JOURNAL OF CHEMICAL PHYSICSVOLUME 110, NUMBER 7

15 FEBRUARY 1999

Thermodynamic expansion of nucleation free-energy barrier and sizeof critical nucleus near the vapor-liquid coexistence
Kenichiro Kogaa) and X. C. Zeng
Department of Chemistry and Center for Materials Research and Analysis, University of Nebraska-Lincoln,
Lincoln, Nebraska 68588~Received 28 September 1998; accepted 10 November 1998!
Nucleation free-energy barrier height and size of the critical nucleus are expanded in powers of thechemical potential difference between the supersaturated vapor ~or expanded liquid! in themetastable state and the saturated vapor-liquid system in the stable equilibrium state at the same
temperature. The coefficients in the expansions are expressed in terms of the thermodynamic
properties at the stable equilibrium state. Comparisons with the results obtained from thedensity-functional calculation for nucleation of the Lennard–Jones fluid show that systematic
improvement in predicting properties of the critical nucleus, either liquid droplet or vapor cavity, isachieved by adding the higher order terms in the expansions. The scaling relations proposed by
McGraw and Laaksonen are found to be good approximations to the general expansion; inparticular, the barrier height displacement appearing in these scaling relations is naturally given as
the second order coefficient in the expansion of the barrier height. © 1999 American Institute ofPhysics. @S0021-9606~99!50707-1#

ticle, McGraw and Laaksonen8 discussed the physical basis
for the barrier height displacement D(T) using a nonuniform
spherical droplet model of the nucleus.
The advanced scaling relations by McGraw and Laaksonen have made an important step towards developing better phenomenological theory of nucleation for describing experiments. Talanquer proposed a simple method to determine* for
D(T), that is, D(T) can be obtained by evaluating W CNT
the critical nucleus at the spinodal. The spinodal is calculated
from the equation of state, obtained by fitting the experimental values of vapor pressure and binodal curve. Talanquer
found that the McGraw–Laaksonen scaling relations improve the description of homogeneous nucleation for both
nonpolar and weakly polar systems. It seems that McGraw–
Laaksonen’s scaling relations indeed provide a simple explanation for reported systematic discrepancies between CNT
and measurements.10
However, it is fair to mention that the McGraw–
Laaksonen scaling relations have been obtained as homogeneous solutions to a differential equation derived from thenucleation theorem ~21! and a general form for W * . 7,11 Since
the homogeneity ansatz has not been proved in general, it
remains unanswered why the scaling relations of McGraw
and Laaksonen are overall successful. Furthermore, the following two questions also need to be addressed: Can thescaling relations be derived and justified based on a more
rigorous theoretical foundation? Can the barrier height displacement D(T) be given in terms of thermodynamic properties at stable equilibrium state, rather than determinedbased on the spinodal condition? In this regard, it is important to pursue the derivation of W * and Dn * rigorously
based on the Gibbs theory of capillarity. In this article we
present thermodynamic expansions of W * and Dn * in powers of the chemical potential difference Dm between the su-

I. INTRODUCTION

A central concern in the development of theory of nucleation, either a phenomenological one or a molecular-based
one, is the nucleation free-energy barrier height W * , for thisquantity primarily controls the rate of nucleation.1 The capillary drop model in the classical nucleation theory2 ~CNT!,
for example, expresses the barrier height in terms of several
measurable macroscopic properties including the tension ofplanner vapor–liquid interface. The latter quantity is assumed to be the same as the surface tension of the critical
nucleus; in other word, the Tolman length is assumed to be
zero in the classical nucleation theory. Modern theory of
nucleation, such as the i/ v cluster theory and its derivative
by Reiss and co-workers3 or the density-functional ~DF!
theory of nucleation by Oxtoby and co-workers,4–6 offers amolecular-based approach to obtain W * ~the Tolman length
can be given from DF approach as well!. These theories
reveal that the assumptions underlying the CNT are, in certain circumstances, invalid. In turn, these molecular-based
theories stimulated new efforts in searching for better phenomenological theories of nucleation.7–9 Recently, McGraw
and Laaksonen7 proposed a set of scaling relations for thesize of critical nucleus Dn * , the nucleation barrier heightW * and their combination. According to their scaling relation the true barrier height W * differs from the CNT predic* by a function only of temperature D(T) ~alsotion W CNT
called the barrier height displacement!, that is, the difference* depends neither on the nucleus size nor the suW * 2W CNT
persaturation ratio S of vapor. Within the framework of DF* only changes aapproach, they confirmed that W * 2W CNT
few percent at various supersaturations while the barrierheight itself varies by an order of magnitude. In a later ara!

Electronic mail: koga@super2.unl.edu

0021-9606/99/110(7)/3466/6/$15.00

3466© 1999 American Institute of Physics

J. Chem. Phys., Vol. 110, No. 7, 15 February 1999

K. Koga and X. C. Zeng

persaturated and saturated vapor at a given temperature. We
show that the McGraw–Laaksonen scaling relations are recovered by considering only the second-order term in the* . Theexpansion of W * as a correction to W CNT
supersaturation-independent displacement D(T) is in fact
just the coefficient of the second-order term. The coefficientsin the expansion including D(T) are expressed in terms of
the bulk and surface properties at the stable equilibrium stateof vapor-liquid coexistence. Among others, these properties
include the Tolman length12 and the fundamental lengthcharacteristic of liquid proposed by Egelstaff and Widom.13
In the next section, we derive the expansions of thermodynamic formulas for the nucleation barrier height W * andthe size of critical nucleus Dn * . In Sec. III, we examine theeffects of truncation of the higher order terms in the expansions within the framework of the density functional theoryof vapor-liquid nucleation. Our concluding remarks are
given in Sec. IV.

3467

`

1
A i i12x ,g ~ x ! 5 C1Bx2Dx 2 2
2
i51 i

(

~3!

where x5D m andC52g ~ 0 ! ,~4!

B5g 8 ~ 0 ! ,~5!

1
D52 g 9 ~ 0 ! ,2~6!

A i 52ig ~ i12 ! ~ 0 ! ,~ i12 ! !

~7!

where 8, 9, and (n) stand for, respectively, the first, second,and nth derivatives with respect to x (5D m ). Once thesecoefficients are obtained in terms of thermodynamic properties at the vapor-liquid coexistence, the nucleation freeenergy barrier height is given as an explicit function of Dm:`

Ai1
W * 5 C ~ D m ! 22 1B ~ D m ! 21 2D2~ D m !i.
2
i51 i

(

II. EXPANSIONS OF NUCLEATION FREE-ENERGYBARRIER HEIGHT AND THE SIZE OF CRITICAL
NUCLEUS IN POWERS OF THE CHEMICALPOTENTIAL

Systematic calculation of the coefficients can be carriedout in the following manner. Let y(x)5Dp and z(x)5y(x)/x for convenience. Then g(x) is rewritten as

We consider vapor-to-liquid and liquid-to-vapor nucleation of a single component fluid under a condition of fixed
temperature T. Here the nucleation barrier height W * , whichis the work of formation of the critical nucleus, and the sizeof the nucleus Dn * , which is the excess number of molecules in the critical nucleus over that present in the same
volume in the surrounding bulk phase, are considered asfunctions only of the chemical potential m of the system or
the difference Dm from that of the saturated vapor. In thissection we show that W * and Dn * can be expanded aroundD m 50 and that the coefficients in the expansion are expressed in terms of the bulk and surface properties at thestable equilibrium state.
A. Nucleation free-energy barrier height

We start from the formally exact expression for the
nucleation free-energy barrier height ~or the work of formation! of the spherical critical nucleus14
W *5

16ps 3s
3 ~ Dp !

2,~1!

where s s is the surface tension with respect to the surface oftension and Dp[ p a 2p b is the pressure difference between
the homogeneous liquid ~a! and vapor phase ~b! having thesame chemical potential; in the case of vapor-to-liquid nucleation (D m .0), phase a is thermodynamically stable
whereas phase b is metastable. Note that Eq. ~1! is valid up
to the smallest critical nucleus. It is clear from Eq. ~1! that
W * , a function of Dm, has the pole of second order at D m
50. Therefore a function of Dm defined by
g5W * ~ D m ! 2

~8!

~2!

does not diverge at D m 50. In general this function can be
expanded asg~ x !516ps 3s3z 2~9!

and in general the nth derivative g (n) (x) is expressed interms of s s , s s8 , . . . , s (n)and z,z 8 , . . . ,z (n) . When we cals
culate the limits of g(x), g 8 (x), . . . ,g (n) (x) at x50, we use
the following relations ~10!–~17!: First, since z5y(x)/x andy(0)50 the limits of the derivatives of z and of y at x50
are related by the mathematical relation:
lim z ~ n ! 5y ~ n11 ! ~ 0 ! / ~ n11 ! ,

~10!

x→0

including the case n50. Second, we have the following thermodynamic relations for the bulk properties:
y 85
y 95

S D
S D
S D

] Dp
5D r ,]m T~11!

] 2 Dp
5D r 2 x ,
]m2 T

~12!

y ~ n12 ! 5] n12 Dp
5 ~ D r 2x !~ n !,] m n12 T~13!

where D r 5 r a 2 r b is the density difference between thetwo homogeneous phases and D r 2 x 5( r a ) 2 x a 2( r b ) 2 x b isthe difference in the product of square of density ( r 2 ) andisothermal compressibility ~x! between the two phases.
Third, we have a thermodynamic relation for the surface
properties,
lim s s8 /D r 52 d ` ,

~14!

x→0

where d ` is the Tolman length originally defined as the limitof distance from the surface of tension to the equimolar di-

3468

J. Chem. Phys., Vol. 110, No. 7, 15 February 1999

K. Koga and X. C. Zeng

viding surface as x→0.12 Equation ~14! stems from two relations about the adsorption at the surface of tension:12,14,15~i! G s 52( ]s s / ] m ) T and ~ii! d ` 5limD m →0 G s /D r . Hereinafter, we denote the surface tension and its derivatives at x
50 @except the first derivative replaced by Eq. ~14!# as s,s 9 , etc. without the subscript s.
If we calculate the limits of g(x), g 8 (x), . . . ,g (n) (x) as
x→0 using the relations ~10!–~14! and then substitute thelimits into Eqs. ~4!–~7!, we obtain the desired thermodynamic expressions for the coefficients of the expansion of thenucleation barrier height. The first five coefficients are given
asC5

32ps 3,3~ Dr !2A 2 52

B52D5216ps
1
A 15

~15!

F

H

GF

G
G F
F

F

GF
GJ

H

sDr x
~ Dr !22~20!

which has a dimension of length. Note that near the triple
point, r a @ r b and ( r a ) 2 x a @( r b ) 2 x b ; therefore D ` ' sx a ,which is the fundamental length characteristic of the liquid
proposed by Egelstaff and Widom.13 The quantities in Eqs.
~15!–~20! are all defined at the stable equilibrium state ofvapor-liquid coexistence (D m 50); the coefficients are
therefore functions only of T.
Since W * is expressed in terms of the bulk property(Dp) and the surface property ~s! of the system, the coefficients of the expansion about D m 50 consist of the derivatives of the bulk and surface properties with respect tochemical potential at D m 50. For example, expression of thecoefficient C includes the first derivative of the bulk pressure
as Dr whereas the coefficient B contains the derivative of thesurface tension as 2 d ` D r and the second derivative of pressure as D r 2 x . In general a higher-order term’s coefficient
has higher-order derivatives of the bulk pressure and surface
tension with respect to the chemical potential.

~17!

,

G

F

GF

2

s3
s - ~ D r 2x !92
2
2~ Dr ! s6D r

GF

3 s 3 s 9 ~ D r 2x !8 2
s3
s 99 2 ~ D r 2 x ! 2
1
2
4~ Dr !2 s
3D r
8~ Dr !2 s
15D r

where the quantity D ` is defined as

FGJ

3 s 2 s 9 ~ D r 2x !81
2
d `1 D `
Dr s
3D r
2

1

G

F

2

1

s~ D r 2x !8s 2 s - ~ D r 2x !91
1
2
d `1 D `@ d ` 1D ` # d ` 1 D ` 2
Dr
2
Dr s
4D r
2

F

1
2

d `1 D `

~16!

2
16p
1
D r @ d ` 12D ` # d ` 1 D `
3
2

F

2
32p ~ D r ! 2
5
1
3s9
3
1
D ` d `1 D ` d `1 D ` 1
d `1 D ` d `1 D `
3
s
4
2
2
2
2

F

G
G

s2
s 9 2~ D r 2x !82
2~ Dr !2 s
9D r

2

D `5

F
HF

16ps 21
d `1 D ` ,
Dr
3

G

GJ

G

,

~18!

G
~19!

,

nucleus defined by introducing a dividing surface, the excess
Dn * is independent of the choice of the dividing surface.
The nucleation theorem16–18 gives a relationship between
Dn * and W * ,

S D

]W*
52Dn * .
]m T

~21!

Note that Eq. ~21! is a general thermodynamic relation asproved by Oxtoby and Kashchiev.17 From the nucleation
theorem ~21!, the expansion of Dn * is immediately obtained
from differentiating Eq. ~8! with respect to D m as:`

Dn * 5C ~ D m !

23

1B ~ D m !

221( A i~ D m ! i21 .

i51

~22!

Note also that the term (D m ) 21 does not exist in the expansion.
Expanding f (x)[Dn * x 3 52W * 8 x 3 as done for g(x),the same coefficients for Dn * are obtained. In particular, wefind f 9 (0)50, which is consistent with that the term (D m ) 21
does not exist in Eq. ~22!. It should be noted, however, thatthe thermodynamic expression ~17! for the coefficient D cannot be obtained from Dn * @Eq. ~22!# to W * @Eq. ~8!# by
integration.

B. Expansion of the size of critical nucleus

The size of critical nucleus Dn * is defined as the excess
number of molecules in the critical nucleus over that present
in the same volume in the homogeneous phase surrounding
the nucleus. Unlike the number of molecules in the critical

C. Relation with other phenomenological approaches

The expansions in Eqs. ~8! and ~22! for W * and Dn * aregeneral thermodynamic results, for they are derived based
solely on the Gibbs theory of capillarity and the nucleation

J. Chem. Phys., Vol. 110, No. 7, 15 February 1999

K. Koga and X. C. Zengtheorem. The expansions are analogous to the virial expansion of the equation of state. One can expect better results as
more higher order terms are used. In fact, we found that theclassical expressions for the barrier height and the size ofcritical nucleus and the scaling relations by McGraw andLaaksonen are special forms of the general results.
First, the classical expression for the nucleation barrierheight is recovered if B50, D50, A i 50 for all i:* 5 21 C ~ D m ! 22 ,W CNT

* 5C ~ D m !
Dn CNT

23.

TABLE I. Thermodynamic properties at stable equilibrium of vapor-liquid
coexistence determining the coefficients in the thermodynamic expansionsof the nucleation free-energy barrier height and the size of critical nucleus.
The primes denotes derivatives with respect to chemical potential at fixed
temperature. The values are obtained from the density functional approachto nucleation of the Lennard–Jones fluid. See the text for details. Temperature is 0.7 in reduced units. All the values are given in reduced units.

~23!

where C is given by Eq. ~15!. ~Dr in C is usually replaced by
r a , which is reasonable near the triple point.! Under thesame condition, the classical expression for Dn * ~the Kelvinrelation! is also recovered:~24!

Therefore, the capillary drop model in CNT is the zerothorder approximation to the general expansions, which is only
exact in the limit as D m →0. This is analogous to the ideal
gas approximation to the virial expansion of the equation ofstate.
Second, the McGraw and Laaksonen scaling relations7
are recovered if B50 and A i 50 for all i. The scaling relation for Dn * is the same as the classical expression ~24!.
From the nucleation theorem ~21!, integrating Eq. ~24! withrespect to Dm gives rise to the scaling relation for W * . Here,2D appears as an integration constant although D does notexist in the CNT expression ~23!. Furthermore, combining
the two scaling relations for Dn * and W * results in thescaling relation for W * /Dn * D m . It has been shown7 that
results of W * and Dn * obtained from DF calculation follow
the scaling relations over a wide range of Dm if D is deter* 2W * . Recently,
mined by the average displacement W CNT
Talanquer9 proposed that the constant D is determined fromthe boundary condition that W * 50 at spinodal. However,since BÞ0 and A i Þ0 in general, the McGraw–Laaksonen
scaling relations are basically particular approximations to
the general expansions. In principle, D should be determinedby Eq. ~17!. The reason why the McGraw–Laaksonen scaling relations are successful for a wide range of Dm will be
discussed in the next section.

III. NUMERICAL RESULTS AND DISCUSSION

In this section, the analytical results in Sec. II are examined by numerical calculation of the Lennard–Jones model
fluid based on the density-functional theory of nucleation.4,5The details of the DF approach to the LJ fluid nucleation
have been given elsewhere.5 The properties of the critical
nucleus are computed over a wide range of Dm ~from 21.2
to 1.8 in reduced units! at a fixed temperature T near the
triple point (T * 50.7 in reduced units, corresponding in thecase of argon to T584 K). Note that the negative Dm corresponds to the liquid-to-vapor nucleation or bubble formation.
Determination of the coefficients in the expansion requires the bulk and interfacial properties at the stable equilibrium state (D m 50). The bulk properties such as D r 2 x
and higher derivatives of D r 2 x with respect to Dm are

3469

a

Coefficient

Value

Property

Value

Property

Value

C
B
D
A1
A2

100.15
11.35
29.00
21.5a
269.2a

s
s8
s9
ss 99

1.287
0.131
20.5200.272
5.784
20.155

Dr
D r 2x(D r 2 x ) 8(D r 2 x ) 9(D r 2 x ) D`

0.844
0.066
20.0380.038
20.0910.120

d`Expected numerical errors are about 61 for A 1 and 640 for A 2 due to theuncertainty in determining s - and s 99 .

obtained from the equation of state for m ( r ,T) of the model
fluid5 through the thermodynamic relations: r 2 x
5( ] m / ]r ) 21 , ( r 2 x ) 8 52( ] 2 m / ]r 2 )( ] m / ]r ) 23 , etc. Calculation of the surface properties requires the density profileof the inhomogeneous system. The surface tension s is obtained by DF calculation of the planner interface. The derivatives of surface tension s 8 , s 9 , s - , and s 99 are evaluatedby fitting a polynomial to a set of data ~Dm, s s ). 19 The
Tolman length d ` is determined from s 8 ~the adsorption
route20 to d ` ). Substituting these results into Eqs. ~15!–~19!
gives the coefficients. Table I summarizes the coefficients
together with the properties determining the coefficients. The
Tolman length d ` and the fundamental length D ` characteristic of the liquid are comparable in magnitude and opposite
in sign.21 Therefore the sum @ d ` 1kD ` # with the factor k oforder of 1 is very small. Thus, the coefficient B containing
the sum is much smaller than C. For the same reason, theterms having @ d ` 1kD ` # in Eqs. ~17!–~19! are negligible fordetermining the coefficients D, A 1 , and A 2 . The coefficient
D is primarily determined by the term having s 9 in Eq. ~17!,
coefficient A 1 by s - in Eq. ~18!, and coefficient A 2 by s 99
in Eq. ~19!.
The expansion formula in Eq. ~8! for W * is examinedthrough comparisons with results obtained from DF calculation for the LJ fluid model. The quantity 2W * (D m ) 2 /C isplotted in Fig. 1 against Dm in reduced units. The horizontal
lines in Fig. 1 correspond to prediction of the classical nucleation theory. The zeroth order approximation works better inthe case of vapor-to-liquid nucleation @Fig. 1~a!# compared toliquid-to-vapor nucleation @Fig. 1~b!# due to the asymmetryin 2W * (D m ) 2 /C. The first-order correction to CNT by introducing the linear term with coefficient B gives the correct
slope at D m 50; however the range of its validity is as limited as that of CNT. Inclusion of the second-order term withcoefficient D makes significant improvement to the firstorder approximation. Because of the parabola-like nature of2W * (D m ) 2 /C, the prediction of the second-order approximation agrees qualitatively well with the result obtained
from DF calculation over a whole range of Dm. The thirdorder approximation by including the cubic term with coefficient A 1 improves the second-order prediction little, because A 1 is much smaller than D when compared in reduced

3470

J. Chem. Phys., Vol. 110, No. 7, 15 February 1999

FIG. 1. Nucleation free-energy barrier height W * multiplied by (D m ) 2 /C:
the density-functional result for the Lennard–Jones critical nucleus ~black
circles!, the classical nucleation theory ~the horizontal lines!, first ~dasheddotted lines!, second ~dotted curves!, third ~dashed curves! and fourth ~solid
curves! order approximation of the expansion formula. The coefficients inthe expansion were obtained from the density functional calculation of thevapor–liquid interface at stable equilibrium and the critical droplets andbubbles. Temperature is 0.7 in reduced units. All the values are in reduced
units.

units. Inclusion of the fourth-order term gives rise to an excellent agreement with the result of DF calculation within therange: u D m * u <0.5 as shown by the solid curves in Figs. 1~a!
and 1~b!. However deviation from the DF result increases
rapidly in the range of large Dm. The scaling relation for W *proposed by McGraw and Laaksonen does not have the termwith coefficient B but does include the important term withcoefficient D. This is the main reason why the scaling relation with a single parameter D ~beside C) significantly improves CNT prediction over a wide range of Dm. However inorder to account for the asymmetrical feature of2W * (D m ) 2 /C, the inclusion of the term with coefficient B
is essential.
The expansion formula in Eq. ~22! for Dn * is also examined through comparison with the DF result. The quantity
Dn * (D m ) 3 /C is plotted against Dm in reduced unit in Fig. 2.
It is found that Dn * (D m ) 3 varies only 15% over a whole
range of Dm. Therefore, the zeroth order approximation for
Dn * ~the Kelvin relation! corresponding to the horizontal
lines in Fig. 2 is a good approximation beyond a small range
close to D m 50. This is in contrast to the zeroth-order ap-

K. Koga and X. C. Zeng

FIG. 2. Size of the critical nucleus Dn * multiplied by (D m ) 3 /C: thedensity-functional result ~black circles!, the classical nucleation theory ~thehorizontal lines!, first ~dashed-dotted lines!, third ~dashed curves!, andfourth ~solid curves! order approximation of the expansion formula. The
conditions are the same as in Fig. 1. All the values are in reduced units.

proximation ~CNT! for W * . Introduction of the first-orderterm with coefficient B improves further the classical prediction and gives the correct slope at D m 50. Improvement
made by the third-order term with coefficient A 1 is found to
be insignificant as in the case for W * . The expansion up to
the fourth-order term with coefficient A 2 betters the result ina certain range up to 60.4 @Figs. 2~a! and 2~b!#, but gives
rise to large errors for large Dm. These results show that thescaling relation for Dm, i.e., extension of the Kelvin relation
to a wide range of Dn * , is reasonable though inexact. The
other scaling relations are also successful because they arederived from the scaling relation for Dn * .
IV. CONCLUSION

The nucleation free-energy barrier height W * and thesize of the critical nucleus Dn * are expressed in powers of
the chemical potential difference Dm from the stable equilibrium state. The coefficients in the thermodynamic expansion
are written in terms of the bulk and surface properties at thestable equilibrium state of the vapor-liquid coexistence.
Through comparisons with the density functional results, wehave observed systematic improvement of the prediction of
the expansion formulas as adding the higher order terms bothfor W * and Dn * . We have found that introduction of the

J. Chem. Phys., Vol. 110, No. 7, 15 February 1999

second order term with coefficient D is crucial to describe
the parabola-like behavior of W * (D m ) 2 over a wide range of
Dm. Asymmetry in the behavior of W * (D m ) 2 between
vapor-to-liquid and liquid-to-vapor nucleation is accounted
for by the first order term with coefficient B, which include
the sum of the Tolman length and the Egelstaff and Widom
fundamental length characteristic of liquid. For the behavior
of Dn * (D m ) 3 , we have found that the zeroth order termalone is a good approximation beyond a small range of Dm.
Inclusion of the first order term with coefficient B accounts
for the asymmetry in Dn * (D m ) 3 and gives rise to more precise prediction in a wide range of Dm.
From the general expansion formulas we can recover thescaling relations for the critical nucleus proposed by
McGraw and Laaksonen via neglecting the first order coefficient B and the third and higher order coefficients. Furthermore the parameter D(T) in the scaling relations is identified
as the second order coefficient in the expansion of W * andcan be given explicitly in terms of the thermodynamic properties at the stable equilibrium. The scaling relation for Dn * ,corresponding to the zeroth order approximation to the expansion, is indeed a reasonable approximation for both the
droplet and bubble formation. Thus other scaling relations
resulted from this relation through the nucleation theorem
are also good approximations given that an appropriate value
for the parameter D(T) is known. However, the scaling relations do not account for any asymmetry in the behavior ofW * (D m ) 2 and Dn * (D m ) 3 ; instead, they predict complete
symmetry of these quantities with respect to D m 50 and result in d ` 52D ` /3, a particular relation between the Tolman
length and the fundamental length characteristic of the liquid. However, W * (D m ) 2 and Dn * (D m ) 3 obtained from theDF calculation of the LJ fluid exhibit asymmetry between
droplet nucleation (D m .0) and bubble formation (D m,0), and the relation between d ` and D ` is rather given byd ` '2D ` . Therefore, in general, the first-order term withcoefficient B is necessary in order to describe asymmetricalfeatures for the critical nucleus.

K. Koga and X. C. Zeng

3471

ACKNOWLEDGMENTS

We thank Dr. A. K. Shchekin for fruitful discussions.
X.C.Z. thanks NSF for support of this work.
1

See, for example, A. Laaksonen, V. Talanquer, and D. W. Oxtoby, Annu.
Rev. Phys. Chem. 46, 489 ~1995!.
2
M. Volmer and A. Weber, Z. Phys. Chem. 119, 277 ~1925!; R. Becker and
W. Döring, Ann. Phys. 24, 719 ~1935!; J. Frenkel, J. Chem. Phys. 7, 538~1939!.
3
H. M. Ellerby and H. Reiss, J. Chem. Phys. 97, 5766 ~1992!; C. L. Weakliem and H. Reiss, J. Chem. Phys. 98, 6408 ~1994!; C. L. Weakliem and
H. Reiss, J. Chem. Phys. 101, 2398 ~1994!.
4
D. W. Oxtoby and R. Evans, J. Chem. Phys. 89, 7521 ~1988!.
5
X. C. Zeng and D. W. Oxtoby, J. Chem. Phys. 94, 4472 ~1991!.
6
V. Talanquer and D. W. Oxtoby, J. Chem. Phys. 100, 5190 ~1993!.
7
R. McGraw and A. Laaksonen, Phys. Rev. Lett. 76, 2754 ~1996!.
8
R. McGraw and A. Laaksonen, J. Chem. Phys. 106, 5284 ~1997!.
9
V. Talanquer, J. Chem. Phys. 106, 9957 ~1997!.
10
G. W. Adams, J. L. Schmitt, and R. A. Zalabsky, J. Chem. Phys. 81, 5074
~1984!; R. Strey, P. E. Wanger, and T. Schmeling, ibid. 84, 2325 ~1986!.
11
We note that if validity of the Kelvin relation beyond the limit D m →0 is
assumed, the McGraw–Laaksonen scaling relations can be also derived
without using the homogeneity ansatz.
12
R. C. Tolman, J. Chem. Phys. 17, 333 ~1949!.
13
P. A. Egelstaff and B. Widom, J. Chem. Phys. 53, 2667 ~1970!.
14
J. W. Gibbs, The Scientific Papers of J. Willard Gibbs ~Ox Bow, Woodbridge, 1993!.
15
J. S. Rowlinson and B. Widom, Molecular Theory of Capillarity ~Oxford
University, New York, 1989!.
16
D. Kashchiev, J. Chem. Phys. 76, 5098 ~1982!.
17
D. W. Oxtoby and D. Kashchiev, J. Chem. Phys. 100, 7665 ~1994!.
18
Y. Viisanen, R. Strey, and H. Reiss, J. Chem. Phys. 99, 4680 ~1993!.
19
Several polynomials of different order ~from 6 to 16! are used to fit a setof data ~Dm, s s) in a range of Dm between 20.5 and 1.2 in reduced units.
The third and fourth order fitting parameters which determine s -8 and s
are relatively sensitive to the choice of polynomial, compared to the firstand second parameters which determine s 8 and s 9 . However we confirmed that our conclusions do not depend on the choice of the polynomial. The results given here are obtained by ten-parameter fitting.
20
K. Koga, X. C. Zeng, and A. K. Shchekin, J. Chem. Phys. 109, 4063
~1998!.
21
There are some theoretical results which imply the close relationship between the Tolman length and the fundamental length. First, within aframework of the gradient theories the Tolman length is given analyticallyas d ` 52D ` . ~Refs. 22, 23! Also DF calculation of LJ and Yukawa
model fluids shows that the relation, d ` (T)52kD ` (T) where k is aconstant close to 1, holds at least for a range of T near the triple point~Ref. 23!.
22
M. Iwamatsu, J. Phys.: Condens. Matter 6, L173 ~1994!.
23
T. V. Bykov and A. K. Shchekin ~unpublished!.