# A mean field approach for computing solid-liquid surface tension for nanoscale interfaces.

**ABSTRACT** The physical properties of a liquid in contact with a solid are largely determined by the solid-liquid surface tension. This is especially true for nanoscale systems with high surface area to volume ratios. While experimental techniques can only measure surface tension indirectly for nanoscale systems, computer simulations offer the possibility of a direct evaluation of solid-liquid surface tension although reliable methods are still under development. Here we show that using a mean field approach yields great physical insight into the calculation of surface tension and into the precise relationship between surface tension and excess solvation free energy per unit surface area for nanoscale interfaces. Previous simulation studies of nanoscale interfaces measure either excess solvation free energy or surface tension, but these two quantities are only equal for macroscopic interfaces. We model the solid as a continuum of uniform density in analogy to Hamaker's treatment of colloidal particles. As a result, the Hamiltonian of the system is imbued with parametric dependence on the size of the solid object through the integration limits for the solid-liquid interaction energy. Since the solid-liquid surface area is a function of the size of the solid, and the surface tension is the derivative of the system free energy with respect to this surface area, we obtain a simple expression for the surface tension of an interface of arbitrary shape. We illustrate our method by modeling a thin nanoribbon and a solid spherical nanoparticle. Although the calculation of solid-liquid surface tension is a demanding task, the method presented herein offers new insight into the problem, and may prove useful in opening new avenues of investigation.

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**ABSTRACT:**A lipid assembly composed of a finite number of lipid molecules can have multiple metastable structures. Using a series of coarse-grained molecular dynamics simulations, we evaluate the free energy profile for the transformation of a small vesicle to a disk-like structure called a bicelle. This free energy is found to be lower than that predicted from continuum elastic theory. For small unilamellar vesicles, the relaxation of the internal structure of the membrane is suggested to play an important role in lowering the free energy barrier for the vesicle-to-bicelle transformation.Soft Matter 01/2011; 7(19):9012-9020. · 4.15 Impact Factor - SourceAvailable from: Udayana Ranatunga[Show abstract] [Hide abstract]

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**ABSTRACT:**Atomistic molecular dynamics calculations of self-assembled monolayers of alkanethiol molecules on gold nanoparticles are carried out to determine the surface per ligand molecule as a function of the size. The molecular footprint is determined by calculating the surface tension starting from its thermodynamic definition in the canonical ensemble. We use the method of Chiu et al. (Chiu, C. C. et al. J. Chem. Phys.2010, 132, 054706) that makes use of a parametric dependence on the system size of the potential energy between the gold surface and the ligand molecules. The role of the different groups in the molecule in the surface tension is studied. An analysis of the dependence of the surface per thiol molecule on the molecular length is carried out showing that, for molecules larger than hexanethiol, this value is independent of the number of carbon groups in the molecule. The surface occupied per molecule on spherical nanoparticles as a function of the curvature is obtained, and the method is applied to flat surfaces, obtaining a very good agreement with the experimental results. A simple model for the surface density per molecule in curved and flat surfaces is developed.The Journal of Physical Chemistry C. 11/2010; 114(49).

Page 1

A mean field approach for computing solid-liquid surface tension

for nanoscale interfaces

Chi-cheng Chiu,1R. J. K. Udayana Ranatunga,1David Torres Flores,2D. Vladimir Pérez,3

Preston B. Moore,3Wataru Shinoda,4and Steven O. Nielsen1,a?

1Department of Chemistry, The University of Texas at Dallas, 800 West Campbell Road,

Richardson, Texas 75080, USA

2Division de Ciencias Exactas, Universidad de Guanajuato, Campus Guanajuato,

Callejon Jalisco, Valenciana, Guanajuato 36240 Gto., Mexico

3Department of Chemistry and Biochemistry, University of the Sciences in Philadelphia,

Philadelphia, Pennsylvania 19104, USA

4Research Institute for Computational Sciences, National Institute of Advanced Industrial Science and

Technology (AIST), Central 2, 1-1-1, Umezono, Tsukuba, Ibaraki 305-8568, Japan

?Received 15 October 2009; accepted 13 January 2010; published online 4 February 2010?

The physical properties of a liquid in contact with a solid are largely determined by the solid-liquid

surface tension. This is especially true for nanoscale systems with high surface area to volume

ratios. While experimental techniques can only measure surface tension indirectly for nanoscale

systems, computer simulations offer the possibility of a direct evaluation of solid-liquid surface

tension although reliable methods are still under development. Here we show that using a mean field

approach yields great physical insight into the calculation of surface tension and into the precise

relationship between surface tension and excess solvation free energy per unit surface area for

nanoscale interfaces. Previous simulation studies of nanoscale interfaces measure either excess

solvation free energy or surface tension, but these two quantities are only equal for macroscopic

interfaces. We model the solid as a continuum of uniform density in analogy to Hamaker’s treatment

of colloidal particles. As a result, the Hamiltonian of the system is imbued with parametric

dependence on the size of the solid object through the integration limits for the solid-liquid

interaction energy. Since the solid-liquid surface area is a function of the size of the solid, and the

surface tension is the derivative of the system free energy with respect to this surface area, we obtain

a simple expression for the surface tension of an interface of arbitrary shape. We illustrate our

method by modeling a thin nanoribbon and a solid spherical nanoparticle. Although the calculation

of solid-liquid surface tension is a demanding task, the method presented herein offers new insight

into the problem, and may prove useful in opening new avenues of investigation. © 2010 American

Institute of Physics. ?doi:10.1063/1.3308625?

I. INTRODUCTION

Most of the physical properties of a liquid in contact

with a solid depend on the solid-liquid surface tension. De-

velopment of methods that allow for the determination of

solid-liquid surface tension using molecular simulation is

still an ongoing task.1Reliable simulation methods are of

crucial importance because experimental techniques cannot

directly measure surface tension at the nanoscale ?e.g., for

small colloids?; instead experimental measurements of con-

tact angles are used in conjunction with a theoretical frame-

work to estimate the surface tension.2However, assignment

of surface tension from contact angle data for nanoscale in-

terfaces is a controversial and unresolved issue.3,4

The calculation of solid-liquid surface tension is un-

doubtably a challenging task. While we make no pretension

here to fully resolve all of the difficulties, we believe that the

method to be described offers new insight into the problem,

and may prove useful in opening up new lines of attack,

for example, when combined with the recent method of

Leroy et al.1

In most simulation studies the surface tension has been

determined by using the virial expression; for a planar inter-

face this involves the difference of pressure tensor compo-

nents normal and tangential to the interface, ?=L??P?−P??,

where L?denotes the simulation unit cell length perpendicu-

lar to the interface.5,6Salomons and Mareschal7rigorously

proved that the virial expression is equivalent to the thermo-

dynamic definition of surface tension ?=??F/?A?NVT, where

F is the Helmholtz free energy and A is the surface area of

the interface. They went on to formulate a novel method

based directly on the thermodynamic definition in which the

free energy of a small isothermal, volume-conserving defor-

mation of the system which increased the interfacial area was

computed. Furthermore, this alternative method was sug-

gested as being applicable to geometries other than planar.

In a similar spirit to Salomons and Mareschal, we intro-

duce here a novel method to compute the surface tension of

a planar interface based directly on the thermodynamic defi-

a?Electronic mail: steven.nielsen@utdallas.edu.

THE JOURNAL OF CHEMICAL PHYSICS 132, 054706 ?2010?

0021-9606/2010/132?5?/054706/7/$30.00© 2010 American Institute of Physics

132, 054706-1

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Page 2

nition. Importantly, our method also applies to curved inter-

faces and reveals the exact relationship between the surface

tension and the excess solvation free energy per unit surface

area. These quantities are not equivalent for nanoscale inter-

faces, yet most simulation studies of nanoscale interfaces

have measured excess solvation free energy and not surface

tension.8–14We present below the formulation for planar in-

terfaces of finite width and spherical interfaces of positive

curvature. Other geometries, including interfaces of negative

curvature, follow in a straightforward manner but are beyond

the scope of the present study.

The methods we present here rely on a mean field treat-

ment of the solid objects. Such an approach has a long his-

tory in science. For example, Hamaker based his 1937 paper

on the van der Waals attraction between solid spherical par-

ticles on such a treatment,15and this treatment underlies the

Derjaguin, Landau, Verwey, and Overbeek theory of col-

loids. In terms of computer simulations, such approaches for

calculating surface tension go back at least to 1976 when

Miyazaki et al. computed the free energy of converting a

solid-liquid interface to a liquid-vapor interface by sliding a

solid wall outwards and integrating the derivative of the free

energy with respect to the position of the wall.16The advan-

tages to using such a mean field approach are fourfold. First,

it allows us to get rid of inessential details of real materials

which might otherwise obscure the universal features of the

curvature and nanoscale size phenomena in question. Sec-

ond, it allows us to use ideal defect-free substrate surfaces,

thus eliminating a lot of practical complications. Third, in

contrast to experiments, it allows us to easily vary param-

eters such as the solid-liquid interaction strength and monitor

the resulting surface tensions. Fourth, it allows us to develop

analytic expressions which reveal the underlying physics and

the relationship between surface tension and excess solvation

free energy.

We begin by casting the thermodynamic expression for

surface tension in a form amenable to a mean field treatment.

Based on this formulation, we outline how one would com-

pute the solid-liquid surface tension without using the mean

field description in order to illustrate the nature of the full

problem. We then proceed to use a mean field description for

the solid and present in detail the treatment of planar and

spherical interfaces. Our analysis shows that the surface ten-

sion and the excess solvation free energy per unit surface

area of a nanoscale interface are in general inequivalent;

their precise relationship is derived for each of the interfaces

we consider and their inequivalence is validated with exten-

sive molecular dynamics ?MD? simulation data.

II. THERMODYNAMIC DEFINITION OF SOLID-LIQUID

SURFACE TENSION

The surface tension, ?, of an interface is defined in ther-

modynamics as

? =??F

NVTNPT

?A?

=??G

?A?

,

?1?

where F is the Helmholtz free energy of the system, G is the

Gibbs free energy, and A is the surface area of the interface.

Consider an interface between a liquid and a solid. Let us

assign a size a with units of length to the solid object in some

manner, and use the chain rule to write ?with an equivalent

formula for F?

?a?dA?a?

?G

?A=?G

?a

da

dA=?G

da?

−1

.

?2?

This formulation accommodates a range of solids, from

spherical where A?a?=4?a2with a the radius, to truncated

icosahedral where A?a?=3a2?10?3+?5?5+2?5? with a the

edge length, to any other shape for which the surface area

can be written as a function of a single parameter.

III. MEAN FIELD MACHINARY

We will use a mean field approach in Secs. VI and VII

which imbues the solid-liquid intermolecular potential en-

ergy with parametric dependence on a. In this section we

wish to explore the consequences of such an approach on Eq.

?2?. As pointed out by Onsager, parameters which appear in

the potential of the intermolecular interactions have essen-

tially the same status as the parameters of external force and

may be manipulated in the same manner.17Hence, we may

write

?a?−1

?G

?a=

?

?ln Q?=1

Q?dx?U

?ae−?H=??U

?a?,

?3?

where Q is the canonical partition function, x represents all

the particle coordinates and momenta, U is the potential en-

ergy of the system, and the angle brackets are understood to

be evaluated in the canonical ensemble. Likewise, we have

?a?−1

??

0

=??U

?G

?a=

?

?ln ??

dV?dx??U

?a?+ P??V

=1

?

?a+ P?V

?a?,

?a?e−?He−?PV

?4?

where ? is the isothermal-isobaric partition function and the

angle brackets are understood to be evaluated in the

isothermal-isobaric ensemble. The PV term is usually negli-

gibly small8but can easily be computed either by simply

monitoring the unit cell size in MD simulations for different

values of a ?keeping the number of liquid particles fixed?, or

by assuming the liquid is incompressible in which case the

known dependence of the solid volume on the size a can be

used. The mean field approach will allow us to compute

??U/?a? from a single equilibrium MD simulation. This is

our machinery: we have turned ?G/?a into an expression

which can be easily computed from an equilibrium MD run

which through Eqs. ?1? and ?2? gives us the surface tension.

IV. HOW TO CALCULATE SOLID-LIQUID SURFACE

TENSION WITHOUT THE MEAN FIELD

APPROACH

The advantage of using the mean field approach can be

illustrated by the problem of not using it. In this context, we

054706-2Chiu et al. J. Chem. Phys. 132, 054706 ?2010?

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Page 3

wish to conduct a thought experiment to directly compute the

surface tension of a solid-liquid interface using MD simula-

tions without any idealizations. From Eqs. ?1?–?3? we are

required to evaluate ??U/?a?. We assume the ergodic hy-

pothesis and replace the ensemble average with a time

average,18

??U

?a?=1

T?

ti=1

T?Uti

?a.

?5?

Consider a particular time ti, sketched in Fig. 1?a?. At this

stage of the argument, since we have not yet replaced the

solid with a mean field, we can evaluate ?U/?a by finite

difference as follows. Assume that the solid particle in Fig.

1?a? is of characteristic size a. As shown in Fig. 1?b?, let us

replace the solid atoms in the range ?a−?a,a?, measured

from the center of the solid, with liquid particles using, for

example, an alchemical free energy perturbation approach.19

These liquid particles are initially out of equilibrium be-

cause, among other reasons, they have the wrong local den-

sity. We can run a short MD simulation to let them diffuse

and come to equilibrium with the existing fluid phase, shown

in Fig. 1?c?. Then, we can form ?U/?a by finite difference

?U

?a?U?c?− U?a?

?a

,

?6?

where the superscripts ?a? and ?c? refer to the systems in

Figs. 1?a? and 1?c?, respectively.

However, there is a numerical problem with this calcu-

lation because of how systems ?a? and ?c? differ from one

another. Clearly the solid-liquid interface is different since

we have explicitly changed it ?e.g., it has a different curva-

ture?. Unfortunately, the noninterfacial solid particles and the

noninterfacial liquid particles have evolved through various

mechanisms such as librations and diffusion. Ultimately such

fluctuations in the noninterfacial parts of the system cancel

out in the equilibrium average of Eq. ?5? and hence do not

contribute to ?. However, their contribution to the individual

measurements on the right hand side of Eq. ?6? dominates

over the interfacial contribution simply due to the respective

number of particles involved; the former ?bulk? region is

three dimensional and the latter ?interfacial? region is two

dimensional. In other words, the signal is swamped by the

noise. The mean field approach, as demonstrated in Secs. VI

and VII, eliminates these fluctuations or, in other words, re-

moves the noise from the signal and allows an accurate es-

timation of ??U/?a?.

V. LIQUID FORCE FIELD AND SIMULATION DETAILS

We used two different liquid models to illustrate the gen-

eral applicability of our mean field method: the Lennard-

Jones ?LJ? model and a coarse grain water model. For the LJ

model, the potential energy between liquid particles is given

by ull?r?=4?ll???ll/r?12−??ll/r?6?. Reduced units are used for

all reported quantities where the unit of length is ?ll, the unit

of energy is ?ll, the unit of mass is the particle mass, and the

unit of temperature is ?ll/kBwhere kBis Boltzmann’s con-

stant. We used a temperature of T=0.75 and a time step of

?=5?10−4.20,21The interaction between LJ particles is trun-

cated at 3?lland no long-range corrections are included for

either the energy or pressure. For the interaction between the

solid strip and the liquid, we take ?sl=?lland measure the

solid-liquid interaction energy ? in units of ?ll?see Sec. VI?.

The interaction between water molecules is modeled us-

ing the recent coarse grain force field of Shinoda.22,23This

model reproduces the experimental liquid-vapor surface ten-

sion which is the most importance physical property of the

solvent related to our study. The potential energy between

watersites isgivenby

−??ll/r?4?, where r is the distance between the molecules and

where ?ll=0.895 kcal/mol and ?ll=4.37 Å. We used a tem-

perature of 303 K and a time step of 3 fs for the water-solid

and short-ranged water-water interactions and 30 fs for long-

ranged ?r?6 Å? water-water interactions. The water-water

interaction is truncated at 15 Å and no long-range energy or

pressure corrections are included.

The MD simulations were performed using the CM3D

and MPDYN MD packages.24,25All simulations were run un-

der the isothermal-isobaric ensemble at the specified tem-

perature and atmospheric pressure. For each of the 10 values

of ? shown in Fig. 2, 3500 liquid particles were simulated for

ull?r?=3?3?1/2?ll/2???ll/r?12

FIG. 1. Schematic plan to measure the surface tension of a solid-liquid

interface with an explicit representation of the solid. At various times i

during an equilibrium MD simulation ?panel a?, the interfacial area is re-

duced using alchemical free energy perturbation techniques ?panel b?, and

then equilibrated ?panel c?. See the text for details.

054706-3 Surface tension of nanoscale interfacesJ. Chem. Phys. 132, 054706 ?2010?

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Page 4

1?107time steps for the virial method and 2.5?106time

steps for the strip method. For the 16 values of a used in Fig.

3, 3500 liquid particles were simulated for 5?106time

steps. For each of the nine values of ? shown in Fig. 5, we

performed 30 equilibrium simulations for 10 ns each with

spheres of radii 1,2,...,30 Å in unit cells with between

5000 and 30 000 water sites. The solid objects are treated as

follows: the planar strip can be viewed as being infinitely

long and is hence assigned an infinite mass; on the other

hand the solid sphere is finite in extent and is assigned a

mass proportional to its volume and is thus displaced during

the MD simulations according to the forces it experiences

from the liquid particles.

VI. PLANAR STRIP

Consider a planar strip of solid material, for example, a

graphene nanoribbon, characterized by a width 2a; assume

for convenience the strip is in the x-y plane, oriented along

the x-axis. In the mean field approach, we replace the lattice

sites which constitute the solid with a continuum in analogy

to Hamaker’s treatment.15To derive the resulting solid-liquid

interaction potential, we suppose that the strip has an atomic

number area density ?=1.6 ?expressed as the number of at-

oms per unit area in dimensionless units; see Sec. V?, and

that the van der Waals interaction energy between one of

these atoms and a liquid atom is given by usl?d?

=4????sl/d?12−??sl/d?6?, where d is the distance between the

solid and liquid atoms, ? is the tunable solid-liquid interac-

tion strength, and ?sl=?ll. The potential energy between a

liquid atom and the entire solid object is then obtained in the

mean field model by integrating over the continuum solid,

U=?A?usldA where A is the area of the solid strip. For pur-

poses of integration, we can without loss of generality place

the liquid atom at position ?0,y0,z0? to obtain

U?y0,z0;a? = 4???

−a

−?

a

dy?

?

dx?

?sl

12

?x2+ ?y − y0?2+ z0

2?6/2?

2?12/2

−

?sl

6

?x2+ ?y − y0?2+ z0

= ???

−a

a

dy?

63??sl

12

64??y − y0?2+ z0

2?5/2?.

2?11/2

−

3??sl

6

2??y − y0?2+ z0

?7?

Notice that ? and ? appear simply as linear factors in Eq.

?7?; choosing a different value for ? yields the same potential

U?y0,z0;a? if ? is rescaled appropriately. The y integral can

be performed to arrive at an analytic expression for

U?y0,z0;a? but the resulting expression is too long to include

here; analytic expressions

?0,−?U/?y0,−?U/?z0? on the liquid particle. The derivative

needed to access the surface tension, ?U/?a, is simply

64?

??a + y0?2+ z0

2?

??a + y0?2+ z0

alsoexistfor theforce

?U

?a=63????sl

12

1

2?11/2+

1

??a − y0?2+ z0

2?11/2?

2?5/2?.−3????sl

6

1

2?5/2+

1

??a − y0?2+ z0

?8?

The strip area is given by

A?a? = 2?a + ??Lx,

?9?

where Lxis the unit cell dimension in the x-direction and the

parameter ? accommodates any particular choice of the

Gibbs dividing surface. From Eqs. ?2? and ?4? we may write

the surface tension as

? = ?2Lx?−1??

i

?Ui

?a?⇒?G

?a= 2Lx?,

?10?

where i indices the liquid particles. The strip surface tension

data for a strip width of 2a=8 is compared in Fig. 2 to the

surface tension of an infinitely extended planar interface

computed using the virial expression. The infinitely extended

interface is described by taking a→?, yielding

U?z0? = 4????

5z0

z0

4?sl

12

10−2?sl

6

4?.

?11?

The agreement is seen to be quantitative because the strip is

wide enough to essentially be free of finite size effects.

Through

00.1 0.20.3 0.4

ε

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

γ

virial method

strip method

FIG. 2. Surface tension of a planar interface computed using the strip

method ?for a strip width of 2a=8? vs the virial ?pressure tensor? method as

a function of the solid-liquid interaction strength ?. All quantities are ex-

pressed in dimensionless units.

01234

strip half-width

0

0.1

0.2

0.3

0.4

0.5

γ

γ~

dγ~

da

FIG. 3. Variation in the solid-liquid surface tension ?, excess free energy per

unit surface area ? ˜, and its derivative d? ˜ /da with the strip half-width a for

a solid-liquid interaction strength of ?=0.16. All quantities are expressed in

dimensionless units.

054706-4 Chiu et al. J. Chem. Phys. 132, 054706 ?2010?

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Page 5

G?a? =?

0

a?G

?wdw,

?12?

we may identify G?a? with the excess solvation free energy

of the solid object where the right hand side of Eq. ?12? is

understood to be evaluated by thermodynamic integration; w

here is a dummy integration variable representing the strip

half-width. G?a? can be thought of as the free energy cost of

growing the solid object in the solvent from a vanishingly

small size to size a, or equivalently, as the free energy cost of

transferring the planar strip of half-width a from an ideal gas

reference state into the solvent.

We define the excess solvation free energy per unit sur-

face area using Eq. ?9? by

? ˜ = G?a?/A?a? = ?2?a + ??Lx?−1G?a?,

?13?

and denote it with ? ˜ to distinguish it from the surface tension

?. From Eq. ?13? we have

?G

?a= 2Lx? ˜ + 2?a + ??Lx

?? ˜

?a.

?14?

Combining this with Eq. ?10? gives

? = ? ˜ + ?a + ???? ˜

?a,

?15?

which is a differential equation relating ? and ? ˜. The surface

tension and the solvation free energy per unit surface area are

seen to be different if ? ˜ depends on a. To simplify the analy-

sis of Eq. ?15?, let us take ?=0 and assume that Lxis inde-

pendent of a, which could be realized by adjusting the num-

ber of liquid particles between simulations to compensate for

the different planar strip excluded volumes. We then see that

a2?a? −?

0

?? ˜

?a=1

a

?dw?,

?16?

from which it is clear that ?? ˜ /?a is nonzero if ? varies with

a over some range of a-values. Specifically, we see from Eq.

?16? that the difference between ? and ? ˜ as per Eq. ?15?

depends on how quickly ? reaches its plateau value; the data

in Fig. 3 shows that this happens very quickly. Nonetheless,

? and ? ˜ are clearly distinguishable for all strip widths con-

sidered in Fig. 3. Moreover, a careful examination of the data

shows that the a=4 value of ? is ever so slightly different

from its a→? extrapolation, which explains the very small

but systematic discrepancy between the two curves of Fig. 2.

VII. SOLID SPHERE

Consider a solid sphere of radius a and suppose that the

van der Waals interaction energy between a lattice site in the

solid and a liquid site is given by usl?d?=27?/4???sl/d?9

−??sl/d?6?, where d is the distance between the sites, ? is the

tunable solid-liquid interaction strength, and ?sl=4.0 Å. The

LJ 9–6 functional form used here is typical of coarse grain

force fields.22,23

The potential energy between a single liquid site and the

entire solid sphere is obtained in the mean field model by

integrating over the continuum solid, U=?V?usldV where V

is the volume of the sphere and ? is the number density of

lattice sites in the solid sphere. For purposes of integration,

we can without loss of generality place the solid at the origin

and the liquid site at ?0,0,R?. An interaction site in the solid

is located at position ?r sin ? cos ?,r sin ? sin ?,r cos ??,

so that the integral can be written explicitly as

U =?

0

00

a?

2??

?

r2sin ??usl???d?d?dr,

?17?

where a is the sphere radius, R is the distance between the

sphere center and a liquid molecule, and ?2=r2−2rR cos ?

+R2.

The integration can be performed analytically, yielding

U?R;a? =9????sl

9a3?3a4+ 42a2R2+ 35R4?

35R?R2− a2?6

−9????sl

?R2− a2?3.

6a3

?18?

We use ?=3.76?10−2Å−3. Once again, notice that ? and ?

appear simply as linear factors in Eq. ?18?; choosing a dif-

ferent value for ? yields the same potential U?R;a? if ? is

rescaled appropriately.

The interfacial area of the sphere is given by

A?a? = 4??a + ??2,

?19?

where the parameter ? accommodates any particular choice

of the Gibbs dividing surface. From Eqs. ?2? and ?4?, we may

write

? = ?8??a + ???−1??

i

where i indices the liquid particles. We define the excess

solvation free energy per unit surface area using Eq. ?19? by

? ˜ = G?a?/A?a? = ?4??a + ??2?−1G?a?,

?Ui

?a?⇒?G

?a= 8??a + ???

?20?

?21?

and denote it with ? ˜ to distinguish it from the surface tension

?, where G?a? is the excess solvation free energy of the solid

sphere defined in an analogous manner to Eq. ?12?. From

Eq. ?21? we have

?G

?a= 8??a + ??? ˜ + 4??a + ??2?? ˜

?a.

?22?

Combining this with Eq. ?20? gives

? = ? ˜ +a + ?

2

?? ˜

?a,

?23?

which is a differential equation relating ? and ? ˜. Our formal-

ism makes this relationship explicit, which is one of its great

strengths. Clearly lima→?? ˜ =lima→?????, which corre-

sponds to the surface tension of a planar solid-liquid

interface.26In Fig. 4 we explicitly compare ? and ? ˜ for three

different values of the solid-liquid interaction strength ? and

two different choices of the dividing surface ?. The

asymptotic value ??is determined by the strength of the

solid-liquid interaction energy ?. There is no restriction on

the sign of ??as long as the liquid-vapor surface tension is

positive.27

054706-5 Surface tension of nanoscale interfacesJ. Chem. Phys. 132, 054706 ?2010?

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Page 6

Let us express ? ˜ in inverse powers of the sphere radius a

? ˜ = ??+ c1a−1+ c2a−2+ ... ,

?24?

Up to second order, Eq. ?23? yields

? = ??+c1

2a−?c1

2a2.

?25?

In their analysis of hard sphere solutes, Huang et al.8trun-

cated the series expansion for ? ˜ at first order and identified

c1/2 as the Tolman length;28one can see from Eq. ?25? that

the coefficient of the 1/a term in the series expansion for ? is

c1/2 because of the relationship between ? and ? ˜ given in

Eq. ?23?. Interestingly, the coefficient c2cancels from the a−2

term in the series expansion of ?; this fact could be used as

justification for truncating the ? ˜ series at first order.

Even though we obtained ? directly using the mean field

approach through Eq. ?20?, in general it is conceptually sim-

pler to measure ? ˜ since the solvation free energy is unam-

biguously defined and can be accurately measured.8–14Since

the mean field approach gives us ready access to both ? and

? ˜, we can assess the accuracy of the route suggested by Eq.

?23? to obtain ? indirectly from ? ˜. In Fig. 5 the surface

tension ? from Eq. ?20? is plotted against its reconstruction

through Eq. ?25? for ?=0 and ?=4 Å. The fidelity of the

reconstruction is quantitative for a?7 Å in the ?=0 case

and a?15 Å in the ?=4 Å case. In order to capture the

surface tension behavior below this size, we would need to

include higher order terms in the series expansion of Eq.

?24?. One can also see from Fig. 5 that the second order term,

which is only present for nonzero ?, makes the reconstructed

surface tension less accurate at small a values but more ac-

curate at large a values. It could be expected that ?=4 would

lead to a more accurate reconstruction than ?=0 because it is

equal to ?sland hence accounts for the excluded volume of

the solid-liquid interaction on the location of the solid-liquid

interface; however we see the actual situation is more com-

plicated than this because there is also a dependence on the

solid-liquid interaction energy ?.

VIII. CONCLUSIONS

The computation of solid-liquid surface tension is chal-

lenging, especially for nanoscale interfaces. We demon-

strated that great physical insight could be achieved by using

a theoretical framework in which the solid was represented

as a continuum of uniform density in analogy to Hamaker’s

treatment of colloidal particles. As a result of this mean field

treatment the Hamiltonian of the system was imbued with

parametric dependence on the size of the solid object through

the integration limits for the solid-liquid interaction energy.

Since the solid-liquid surface area is a function of the size of

the solid, and the surface tension is the derivative of the

system free energy with respect to this surface area, we ob-

tained a simple expression for the surface tension ? of an

interfaceofarbitraryshape.

=??G/?A?NPT=???U/?a?+P??V/?a???dA/da?−1where a is

the size of the solid object, U is the solid-liquid interaction

energy, A is the surface area of the interface, and the angle

brackets are understood to be evaluated in the isothermal-

isobaric ensemble; equivalently ?=??F/?A?NVT=??U/?a?

??dA/da?−1in the canonical ensemble.

Moreover, our formalism yielded a differential equation

relating the surface tension ? to the excess solvation free

energy per unit surface area ? ˜; these quantities converge to a

common value for macroscopic interfaces but are distinct on

the nanoscale. We explored in detail the relationship between

these two quantities for a thin nanoribbon and a solid spheri-

cal nanoparticle immersed in liquid using MD simulations.

Knowledge of the precise relationship between ? and ? ˜ is

important because most simulation studies report one or the

This expressionis

?

0

0.05

0.1

γ~

γ

5

10

15

20

25

30

sphere radius (Å)

0

0.05

0.1

γ or γ~(kcal/mol/Å2)

δ=0

ε=0.194

ε=0.336

ε=0.477

δ=4

ε=0.194

ε=0.336

ε=0.477

FIG. 4. For a spherical solid-liquid interface, the excess free energy per unit

surface area ? ˜ and the surface tension ? are plotted for three different values

of the solid-liquid interaction strength ? ?kcal/mol? and two different values

of the dividing surface ? ?Å?. See Eqs. ?20? and ?21? and the accompanying

text for details.

0

0.04

0.08

5

10

15

20

25

30

sphere radius (Å)

0

0.04

0.08

γ (kcal/mol/Å2)

δ=0

δ=4

ε=0.194

ε=0.230

ε=0.265

ε=0.300

ε=0.336

ε=0.371

ε=0.406

ε=0.442

ε=0.477

ε=0.194

ε=0.265

ε=0.336

ε=0.406

ε=0.477

ε=0.230

ε=0.300

ε=0.371

ε=0.442

FIG. 5. For a spherical solid-liquid interface, the surface tension ? from Eq.

?20? ?shown with solid line? is plotted against its reconstruction, Eq. ?25?

?shown with dashed line?, for two different choices of the dividing surface ?

?Å?. The different curves correspond to different values of the solid-liquid

interaction strength ? ?kcal/mol?.

054706-6 Chiu et al.J. Chem. Phys. 132, 054706 ?2010?

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Page 7

other but not both. Intriguingly, we showed that assignment

of the Tolman length depends on which quantity is analyzed.

In summary, we have formulated an approach to the cal-

culation of solid-liquid surface tension and excess solvation

free energy per unit surface area by introducing a size pa-

rameter a of the solid object into the system Hamiltonian

through a mean field treatment. If desired, information about

the actual solid-liquid interface ?which, for example, could

have defect sites? could be recovered by using algorithms

from the literature. The size parameter a can encode for a

variety of nanoscale interfaces, for example, spherical

?where a is the radius? or ribbonlike ?where a is the ribbon

width? as considered here, or polyhedral ?where a can be the

edge length?, cylindrical ?where a can be the radius?, conical

?where a can be the cone angle?, or of negative curvature

such as a spherical or cylindrical cavity.

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054706-7Surface tension of nanoscale interfacesJ. Chem. Phys. 132, 054706 ?2010?

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