11586
J. Phys. Chem. B 2001, 105, 11586-11594
Statistical Mechanics of Surface Tension and Tolman Length of Dipolar Fluids† T. V. Bykov and X. C. Zeng* Department of Chemistry, UniVersity of NebraskasLincoln, Lincoln, Nebraska 68588 ReceiVed: March 7, 2001
The Tolman length (δ∞) characterizes the first-order correction to the surface tension (σ∞) due to curvature of the interface. Statistical mechanical expressions for σ∞ and δ∞ were derived previously by Blokhuis and Bedeaux [Physica A 1992, 184, 42] for nonpolar fluids. Here Blokhuis and Bedeaux’s approach is extended to dipolar fluids. Two approximations to the pair correlation function are employed, and their effects on σ∞ and δ∞ are examined. The dependence of σ∞ and δ∞ on temperature and dipole moment is also explored. It is found that σ∞ increases with the dipole moment while |δ∞| decreases as the dipole moment is increased. The sign of δ∞ is, however, negative.
I. Introduction Surface tension is a fundamental property of liquid droplets. It can be strongly dependent on the size of droplets when the size is very small. Indeed, if the radius of a spherical droplet approaches zero, the value of the surface tension diminishes. This result can be shown from thermodynamics1 and densityfunctional theory as well.2-6 In the classical theory of (droplet) nucleation,7 the surface tension appears explicitly in the expression of the activation barrier to nucleation and, thus, directly affects the nucleation rate. Since the classical theory itself cannot give the surface tension of droplets, in practice, the surface tension of planar liquid-vapor interfaces is usually invoked to calculate the nucleation rate (capillarity approximation7). Clearly, to achieve more accurate evaluation of the rate, the correction to the surface tension due to the curvature should be accounted for. Common methods for calculating the curvature dependence of the surface tension include density-functional theory (DFT)2-6 and molecular dynamics simulation.8,9 For a sufficiently large droplet, the surface tension can be expanded in powers of the curvature, which can be described by the Tolman’s formula10
[
σ(R) ) σ∞ 1 -
]
2δ(R) R
(1)
where σ(R) is the surface tension of the droplet, R is the radius of the dividing surface, σ∞ is the surface tension of a planar liquid-vapor interface, and δ(R) is the Tolman length of the droplet. Because experimental measurements of the Tolman length are still lacking, “quantitative measurements” of the Tolman length are mainly from computer simulation of model fluids.11,12 On the theoretical development, we note that Mandell and Reiss were among the early workers to study the correction to the interfacial tension due to curvature of the interface. They reported a thermodynamical approach13 combined with the scaled-particle theory14 to treat curved boundary-layer problems. Theoretical development up to mid-nineties has been reviewed in ref 15. In the past decade, various DFTs have been developed to evaluate σ and δ, with the approach either from treating a large spherical droplets3-5 or a planar liquid-vapor interface.16-18 †
Part of the special issue “Howard Reiss Festschrift”.
A more simplified gradient-expansion based DFT has also been developed.2,19 Analytical expressions of σ and δ have been obtained6,20-22 on the basis of various approximations. In this paper, we mainly study the behavior of surface tension σ∞ and the asymptotic Tolman length in the limit of R f ∞, δ∞. In this limit, δ∞ is independent of the choice of the dividing surface. Most studies of the curvature dependence of σ∞ and δ∞ thus far have been devoted on simple nonpolar fluids such as Yukawa or Lennard-Jones. To our knowledge, δ∞ of dipolar fluids has not yet been reported. To evaluate σ∞ and δ∞ for dipolar fluids involving anisotropic dipole-dipole interaction, we extend the statistical mechanical approach of Blokhuis and Bedeaux16 for nonpolar fluids. A new expression of σ∞ and δ∞ for the dipolar fluids are obtained. We applied these expression to a prototype model dipolar fluid whose bulk and interfacial properties have been reported by several groups.23-25 In section II, we present a theoretical formulation for dipolar fluids. We then derive a general expression for σ∞ and δ∞. Application of the two expressions to the dipolar fluid is shown in section III. Numerical results of σ∞ and δ∞ and discussions are given in section IV. II. Statistical Mechanical Formula for Surface Tension and the Tolman Length In this section, we extend the Blokhuis-Bedeaux approach16 to derive a statistical mechanical expression of the surface tension of a planar liquid-vapor interface σ∞ and the Tolman length δ∞ for simple dipolar fluids. In these systems, the interparticle interaction can be described by the potential u(r12,θ,ω1,ω2), where r12 ) |r b2 - b r1| is the interparticle distance, ωi ) (θi,φi) is the orientation of the point dipole, and θ (defined below) characterizes the relative orientation between the two dipoles in a space-fixed coordinate. To begin with, we consider a portion of a spherical droplet in the form of a spherical cone as shown in Figure 1, where R is the radius of a dividing surface of the droplet and γ is the apex angle. Axis z is the symmetry axis of the cone, and the point z ) 0 is on the dividing surface. Vectors b r1 and b r2 denote the positions of two particles of the system, and θ is the angle between the vector b r12 and the z axis. For a dipolar fluid having a number density F(r b,ω), the free energy functional can be expressed as23
10.1021/jp0108723 CCC: $20.00 © 2001 American Chemical Society Published on Web 10/12/2001
Surface Tension and Tolman Length
J. Phys. Chem. B, Vol. 105, No. 47, 2001 11587 Suppose that the droplet is critical (in the sense of critical nucleus of nucleation); then the grand canonical potential Ω[F] should satisfy the variational condition
δΩ[F] δ(F[F] - µN) ) )0 δF δF
(5)
Combining eqs 2, 4, and 5 yields (see Appendix A)
∫
∫
∫
∫
∫
1 1 dR db r 1 db r 2 dω1 dω2 2 0 δ[up(r12,θ,ω1,ω2)g(r12,θ,ω1,ω2;R)]F(r1,ω1)F(r2,ω2) (6)
δF ) µδN +
and substituting eq 3 into eq 6 with taking integration by parts for the R integral gives
∫
∫
∫
∫
1 db r 1 db r 2 dω1 dω2 × 2 ∂up(r12,θ,ω1,ω2) ‚δb r 12 g(r12,θ,ω1,ω2;1)F(r1,ω1)F(r2,ω2) (7) ∂b r 12
δF ) µδN +
Figure 1. A portion of the spherical droplet in the form of a spherical cone.
F[F(b,ω)] r ) Fref[F(b,ω)] r + 1 1 dR db r 1 db r 2 dω1 dω2 up(r12,θ,ω1,ω2) 2 0 g(r12,θ,ω1,ω2;R)F(r1,ω1)F(r2,ω2) (2)
∫
∫
∫
∫
∫
Here Fref[F(r b,ω)] is the free energy of a reference system with interparticle potential uref(r12,θ,ω1,ω2); g(r12,θ,ω1,ω2;R) is the pair correlation function of the fluid with the interparticle potential given by
uR(r12,θ,ω1,ω2) ) uref(r12,θ,ω1,ω2) + R[u(r12,θ,ω1,ω2) uref(r12,θ,ω1,ω2)] (3) where R is a coupling parameter ranging from 0 to 1. In writing b2,ω1,ω2) eqs 2 and 3, we have made an approximation that g(r b1,r ≈ g(r12,θ,ω1,ω2). The latter is the pair correlation function of a homogeneous bulk fluid with interparticle potential u(r12,θ,ω1,ω2). Another approximation we have made is the local-density approximation to the free energy of the reference, that is
r ) Fref[F(b,ω)]
∫ dbr ∫ dω fref(F(r,ω))
(4)
Next, we consider two independent transformations of the coordinates b r f b′ r )b r + δr b, undertaken in such a way that the volume of the spherical cone remains unchanged. This means that only surface contributions to the change in free energy, δF, can take place in these transformations. The free-energy change can be written as δF ) µδN + σ(R) dA + AC(R) dR, where µ is the chemical potential, N is the number of particles, A is the surface area, σ(R) ) ∂F/∂A, and AC(R) ) ∂F/∂R. Since the two surface terms σ(R) and C(R) are not known, one needs to consider the two independent transformations in order to calculate them.
(
)
b12 and δr b12 denotes a where the dot between ∂up(r12,θ,ω1,ω2)/∂r scalar product between the two vectors. Rewriting eq 7 in the spherical coordinate gives
δF ) µδN + π 2π 1 db r 1 dω1 dω2 r212dr12 0 sin θ dθ 0 dφ12 2 ∂up(r12,θ,ω1,ω2) ∂up(r12,θ,ω1,ω2) ((δb r 2 - δb r 1)·e b r) + × ∂r12 ∂θ δb r 2 - δb r1 ·e bθ g(r12,θ,ω1,ω2;1)F(r1,ω1)F(r2,ω2) (8) r12
[
∫
∫
(
∫
∫
∫
∫
)]
where b er and b eθ are the conventional unit vector in the spherical coordinate. Now, following Blokhuis and Bedeaux,16 we employ the transformation of coordinates without changing the volume of the spherical cone.16 First, we consider
(
1-cosΘ ,φ sinΘ
b r ) (r,Θ,φ) f b′ r ) r+r,Θ-3
)
(9)
where is an infinitesimally small parameter. Under this transformation, the cone transforms into a new one in such a way that the radius of the cone becomes larger but the apex angle becomes smaller, which keeps the volume of the cone fixed, that is, δA/A ) - , δR/R ) and δV/V ) 0. Thus, the free-energy change is given by
δF1 ) µδN1 - A[σ(R) - C(R)R]
(10)
where the subscript 1 refers to the first transformation, A ) 2πR2(1 - cosγ) is the surface area, and C(R) ) ∂σ(R)/∂R. Combining eqs 8-10 yields (see Appendix B for detailed derivation)
11588 J. Phys. Chem. B, Vol. 105, No. 47, 2001
Bykov and Zeng
δF1 ) µδN1 + 1 2π A dz1 dω1 dω2 r212 dr12 -1 ds 0 dφ12 4 ∂up ∂up r12(3s2 - 1) + (1 - s2)3s g(r12,θ,ω1,ω2;1) + ∂r12 ∂s
Substituting eq 15 into eq 1 and keeping only the zero-order terms in 1/R gives
∂up 1 ∂up r12(3s2 - 1)(2z1 + sr12) + (1 - s2)3 2z1s + (1 + R ∂r12 ∂s
where the subscript (0) refers to the zero order in 1/R. Keeping the first-order terms gives
∫
( {[
∫
∫
∫
)
∫
)
∫
(
2 2 r12 ∂2up 3 ∂ up 2 (3s - 1) (1 3s ) r (1 - s ) r12 2 ∂r12∂s 12 2 2 ∂s2 2
]
s2)2sr12 g(r12,θ,ω1,ω2;1) s2)3s
]
[
∂up ∂up r12(3s2 - 1) + (1 ∂r12 ∂s
}
(1 - s ) ∂g(r12,θ,ω1,ω2;1) r12 F(z1,ω1)F(z1 + sr12,ω2) 2 ∂s (11) 2
Note that in deriving eq 11, we have changed the variable r1 (the spherical radius) to z1 ) r1 - R (the distance from the planar liquid-vapor interface); we also replaced the expansion over r12/r1 to r12/R. The second transformation16 is given by
(
R3 b r ) (r,Θ,φ) f b′ r ) r + 2 ,Θ,φ r
)
(12)
Since this transformation at r ) 0 is undefined, one needs to slightly modify the spherical cone by excluding an infinitesimally small portion near r ) 0. Under this transformation, we have δA/A ) 2, δR/R ) , and δV/V ) 0. The free-energy change in this case is given by
δF2 ) µδN2 + 2A[σ(R) + C(R)R/2]
[
σ∞ )
δ∞ )
]
δF2 - µδN2 - (δF1 - µδN1) 3A
[
(0)
]
δF2 - µδN2 + 2(δF1 - µδN1) 6Aσ∞
where the subscript (1) refers to the first order in 1/Re and Rerefers to the radius of equimolar dividing surface. Finally, we introduced a functional expansion of the density profile over the (small) curvature 1/R
1 F(z,ω) ) F(0)(z,ω) + F(1)(z,ω) + ... R
∫
∫
∫
∫
∫
∫
1 2π 1 dz1 dω1 dω2 r212 dr12 -1 ds 0 dφ12 4 ∂up ∂up (1 - s2)3s g(r12,θ,ω1,ω2;1) × r (1 - 3s2) ∂r12 12 ∂s F(0)(z1,ω1)F(0)(z1 + sr12,ω2) (19)
σ∞ )
[
(13)
]
δF2 ) µδN2 + 1 2π A dz1 dω1 dω2 r212 dr12 -1 ds 0 dφ12 2 ∂up ∂up r12(1 - 3s2) (1 - s2)3s g(r12,θ,ω1,ω2;1) + ∂r12 ∂s
δ∞ ) -
[
{[
∫
∫
(
∫
]
)
∫
∂up sr12 1 ∂up r12(3s2 - 1) + z1 + (1 - s2)3sz1 R ∂r12 2 ∂s 2
2
]
∂ up (1 - 3s2) 3 ∂ up r12(1 - s2) r12 + (1 - s2)2sr12 × ∂r12∂s 2 2 ∂s2 ∂up ∂up (1 r (1 - 3s2) g(r12,θ,ω1,ω2;1) ∂r12 12 ∂s
[
]
s2)3s
}
(1 - s2) ∂g(r12,θ,ω1,ω2;1) F(z1,ω1)F(z1 + sr12,ω2) r12 2 ∂s (14)
δF2 - µδN2 + 2(δF1 - µδN1) 3AR
∫ dz1 ∫ dω1 ∫ dω2 ∫r212 dr12 ∫-11 ds ∫02π dφ12
∂up ∂up (1 - s2)(6z1s + r12(1 + r12(1 - 3s2)(2z1 + sr12) ∂r12 ∂s
]
3s2)) g(r12,θ,ω1,ω2;1)F(0)(z1,ω1)F(0)(z1 + sr12,ω2) (20) Note that eqs 19 and 20 recover the formula of σ∞ and δ∞ for nonpolar fluids16 if ∂up/∂s ) 0. Because σ∞ is independent of the choice of the dividing surface in general, eq 19 can be applicable to any dividing surface. δ∞ is also independent of the choice of the dividing surface; however, eq 20 is applicable only when the origin of the z1 axis is taken to be the equimolar dividing surface ze ) 0. Additional terms are needed otherwise. Equations 19 and 20 can be further simplified by defining a function
∫01 dR g(r12,θ,ω1,ω2;R)up(r12,θ,ω1,ω2) (21)
δF2 - µδN2 - (δF1 - µδN1) σ(R) ) 3A C(R) )
[
1 8σ∞
w(r12, s,ω1,ω2) )
The combination of eqs 10 and 13 results in
(18)
where F(0)(z,ω) is the density profile of the planar liquid-vapor interface and F(1)(z,ω) is the first-order correction to F(0)(z,ω) due to curvature of the interface. Substituting eqs 11, 14, and 18 into eq 16 and eq 17 yields, respectively, the following expression for the surface tension of a planar liquid-vapor interface
and the following expression for the Tolman length
∫
(17)
(1)
where the subscript 2 refers to the second transformation. Combining eqs 8 and 12 (see Appendix B) yields
∫
(16)
(15)
Thus, ∂w/∂r12 ) g(r12,θ,ω1,ω2;1)∂up/∂r12, and ∂w/∂s ) g(r12,θ,ω1,ω2;1) ∂up/∂s. With these two derivatives and taking integration by parts over variables r12 and s, eq 19 can be rewritten as
Surface Tension and Tolman Length
σ∞ )
J. Phys. Chem. B, Vol. 105, No. 47, 2001 11589
∫ dz1 ∫ dω1 ∫ dω2 ∫ r212 dr12 ∫-11ds∫02πdφ12
1 2
interaction potential. The latter can be written as a sum of spherical harmonics26
w(r12,s,ω1,ω2)sr12F(0)(z1,ω1)F′(0)(z2,ω2) (22) where z2 ) z1 + sr12 and the symbol ′ denotes a derivation over z2. Similarly, eq 20 can be rewritten as
∫ ∫ dω1 ∫ dω2 ∫
1 dz1 δ∞ ) 2σ∞
[
r212
dr12
]
∫-1 ds ∫0 1
2π
udd ) -
µ20 r312
4π
() 8π 15
1/2
∑
m1,m2,m3
C(112;m1,m2,m3)Y1m1(ω1) × / (ω) (30) Y1m2(ω2)Y2m 3
dφ12
r212(1 + s2) w(r12, s,ω1,ω2) z1sr12 + F(0)(z1,ω1)F′(0)(z2,ω2) 4 (23) With the random-phase approximation and assuming that up is independent of θ, eq 22 recovers the expression σ∞ for nonpolar fluids.17 III. Application to a Model Dipolar Fluid The newly derived statistical formulas for σ∞ and δ∞ (eqs 19 and 20 or eqs 22 and 23) are general. To apply these formulas to particular model dipolar fluid requires the knowledge of the intermolecular potential u(r12,θ,ω1,ω2) and the pair correlation function g(r12,θ,ω1,ω2;R). A common approximation used to calculate g(r12,θ,ω1,ω2;R), given the perturbation potential function up(r12,θ,ω1,ω2), is the so-called modified mean-field (MMF)23 approximation, that is
g(r12,θ,ω1,ω2;R) ) exp[-βuR(r12,θ,ω1,ω2)]
(24)
where β ) 1/kBT. Then, according to eq 21
w(r12, s,ω1,ω2) ) exp[-βuref(r12,θ,ω1,ω2)]kBT[1 exp(-βup(r12,θ,ω1,ω2))] (25) Furthermore, the second exponential term in eq 25 can be expanded in powers of βup. If the expansion is truncated at the first order of up, MMF becomes the random-phase approximation (RPA), i.e.
where µ0 is the magnitude of the molecular dipole moment, Ylm is the spherical harmonics, and C(112;m1,m2,m3) is the Clebsch-Gordan coefficient in the Rose convention.27 The angle ω ) (θ,φ12) describes the orientation of the vector b r12 in the space-fixed coordinate. Since many workers23,25 have shown that the RPA is inadequate to describe dipolar fluids, we have only used the RPA in the case of zero-dipole moment (µ0 ) 0). The TMMFA, however, is used in both cases of zero- and nonzero-dipole moment. For strongly dipolar fluids, more sophisticate approximations25,28 are needed because the second exponential function in eq 25 cannot be expanded. However, for weakly dipolar fluids (when the scaled dipole moment µ/0 ≡ µ0/ (d3LJ)1/2 e 1), the TMMFA is quite sufficient and easy to implement. Van Leeuwen29 has fitted the scaled dipole moment µ/0 to many dipolar systems on the basis of the Stockmayer model. For most dipolar molecular fluids, he found the value of µ/0 is less than or close to 1. Note that the potential function (eqs 28 and 29) of the model dipolar fluid has a discontinuity at r12 ) d, and for certain calculations, extra terms have to be included in eqs 19 and 20 to address this discontinuity. However, these extra terms may not be considered in eqs 22 and 23 because they can be removed via the integration by parts. Thus, it is convenient to use eqs 22 and 23, rather than eqs 19 and 20, to calculate σ∞ and δ∞. The calculation of σ∞ and δ∞ on the basis of eqs 22 and 23 requires the density profile F(0)(z,ω). Here, we use DFT to achieve this information. We start with expanding the number density F(0)(z,ω) in terms of orientational order parameters,
wRPA(r12, s,ω1,ω2) ) exp[-βuref(r12,θ,ω1,ω2)]up(r12,θ,ω1,ω2) (26)
1 F(0)(z,ω) ) F(0)(z) [1 + 3η1(z)P1(cosΘ) + 4π 5η2(z)P2(cosΘ) + ...] (31)
If the expansion is truncated at the second order of up, the MMF becomes the truncated modified mean-field approximation (TMMFA) introduced originally by Teixeira and Telo da Gamma,23 i.e.
where F(0)(z) is the orientationally averaged number density, Pi(cosΘ) is the Legendre polynomials, and ηi(z) is the orientational order parameters. The first two orientational order parameters in eq 31 are given by
wTMMFA(r12, s,ω1,ω2) ) exp[-βuref(r12,θ,ω1,ω2)] × β up(r12,θ,ω1,ω2) - u2p(r12,θ,ω1,ω2) (27) 2
[
]
In this paper, we consider the following model dipolar fluid with the reference and perturbation potential function given by23
uref(r12) )
{
+∞ r12 e d r12 > d 0
(28)
and
up(r12,θ,ω1,ω2) )
{
r12 e d 0 (29) -4LJ(d/r12)6 + udd(r12,s,ω1,ω2) r12 > d
Here d is the hard-sphere diameter, LJ is the energy parameter of Lennard-Jones potential, and udd is the dipole-dipole
η1(z) )
1 4πF(0)(z)
∫ dω P1(cosΘ)F(0)(z,ω)
η2(z) )
1 4πF(0)(z)
∫ dω P2(cosΘ)F(0)(z,ω)
(32)
Also, we truncated the expansion in eq 31 at the η2(z) term because η1 ) 023,25,31 due to the surface dipoles have no preferred orientation. The density profile F(0)(z) and the orientational order parameter η2(z) have been obtained previously31 via DFT. Keeping up to the second order of η2 and using the condition η2(z) , 1/5, we have (see eqs 20 and 22 in ref 31)
µh(F(0)(z1)) 5 2 1 µ dz2 [u0(z2 - z1) ) + η2(z1) + kBT kBT 2 k BT η2(z1)η2(z2)u1(z2 - z1) - (η2(z1) + η2(z2))u2(z2 - z1)]F(0)(z2)
∫
11590 J. Phys. Chem. B, Vol. 105, No. 47, 2001
η2(z1) )
1 5kBT
Bykov and Zeng
∫ dz2 F(0)(z2)(η2(z2)u1(z2 - z1) + u2(z2 - z1)) (33)
where µh is the chemical potential of the hard-sphere fluid in the Carnahan-Starling form.32 The explicit form of the function u0(z), u1(z), and u2(z) in eq 33 can be found in ref 23. Finally, substituting eqs 27 and 29-31 into eq 22 results in
σ∞ )
∫ dz1 F(0)(z1) ∫ dz2 F(0)(z2)[σ0(z2 - z1) -
η2(z1)η2(z2)σ1(z2 - z1) - (η2(z1) + η2(z2))σ2(z2 - z1)] (34) where the explicit form of the function σ0(z), σ1(z), and σ2(z) under the RPA and TMMFA is given in Appendix C; substituting those equations into eq 23 results in
∫
∫
1 dz1 F(0)(z1) dz2 F(0)(z2)[(z1σ0(z2 - z1) + σ∞ δ0(z2 - z1)) - η2(z1)η2(z2) (z1σ1(z2 - z1) + δ1(z2 - z1)) (η2(z1) + η2(z2))(z1σ2(z2 - z1) + δ2(z2 - z1))] (35)
δ∞ ) -
where the explicit form of the function δ0(z), δ1(z), and δ2(z) is also given in Appendix C. IV. Numerical Results and Discussions In this section, results of of the critical density Fc, temperature Tc, surface tension σ∞, and the Tolman length δ∞ of the model dipolar fluid are presented. In the case of zero-dipole moment (µ0) 0), both the RPA and TMMFA are examined, whereas in the case of nonzero-dipole moment (µ0* 0), only the TMMFA is considered. A. Critical Density and Temperature. Since the hard-sphere diameter is temperature-independent, the critical density Fc, determined via the equation ∂2µh/∂F2(Fc) ) 0, cannot be affected by either the RPA or the TMMFA if the calculation is within the framework of DFT.31 On the other hand, the critical temperature Tc, determined via the equation β∂µh/∂F(Fc) ψ(1/Tc)Fc ) 0, is strongly sensitive to the approximation to the pair correlation function. The function ψ(1/T) is related to the volume integration of function u0, as was defined previously in the case of TMMFA.31 It is a quadratic function of 1/T in that case but becomes a linear function of 1/T under the RPA. As a result, the scaled critical temperature T/c ≡ kBTc/LJ ) 1.509 using the RPA, but Tc) 2.009 using the TMMFA. The difference in T/c amounts to 25%. Note that the TMMFA is basically a first-order correction to the RPA. Thus, the TMMFA also provides a first-order correction to T/c calculated on the basis of the RPA. Frodl and Dietrich25 have shown that if the second exponential term in eq 25 is fully considered without taking any expansion (e.g., eq 27), the calculated T/c is 2.204. Therefore, the difference in T/c from that using the nonperturbation theory and the TMMFA is only about 10%. It turns out25 that this difference in T/c remains less than or about 10% as long as µ/0 e 1 (i.e., for weakly dipolar systems). We also find that it is the pair correlation function that dictates the change in T/c . In fact, if the perturbation potential up is taken to be a different one (e.g., the Weeks-Chandler-Andersen perturbation potential for Lennard-Jones with a temperature-dependent hard-sphere diameter), the RPA only changes T/c slightly (from 1.509 to 1.488). B. Surface Tension and the Tolman Length. We first examine the extent to which different approximation to the pair correlation function (either the RPA or TMMFA) affects values
Figure 2. Temperature dependence of the scaled surface tension σ/∞ ) σ∞d2/LJ under different approximations to the pair correlation function and in the case of µ0 ) 0. The temperature scales are (a) T* ) kBT/LJ and (b) T* - T/c ) kB(T - Tc)/LJ.
of σ∞ and δ∞. Figure 2a shows the temperature dependence of σ/∞(≡ σ∞d2/LJ) on the basis of eq 34 using either the RPA or TMMFA. It can be seen that the two σ/∞ curves exhibit nearly the same shape and slope. Presumably, the effect of the TMMFA is just a horizontal shift of the RPA’s σ/∞ curve along the T* axis. This shift follows basically that of T/c due to the difference in the RPA and TMMFA. To see this more clearly, the T* - T/c instead of T* is used as the temperature axis (see Figure 2b). Indeed, the difference between the two σ/∞ curves ranges from only a few percent at high temperatures to about 20% at low temperatures, indicating a good agreement between σ/∞ from the RPA and TMMFA. We therefore conclude that σ/∞ is less sensitive to the approximation to the pair correlation function than T/c . In Figure 3a, the temperature dependence of δ/∞(≡ δ∞/d) based on eq 35 is shown. It appears that neither the two approximations (the RPA and TMMFA) nor the perturbation scheme for the intermolecular potential have much effect on δ/∞. As shown previously for the Lennard-Jones fluid,17,18,22 δ/∞ is negative with an absolute value about 0.2. |δ/∞| also slowly increases with the temperature and seems to be divergent near the critical point. The trend is the same regardless whether the RPA or TMMFA are used. Indeed, the shape and slope of the two δ/∞ curves are nearly the same, other than a shift between the two curves along the T* axis (again, consistent with the shift of T/c ). In fact, the difference between the RPA’s
Surface Tension and Tolman Length
J. Phys. Chem. B, Vol. 105, No. 47, 2001 11591
Figure 4. Temperature dependence of the scaled surface tension σ/∞ ) σ∞d2/LJ (under TMMFA) for given various scaled dipole moment µ/0 ) µ0/(kBTcd3)1/2. The temperature scale is T* ) kBT/LJ.
Figure 3. Temperature dependence of the scaled Tolman length δ/∞ ) δ∞/d under different approximation to the pair correlation function and in the case of µ0 ) 0. The temperature scales are (a) T* ) kBT/LJ and (b) T* - T/c ) kB(T - Tc)/LJ.
and TMMFA’s δ/∞ becomes very small (within 6%) if the T* - T/c instead of T* is used as the temperature axis. In this sense, we can conclude that δ*∞ is even less sensitive to the approximation of correlation function than σ*∞. In passing, we note that the TMMFA yields the same sign of δ*∞ as the RPA, i.e., δ*∞ is still negative. As mentioned above, the TMMFA can be considered as a first-order correction to the RPA. The higher-order terms in the expansion (for the second exponential term in eq 25) is expected to have even less effect on the interfacial properties. Thus, it is likely that the negative sign of the δ*∞ is not an artifact of the RPA but a general result within the framework of perturbative DFT. Of course, effects of other approximations, e.g., the local-density approximation to the free energy of the reference system, on the sign of δ/∞ remain to be examined. In Figure 4, the temperature dependence of σ/∞ is shown for various given dipole moments µ/0. We have also included µ/0 ) 1.25 in the figure to show a more pronounced effect of the larger dipole moment on σ/∞. However, in this case, the obtained σ/∞ should be viewed only qualitative. As shown in Figure 4, increasing the dipole moment can appreciably increase σ/∞, a result consistent with many previous studies23,24,28 even though different expressions of surface tension were employed. In Figure 5, the temperature dependence of δ/∞ is shown for various given dipole moments. Clearly, increasing the dipole moment reduces the absolute value of |δ/∞|, but the sign of δ/∞
Figure 5. Temperature dependence of the scaled Tolman length δ/∞ ) δ∞/d (under TMMFA) for given various scaled dipole moment µ/0 ) µ0/(kBTcd3)1/2. The temperature scale is T* ) kBT/LJ.
remains negative. This result is a main conclusion of this work. We also find δ/∞ is not as sensitive to the change of µ/0 as σ/∞. For example, |δ/∞| decreases only about 5%, whereas σ/∞ increases about 30% as µ/0 is changed from 0 to 1.25. Another noteworthy result is that the improvement of approximation (i.e., from the RPA to TMMFA) also leads to a reduction of |δ/∞|. In summary, we have shown two different ways to reduce |δ/∞|: (i) improvement in the approximation to the radial correlation function and (ii) with a higher molecular dipole moment. As shown previously using DFT,4,5 σ/∞ is not a monotonic function of R. It has a maximum in the region of large droplets due to the negative sign of δ/∞. This maximum is very difficulty to detect since δ/∞ is such a small quantity (about 0.2). In most previous molecular dynamics simulations, this nonmonotonic behavior cannot be observed due to the limitation of the size of the system. A possible explanation, based on summary (i), is that |δ/∞| is perhaps even smaller than 0.2 if a better approximation to the correlation function is used. Because molecular dynamics simulation is essentially exact in the sense that no approximations to the correlation function is required, δ/∞ may be so small that it is almost impossible to observe the nonmonotonic behavior of σ/∞, particularly when the system is not large enough.
11592 J. Phys. Chem. B, Vol. 105, No. 47, 2001
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On the basis of the summary (ii), it is reasonable to expect that for strongly dipolar fluids such as water, |δ∞| could be much smaller than the weakly dipolar fluid |δ∞| calculated here. This, of course, must await for future computer-simulation and experimental confirmation, or calculation of δ∞ from the DFT for water. In the context of homogeneous nucleation which requires quantitative estimation of σ as a function of R, the small value of |δ∞| indicates that the next coefficient in the expansion of σ over 1/R may be needed to achieve better accuracy. One of the authors18 has shown that including second term in the expansion along with the first-order Tolman correction can give very accurate calculation of σ/∞ over a wide range of R. In the context of heterogeneous nucleation on macroscopic nucleus, on the other hand, the first-order Tolman correction itself can become dominant because of the involved, large size of macroscopic nucleus.33,34 The present study is particularly relevant to the ion-induced nucleation, where the nucleating species is polar. A quantitative calculation of δ∞ for dipolar fluids will provide better estimation of the rate of ioninduced nucleation over that based on the capillarity approximation.7 Acknowledgment. We are grateful Dr. Vadim Warshavsky for useful discussions. This work is supported by the National Science Foundation. Appendix A Here, we derive eq 6. By substituting eqs 2 and 4 in eq 5, we have
δΩ[F] δFref[F(r,ω)] ) + δF δF δ 1 1 dR db r 1 db r2 δF 2 0
{∫
∫
∫
∫ dω1 ∫ dω2 up(r12,θ,ω1,ω2) ×
}
g(r12,θ,ω1,ω2;R)F(r1,ω1)F(r2,ω2) δ µ δF
∫ dbr ∫ dω F(r,ω) ) 0
(A1)
that is
∂fref(F(r1,ω1)) 1 r 2 dω2 F(r2,ω2) × + 0 dR db ∂F up(r12,θ,ω1,ω2)g(r12,θ,ω1,ω2;R) - µ ) 0 (A2)
∫
∫
∫
Equation A2 is a general equation for the density profile of the system. Under the coordinate transformation b r f b′ r )b r + δr b without changing the volume of the system, the variation of the free energy (eq 2) can be expressed as
δF )
∂f (F(r ,ω ))
∫ dbr 1 ∫ dω1 ref ∂F1 1 δF(r1,ω1) + ∫01 dR ∫ dbr 1 ∫ dω1 ∫ dbr 2 ∫ dω2 F(r2,ω2) up(r12,θ,ω1,ω2)g(r12,θ,ω1,ω2;R)δF(r1,ω1) +
∫01 dR ∫ dbr 1 ∫ dω1 ∫ dbr 2 ∫ dω2 δ[up(r12,θ,ω1,ω2) ×
1 2
g(r12,θ,ω1,ω2;R)]F(r1,ω1)F(r2,ω2) (A3) while the variation of the number of particles can be written as
δN )
∫ dbr ∫ dω δF(r,ω)
(A4)
Substituting eq A2 into eq A3 with using eq A4, we arrive at
∫
∫
∫
∫
∫
1 1 dR db r 1 dω1 db r 2 dω2 2 0 δ[up(r12,θ,ω1,ω2)g(r12,θ,ω1,ω2;R)]F(r1,ω1)F(r2,ω2) (A5)
δF ) µδN +
that is, eq 6. Appendix B Here, we derive the free-energy change, δF, under the two independent transformations of coordinates described by eqs 9 and 12. The general equation for δF is given by eq 8, which can be written in a slightly different form as
δF ) µδN + 1 2π 1 db r 1 dω1 dω2 r212dr12 -1 ds 0 dφ12 2 ∂up(r12,θ,ω1,ω2) ∂up(r12,θ,ω1,ω2) ((δb r 2 - δb r 1)·e br) sin θ ∂r12 ∂s δb r 2 - δb r1 ·e bθ g(r12,θ,ω1,ω2;1)F(r1,ω1)F(r2,ω2) (B1) r12
[
∫
∫
∫
∫
∫
∫
)]
(
where s ) cos θ. Keeping in mind that δ∞ is the first-order correction to the surface tension due to curvature of the interface, we start with expanding δF in powers of the curvature 1/R. For this, we need to carry out several function expansions in powers of r12/r1. These expansions are conceivable because the integration over r1 in eq B1 is mainly contributed by that over the interfacial region at which r1 ≈ R. On the other hand, the integration over r12 is mainly contributed by that over a small range of r12 within which up(r12,θ,ω1,ω2) is not negligible. That range of r12 is much less than R for very large droplets. First, we expand r2 about r1, i.e.
[
()
() ]
r12 1 r12 2 1 r12 3 + (1 - s2) - s(1 - s2) + ... r1 2 r1 2 r1
r 2 ) r1 1 + s
(B2) Next, we expand the density F(r2,ω2) around the point r1 + sr12, i.e.
F(r2,ω2) ) F(r1 + sr12,ω2) + ∂F(r1 + sr12,ω2) r12 1 + ... (B3) (1 - s2) 2 ∂s r1 Third, we make the expansion of the two scalar products in eq B1: (1) For the first transformation (eq 9), the two scalar products
Surface Tension and Tolman Length
J. Phys. Chem. B, Vol. 105, No. 47, 2001 11593
δF2 ) µδN2 +
can be written respectively as
((δb r 2 - δb r 1)·e b r) )
(
)
A 2
(3r2 + 3sr1r12 + r212 - 3r1r2) r12 1
δb r 2 - δb r1 3 ·e bθ ) 2 (r1 + sr12 - r2)(r12 + sr1) (B4) r12 r12 sin θ
Expanding eq B4 in powers of r12/r1 gives
r 1)·e b r) ) ((δb r 2 - δb
(
)
(
r12 2 r12 3s - 1 + 3s(1 - s2) + ... 2 r1
(
)
)
δF1 ) µδN1 +
[
() r
2
∫ dω1 ∫ dω2 ∫ r212 dr12 ∫-11 ds ∫02π dφ12
dr1
{[
r12 ∂up ∂ 2 up (1 - 3s2) s r12(3s2 - 1) r12(1 - s2) + r1 ∂r12 2 ∂r12∂s 2
]
To calculate σ∞ and δ∞, we first rewrite eqs 22 and 23 in the cylindrical coordinates (see Figure 1) rather than the spherical coordinates, i.e.
1 2
∫ dz1 ∫ dω1 ∫ dω2 ∫ dz ∫y dy ∫02πdφ12z
s2)3s
[
]
}
(1 - s ) ∂g(r12,θ,ω1,ω2;1) F(r1,ω1)F(r1 + sr12,ω2) 2 ∂s (B6)
(2) For the second transformation (eq 12), the two scalar products can be written respectively as
[( )
R3 r1 3 r 1)·e br) ) 3 (r12 + sr1) - sr1 ((δb r 2 - δb r1 r2
(
)
( )
]
δb r 2 - δb r1 1 1 R3 ·e bθ ) r sin θ 3 - 3 r12 r12 1 r1 r2
(B7)
Expanding eq B7 in powers of r12/r1 gives
r 1)·e br) ) r12 ((δb r 2 - δb
(
)
[
r12 3 R3 1 - 3s2 - s(3 - 5s2) + ... 3 2 r1 r1
[
r12 δb r 2 - δb r1 R3 3 ·e bθ ) 3 sin θ 3s + (1 - 5s2) + ... r12 2 r1 r1
]
1 2σ∞
∫ dz1 ∫ dω1 ∫ dω2 ∫ dz ∫y dy ∫02π dφ12
[
w(y, z,ω1,ω2) z1z +
∂ up (3s - 1) 3 ∂ up (1 - s2)2s × r12(1 - s2) ∂r12∂s 2 2 ∂s2 ∂up ∂up g(r12,θ,ω1,ω2;1) (1 r12(3s2 - 1) + ∂r12 ∂s 2
}
Appendix C
δ∞ ) -
]
2
[
2
w(y, z,ω1,ω2)F(0)(z1,ω1)F′(0)(z1 + z,ω2) (C1)
r12 ∂up 3 ∂up r12(3s2 - 1)s + (1 - s2)(1 + 3s2) r1 ∂r12 2 ∂s 2
]
∂up (1 - s ) ∂g(r12,θ,ω1,ω2;1) (1 - s2)3s F(r1,ω1)F(r1 + ∂s 2 ∂s sr12,ω2) (B9)
σ∞ )
]
2
]
∂up ∂up (1 - s2)3s g(r12,θ,ω1,ω2;1) + r12(1 - 3s2) ∂r12 ∂s
∂up ∂up r (3s2 - 1) + (1 - s2)3s g(r12,θ,ω1,ω2;1) + ∂r12 12 ∂s
{[
∫ dω1 ∫ dω2 ∫ r212 dr12 ∫-11 ds ∫02π dφ12
∂up 3 ∂ up (1 - s2)2s g(r12,θ,ω1,ω2;1) r (1 - 3s2) 2 ∂s2 ∂r12 12
Combining eqs B1, B3, and B5 and taking integration by parts over the variable s yield
∫ R1
dr1
2
r12 δb r 2 - δb r1 3(1 - s2) ·e bθ ) -s - (1 - s2) + ... (B5) r12 2 sin θ r1
A 4
[
()
∫ rR1
]
]
(y2 + 2z2) F(0)(z1,ω1)F′(0)(z1 + z,ω2) 4 (C2)
Next, we take the integration over variables y and φ12. Under the two approximationssRPA and TMMFAseqs 26 and 27 can be rewritten respectively as
wRPA(y, z,ω1,ω2) )
wTMMFA(y, z,ω1,ω2) )
{
{
r12 e d 0 up(r12,s,ω1,ω2) r12 > d
(C3)
r12 e d β 2 wRPA - up (r12,s,ω1,ω2) r12 > d 2 (C4)
0
The integration appear in eqs C1 and C2 can be carried out in the same way as that described in ref 23 by using eqs 29 and 30 and by using the equations r12 ) (y2 + z2)1/2 and s ) z/(y2 + z2)1/2. Note that after taking the integration by parts, a term in eq C2 becomes divergent. However, this term will disappear once the integration over the variables ω1 and ω2 is taken because η1 ) 0. Carrying out the integration (in eq C1) over ω1 and ω2 using eqs 31 and 32 under the RPA, we arrive at eq 34, in which
σ0(z) ) A(z) (B8)
Last, combining eqs B1, B3, and B8 and taking integration by parts over the variable s yield
σ1(z) ) 0 σ2(z) ) 0 Using the TMMFA, we obtain
(C5)
11594 J. Phys. Chem. B, Vol. 105, No. 47, 2001
Bykov and Zeng
σ0(z) ) A(z) + βC(z) - 4βD(z) + 2βE(z) σ1(z) ) βD(z) - 3βE(z) + 8βF(z) - βG(z) σ2(z) ) 2βD(z) + βE(z) - βF(z)
(C6)
{
{
-
C(z) )
36 2 d12 π z>d 5 LJ z10
4 2 2 π d 5 LJ
(C7) E ˜ (z) )
(C8)
zed
{ {
1 41 πµ z>d D(z) ) 16 0z4 0 zed 0 z>d E(z) ) 1 πµ4 1 24 0d4 z e d
{ {
(C9)
(C10)
(C11)
0 z>d 4 G(z) ) 45 4 z πµ zed 16 0d8
(C12)
Similarly, using the RPA, we obtain eq 35, in which
δ0(z) ) A ˜ (z) δ1(z) ) 0 (C13)
Using the TMMFA, we obtain
δ0(z) ) A ˜ (z) + βC ˜ (z) - 12βD ˜ (z) + βE ˜ (z) δ1(z) ) βD ˜ (z) + 212βE ˜ (z) - βF ˜ (z) - βG ˜ (z) ˜ (z) - βE ˜ (z) - βF ˜ (z) δ2(z) ) 4βD where
{
d6 3 - πLJ 3 z > d 2 z A(z) ) 1 πLJzd2 z e d 2
{ {
0 z>d 1 4z πµ 24 0d4 z e d
{ {
(C16)
(C17)
(C18)
0 z>d 1 4 z3 πµ zed 12 0d6
(C19)
0 z>d 5 G ˜ (z) ) 27 4 z πµ zed 32 0d8
(C20)
F ˜ (z) )
References and Notes
0 z>d 2 F(z) ) 1 4 z πµ zed 4 0d6
δ2(z) ) 0
{
2 2 2 π d z zed 5 LJ 1 41 πµ z>d D ˜ (z) ) 96 0z3 0 zed
where
d6 -3πLJ 4 z > d A(z) ) z πLJd2 zed
C(z) )
18 2 d12 π z>d 5 LJ z9
-
(C14)
(C15)
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