical nucleusfor the prediction of the classical nucleation theory as 6.8and 25.4, respectively. The nucleation rate J is calculatedfrom these data by counting the number of clusters largerthan some threshold. We adopt six different threshold sizes(nt�10,20,...,60) and the results are plotted against the time�middle and bottom figure�. In Stage I �time �600�, the in-crease of the number of clusters has similar inclination fornt�30– 60 �bottom figure�, while the increase for nt�10and 20 �bottom figure� is much faster. This suggests thatmany clusters smaller than 20 are formed in this stage, butonly clusters larger than 30 are able to grow. From the incli-nation for nt�30– 60, we can calculate the nucleation rate ofthis simulation Jsim as 9.71�10�8(��3 �1). Comparing itwith a theoretical value Jcl�3.28�10�15 estimated with Eq.

�8�, we conclude that the nucleation rate in our MD simula-tion is almost seven orders of magnitude larger than the the-oretical prediction.

In Stage II, the number of clusters shows a plateau,which suggests that generation of nuclei stops because thesupersaturation ratio gradually decreases.

It should be noted that there is a certain induction period K before which no nuclei larger than the critical nuclei areobserved. From a kinetic point of view, it roughly takes K

for the system to establish the steady state after beingquenched. From Fig. 9, we can obtain K�200. In the clas-sical theory, K is given as16

K�Z�2�n*��1 �12�

and estimated to be 1480 under the conditions of our simu-lation; the simulational result is seven times smaller than thetheoretical one.

D. Critical nucleus

From the above analysis, the size of the critical nucleusis roughly estimated as 20–30. From a kinetic point of view,the growth and the decay of clusters are balanced at this size.In order to investigate the critical nucleus in more detail, wecalculate the rate of cluster size change. By comparing twosequential configurations �at time t and t��t , where �t�0.15 is the time step for storing configurations�, we evalu-ate the transition probability n ,n�k(t) which is the probabil-

FIG. 8. Time development of cluster size distribution.

FIG. 9. Time development of the number of monomers and clusters largerthan a threshold size. In the bottom figure, the increase rate �shown byleast-squares fitted lines� in the early stage seems independent of the thresh-old, from which the nucleation rate is evaluated.



ity that a cluster of size n at t becomes a cluster of size n�k at t��t , where k� . . . , �2,�1,0,1,2... . The normalizingcondition is �kn ,n�k(t)�1 for all n.

After taking time average �n ,n�k(t)� t�n ,n�k , we de-fine the mean increase rate �k�� , the mean decrease rate�k�� , and the mean size change �k� as follows:

�k��n �� k�1

�

kn ,n�k , �13�

�k��n �� k��

�1

kn ,n�k , �14�

�k��n ��k��k�� . �15�

The results are shown in Fig. 10. The size of the kineti-cally defined critical nucleus nK is defined as n which satis-fies �k�(n)�0, and is estimated to be 30–40. The classicalnucleation theory, Eq. �2�, gives the thermodynamically de-fined critical nucleus as n*�25.4. The simulation result is aslarge as the theoretical one. On the other hand, the size ofcritical nucleus obtained from the density functional calcula-tions is in good agreement with classical theorypredictions.17,18 Our results support this agreement of densityfunctional calculations and classical theory prediction.

Recently, Nishioka and his coworker theoreticallyshowed19,20 that the thermodynamically defined n* and thekinetically defined nK is same except special region of hightemperature. Thus, our simulational result of nK suggests n*as large as the above prediction of 25.4. Considering thecurvature effect, one might be tempted to modify the surfacetension � in Eq. �2�. However, if we use � smaller than thatof flat surface to attain larger J, n* becomes even smallerand the discrepancy between simulational results and the the-oretical prediction enlarges. Thus, the classical theory withthe free energy difference of the form Eq. �1� should bereconsidered; the detail will be discussed later in Sec. V.

IV. MODEL CALCULATION OF CLUSTER SIZECHANGE

As shown in the preceding section, the kinetically de-fined critical nucleus is described in terms of the mean ratesof growth �k�� and decay �k�� . In order to estimate theserates, it is necessary to investigate the mechanism of clustersize change in more detail. In this section, we look into themechanism by constructing simple molecular models.

A. Growth mechanism

Cluster growth takes place mainly when a monomermolecule collides with the cluster; considering the numberratio of clusters to monomers, cluster–cluster collision hasonly a minor effect. Thus, evaluation of condensation flux ona cluster surface is sufficient to estimate the growth rate.

When the vapor can be regarded as an ideal gas, thecollision flux on a ‘‘flat’’ surface is described by well-knownHeltz–Knudsen formula,12

Jcoll�� kBT

2�m, �16�

where � is the number density of vapor and m the molecularmass. If we can neglect the curvature effect, the growth rateC� of a cluster with the surface area A is expressed as

C��Jcoll�A , �17�

where � is the condensation coefficient, i.e., the ratio of con-densation flux to collision one. Although � is less than unityeven for simple fluids,12 we use ��1 here, which gives themaximum condensation flux.

Therefore, it is essential to evaluate A for a cluster ofeach size. The first choice is that we use the Gibbs dividingsurface, which means that the cluster is a rigid sphere of bulkliquid density � l . If the cluster size is n, the area is

�Model 1� A��1/3�6n/� l�2/3. �18�

In Fig. 11, the model calculation �dotted line� of the growthrate is compared with the simulation results �triangles�. It isapparent that the model calculation largely underestimatesthe rate.

Next, let us take account of the range of attractionaround a cluster, which will effectively increase A. From themodel calculation given in Appendix A, the effective area isgiven as

FIG. 10. Cluster size dependence of the rate of cluster size change in thesimulation. The triangles indicate the growth rate, the crosses the decay rateand the circles the total rate which is the sum of two rates. See Eqs. �13�,�14�, and �15� for the definitions. The growth and decay rates are balanced atthe size of the kinetically defined critical nucleus.

FIG. 11. Growth rate of clusters. The triangles show the simulation data�same as in Fig. 10�. Lines are the model calculations; Model 1 �dotted line�,model 2 �broken�, and model 3 �solid�.



�Model 2� A�M

kBT �0

�

dv04�dc2v0 exp� �

Mv02

2kBT � ,

�19�

where M is the effective mass (�n/(n�1)�m), and thethreshold of collision parameter dc is determined as a func-tion of v0 by Eq. �A3�. The broken line in Fig. 11 shows theresult, which is still smaller than the simulation data.

Another possibility is taking account of the cluster struc-ture, i.e., the long tail of the density profile described in Sec.III B. We make a rough estimation that the density profileextends to rc�1.0 from cluster surface thickness �Fig. 7�,where rc is the position of the Gibbs dividing surface.

�Model 3� Similar to Model 2,

but with rc �1.0 instead of rc . �20�

The results is also shown in Fig. 11 �solid line�, and theagreement with the simulation data is fairly good.

There is still a discrepancy for small cluster region. Thereason is not fully understood, but one possibility is that thecondensation coefficient � becomes small for small clusters.We found12 that ‘‘molecular exchange,’’ i.e., evaporationstimulated by condensation molecules, is significant even forsimple fluids. Since the surface of small clusters are not en-ergetically stable, the molecular exchange of small clusterscan be larger than that of larger ones, resulting in the smallgrowth rate.

B. Decay mechanism

In order to estimate the rate of cluster decay �k�� , wehave to investigate the mechanism of evaporation from clus-ter surface.

The starting point is the Laplace equation,

P�Pe�2�

r, �21�

which relates the saturated vapor pressure Pe and the vaporpressure of a cluster �radius r� P. Here � is the surface ten-sion. In other words, an equilibrium will be attained if thecluster is placed in �supersaturated� vapor of pressure P.

Therefore, we can estimated the evaporation flux Jevap

from the condensation flux in the vapor of pressure P, be-cause these two fluxes should be balanced. From Eq. �16�,we obtain

Jevap�� e�2�

r

1

kBT �� kBT

2�m, �22�

where �e is the number density of saturated vapor. The decayrate C� is estimated as JevapA , where A is the surface area ofthe cluster by use of the Gibbs dividing surface.

In Fig. 12, simulation data of decay rate �crosses� arecompared with the model calculations �lines�. If we use thesurface tension of a flat surface for �, the result �solid line�greatly overestimates the decay rate. On the other hand, ne-glecting the curvature effect, i.e., r→� in Eq. �22�, results ina very weak dependence �dashed line� on the cluster size.Thompson et al.6 showed that �P�2�/r is almost indepen-dent of r when the size of the cluster is less than �500; in

dimensionless unit, 2�/r�0.2 near the triple point. Usingthis value �dotted line�, we have a reasonable agreement withthe simulation data.

Apart from the theoretical calculation using the Laplaceequation, we also try to estimate C� with direct MD simu-lation of a cluster evaporating into the vacuum. Dumontet al.21 investigated evaporation of argon clusters in thevacuum via MD simulation. They focused on the coolingprocess during the evaporation, and the initial evaporationrate, which should be dependent on the cluster size, is notclear. Thus we have executed MD simulation of clusterevaporation for several sizes of clusters, n�8, 10, 15, 20, 30,40, and 70. Initially, we confined each cluster in a externalfield of soft repulsion, �(r)�(r�r�)12, r�r�, where r� ischosen as a little larger than Gibbs dividing surface, to pre-vent the cluster from evaporation during the equilibrationprocess; the temperature is controlled to be the one observedin our nucleation simulation �Fig. 5�. At time�0, we removethe external force field, letting the molecules to freely evapo-rate, and measure the waiting time for a first molecule toevaporate from the cluster. The distribution of the waitingtime t is fitted fairly well to a single exponential function oft as � exp(��t), which suggests that the evaporation is re-garded as a Markov process. The decay rate is estimated as�, the results of which are shown in Fig. 12 as filled circles.Both simulation data agree fairly for large clusters (n�40), but there seems to be discrepancies in smaller sizeregion. The decay rate in the supersaturated vapor �crosses�is larger than that in the vacuum �filled circles� for 10�n�40, which suggests that the evaporation from clusters isstimulated by the condensation �or collision� of vapor mol-ecules. Therefore, the correlation between evaporation andcondensation at the cluster surface is important, which hasbeen neglected in most theoretical treatments.

V. CLUSTER FORMATION FREE ENERGY

In Sec. III D, we pointed out that the cluster formationfree energy �G of Eq. �1� is questionable. Thus, we try toevaluate �G directly from the simulation data.

Since the concentration c(n) of clusters of each size iscalculated from the simulation, �G is expressed from Eq. �7�as

FIG. 12. Decay rate of clusters. The crosses show the simulation data �sameas in Fig. 10�. Lines are the model calculations from the Laplace equation;with the surface tension of a flat surface �solid line�, without the curvatureeffect �dashed�, and taking account of size dependence of the surface tension�dotted�. Filled circles are the rate of cluster evaporation into the vacuum.



�G��kBT ln�c�n �/�� , �23�

where ��c(1) is the number density of monomers. Equa-tion �23� is applicable only to the clusters smaller than thecritical nucleus n*, because Eq. �7� assumes equilibrium; forclusters larger than n*, the equilibrium is never attained, and�G calculated with Eq. �23� is to level off. In Fig. 13�a�, theresult of �G is shown as a function of n. In this calculation,the data between 300�time�600 are used because the con-stant nucleation rate is observed in the time range, whichsuggests that the concentration of clusters smaller than n* isat �quasi� equilibrium.

As expected, �G levels off near n�30. Comparing itwith the theoretical prediction Eq. �1� shown by solid line,the n* becomes larger and �G*��G(n*) is smaller. Withsubtracting the bulk phase contribution �gV from �G , weestimate the surface contribution, which is plotted in Fig.13�b�. The contribution has a functional form of n� with anexponent ��0.88, while ��2/3 is expected in the classicaltheory �solid line�.

Thus, we assume the following form for �G:

�G�an0.88��gV , �24�

where a is a fitting parameter. The data of n�30 in Fig. 13are fit to Eq. �24� to obtain �G* and the Zeldovich factor Zdefined in Eq. �5�. Assuming that the forward rate equalsto �k�� /�t in Sec. III D, we obtain from Eq. �4� the nucle-ation rate J�8.3�10�8, which is well compared with thesimulational result Jsim�9.71�10�8.

Therefore, we conclude that the large discrepancy be-tween the classical theory and the simulational result ismainly due to the failure in estimating the cluster size depen-dence of �G .

The induction period K defined in Eq. �12� is also re-calculated with Eq. �24�. The result is K�260. By compari-son with 1480 in the classical theory, we find that K hasonly a weak dependence on �G and .

One thing should be noted. Strictly speaking, c(n) ob-served in the simulation is a steady state distribution, not anequilibrium one; these two distributions in general differeach other. From our preliminary analysis, however, the dif-ference is found negligible even for our system of largenucleation rate, as shown in Appendix B.

VI. FUSION OF CLUSTERS

In the model analyses of Sec. IV, only ‘‘step-by-step’’processes were treated, in which condensation of monomersto, and evaporation of monomers from, clusters are relevantin considering the cluster size change mechanism. However,the density of nuclei is rather large due to the large super-saturation ratio, and fusion of clusters is sometimes ob-served.

Figure 14 shows an example of the fusion process,where two clusters of size �150 and �120 �much larger thanthe critical nucleus� collided each other to form a larger clus-

FIG. 13. �a� Cluster formation free energy and �b� its surface contributioncalculated from the cluster size distribution. Solid lines indicate the classicaltheory results.

FIG. 14. Snapshots of a cluster fusion process observed in the MD simula-tion. Here ‘‘cluster’’ is defined as a group of mutually connected moleculeswith the criteria r0�1.5 and t0�0 as described in Sec. III B2.



ter. This process is monitored from viewpoints of clusterstructure and thermodynamic properties as shown in Fig. 15.

In Fig. 15�a�, the time development of cluster size isshown. The cluster fusion takes place about time�915. Alsoin the figure, the number of newly coming molecules nnew

which did not belong to either clusters at time�900 is plot-ted. Although the total number is kept almost constant��270�, nnew gradually increases, which suggests an activeexchange of molecules between the clusters and the sur-rounding vapor through evaporation and condensation.

Concerning the cluster structure, the time developmentof the largest principal value of gyration tensor RL is shownin Fig. 15�b�. Before the fusion, the two clusters is spherical�see Fig. 14�, and their size is similar. At the instant of fu-sion, RL becomes twice, and the fused cluster takes a sphe-roidal shape. This elongated shape relaxes with time, thecharacteristic time of which seems �20.

During the relaxation process, the kinetic energy in-creases as shown in Fig. 15�c�, after which it gradually de-creases. The behavior of the potential energy �Fig. 15�d�� isdifferent from it. The potential energy shows little changeduring the shape relaxation. Its relaxation begins after thefused cluster takes a near-spherical shape.

VII. CONCLUSION

Using large-scale molecular dynamics computer simula-tion, we studied a homogeneous nucleation process in thevapor phase. The system is a mixture of Lennard-Jones fluid

�5000 molecules� and soft-core carrier gas �5000�, and thetemperature is controlled through the carrier gas to be nearthe triple point of the LJ fluid. The supersaturation ratio is�6.8.

The time development of the system was clearly dividedinto two stages. At the earlier stage, formation of many nu-clei was observed, and the estimated nucleation rate wasseven orders of magnitude larger than a prediction of a clas-sical nucleation theory. The later is the cluster growth stage.

From the structural and thermodynamical analyses ofclusters, it was found that the surface excess of energy andentropy takes a bulk value for clusters larger than �50.

Examining the rate of cluster size change, the criticalnucleus was kinetically defined, and found to be 30–40. Thesize change was analyzed as a growth process and a decayprocess separately; the growth process can be understoodmainly as successive condensation of monomers, except forsmall clusters. The decay can also be understood as simpleevaporation process when size dependence of the surfacetension are taken into account.

There seems to be no satisfactory theories to predict theobserved fast nucleation rate. One prospecting approachwould be a modeling of the transition probability n ,n�k �seeSec. III D for definition�; once n ,n�k are calculated withcertain precision, the nucleation rate can be evaluated byassuming a Markovian process. Our simple analyses de-scribed in Sec. IV are apparently insufficient for clusterssmaller than the kinetically defined critical nucleus. It is thusneeded to investigate the properties �dynamic as well asstatic� of small clusters in more detail, and the study is underway.

Using the cluster concentration expression Eq. �7� re-versely, the free energy of cluster formation �G is esti-mated. The surface contribution term has a peculiar depen-dence on the cluster size n as �n0.88. With this �G , thepredicted nucleation rate agrees with the simulation result.

The reason behind this behavior of surface contributionis not fully understood yet, but it should be closely related tothe structure and properties of small clusters. As described inthe following paper, nucleation process of water vapor showsa similar behavior; thus, it is considered that this form of �Gfor small clusters is general to some extent. In the future, wewill investigate the relation between �G and various clusterproperties.

ACKNOWLEDGMENTS

We are grateful to Professor K. Nishioka �TokusimaUniversity� for stimulating discussion and constant encour-agement. A part of the calculation was carried out at theComputer Center of the Institute for Molecular Science andat the Institute of Solid State Physics, University of Tokyo.

APPENDIX A

Using a simple two-body collision model, we estimatethe rate of vapor collision with a cluster by taking account ofattractive interaction between the vapor and the cluster. Thesituation is shown in Fig. 16, where a vapor molecule �filledcircle� approaches a cluster �open circle�. First, let us express

FIG. 15. Structural and thermodynamical properties are plotted during acluster fusion process in Fig. 14. �a� Cluster size and the number of newlycoming molecules, �b� the largest principal value of gyration tensor, �c�kinetic energy of each cluster, and �d� potential energy of each cluster.



the speed v of the molecule at the nearest distance d in termsof the initial speed v0 and the collision parameter d0 . Usingthe conservation laws of energy and angular momentum, weobtain the following expression:

M

2v0

2�M

2 � d0v0

d � 2

��d �, �A1�

where �(d) is the potential energy between the cluster andthe molecule, and M is the effective mass. To proceed fur-ther, we evaluate �(d) by assuming that the Lennard-Jonesfluid uniformly fills the cluster region of radius rc�d , whererc corresponds to the Gibbs dividing surface. When only anattractive part is considered, a simple volume integrationgives

��d ��16

3�� l

rc3

�d�rc�3�d�rc�

3 , �A2�

where � l is the number density of bulk liquid.We determine that the vapor collides with the cluster in

case of d�rc�r0 , where r0 is the criterion of ‘‘molecularconnection’’ described in Sec. III B. The threshold value ofd0 is determined by coupling Eq. �A1� with the condition d�rc�r0 . The result has v0 dependence as

dc�v0�1�v0

2�2��rc�r0�/M �rc�r0�. �A3�

The v0 dependent surface area is thus 4�dc2, and the

total collision rate C� is evaluated by averaging it with theMaxwell distribution as

C�� M

2�kBT�0

�

dv04�dc2v0 exp� �

Mv02

2kBT � , �A4�

where � is the number density of vapor.In actual integration with respect to v0 , using Eq. �A3�

causes a divergence at v0�0 because the attraction Eq. �A2�exists at long distance. Thus, we cut off �(d) at d�rc

�rt , where rt�4.5 is the cut-off length in our MD simula-tion.

APPENDIX B

The cluster concentration c(n) in a steady state is ingeneral different from that in an equilibrium state. In thisAppendix, we give a formula for the difference and showthat the observed c(n) is very close to the equilibrium con-centration.

For the simplicity, we consider a one-dimensionalFokker–Planck �FP� equation in continuum space x instead

of a one-step process in discrete space n �cluster size�. Theprobability P(x ,t), which is proportional to the concentra-tion c(x ,t), is assumed to be governed by the following FPequation:

P

t��

xJ�x ,t �

��

x �A�x �P�x ,t ��

xB�x �P�x ,t �� , �B1�

where J(x ,t) is the probability flux. In a steady state,P(x ,t)/t�0 and J(x ,t) should be constant J. Thus weobtain the equation of PJ(x), which is independent of t, as

A�x �PJ�x ��d

dxB�x �PJ�x ��J . �B2�

A special case is the equilibrium solution, i.e., J�0, andEq. �B2� gives

P0�x ��P0�a �exp� �a

x A�x ��B��x �

B�x �dx� , �B3�

where B�(x) is a first derivative of B(x) with respect to xand a constant a is the lower boundary of x corresponding ton�1 �monomer�.

In the case of J�0, the steady state solution with thesame boundary condition PJ(a)�P0(a), i.e., constantmonomer density, is expressed in terms of P0(x) as

PJ�x �

P0�x ��1�J�

a

x 1

B�x �P0�x �dx . �B4�

Inversely, P0(x) can be expressed in term of PJ(x) as

P0�x ��PJ�x �exp� J�a

x 1

B�x �PJ�x �dx� . �B5�

Our MD simulation directly gives PJ(x) �proportional tothe cluster concentration� and J �nucleation rate�. The diffu-sion coefficient B(x) is related to the transition probabilityn ,n�k in Sec. III D, and is estimated as (�k��k��)/2with the assumption of one-step process.

The results of P0(x) and PJ(x) are shown in Fig. 17; thedifference of concentrations in equilibrium and steady statesis small even in the case of extremely large J.

FIG. 16. Two-body collision model used to calculate the collision rate of avapor molecule and a cluster when an interaction exists.

FIG. 17. Probability density in the steady state PJ �solid� and equilibriumstate P0 �dashed�. The unit of ordinate axis is arbitrary.



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