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Performance of some nucleation theories with a nonsharp droplet-vapor

interface

Ismo Napari,a?Jan Julin, and Hanna Vehkamäki

Department of Physics, P.O. Box 64, University of Helsinki, Helsinki 00014, Finland

?Received 30 June 2010; accepted 24 September 2010; published online 15 October 2010?

Nucleation theories involving the concept of nonsharp boundary between the droplet and vapor are

compared to recent molecular dynamics ?MD? simulation data of Lennard-Jones vapors at

temperatures above the triple point. The theories are diffuse interface theory ?DIT?, extended

modified liquid drop-dynamical nucleation theory ?EMLD-DNT?, square gradient theory ?SGT?, and

density functional theory ?DFT?. Particular attention is paid to thermodynamic consistency in the

comparison: the applied theories either use or, with a proper parameter adjustment, result in the

same values of equilibrium vapor pressure, bulk liquid density, and surface tension as the MD

simulations. Realistic pressure-density correlations are also used. The best agreement between the

simulated nucleation rates and calculations is obtained from DFT, SGT, and EMLD-DNT, all of

which, in the studied temperature range, show deviations of less than one order of magnitude in the

nucleation rate. DIT underestimates the nucleation rate by up to two orders of magnitude. DFT and

SGT give the best estimate of the molecular content of the critical nuclei. Overall, at the vapor

conditions of this study, all the investigated theories perform better than classical nucleation theory

in predicting nucleation rates. © 2010 American Institute of Physics. ?doi:10.1063/1.3502643?

I. INTRODUCTION

Classical nucleation theory ?CNT?1,2has been a mainstay

of atmospheric nucleation studies and technological applica-

tions for decades. CNT offers mathematical simplicity and

very little computational challenge, but it gives poor perfor-

mance in many cases.3Many new theoretical approaches

have been invented during the recent decades in attempts to

surpass and replace CNT but the success of these theories

has been limited. For example, they may fail in some ambi-

ent conditions, they can only be applied to a certain class of

substances, or they require data that are not easily available.

The quality and limitations of the theories can only be as-

sessed by comparing them to experiments; however, in some

cases the observed nucleation rates are found to depend on

the experimental method4and the approximations when data

is analyzed.5

In this paper, we try to circumvent the issues concerning

the experimental uncertainties by comparing nucleation theo-

ries to recent molecular dynamics ?MD? simulations, where

the condition of the nucleating vapor is controlled more eas-

ily and exactly than in experiments. For example, in MD, the

nucleating vapor can be connected directly to an artificial

thermostat for a stricter regulation of temperature than could

be achieved experimentally by a carrier gas. The nucleation

rate is also accurately obtained by following the formation

and growth of the clusters in the vapor, a task that is, in

practice, impossible in an experiment. MD is unfortunately

computationally demanding and in most of the nucleation

studies by MD, simple Lennard-Jones ?LJ? systems have

been investigated. For the purpose of this study, we have

chosen very recent sets of MD nucleation data in LJ

systems.6–8

Nucleation of molecular substances might

present a more stringent test of a theory, but lacking MD data

at various temperatures and vapor conditions, such a study is

not possible at the moment.

The investigated nucleation theories are diffuse interface

theory ?DIT?,9extended modified liquid drop model com-

bined with dynamical nucleation theory ?EMLD-DNT?,10,11

square gradient theory ?SGT?,12,13and density functional

theory ?DFT?.14These theories are connected by the fact that

they incorporate, explicitly or implicitly, a nonsharp bound-

ary between the nucleating cluster and the vapor. There are,

however, differences in the input data needed to apply these

theories to nucleation. DIT requires the heat of evaporation

in addition to the same set of thermophysical quantities as

CNT. If the vapor can be considered ideal, the same quanti-

ties as in CNT are all that is needed to apply EMLD-DNT.

However, in the case of a nonideal vapor, EMLD-DNT also

requires an equation of state ?EoS? for the vapor. SGT must

always be supplemented with a complete EoS and DFT

needs the exact molecular interaction potential. A short re-

view of the theories is presented in Sec. II.

We aim to attain internal consistency to the comparison

between theory and simulation by using the MD values for

certain thermodynamic bulk properties in the theories. In

some cases, this is achieved by a fitting procedure, as ex-

plained in Sec. III in more detail. The thermodynamic con-

sistency at the bulk level will help in revealing the true dif-

ferences between the theories when compared to the MD

nucleation simulations. Thus, although the applied theories

are all well known and they do not contain any new features,

a better insight to the validity of the theories will be found.

Our work parallels to that by Kalikmanov et al.,15where

a?Electronic mail: ismo.napari@helsinki.fi.

THE JOURNAL OF CHEMICAL PHYSICS 133, 154503 ?2010?

0021-9606/2010/133?15?/154503/7/$30.00© 2010 American Institute of Physics

133, 154503-1

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several theoretical approaches to nucleation were compared

to MD simulations of LJ vapors. However, all the investi-

gated theories in our work are not the same as in Ref. 15 and

different sets of MD simulations are used as references. A

further noteworthy difference between Ref. 15 and our work

is the temperature range: in Ref. 15, nucleation was mainly

studied below the triple point, whereas we consider tempera-

tures above it. This ensures that the critical droplets are liq-

uidlike. The high temperatures necessitate the use of a real-

istic EoS instead of ideal gas law because MD simulations

are restricted to fairly high vapor saturation ratios, where

significant departure from ideal gas is observed especially

above triple point ?see Sec. III?.

The objective of this work is to determine which of the

theories listed above is the best in reproducing the nucleation

properties observed in MD simulations of LJ fluid. We are

mainly interested in the key observable quantity in nucle-

ation, the nucleation rate ?Sec. IV A?. Nevertheless, using the

nucleation rate data, we also evaluate the critical cluster sizes

and compare them to the cluster sizes obtained from the in-

vestigated theories ?Sec. IV B?. This study thus also consti-

tutes a follow-up of our previous work,7where cluster sizes

in MD simulations were compared according to various clus-

ter definitions. Finally, in Sec. IV C, we discuss the effect of

EoS on the theoretical nucleation rates and cluster sizes and

in Sec. V we present our conclusions.

II. A SHORT REVIEW OF THE THEORIES

A. DIT

Diffuse interface theory9is based on a parametrization of

radial enthalpy ?h?r? and entropy ?s?r? profiles of the drop-

let. The work of formation of the droplet ?that is, the grand

potential difference ?? in an open system? can be expressed

in terms of step profiles ?h?0? and ?s?0? as

?? =4?

3?RH

3?h?0? − RS

3T?s?0??,

?1?

where RHand RSare the respective positions of the steps and

RH=RS−?, where ? is the interfacial thickness. If the bulk

properties are assumed to prevail at least at the center of the

droplet, ?h?0? and ?s?0? can be replaced by their bulk liquid

values ?h and ?s. Furthermore, if ? is assumed independent

of saturation ratio at fixed temperature, one can estimate ?

=−??/?hf, where ??is the surface tension of the planar

interface and ?hfis the volumetric heat of fusion.

The work of formation of the critical cluster is found

from the maximum of ??. With the above assumptions, the

maximum of Eq. ?1? is given by

???= −4?

3?3?g?,

?2?

where ?g=?h−T?s, ?=2?1+q??−3−?3+2q??−2+?−1, q

=?1−??1/2, and ?=?g/?h. The Gibbs free energy density

difference ?g is obtained from ?g=−?l??, where ?? is the

chemical potential difference between supersaturated and

saturated vapor and ?lis the bulk liquid density.

Compared to CNT, DIT has been shown to give im-

proved predictions of nucleation rates of nonpolar, weakly

polar, and metallic substances.9For polar substances ?such as

water?, the theory is less successful.

B. EMLD-DNT

The extended liquid drop model combined with the dy-

namical nucleation theory10,11is in principle an extension of

CNT, but it has many features which makes it a more real-

istic theory than CNT. EMLD-DNT is formulated for a sys-

tem of N molecules in a spherical container of volume V

?so-called N,V-cluster?. The container encloses a liquid drop

of n molecules and N−n vapor molecules. The Helmholtz

free energy differential for such a system is written as16

?dF?N,V,T= −?Pl− Pv−2??

r?dVl+ ??l− ?v?dn,

?3?

where Pland Pvare the pressures inside the drop and in the

vapor, ?land ?vare the corresponding chemical potentials,

Vlis volume of the drop, and r is the radius of the drop.

Assuming incompressible liquid Eq. ?3? can be integrated to

give the formation energy in the closed container

?F?n? = F?n? − F?0?

=?

0

n

?Pv− Pe?vldn? −?

0

n

??v− ?e?dn? + ??A?n?,

?4?

where Pvand ?vare the pressure and the chemical potential

at the density ?N−n?/?V−nvl?, Peis the pressure of saturated

vapor, vl=1/?lis the molecular volume in the liquid, and A

is the surface area of the cluster.

The essential idea in the EMLD model is the fluctuation

of cluster size.16The drop size does not have a fixed value

inside the container but instead it fluctuates with the relative

probability of a fluctuation of size n given by f?n?

?exp?−?F?n?/kBT?. Knowing the free energies ?F?n? for all

n=0,...,N, the canonical partition function Z?N,V,T?

=−kBT ln??n=0

ence ?F=−kBT ln Z can be calculated.

Another important essence of the EMLD model is the

incorporation of the dynamical nucleation theory, which to-

gether with the variational transition state theory gives a

unique cluster definition: the volume V of the N,V-cluster is

such that the evaporation rate is minimized. This condition

amounts to finding the volume which minimizes the vapor

pressure in the container. Note that due to the fluctuating

cluster size, the pressure must be calculated as an ensemble

average. Furthermore, since the cluster ?considered as a

single hard-sphere particle? is free to move inside container,

a translation correction must be added to the pressure.16

Once the minimum pressure Pminand the corresponding

volume Vminare found for a given critical size N?, the for-

mation free energy of the cluster in the vapor at pressure

P?=Pminis obtained from11

???= ?F?N?,Vmin? − Vmin?P0− P?? + N???0

where P0=P??0?, ??0

Nexp??F?n?/kBT?? and the free energy differ-

?,

?5?

?=??P0?−??Pmin?, and ?0=N/Vmin.

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Note that although EMLD-DNT uses the concept of

classical drop with a sharp boundary, the translating drop and

its fluctuating size produces an averaged density profile.17

EMLD-DNT has been shown to be in a very good agree-

ment with Monte Carlo simulation,11MD simulation,18and

1-pentanol data19on nucleation.

C. SGT

In the square gradient theory, the free energy of cluster

in a vapor characterized by chemical potential ? is given

by12,13,20

????r?? =??f0???r?? +c

where f0??? is the local free energy density of the uniform

fluid and c is a ?temperature-dependent? constant. The salient

feature of SGT is the dependence of the free energy on slope

of the density profile ??r? ?the gradient term in Eq. ?6??. The

actualdensityprofileis

?????r??/???r?=0, which gives a variational equation for

??r?. Using the density profile, the formation free energy is

obtained by subtracting the free energy of the supersaturated

vapor from Eq. ?6?. The drawback of SGT is that the local

free energy ?that is EoS? must be known at all densities,

including the unstable regions. In this work we have used the

DFT EoS, which is described in Sec. III.

SGT with a parameter fitting has been applied to de-

scribe nucleation of LJ fluid,21nonane,22and polar fluids23

with some success.

2????r??2− ???r??dr,

?6?

obtainedfromcondition

D. DFT

Density functional theory can be considered the most

refined of the theories reviewed here. The free-energy is

given by14

????r?? =?drfh???r?? +1

− ??dr??r?.

2??drdr????r − r?????r???r??

?7?

The approach of Eq. ?7? is perturbative: a hard-sphere fluid

with the free-energy density fh??? is chosen as a reference

system and a perturbation ?the second term on the right-hand

side of Eq. ?7?? is added to account for the attractive inter-

actions. The essential difference between SGT and DFT is

that SGT is a purely local theory, whereas in DFT the density

at a given point depends on the interactions accounted for

over all space, although in Eq. ?7? this is done in the mean-

field level. For DFT, an interaction potential ? must be de-

fined, in our case the LJ potential. The calculation of the

density profile and the formation free energy is done as in

SGT.

In our earlier study,21we showed that DFT with a fitting

procedure results in an excellent agreement with MD nucle-

ation simulations of LJ vapors at a relatively low tempera-

ture. The same approach was used much earlier by Nyquist

et al.24to nonane and toluene nucleation and considerable

improvement over CNT results were found.

The considered theories are all thermodynamic ap-

proaches to nucleation which give the formation free energy

???of the critical nucleus. The output quantity of MD simu-

lations, however, is the nucleation rate. The relation between

the formation energy and the nucleation rate J is

J = K exp?− ???/kBT?,

?8?

where K is a kinetic prefactor. Following Ref. 15, we use the

classical formula for K

K = vl?

Pv

kBT?

2?2??

?m,

?9?

where m is the molecular mass.

The classical prefactor is only an approximation, but one

that works quite well. A MD study25of a LJ system showed

that when the formation free energy and nucleation rate were

obtained from the simulation independently, the simulated

formation free energy combined with the classical prefactor

gave an accurate estimate of the simulated nucleation rate. In

another work, SGT formation free energy and classical pref-

actor gave only 10% higher nucleation rates than an accurate

kinetic calculation.26However, a MD analysis of the nucle-

ation kinetics of a LJ system suggested that the classical

prefactor can underestimate the numerical value by one order

of magnitude.27

III. MD DATA SETS AND THERMODYNAMIC

CONSISTENCY

The reference MD data set is comprised of three recent

nucleation studies in LJ vapors,6–8where the actual nucle-

ation event is observed in a simulation box ?so-called direct

nucleation simulation?. All these studies consider a LJ poten-

tial that is truncated and shifted at rcut=2.5?, where ? is the

LJ length parameter. A notable difference exists in the

method by which the nucleation rate is extracted from the

simulations. Refs. 6 and 8 use the method by Yasuoka and

Matsumoto ?YM?,28whereas in Ref. 7 the mean first passage

time ?MFPT? analysis29is applied. In Ref. 8, the simulations

are performed in the grand-canonical ensemble and the

nucleation rates are shown to agree with those from canoni-

cal simulations. The simulations cover temperatures T

=0.65–1.0, but the data presented at T=0.95 and T=1.0 are

excluded from this study because the reported rates at these

temperatures are not probably valid approximations of the

actual nucleation rates.6Here and throughout the rest of the

article, temperatures are scaled with kB/?, where ? is the LJ

energy parameter. The critical temperature of the fluid is at

Tc=1.0779.30There exists some uncertainty about the triple

point temperature: Ref. 31 gives Ttr=0.65, whereas in Ref.

32 the value Ttr=0.618 is reported. In any case, the MD

simulations of Refs. 6–8 describe nucleation above the triple

point.

The successful application of any nucleation theory re-

quires that the bulk thermodynamic properties used as input

are those of the nucleating fluid. Fortunately, simulation-

based thermodynamic data are given in Ref. 30 for the par-

ticular LJ fluid studied here. The reported properties are the

saturated values of pressure and vapor and liquid densities,

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surface tension, and the enthalpy of vaporization. The tem-

perature correlations for these quantities given in Ref. 30 are

valid for T=0.64–1.05, which nicely covers the range of the

nucleation simulations.

Nucleation theories also need information on the vapor

condition in supersaturated states. If ideal gas is assumed,

equilibrium vapor pressure Petogether with the ideal gas law

suffice to supply this information, but for a highly nonideal

LJ vapor, a realistic EoS should be provided. To our knowl-

edge, there does not exist an EoS for the LJ system truncated

and shifted at rcut=2.5?. Using an EoS of the full-potential

LJ system and correcting for the cutoff is not recommended

when the cutoff value is small.33To be fully consistent in the

comparison of the theories, we chose the DFT EoS to obtain

pressure and chemical potential in all our calculations. DFT

EoS is given by the DFT formalism at the limit of homoge-

neous fluid as

P = Ph−1

2??2,

?10?

where Ph is the hard-sphere pressure according to the

Carnahan–Starling formula34and ?=−?dr??r?. Knowing the

EoS, we describe the state of the vapor in terms of chemical

potential rather than density or pressure. Thus, for example,

saturation ratio is defined by

S = e??/kBT,

?11?

where ?? is the chemical potential difference between the

supersaturated and saturated vapor according to the EoS.

The only input the CNT and EMLD-DNT require are

surface tension and liquid density in addition to EoS. The

latent heat of vaporization needed in DIT were obtained from

Ref. 30. It should be kept in mind that in CNT and DIT, ??

is an external parameter ?the “? form” of CNT in Ref. 35?

and EoS does not directly enter to the calculation, whereas

EMLD-DNT uses the actual EoS.

While CNT, DIT, and EMLD-DNT are purely theories of

nucleation, SGT and DFT are more general theories of fluid

state which can be applied in the planar geometry as well.

They thus provide surface tension, which depends on the

parameter c ?SGT? or the potential parameters ?DFT?. As in

our earlier paper,21the parameter c in SGT was fitted to give

the MD surface tension. The free parameters in DFT are the

two LJ parameters ?? and ?? and the hard-sphere diameter

?d?. These were fitted to give the MD values of Pe, ?l, and ??

at each studied temperature and the DFT EoS with the same

parameters were also used in the other theories. In the tem-

perature range T=0.65–0.9 we obtain ??/?=1.031–1.064,

??/?=0.996–1.006, and d?/d=1.026–1.027, where the

primed quantities denote the new set of parameters.

To correlate the vapor densities in MD simulations of

Ref. 7 with the corresponding chemical potential differences

??, we used at T=0.65 and T=0.8 the same simulated

pressure-vapor density correlations as in Ref. 7. This EoS

was also applied to convert the pressure ratios Pv/Pere-

ported in Ref. 6 to chemical potential differences ?tempera-

tures 0.65 and 0.8?. Finally, the saturation ratios given in Ref.

8 were used as such to calculate ?? according to Eq. ?11?.

The use of consistent set of bulk thermodynamic param-

eters, the same EoS, and, if necessary, a fitting procedure sets

the theories on equal footing: they are all identical at the bulk

level ?at least as far as input data for nucleation is concerned?

and thus the differences in calculated nucleation rates better

reflect the true differences between the theories.

IV. RESULTS AND DISCUSSION

A. Nucleation rate

Logarithmic nucleation rates as a function of ??

=kBT ln S are depicted in Fig. 1 at T=0.65 and T=0.8. The

figure shows the MD nucleation rate data6–8together with

nucleation rates from the theories reviewed in Sec. II, includ-

ing CNT. There is a small difference between the MD data

sets at T=0.65 and the difference increases up to one order of

magnitude at T=0.8, although at the latter temperature the

scatter in the data of Ref. 6 makes it difficult to evaluate the

exact deviation. It seems that the simulations using the YM

method yield somewhat higher nucleation rates than those

using the MFPT method. Chkonia et al.36have shown that

these methods should result in similar nucleation rates; the

conclusions, however, were based on simulations at the triple

point temperature and in the present study we are above it ?at

T=0.65, the difference between the methods is almost neg-

ligible?. Whether there really is a methodological difference

or the source for the discrepancy lies elsewhere remains un-

clear ?see also Ref. 37?. We do not pursue the matter further

here.

According to Fig. 1, the MD nucleation rates are repro-

duced with good accuracy by DFT, SGT, and EMLD-DNT.

At T=0.8, the nucleation rates from these theories practically

coincide. DIT underestimates the MD rates, especially at T

=0.65. Not unexpectedly, CNT gives the worst results. The

overall deviation of the theoretical nucleation rates from the

MD data is less than one order of magnitude, excluding CNT

and DIT.

Figure 1 shows that the difference between the simulated

nucleation rates JMDand theoretical nucleation rates Jtheor

depends on the saturation ratio somewhat. Regardless, it is

worthwhile to consider a situation where this dependence is

ignored and to investigate the performance of the theories at

different temperatures by plotting the ratios log10?JMD/Jtheor?

−10

−9

−8

−7

−6

0.3 0.40.5 0.60.70.8 0.91

log10(J)

∆µ

T = 0.65 T = 0.8

MD (Ref. 6)

MD (Ref. 8)

MD (Ref. 7)

DFT

SGT

EMLD−DNT

DIT

CNT

FIG. 1. Nucleation rates for a LJ fluid truncated and shifted at 2.5? as

function of the chemical potential difference between supersaturated and

saturated vapors. Shown are results from MD simulations and theoretical

calculations. Temperatures are given in units of ?/kBand chemical potential

in units of ?.

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at each studied temperature, although some scatter in the

vertical direction ensues. Such a plot is shown in Fig. 2. The

larger symbols relate to simulations of Refs. 6 and 8 and the

smaller symbols to the simulations of Ref. 7. Again we see

that DFT, SGT, and EMLD-DNT manage best in predicting

the MD results with the nucleation rate always within one

order of magnitude from the simulated one. EMLD-DNT,

however, shows a little more deviation than DFT and SGT.

DIT, which on the basis of Fig. 1 showed considerable de-

viation at T=0.65, actually fares much better at higher tem-

peratures. At T=0.9, CNT seems to indicate an almost per-

fect match between theory and simulation, but experiments

show that there is usually one temperature where the nucle-

ation rates from CNT intersect the experimental ones.38

The main trend in Fig. 2 seems to be that for all theories,

log10?JMD/Jtheor? decreases with increasing temperature; in

other words, the theoretical nucleation rate increases with

respect to the simulated rate. Only at T=0.8 the ratios shown

as large symbols are somewhat elevated; this may be a real

effect or an artifact caused by incompatibility of the MD EoS

of Ref. 7 with the pressure data of Ref. 6

Kalikmanov et al.15compared mean-field kinetic nucle-

ation theory and EMLD-DNT to MD simulations of Refs. 39

and 18 and to DFT calculations of Ref. 14. Our Fig. 2 should

be compared with Fig. 2 of Ref. 15. In accordance with our

results, Kalikmanov et al. found that EMLD-DNT closely

predicts the MD nucleation rates. However, the DFT rates in

Ref. 15 are three to four orders of magnitude higher than

EMLD-DNT rates, whereas our calculations show that the

EMLD-DNT and DFT rates differ less than one order of

magnitude. The likely reason for this discrepancy is that in

Ref. 15 the DFT data of Ref. 14 were used without any

parameter fitting to the bulk data.

B. Critical cluster size

There is some interest to appraise the critical cluster

sizes from theoretical approaches. The comparison to simu-

lations is not straightforward because the critical cluster size

is not a uniquely defined quantity. There exist cluster defini-

tions based on the thermodynamics of surfaces, geometric

definitions, and energetic rules. The most generally appli-

cable definition is given by the nucleation theorem ?NT?,40

which gives the excess number of particles in the critical

cluster as

?N?? kBT? ln J

???− 1.

?12?

Only the nucleation rate data are required to calculate the NT

size of the cluster in a given vapor. The uncertainty in ?N?in

Eq. ?12? is about one particle.

The cluster definition based on Eq. ?12? is adopted in

this study. To obtain the cluster sizes from the simulation

data, we have fitted the MD nucleation rates at each tempera-

ture to a simple CNT-like function ln J=a?T?????−2+b?T?,

which, partly owing to the narrow range of MD data in the

?? space, gives a good estimate of the actual nucleation

rates. Note, however, that a cluster definition is already

needed to obtain the MD nucleation rates from MFPT analy-

sis and NT cannot be used for this purpose. We have reported

nucleation rates according to the Stillinger definition41and

the ten Wolde–Frenkel ?TWF?42definition in our previous

paper.7The two cluster definitions yield practically the same

nucleation rates, but a slight difference in ??-dependence of

the nucleation rate curves causes a noticeable change in the

cluster sizes. The TWF-based rates are used in this paper.

Applying the Stillinger definition in MFPT analysis instead

of the TWF definition decreases the NT cluster sizes 2.3% at

T=0.65 and 4.3% at T=0.8.

In EMLD-DNT and DIT the critical cluster sizes are

obtained via NT by calculating numerical derivatives of the

nucleation rate data. DFT and SGT give the excess number

of particles in the cluster as an integrated quantity from the

density profile, but this quantity agrees with NT.

Figure 3 shows the critical cluster sizes according to the

theories and simulations at T=0.65, T=0.7, and T=0.8. For

comparison, we also show CNT values and the particle num-

ber excesses obtained from the cluster-vapor equilibrium

simulations of our previous study.7DFT seems to give the

best theoretical results, although at T=0.8 the sizes are over-

estimated. DFT is closely matched by SGT with only slightly

larger sizes. CNT and especially DIT underestimate the MD

sizes and EMLD-DNT, in the range of supersaturations of

the MD data sets, overestimates them. We note that due to

the scatter in the MD data of Ref. 6 at T=0.8, the fitting of

the nucleation rate and resulting cluster sizes are unreliable.

The EMLD-DNT sizes are too large at higher saturation

ratios and the dependence of size on ????−3is clearly wrong.

While EMLD-DNT predicts MD nucleation rates quite well

according to Fig. 1, the slopes of the EMLD-DNT nucleation

rate curves are slightly different from DFT, SGT, and MD

curves, which produces a considerable effect in Fig. 3 when

NT is used to calculate the cluster sizes. At lower saturation

ratios, where no MD simulations are available, the EMLD-

DNT sizes approach the CNT line. The problems of EMLD-

DNT at high saturation ratios have already been addressed

by Reguera and Reiss in Ref. 11. The pressure of the vapor

in the small container in EMLD-DNT does not correspond to

that given by EoS, which distorts the relation between Pmin

and Vmin.

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

0.6 0.650.70.75 0.8 0.850.90.95

log10(JMD/Jtheor)

Temperature

CNT

DIT

EMLD−DNT

SGT

DFT

FIG. 2. The logarithmic ratios of simulated and theoretical nucleation rates

at different temperatures. The larger symbols refer to simulations of Refs. 6

and 8 and the smaller symbols to the simulations of Ref. 7.

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C. Effect of equation of state

We have used DFT EoS to achieve consistency between

the nucleation theories in metastable vapor states. However,

in SGT and EMLD-DNT one is free to choose any EoS. In

our earlier work21we used Peng–Robinson ?PR? EoS ?Ref.

43? to calculate SGT rates. Motivated by the apparent suc-

cess of SGT in that paper, we have also performed the SGT

and EMLD-DNT calculations with PR EoS.

Resembling van der Waals EoS, PR EoS is a two-

parameter model. We fitted the parameters to reproduce the

MD equilibrium vapor pressure and liquid density. The re-

sulting EoS gives a better approximation of the correlation

between the density and pressure of the metastable LJ vapor

than DFT EoS, which overestimates the pressure. However,

the calculated nucleation rates deviate more from the MD

values than the rates obtained using DFT EoS: compared to

the results presented in Figs. 1 and 2, the SGT rates increase

one to two orders of magnitude and the EMLD-DNT one

order of magnitude at high supersaturations. EMLD-DNT

nucleation rates are close to MD rates at the lower end of the

vapor density range, but the slope of the nucleation rate

curves is erroneous. When NT is used, the wrong slope trans-

lates to much larger cluster sizes than those shown in Fig. 3.

The SGT sizes are also somewhat larger.

PR EoS also causes a problem in EMLD-DNT calcula-

tions. At high saturation ratios, there may not be a minimum

on a P?V? curve, even though a ?hypothetical? small-system

EoS would indicate otherwise. In this study, we did not find

the critical cluster at high saturation ratios at T=0.85 and T

=0.9 when PR EoS was used.

V. CONCLUSIONS

We have performed an assessment of several nucleation

theories by comparing the calculated nucleation rates and

critical cluster sizes to MD simulation values in the nucle-

ation of LJ vapors. The comparison has been done so that the

theories either use or predict the same bulk thermodynamic

properties as MD. Realistic EoS has been used to account for

the nonideality of the vapor. The studied temperatures are

above the triple point of the LJ fluid and the saturation ratios

correspond to vapor conditions where the nucleation rates

are high.

Ourresults supportthe

nucleation11,15,18,21and they also seem to be in accordance

with the studies where the theories were compared to real

nucleation experiments,9,19,24with the possible exception of

SGT.22,23It is therefore justified to draw general conclusions

based on the results of this study.

The best results are obtained from DFT, which repro-

duces both the MD nucleation rates and critical cluster sizes

rather well. DFT is unfortunately limited to rather simple

fluids, where the intermolecular interactions can be described

by a spherically symmetric potentials unless a more compli-

cated version of the theory, such as interaction site model,44

is applied. SGT does not need an interaction potential and

the SGT nucleation rates and cluster sizes are close to DFT

values. Unfortunately, SGT is quite sensitive to the choice of

EoS. DIT somewhat underestimates the nucleation rate, but it

is still much better than CNT when predicting nucleation

rates. DIT is easy to use because it is almost as simple as

CNT and needs only the heat of evaporation as an additional

input, butitsapplicability

questionable.9EMLD-DNT almost equals DFT in predicting

nucleation rates, but the dependence of the nucleation rate on

saturation ratio is slightly wrong, which results in erroneous

cluster sizes. This problem arises at high vapor densities,

where EMLD-DNT is also highly dependent on EoS.

EMLD-DNT is thus at its best when applied to nearly ideal

vapors.

previousresultson LJ

topolar substances is

ACKNOWLEDGMENTS

This research was supported by the Academy of Finland

Center of Excellence program ?Project No. 1118615?.

10

20

30

40

50

60

70

80

1 1.52 2.533.5

∆N*

(∆µ)−3

T = 0.65

EMLD−DNT

MD profile

SGT

MD (Refs. 6 and 8)

MD (Ref. 7)

DFT

CNT

DIT

15

20

25

30

35

40

45

50

55

60

22.53 3.544.55

∆N*

(∆µ)−3

T = 0.7

10

20

30

40

50

60

70

80

90

100

110

10 1214

(∆µ)−3

16 1820

∆N*

T = 0.8

FIG. 3. The theoretical and simulated excess number of particles in the

critical cluster at T=0.65, T=0.7, and T=0.8. The MD sizes are based on

fitting the nucleation rate data of Refs. 6–8 and using the nucleation theo-

rem. The MD profile size is obtained from the cluster-vapor equilibrium

profiles of Ref. 7.

154503-6Napari, Julin, and VehkamäkiJ. Chem. Phys. 133, 154503 ?2010?

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Page 7

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