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arXiv:1204.6050v1 [cond-mat.stat-mech] 26 Apr 2012

Wall-liquid and wall-crystal interfacial free energies via thermodynamic integration: A

molecular dynamics simulation study

Ronald Benjamin1,2and J¨ urgen Horbach2

1Institut f¨ ur Materialphysik im Weltraum, Deutsches Zentrum f¨ ur Luft- und Raumfahrt (DLR), 51170 K¨ oln, Germany

2Institut f¨ ur Theoretische Physik II, Universit¨ at D¨ usseldorf,

Universit¨ atsstraße 1, 40225 D¨ usseldorf, Germany

A method is proposed to compute the interfacial free energy of a Lennard-Jones system in contact

with a structured wall by molecular dynamics simulation. Both the bulk liquid and bulk face-

centered-cubic crystal phase along the (111) orientation are considered. Our approach is based

on a thermodynamic integration scheme where first the bulk Lennard-Jones system is reversibly

transformed to a state where it interacts with a structureless flat wall. In a second step, the flat

structureless wall is reversibly transformed into an atomistic wall with crystalline structure. The

dependence of the interfacial free energy on various parameters such as the wall potential, the

density and orientation of the wall is investigated. The conditions are indicated under which a

Lennard-Jones crystal partially wets a flat wall.

I.INTRODUCTION

Knowledge of the interfacial free energy between a

crystal or liquid in contact with a solid wall is crucial

to the understanding of heterogeneous nucleation and

wetting phenomena [1–5]. However, interfacial free en-

ergies are hardly accessible in experiments and in fact

only a few measurements have been reported so far (see

e.g. [2, 6, 7]).

Due to the lack of experimental data, particle-based

simulation techniques such as Molecular Dynamics (MD)

and Monte Carlo (MC) [8, 9] are of special importance to

understand the properties of wall-liquid and wall-crystal

interfaces and to rationalize calculations in the frame-

work of density functional theory [10–12]. In this con-

text, MC and MD simulations have been used to un-

derstand the microscopic mechanism of fluid wetting on

solid surfaces [6, 13–16] as well as the wetting and dry-

ing transition of a fluid at liquid-vapor coexistence and

in contact with a solid wall [15–18]. The question of how

the wall structure affects the interfacial tension with re-

spect to liquid, vapor and solid phases has been also ad-

dressed [16, 19].

On a macroscopic scale, a crystal that partially wets

a wall might be described as a spherical cap. Then, the

contact angle θc of the cap with the wall is given by

Young’s equation [5],

γwc+ γclcosθc= γwl

(1)

with γwcthe wall-crystal, γclthe crystal-liquid, and γwl

the wall-liquid interfacial free energy. Equation (1) de-

scribes the condition of a spherical crystal droplet resting

on a wall, being in coexistence with the liquid phase. In-

complete wetting corresponds to contact angles 0 < θc<

π.

On a nanoscopic scale, deviations from Young’s equa-

tion can be expected, e.g. due to the contribution of line

tension effects [1, 5, 20]. To quantify the latter devia-

tions, reliable estimates of γwc, γcland γwlare required.

Then, the contact angle can be obtained via Eq. (1) and

compared to a direct measurement of θc.

In this paper, we propose a thermodynamic integration

(TI) [21] scheme for the calculation of γwc and γwl. To

obtain γwl, most previous studies have used the mechan-

ical approach of calculating the normal and tangential

pressure components at the wall and integrating over the

pressure anisotropy (PA) [6, 13, 15–17]. While the PA

method is valid for planar wall-liquid or liquid-vapor in-

terfaces it fails in case of small liquid drops in contact

with a solid wall [22]. Moreover, its use is justified only

for systems where the interfacial tension equals the inter-

facial free energy [23]. This is true for a wall represented

by a time-independent external field [13, 24, 25] or a wall

made of particles rigidly fixed at the sites of an ideal

lattice [15–18, 26, 27]. However, for systems which can

support stress, such as a wall consisting of a “fully in-

teracting solid phase” [28], this method is invalid. For

the same reason, the PA technique cannot be used to de-

termine γwc[23]. Even for wall-liquid interfaces, the PA

method can yield results with acceptable precision only

with huge computational effort.

based on the PA technique yielded results of low accu-

racy and the values of the interfacial tension reported in

the literature differ widely, even for simple systems.

Most previous works

Due to the obvious disadvantage in using the PA

method, a few thermodynamic approaches have been de-

veloped to evaluate the wall-liquid and wall-crystal in-

terfacial free energies with improved precision. Heni and

L¨ owen [29] combined MC simulations and thermody-

namic integration to determine the interfacial free en-

ergies of hard sphere liquids and solids near a planar

structureless wall over a whole range of bulk densities

including the solid-liquid coexistence density. In their

thermodynamic integration scheme, a bulk hard sphere

system was reversibly transformed into a system interact-

ing with a more and more impenetrable wall and finally

a hard wall. Fortini and Djikstra [30] used a thermo-

dynamic integration scheme based on exponential poten-

tials to calculate γwland γwcat bulk coexistence condi-

tions. Their results were in good agreement with those

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of Heni and L¨ owen but obtained with significantly higher

precision. Due to precrystallization of the hard spheres

near the wall close to the bulk freezing transition, both

Heni and L¨ owen and Fortini and Dijkstra extrapolated

the value of the interfacial tensions at coexistence from

the data at lower densities.

Laird and Davidchack [31] developed a TI method by

the use of “cleaving potentials”, to obtain γwland γwcfor

hard sphere systems at coexistence. In another work [28],

they used the “Gibbs-Cahn integration” method, to ob-

tain wall-fluid interfacial free energies for hard sphere

systems.This method yielded results consistent with

the TI method with “cleaving potentials” but were ob-

tained with significantly less computational effort. How-

ever, “Gibbs-Cahn integration” requires that one knows

already the interfacial free energy at one point. Deb et

al. [32, 33] compared different methods to obtain wall-

fluid and wall-crystal interfacial free energies for hard

sphere systems confined by hard walls or, soft walls de-

scribed by the Weeks-Chandler-Anderson (WCA) poten-

tial. They introduced a scheme similar to Wang-Landau

sampling [34], known as the “ensemble mixing” method,

to perform a TI from a system without walls to a sys-

tem confined by walls. For hard spheres, Deb et al.

obtained good agreement with the results of Laird and

Davidchack.

In contrast to these few works on hard-sphere systems,

there is a dearth of results on the interfacial free energies

of systems with continuous potentials, such as Lennard-

Jones (LJ) systems. Recently, Leroy et al. [19] obtained

γwl for a LJ liquid in contact with a flexible LJ struc-

tured wall by the use of a TI technique, known as the

“phantom wall” method. In this approach, the struc-

tured wall interacting with the liquid is gradually moved

away from the liquid, while a structureless flat wall is

moved towards it such that in the final state, the liquid

interacts only with the structureless wall. Computing the

free energy difference during this transformation, along

with the interfacial free energy of liquid in contact with

the structureless wall, gives γwl. γwl for the liquid-flat

wall system, which serves as the reference state for their

system was obtained using the PA technique. Since the

PA technique fails in case of crystal-wall interfaces [23]

one cannot use their scheme to determine γwcfor crystal

in contact with a structured wall. In fact, much less is

known about γwcfor LJ systems in contact with a wall.

Grochola et al. [35] developed another TI technique

which they have called “λ-integration”, to determine the

surface free energies of solids. In principle, this technique

could be also applied to wall-crystal or wall-liquid inter-

faces, but the method has not been worked out yet for

such interfaces.

A straightforward and comprehensive method is thus

needed to compute the interfacial free energies of LJ sys-

tems in contact with a wall. In the present work, a novel

TI scheme is introduced to compute the interfacial free

energy of a LJ system confined between walls. We con-

sider both the liquid as well as the fcc crystal phase along

the (111) orientation near the wall. While most previous

works employing TI methods to obtain γwl or γwc are

limited to structureless walls, here we specifically con-

sider the case of a structured wall, consisting of particles

rigidly attached to the sites of an ideal fcc lattice. Our

scheme consists of TI in two steps, providing a reversible

thermodynamic path that transforms the bulk LJ system

into a LJ fluid or crystal interacting with a structured

wall. In the first step, a thermodynamic path is devised

to reversibly transform the bulk LJ system without walls

and periodic boundary conditions in all directions to a

state where it interacts with the structureless wall. This

is accomplished by gradually modifying the interaction

potential between the wall and the LJ particles along the

thermodynamic path. The technique is inspired by the

method proposed by Heni and L¨ owen [29] to compute the

interfacial free energy of hard sphere fluids and crystals

in contact with a hard wall.

The LJ system interacting with the flat wall serves as

the reference state to calculate the interfacial free energy

of the LJ liquid or crystal in contact with a structured

wall. In the second step, another TI scheme reversibly

changes the structureless wall interacting with the LJ

system into a structured wall. This is done by gradually

switching off the flat walls and simultaneously switching

on the structured walls. While previous methods based

on TI techniques for the calculation of γwl make use of

“cleaving potentials” [31] or “phantom walls ” [19], here

we directly modify the interaction potential to make the

transformation from the reference state to the final state

in each of the two steps. Though this TI scheme is specif-

ically developed for a LJ potential, it can be easily gen-

eralized to more complex potentials.

The wetting behavior of a liquid or crystal in contact

with a structured solid wall will be affected by various

parameters. In this work we focus on three parameters:

i) the interaction strength between the wall and the LJ

system, ii) the density of the structured wall, and iii)

the orientation of the structured wall with respect to the

interface normal. For these cases, wall-crystal interfacial

free energies only for the (111) orientation of the crystal

are considered.

Since the PA method has been widely applied in the

past to evaluate the wall-liquid interfacial free energy, we

will compare results obtained from it with those yielded

by the TI method for both flat and structured walls. In

addition, we will also show that the interfacial free ener-

gies of the LJ system interacting with a flat wall can be

obtained directly at coexistence, without any extrapola-

tion from data at low densities, enabling us to investigate

its wetting behavior.

Furthermore, to compare the estimates of interfacial

free energy yielded by our TI scheme with that obtained

in previous works, we apply our technique to a model

system studied by Tang and Harris (TH) [16] using the

mechanical definition of the interfacial free energy. Their

system consisted of a LJ fluid confined between identical

rigid structured walls oriented along the (100) orienta-

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tion, under conditions of liquid-vapor coexistence. Later,

Grzelak and Errington (GE) [36] investigated the same

system using Grand Canonical Transition Matrix Monte

Carlo (GCTMMC) simulations. They computed the in-

terfacial free energy profile as a function of the surface

density at bulk liquid-vapor saturation condition, to ob-

tain the contact angle and the solid-vapor and solid-liquid

interfacial tensions. For this system, we will examine the

variation of γwl as a function of the wall-liquid interac-

tion strength and compare estimates of γwl from these

two studies. Due to the paucity of studies on the crystal-

wall interfacial free energy, we will restrict this compari-

son with previous works only to the wall-liquid interfacial

free energy.

In the following, we introduce the details of the model

potentials considered in this work (Sec. II), give the vari-

ous definitions of interfacial free energies, outline the PA

method, describe the proposed TI scheme, and provide

the main details of the simulation (Sec. III). Then, we

present the results (Sec. IV) and finally draw some con-

clusions (Sec. V).

II.MODEL POTENTIAL

The MD system to determine the interfacial excess free

energy of a LJ system in contact with a structured wall

consists of N identical particles interacting with each

other and with the structured wall via a shifted-force

LJ [8] potential. If two particles i and j of types α and β

are separated by a distance rij, the interaction potential

is written as

uαβ(rij) =

φαβ(rij) − φαβ(rc) − φ′

for 0 < rij≤ rc,

0 for rij> rc,

αβ(rij= rc)[rij− rc]

(2)

where the prime in φ′

spect to r and

αβdenotes the derivative with re-

φαβ(rij) = 4ǫαβ

??σαβ

rij

?12

−

?σαβ

rij

?6?

. (3)

In Eq. (2), α or β can represent a LJ particle (p) or a

structured wall particle (w). The parameters ǫαβ and

σαβ have units of energy and length, respectively. The

cut-off distance is set to rc= 2.5σαβ.

In the following, energies, lengths and masses are given

in units of ǫpp, σppand mp, respectively. Thus, tempera-

ture, pressure and interfacial free energy are expressed in

units of ǫpp/kB, ǫpp/σ3

is made dimensionless by reducing it with respect to the

characteristic time scale

?

choose σwp= σpp.

The N identical liquid or crystal particles are enclosed

within a simulation box of size Lx×Ly×Lz, using periodic

boundary conditions in the x and y directions. In the

ppand ǫpp/σ2

pp, respectively. Time

mpσ2

pp/ǫpp. For simplicity, we

z direction the particles are confined by the structured

wall, between z = zbat the top and z = ztat the bottom.

The system thus consists of two planar wall-liquid (or

wall-crystal) interfaces with a total area of A = 2LxLy.

The structured wall is arranged in a manner such that the

wall layers closest to the LJ system are positioned at zb=

−Lz/2 and zt= Lz/2. Also, an integer number of unit

cells was chosen for the structured wall such that the wall

is exactly adapted to the lateral size of the simulation cell.

The width of the structured wall is chosen large enough

to avoid LJ particles on opposite sides of the wall from

interacting with each other since the determination of

interfacial free energy by TI or PA methods is built on

the assumption of two independent wall-liquid (or wall-

crystal) interfaces.

The TI scheme adopted in this work consists of two

steps. First, a bulk LJ system with periodic boundary

conditions is transformed into a state where the LJ sys-

tem interacts with impenetrable flat walls. Then, in the

second step, the flat walls are reversibly transformed into

structured walls. The structureless flat wall (fw) is taken

to be a purely repulsive potential interacting along the

z direction with the LJ particles and is described by a

WCA potential,

ufw(z) =

4ǫw

??σpp

for 0 < z ≤ zcw,

for z > zcw

z

?12−?σpp

z

?6+1

4

?

× w(z)

0

(4)

with the cut-off zcw= 21/6σppand z = zi−Z the distance

of particle i at zi to one of the flat walls at Z = zb or

Z = zt.The function w(z) ensures that ufw(z) goes

smoothly to zero at z = zcwand is given by

w(z) =

1

h4+ (1/(z − zcw)4), (5)

where the dimensionless parameter h is set to 0.005.

To compare to the results of Tang and Harris [16] and

Grzelak and Errington [36], we also consider a truncated

and shifted LJ potential for the particle-particle (pp) and

particle-structured wall (pw) interactions,

u∗

αβ(rij) =

?

φαβ(rij) − φαβ(rc)

0

for rij< rc,

for rij≥ rc

(6)

with αβ = pp,pw and the cut-off radius rc= 2.5σpp.

Moreover, we choose σpw= 1.1σppand vary the parame-

ter ǫpwin units of ǫppin order to determine the interfacial

free energy γwl as a function of the strength of the pw

interactions. As Tang and Harris [16], we use a substrate

consisting of three layers of atoms rigidly fixed to fcc lat-

tice sites, with the (100) orientation of the wall facing the

liquid along the z direction. The average number density

of the liquid is set to ρp = 0.661 σ−3

substrate to ρw= 0.59 σ−3

the system fixed at T = 0.9kB/ǫpp.

ppand that of the

ppkeeping the temperature of

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III. CALCULATION OF INTERFACIAL FREE

ENERGIES

A.Definitions

The Hamiltonian of our model, corresponding to the

LJ system interacting with a solid wall, can be written

as

H(r,p) =

Np

?

i=1

1

2mip2

i+

Np

?

i=1

Np

?

j=i+1

upp(rij) + Uwall

(7)

with Npthe total number of LJ particles and Uwallthe

wall-particle potential. For interactions of the LJ system

with a flat wall, Uwall=?Np

?Np

for the walls, since the flat walls are considered to be

of infinite mass and immovable; similarly, the structured

wall particles are considered to be immobile.

Our simulations are performed in the NPNAT ensem-

ble, where the number of particles N, surface area A and

temperature T are kept constant and the length of the

simulation box along the z direction is allowed to fluc-

tuate in order to maintain a constant normal pressure

PN. The use of the NPNAT ensemble is necessary to

maintain a constant bulk density of the system when TI

is applied (see below). Moreover, any stress present in

the crystal due to interaction with the walls can relax

during the NPNAT simulation. The determination of

the interfacial free energy by thermodynamic or mechan-

ical approaches demands that there is a bulk region in the

middle of the simulation box where the density is equal to

the bulk density of the homogeneous system. Hence, the

system size along the z direction must be large enough

to prevent the two walls on either side of the LJ system

from influencing each other.

The isothermal-isobaric partition function correspond-

ing to the Hamiltonian (7) is

i=1ufw(z = zi− Z), and for

the system in contact with a structured wall, Uwall =

?Nw

i=1

j=1upw(rij) (with Nw the total number of wall

particles). In Eq. (7), there is no kinetic energy term

QNPNAT=

1

h3NN!

× AdLzdrNdpN

? ? ?

exp

?

−H(r,p) + PNALz

kBT

?

(8)

where r and p denote respectively the positions and

momenta of the particles and h is the Planck con-

stant.The Gibbs free energy G of the confined liq-

uid or crystal is related to the partition function (8) by

G = −kBT lnQNPNAT.

The derivative of Gibbs free energy with respect to the

surface area defines the interfacial tension:

γ′=

?∂G

∂A

?

NPNAT

. (9)

This thermodynamic definition of the interfacial tension

is equivalent to the mechanical definition [37]:

γ′=1

2

?zt

zb

[PN(z) − PT(z)](10)

where PN(z) and PT(z) are respectively the normal and

tangential pressure profiles of the liquid and the factor

1/2 is introduced to account for the fact that the liquid

is confined between two identical walls. The local pres-

sure tensor components PN(z) and PT(z) are defined in

Eqs. (13), (16) and (14) (see next section).

The interfacial tension γ′is related to the interfacial

free energy γ as [38]

γ′= γ + A∂γ

∂A.

(11)

If a liquid is in contact with a dynamic structured wall,

which can support stress, the interfacial excess free en-

ergy will vary with the area of the interface. However, in

this work we consider rigid substrates and structureless

flat walls, which do not support stress and hence for the

liquid-wall interface, the interfacial tension will be equal

to the interfacial free energy validating the use of the PA

method. For a crystal-wall interface, however, the second

term in Eq. (11) will be a relevant quantity. In this work,

we will restrict our attention only to the determination

of the interfacial free energy.

The interfacial free energy of an inhomogeneous system

with walls can be defined as a Gibbs excess free energy

per area,

γ =Gsystem− Gbulk

A

(12)

with Gsystem and Gbulk the Gibbs free energies of the

inhomogeneous system and the bulk phase of the system,

respectively. We will use this definition to calculate the

interfacial free energy using TI.

B.

γ from PA

Determination of the interfacial free energy by the PA

method is only valid if the interfacial tension equals the

interfacial free energy. This holds, e.g., for interfaces be-

tween a liquid and a flat wall or rigid substrate. Hence,

we will use the PA technique to obtain the wall-liquid in-

terfacial free energy, and compare it with results obtained

from TI.

To obtain interfacial free energies from the mechanical

approach, the local tangential and normal pressure ten-

sor components have to be computed. There is no unique

microscopic definition for these local pressure tensor com-

ponents and different expressions lead to the same value

for the interfacial tension [24]. Mechanical stability, how-

ever, requires that the normal component of the pressure

tensor is independent of the distance from the wall and

furthermore the two tangential components along the x

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and y directions are equal to each other. In the literature,

it is only the Irving and Kirkwood (IK) definition of the

pressure tensor that satisfies these properties [24, 25, 39].

According to the IK definition, contributions to the nor-

mal and tangential components of the pressure tensor

from any two particles i and j at ziand zj, respectively,

can be written as

PIK

N(z) = ρ(z)kBT

−1

A

??

?Np

i=1

i<j

zij

riju′

pp(rij)Θ

?z − zi

zij

?

Θ

?zj− z

zij

??

−1

A

?

Nw

?

j=1

zij

riju′

pw(rij)Θ

?z − zi

zij

?

Θ

?zj− z

zij

??

(13)

and

PIK

T(z) =ρ(z)kBT −

1

2A

??

?z − zi

i<j

x2

ij+ y2

rij

ij

u′

| zij|

pp(r)

×Θ

?z − zi

zij

?

Θ

zij

??

,(14)

where θ is the Heavyside step function, zij= zj−zi, and

ρ(z) is the local density given by

ρ(z) =

N(z)

(A/2) × ∆z.

(15)

Here, ∆z is the bin width used to obtain the pressure

profiles and N(z) is the number of liquid particles in the

bin between z and z+∆z. This contribution to the local

pressure tensor is added to all bins between ziand zj. It

is to be noted that the liquid-structured wall interaction

has no contribution to the tangential component of the

pressure tensor due to the periodicity of our system in

the lateral direction [15, 17].

The contribution to the pressure tensor from the struc-

tureless walls can also be taken into account by the IK

method [24, 25] by considering the walls at zb and zt

to be particles of infinite mass. From Eq. (13), we thus

obtain

Pfw

N(z) =1

A

?N

i=1

?N

i=1

?

Ffw(zi− zb)Θ(zi− z)

?

−

1

A

?

Ffw(zt− zi)Θ(z − zi)

?

,

(16)

with Ffw(z) = −dUfw(z)/dz.

From Eqs. (13) and (14), it is clear that if two particles

in a bin are located on the same side of z, their contri-

bution to the local pressure tensor cannot be taken into

account by the IK method. To minimize the number of

such cases, we must choose the bin width to be compara-

ble to the shortest distance between between the particles

in the z direction. On the other hand if the bin width is

too small, there will be larger fluctuations in the pressure

tensor and the average must be taken over many more

configurations to get a smooth profile, thus increasing

the computational time. In our simulations we choose a

bin-width of ∆z = 0.05.

Equation (10) being the difference between two sim-

ilar numerical values is subject to large relative errors.

Moreover, at large densities near the wall, the density

and pressure profiles show rapid oscillations and hence

resolving them with high precision requires a huge com-

putational effort. Below, the accuracy of the PA method

is studied in detail via a direct comparison to the data

obtained from TI.

C.

γ from TI

In a TI, the free energy of a state of interest is com-

puted with respect to a reference state [21]. A parameter

λ, which couples to the interaction potential, is gradu-

ally changed such that the reference state is reversibly

transformed into the final state of interest.

To calculate the interfacial free energy of the LJ sys-

tem in contact with a structured wall, the TI scheme is

carried out in two steps. In the first step, a bulk LJ sys-

tem without walls and periodic boundary conditions in

all directions is reversibly transformed into a LJ system

in contact with a structureless flat wall along the z di-

rection. In the second step, the flat wall interacting with

the LJ system is reversibly transformed into a structured

wall. To ensure reversibility of the thermodynamic path,

periodic boundary conditions are applied in x, y and z

direction. Calculating the free energy change in the two

steps yields the required interfacial free energy.

To obtain γ for a hard-sphere system in contact with

a hard structureless wall via TI, Heni and L¨ owen [29]

have used a scheme, where a bulk hard sphere system

is reversibly transformed into a system interacting with

an impenetrable hard wall. In this work, we generalize

the scheme of Heni and L¨ owen to continuous wall po-

tentials. To this end, the wall potential is parametrized

by a parameter λ such that the wall changes smoothly

from a penetrable to an impenetrable wall as λ increases.

The following parametrization of the wall potential is

adopted:

ufw(λ,z) =λ24ǫw

??

σpp

z + (1 − λ)zcw

?12

+1

−

?

σpp

z + (1 − λ)zcw

?6

4

?

× w(z).

(17)

Figure 1, shows the parametrized wall potential at dif-

ferent values of λ. At λ = 0 a bulk LJ system can freely

cross the boundaries. For small values of λ the barrier

height at z = 0 is of the same order as kBT and the LJ

particles can penetrate the barrier. As λ increases, the

wall becomes more and more impenetrable and finally an

impenetrable WCA wall is obtained at λ = 1.

Page 6

6

Since, the interfacial excess free energy of the LJ sys-

tem in presence of walls is calculated with respect to a

bulk LJ crystal, it is important that the bulk density is

maintained as the parameter λ is varied during the trans-

formation. This is particularly important for a LJ liquid

close to coexistence, since an increase in the bulk density

in the presence of walls could lead to a precrystallization

of the bulk liquid during the transformation, thus mak-

ing it irreversible. A constant bulk density also ensures

that our system is large enough such that there are no

mutual influences between the walls on either side of the

bulk LJ system. To maintain a constant bulk density one

must keep the normal pressure PNconstant and change

the volume, as is achieved by carrying out simulation in

the NPNAT ensemble.

The system Hamiltonian now depends on λ and is

given by

H(r,p,λ) =

Np

?

i=1

1

2mip2

i+

Np

?

i=1

Np

?

j=i+1

upp(rij)+

Np

?

i=1

ufw(λ,z = zi− Z)

.(18)

and thus the partition function can be written as

Q(λ) =

1

h3NN!

? ? ?

exp

?

−H(r,p,λ) + PzALz

kBT

×AdLzdpNdpN.

?

(19)

The derivative of the Gibbs free energy with respect to

00.51

z

0

0.25

0.5

0.75

1

Ufw(z,λ)

0.1

0.2

0.3

0.5

1.0

λ=

FIG. 1: (Color online) Variation of the WCA wall potential as

a function of λ during the transformation of the bulk LJ liquid

or crystal into an impenetrable flat wall interacting with the

LJ system. From no wall at λ = 0, we have a wall with a

finite barrier at small values of λ. With increasing λ, the wall

becomes more and more impenetrable. At λ = 1, there is an

impenetrable wall represented by the WCA potential.

λ is

∂G(λ)

∂λ

= −kBT

Q(λ)

?∂Q(λ)

∂λ

?

=

?∂H(λ)

∂λ

?

λ

(20)

where the angular brackets denote the ensemble average

at a particular value of λ in the NPNAT ensemble.

The Gibbs free energy difference between the two ini-

tial and final state can then be obtained as

∆G = G(λ = 1) − G(λ = 0) =

?1

0

?∂G(λ)

∂λ

?

λ

dλ (21)

=

?1

0

?∂H(λ)

∂λ

?

λ

dλ . (22)

To compute ∆G from molecular simulations, indepen-

dent simulations runs are carried out at Nλdiscrete in-

tervals between λ = 0 and λ = 1. Alternatively, one

can also calculate the free energy difference in a single

simulation by varying λ step by step such that the final

configuration at a value of λ = λiis the initial configura-

tion for the next value at λ = λi+1. In both methods, the

system is equilibrated at each λ = λi, and then the time

average of the quantity ∂H(λ)/∂λ is calculated. The nu-

merical integration of Eq. (22) is carried out using the

trapezoidal rule:

∆G =

Nλ−1

?

i=1

1

2

??∂H/∂λ?i+ ?∂H/∂λ?i+1

?(λi+1− λi) .

(23)

The partial derivative of H(λ) with respect to λ is given

by

∂H(λ)

∂λ

=∂ufw(z,λ)

∂λ

=2

λufw(λ,z)+

zcw

z + (1 − λ)zcwufw(z,λ) .

(24)

The above TI scheme leads to a wall which is not fully

impenetrable and hence does not correspond to the de-

sired state of interest at the end of the integration path.

While the LJ particles cannot cross the wall at λ = 1,

two particles near the boundary but on opposite sides of

the wall can still interact with each other. To overcome

this problem another TI step is carried out to bring the

system to a state where the LJ particles are in contact

with a fully impenetrable wall excluding such spurious

interactions. This is achieved by parametrizing upp(r)

by a factor µ,

upp(µ,rij) = u(1)

pp(rij) + (1 − µ)u(2)

pp(rij)

(25)

where, u(1)

on same side of the wall, while U(2)

interaction between particles near the boundary but on

opposite sides of the wall, i.e. the separation between par-

ticles is greater than Lz/2. At µ = 0, all such spurious

interactions are taken into account. As µ increases such

pp(rij) denotes interaction between LJ particles

pp(rij) corresponds to

Page 7

7

interactions are reduced by the factor 1 − µ and finally

at µ = 1, these spurious interactions are completely ne-

glected. The µ dependent Hamiltonian for this step can

be written as,

H(r,p,µ) =

Np

?

i=1

1

2mip2

i+

Np

?

i=1

Np

?

j=i+1

?

u(1)

pp(rij) + (1 − µ)u(2)

pp(rij)

?

+

Np

?

i=1

ufw(λ = 1,z = zi− Z) .

(26)

The thermodynamic integrand in this step is

∂H/∂µ = ∂upp(r,µ)/∂µ = −u(2)

pp(rij).(27)

and thus the free energy difference can be expressed as

∆Gfw→fw∗ =

?1

0

?∂upp(r,µ)/∂µ?µdµ.(28)

Our simulations

∆Gfw→fw∗ is very minor, i.e. about 0.1% of ∆G from

Eq. (23) and hence can be neglected.

Using Eqs. (12), (21), and (24), the interfacial free en-

ergy of a LJ system interacting with a flat wall can be

written as

showed that the contribution of

γ =Gfw− Gbulk

A

=∆Gbulk→fw

A

(29)

with

∆Gbulk→fw=

?1

0

?∂ufw(z,λ)

∂λ

?

λ

dλ.(30)

In the second step of our TI scheme, the flat wall is

reversibly transformed into a structured wall in contact

with the LJ system. During this change, the flat walls

are positioned at the same location as the structured wall

layer closest to the LJ liquid or crystal and there is no

interaction between the flat and structured walls. The

transformation from flat walls to structured walls is ac-

complished by parametrizing the wall potential as:

Uwall(rij,λ) = (1−λ)2?

i

ufw(z = zi−Z)+λ2?

i,j

upw(rij) .

(31)

Now, the λ-dependent Hamiltonian is

H(r,p,λ) =

Np

?

i=1

1

2mip2

i+

Np

?

i=1

Np

?

Np

?

j=i+1

upp(rij)+

(1 − λ2)

Np

?

i=1

ufw(z = zi− Z,λ) + λ2

i=1

Nw

?

j=1

upw(rij)

(32)

and the derivative of the Hamiltonian with respect to λ

∂H

∂λ=∂Uwall

∂λ

=2

(λ − 1)

?

i

ufw(z = zi− Z) + λ

Np

?

i=1

Nw

?

j=1

upw(rij)

.

(33)

So, finally the interfacial free energy of the LJ system in

contact with a structured wall (sw) is given by

γwc=Gsw− Gbulk

A

=∆Gbulk→fw

A

+∆Gfw→sw

A

(34)

with

∆Gfw→sw=

?1

0

?∂Uwall(λ)

∂λ

?

λ

dλ.(35)

D.Simulations

To integrate the equations of motion, the velocity

form of the Verlet algorithm was used with a time step

τ = 0.005 and, to maintain constant normal pressure,

the Andersen barostat algorithm [40] was chosen. Peri-

odic boundary conditions are employed in the x, y and z

directions for the first step of the TI method where flat

walls are considered. In the second step periodic bound-

ary conditions are only used along the x and y direc-

tions. The PA simulations are carried out with periodic

boundary conditions only along the x and y directions.

The temperature was kept constant by drawing every 200

steps the velocity of the LJ particles from the Maxwell-

Boltzmann distribution at the desired temperature.

During the NPNAT simulations, the position of the

flat or structured walls must be modified keeping the nor-

mal pressure PNconstant. To ensure this, the flat walls

are treated as particles of infinite mass and, at each time

step, the wall position zfwis rescaled according to

Z(t + ∆t) = Z(t) × Lz(t + ∆t)/Lz(t) . (36)

Note that this method is similar to the “fluctuating wall”

method proposed by Lupowski and van Swol [41], main-

taining a constant normal pressure in a MC simulation

of LJ particles in presence of a structureless wall.

When a rigid structured wall interacts with the LJ sys-

tem, the wall particles must not change their positions

relative to each other, thereby changing the wall density.

To circumvent this problem, the center of mass of the

wall is changed at every time step according to Eq. (36).

The position of the individual particles of the wall is then

shifted such that they are at the same relative distance

from the center of mass as at the beginning of the simu-

lation.

To calculate γwlwe consider systems of 4000 particles.

The structured walls contain between 200-1200 particles,

Page 8

8

depending on the orientation and the density of the wall.

The total surface area of the simulation cell is about

A = 200, yielding a length along the z direction of about

Lz= 65 at the various wall-liquid interaction strengths

ǫpwand structured wall densities ρw. γwlis computed at

a normal pressure of PN= 3 and temperature T = 2. At

the start of the simulation, the LJ particles were placed

on ideal fcc lattice sites and the walls were inserted si-

multaneously. Then the system was allowed to melt and

equilibrate at the desired pressure and temperature, be-

fore the calculations were performed.

To test for the presence of any finite size effects, we

also performed simulations with up to 12000 particles

and a total surface area of about A = 340, but obtained

identical results compared to the simulations carried out

with the smaller system size. This shows that systems of

4000 particles are large enough to avoid finite size effects

in the calculation of interfacial free energies.

From previous works pertaining to hard sphere sys-

tems, it is well known that the (111) orientation of the

crystal in contact with a planar hard wall (or a soft WCA

wall) gives the lowest interfacial tension as compared to

the (100) or (110) orientations [42]. At small undercool-

ings, the hard sphere fluid freezes into the (111) crystal

near the wall [43]. Hence, we obtain interfacial free ener-

gies only for the (111) orientation of the fcc crystal phase

in contact with the walls. Unlike the liquid, the crystal

has a long-range order and, in order to prevent deforma-

tion of the crystal, the system size must be compatible

with this order.

For the determination of γwc, systems of 7056 particles

and area around A = 450 are considered. The number of

structured wall particles ranges from 800 to about 1200,

depending on the different wall densities. Only the (111)

orientation of the crystal in contact with the (111) orien-

tation of the structured wall along the interface normal

was considered. The corresponding simulations to obtain

the interfacial free energy of a crystal in contact with a

flat wall are carried out with 3960 particles with an area

of around A = 200. Simulations were also carried out

with a system size of 6006 particles and a total area of

A = 300 and there was only a marginal deviation (< 1%)

in the value of γwcas compared to the smaller system.

For comparing results obtained by our approach with

that of Tang-Harris [16] and Grzelak-Errington [36], we

performed simulations for their system with 4000 liquid

particles and 392 structured wall particles at the tem-

perature T = 0.9. We considered a lateral system size of

10×10 and the length of the box along the z direction was

kept at 60.5134 to obtain a bulk liquid density of 0.661,

the value reported by Tang and Harris [16] for their simu-

lations. Simulations were performed at this fixed density

in the NV T ensemble. With this system size, the finite

size effects were negligible. The liquid in contact with

the flat wall [Eq. (4)] was used as the reference state to

calculate γwlfor the liquid in contact with the structured

wall at the same bulk density and temperature.

A simulation at constant normal pressure leads to fluc-

tuations of the length of the simulation cell in the z di-

rection, Lz. However, in order to compute the density

and pressure profiles necessary for the PA method, it is

more suitable to keep Lzconstant. Hence, to obtain γwl

via the PA method, we first equilibrate the system in the

NPNAT ensemble for 5 × 105time steps. After equilib-

rium is reached, the simulations continue for 4.5 × 106

time steps, from which the average length of the box in

the z direction is calculated. Lz is set to this average

value and the particle coordinates are rescaled by the

factor ?Lz?/Lz(tf), tfdenoting the time at the end of this

equilibration run. An equilibration run is then carried

out in the NV T ensemble for 5×105time steps and the

final production run consists of 4.5×106steps, when we

accumulate data for the density, energy, temperature and

pressure profiles every 5 time steps, averaging the profiles

over 9×105sample configurations. In our simulations, we

observe a drift of 0.5−2.5% in the normal pressure profile

from the given external pressure PN. This drift can be

reduced by averaging the length of the box for a longer

simulation time or over a large number of realizations.

To calculate the interfacial free energy via TI, we used

around 40 intervals between λ = 0 and λ = 1 to nu-

merically compute Eq. (23). Independent equilibration

runs were carried out at each value of λ, in the NPNAT

ensemble for about 5 × 105− 1 × 106time steps. After

the completion of the equilibration run, production runs

were performed for 5 × 105steps in order to accumulate

data. The same TI scheme and simulation procedure has

been adopted to determine the interfacial free energy of

the system investigated by Tang-Harris [16] and Grzelak-

Errington [36], but in the NV T ensemble at a fixed liquid

density.

010203040 5060

z

0

0.5

1

1.5

2

ρ(z)

fw, λ=0.0

fw, λ=0.5

fw, λ=1.0

sw

10

2030

z

40

50

0.64

0.65

ρ(z)

FIG. 2: (Color online) Density profile of the system configu-

ration for different values of λ at the temperature T = 2.0,

the normal pressure PN = 3.0, and the wall-liquid interaction

strength ǫw = 1. The corresponding density profile for a liq-

uid in contact with the (100) orientation of a structured wall

of density ρw = 1.371 at ǫpw = 1.0 is also shown. The inset

shows the density profiles in the bulk region on a magnified

scale.

Page 9

9

0 0.20.40.60.81

λ

0

0.2

0.4

0.6

0.8

<∂Ufw(z,λ) /∂λ>

Liquid

Crystal

a)

0 0.20.4 0.60.81

λ

-0.25

-0.2

-0.15

-0.1

-0.05

0

<∂Uwall(r,λ)/∂λ>

Liquid

Crystal

b)

FIG. 3: (Color online) (a) ?∂ufw(z,λ)/∂λ? as a function of λ computed from simulations at PN = 3 and T = 2 for the liquid and

at PN = 3.0 and T = 0.5 in case of the crystal. To determine γwl, 4000 liquid particles were enclosed in a simulation box of area

A = 200 with the wall-liquid interaction strength ǫw = 1. Corresponding simulations to calculate γwc were carried out with

3960 particles and and area A = 235.22. (b) ?∂[Uwall]/∂λ?, corresponding to transformation of the flat wall into a structured

wall. In case of the liquid, the structured wall consists of 392 particles rigidly fixed to fcc lattice sites, with the (100) orientation

of the wall facing the liquid. For the crystal, the structured wall consisted of 432 wall particles with the (111) orientation of the

wall in contact with the crystal. The density of the structured wall was ρw = 1.371 for the liquid (ρrm= 0.647 for the crystal)

and the wall-particle interaction strength for the liquid -wall simulations was kept at ǫpw = 1, while for the crystal ǫpw = 0.5.

Other parameters are same as in (a).

For our TI method to be valid, there must be a bulk

region unaffected by the wall. Figure 2 shows the density

profile of liquid in contact with the parametrized flat wall

represented by Eq. (17) at various values of λ, with the

wall-liquid interaction strength ǫw = 1. Also shown in

Fig. 2 is the density profile of the liquid in contact with

the (100) orientation of a structured wall of density ρw=

1.371 and with interaction strength ǫpw= 1. The inset

shows a magnified view of the density profiles in the bulk

region. Clearly, all the density profiles overlap with each

other indicating that the bulk region is unaffected by the

walls.

Figure

?∂ufw(z,λ)/∂λ? as a function of λ, during the transforma-

tion of a bulk liquid (crystal) to a confined liquid (crys-

tal), interacting with flat walls. The integrand is smooth,

thus allowing for an accurate determination of the inter-

facial free energy. Figure 3b shows the integrand as a

function of λ for the second step of the thermodynamic

integration when the flat wall is transformed into a struc-

tured wall. The integrand is always negative, implying

that the interfacial free energy of a LJ liquid (crystal) in

contact with a rigid structured wall is smaller than for

the case where the liquid (crystal) is in contact with a

structureless flat wall.

3a showsthe thermodynamic integrand

IV.RESULTS

A.

γwl

Using TI, we first determine the liquid-flat wall interfa-

cial free energy γwlat several temperatures and pressures.

In Fig. 4, we plot the data obtained from TI along with

the estimate for γwl from the PA technique, as a func-

tion of temperature and pressure, respectively. The error

bars in γwlobtained from TI are smaller than the symbol

size and hence are not reported. It is evident from Fig. 4

that there is good agreement between the two methods

within the statistical error. However, Fig. 4a shows that

γwlobtained from TI is smoothly varying, while the PA

data is less systematic. Relative differences between the

two methods are between 0.3 and 1.8%.

This small disagreement between the two methods is

due to the large fluctuations in the local pressure pro-

files as obtained from Eqs. (13) and (14) (see Fig. 5).

The inset in Fig. 5 clearly shows the large fluctuations

in the normal pressure profile near the wall and in both

the normal and tangential pressure profiles in the bulk

region. Since Eq. (10) represents the difference between

two pressure profiles: Any lack of precision in the numer-

ical measurements magnifies the relative error. The TI

data is more accurate and less computationally expen-

sive compared to what would be required to obtain more

precise values from the PA method.

The liquid-flat wall system can now be used as the ref-

erence system to calculate the interfacial free energy of

the liquid in contact with a rigid structured wall. Some

Page 10

10

0.0 2.0 4.0 6.0 8.0 10.0

PN

0.0

2.0

4.0

6.0

8.0

γwl

1.0 1.5 2.0

T

2.5 3.0

2.8

2.9

3.0

3.1

3.2

γwl

a) b)

FIG. 4: (Color online) (a) Interfacial free energy of liquid in

contact with the flat wall, γwl, as a function of temperature.

NPNAT simulations were carried out at a normal external

pressure PN = 3.0, with 4000 liquid particles and total surface

area of A = 200. The wall-liquid interaction strength is ǫw =

1. Filled squares correspond to data obtained from TI, while

estimates from PA are represented by open circles with error

bars. Uncertainty in data computed by TI is less than the

symbol size. (b) γwl as a function of pressure at T = 2.5.

Other parameters and symbols representing the TI and PA

data are same as in (a).

0204060

z

3

3.04

3.08

P(z)

0102030405060

z

0

1

2

3

4

5

6

7

8

9

10

P(z)

PN(z)

PT(z)

FIG. 5: (Color online) Normal and tangential components of

the pressure profiles of liquid in contact with a flat wall at

PN = 3 and T = 2, with 4000 liquid particles. The wall liquid

interaction strength, ǫw = 1. Inset shows the pressure profiles

on a magnified scale close to the magnitude of the external

pressure PN.

-30 -20 -100

z

102030

0

0.5

1

1.5

2

ρ(z)

Structured

Wall

Structured

Wall

Liquid

x

y

z

FIG. 6: (Color online) a) A two-dimensional projection onto

the zy plane of a sample configuration of liquid in contact

with the (100) orientation of the structured wall at PN = 3

and T = 2. 4000 liquid particles are enclosed in a simulation

box of area A = 200. The density of the structured wall is

ρw = 1.371. The wall-liquid interaction strength is set to

ǫpw = 1. b) Density profile of the liquid, averaged over many

configurations, showing pronounced layering at the structured

wall-liquid interface.

properties of the structured wall such as the wall-liquid

interaction strength, density of the structured wall and

its orientation along the interface will affect the inter-

facial free energy and consequently the wetting behav-

ior of the liquid. We will investigate γwl for the (100),

(110) and, (111) orientations of the structured wall in

contact with the liquid at different wall-liquid interac-

tion strengths. Effects of density of the structured wall

on the interfacial free energy will also be studied. Un-

less otherwise indicated, the external pressure is set to

PN= 3 and the temperature to T = 2.

Figure 6 shows a sample configuration of the liquid in

contact with the (100) orientation of a structured wall

in the zy plane and the corresponding density profile.

We observe layering of the particles near the wall. Away

from the walls a bulk region forms, where the density is

constant.

In Fig. 7a, γwlis displayed as a function of ǫpw, as ob-

tained from the PA and TI method. The error bars in the

PA method were calculated from 2 − 3 realizations. For

TI, the error bars are smaller than the size of the sym-

bols and hence they are not reported. We observe that

γwldecreases with ǫpw, which is in agreement with previ-

ous studies carried out using the mechanical route [15, 16]

and other thermodynamic methods [19, 36]. High values

of ǫpwrepresent stronger attraction between the wall and

liquid particles. This reduces the free energy needed to

move the liquid from the bulk to the surface resulting in

a lower interfacial free energy. Data from the two meth-

ods agree qualitatively but the percentage difference of

the PA results with respect to the TI data increases at

higher values of ǫpw.

Apart from the fluctuations in the local pressure pro-

files, the strong layering near the interface at large ǫpw

also reduces the numerical accuracy of the PA method.

Figure 7b shows the liquid density profiles at various in-

teraction strengths along with the corresponding profile

Page 11

11

0.0 0.20.40.6 0.81.0

εpw

1.0

1.5

2.0

2.5

γwl

a)

0.01.0 2.03.04.05.0

z

0.0

0.5

1.0

1.5

2.0

ρ(z)

012

z

34

0

0.5

1

1.5

2

ρ(z)

fw

sw (100) εpw=0.1

sw (100) εpw=0.5

sw (100) εpw=1.0

sw (110) εpw=1.0

sw (111) εpw=1.0

b)

FIG. 7: (Color online) a) Interfacial free energy of liquid in contact with the (100) orientation of the structured wall vs. ǫpw.

All parameters are the same as in Fig. 6. Open squares with solid line represent data from the TI method and open circles with

dashed line represent estimates yielded by the PA method. Uncertainties in data from TI are less than the symbol size. The

wall-liquid interaction strength is ǫpw = 1. b) Liquid density profiles at different wall-liquid interaction strengths. The inset

shows the profiles corresponding to the (100), (110) and (111) orientations of the structured wall in contact with the liquid at

ǫpw = 1. For comparison the density profile for liquid in contact with the flat wall is also shown in both a) and b). All system

parameters are same as in a).

in presence of a flat wall. The first peak in the density

profile corresponding to the flat wall occurs at a greater

distance from the wall as compared to the peaks arising

out of the liquid-structured wall interaction. This can be

attributed to the purely repulsive flat wall, which pushes

the liquid further away from the walls compared to the

structured walls.

The interfacial free energy of the crystal-melt interface

is influenced by the orientation of the crystal in contact

with the melt [1]. Similarly, it might be expected that

different orientations of the structured wall in contact

with the liquid will affect the wall-liquid interfacial free

energy. In the inset of Fig. 7b, we plot the density profiles

near the wall for the (111), (110) and (100) orientations of

the structured wall in contact with the liquid at ǫpw= 1.

The density of the structured wall ρw= 1.371 and, the

lateral dimensions of the system corresponding to the

(100), (110) and (111) orientations of the wall are 10×10,

10 × 10.102 and 9.623 × 10.102, with 392, 420 and 330

wall particles, respectively. Figure 7b shows the layering

of the density profile to be most pronounced for the (111)

orientation owing to the closely packed atoms exerting

a greater repulsive force on the liquid. In contrast, the

layering for the (110) orientation is much less pronounced

and the first peak in the density profile also occurs closer

to the wall as compared to the (111) or (100) orientations.

In Table I, we report γwl, obtained from the PA and

TI methods, at various ǫpw, for the (100), (110) and

(111) orientations of the structured wall in contact with

the liquid. In general, we find γwlcorresponding to the

(111) and (100) orientations of the wall to be larger com-

pared to the (110) orientation. This can be attributed

to the stronger repulsive forces exerted on the liquid by

the more close packed (111) and (100) planes, as com-

pared to the more loosely packed (110) plane. Also, rela-

tively better agreement is observed between the PA and

TI methods for the (100) and (111) orientations of the

wall as compared to the (110) orientation.

Simulations were also carried out for large system sizes

at ǫpw = 1, with up to 12000 particles, and large sur-

face area and wall separations. However, no systematic

change was observed as compared to the smaller system

size. Clearly, to obtain accurate values of the interfacial

free energy, the pressure profiles need to be determined

with far greater numerical accuracy. This has also been

recently pointed out by D. Deb et al. [33] for hard sphere

systems. Computing the pressure profiles with high pre-

cision is computationally expensive and since accurate

values can be obtained by the TI method with much less

computational effort, use of the PA technique seems to

be unjustified. In the remainder of the discussion on our

model, we report results obtained with the TI method

only.

The density of the structured wall will also have an

impact on the wetting behavior of the liquid in contact

with it. We have carried out simulations at several den-

sities ρw corresponding to different lattice constants of

the ideal fcc lattice structure of the wall. In Fig. 8a,

we report the TI results for γwl as a function of ρw at

three different ǫpw’s. At ǫpw= 1, the interfacial free en-

ergy decreases with the density of the wall. The larger

number of wall particles at greater densities exert strong

attractive forces on the liquid, reducing the interfacial

free energy. At extremely large densities the interfacial

free energy becomes negative indicating that the liquid

completely wets the wall. No data for very low densities

Page 12

12

(100)(110)(111)

ǫpw

γTI

wl

γPA

wl

γTI

wl

γPA

wl

γTI

wl

γPA

wl

0.10 2.115 2.165±0.034 1.865 1.902±0.025 2.149 2.189±0.010

0.25 2.039 2.067±0.034 1.808 1.860±0.033 2.056 2.073±0.004

0.50 1.742 1.775±0.012 1.521 1.587±0.003 1.740 1.806±0.002

0.75 1.364 1.343±0.007 1.144 1.252±0.003 1.348 1.335±0.002

1.00 0.929 0.862±0.012 0.705 0.868±0.012 0.906 0.902±0.012

TABLE I: Interfacial free energy γwl at different wall-liquid interaction strengths, for the (100), (110) and (111) orientations

of the structured wall in contact with the liquid. Data computed from both TI and PA are shown. Simulations are carried out

at PN = 3 and T = 2. The density of the structured wall ρw = 1.371.

0.0 1.02.03.04.0

ρw

-1.0

0.0

1.0

2.0

3.0

γwl

εpw = 0.5

εpw = 1.0

εpw = 0.1

a)

0.01.02.03.04.0 5.0

z

0.0

1.0

2.0

3.0

ρ(z)

ρw = 4

ρw = 2.048

ρw = 1.372

ρw = 0.863

b)

FIG. 8: (Color online) a) Interfacial free energy of a liquid in contact with a structured wall vs. density of the wall at several

wall-liquid interaction strengths ǫpw. Results were obtained from TI. Simulations were carried out at a constant normal pressure

PN = 3 and temperature T = 2.0. b) Density profiles of the liquid in contact with the structured wall at various densities ρw.

The wall-liquid interaction strength is ǫpw = 1. Other parameters are same as in a).

0.30.4 0.5 0.60.7 0.8

εpw

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

γwl

TI

PA

GCTMMC (1-wall)

GCTMMC (2-wall)

FIG. 9: (Color online) γwl vs. ǫpw for the model specified by

Eq. (6). Squares correspond to our results computed via TI.

Filled circles represent data from the studies of Tang-Harris

[16], while diamonds and filled right triangles are the data

obtained by Grzelak-Errington [36] using the one-wall and

two-wall approaches, respectively.

of the structured wall are shown in Fig. 8a since the liq-

uid particles penetrate the wall at low densities and the

interfacial region is no longer well defined.

At ǫpw= 0.5, γwlshows a weak maximum and at large

wall densities decreases with ρwmore gradually as com-

pared to the situation when ǫpw = 1.

wall-liquid interaction strength (ǫpw = 0.1), γwl has a

weak dependence on the density of the structured wall

and remains almost constant in the range of ρwshown in

Fig. 8a.

In Fig. 8b, we show the density profiles of the liquid

corresponding to several densities of the structured wall

ρw at ǫpw = 1.It is observed that the layering gets

more pronounced and the first peak in the density profiles

occurs further away from the walls as ρw increases. A

similar behavior of the density profile was observed when

increasing ǫpwat fixed ρw. The variation in γwland the

nature of the density profiles indicate that increasing ρw

at ǫpw= 1 has a similar effect on γwl as increasing ǫpw

at a fixed value of the wall density.

Finally, to compare results from our TI technique with

those obtained by other methods, we consider the model

defined by Eq. (6), which was first studied by Tang and

At still lower

Page 13

13

Harris using a PA technique [16].

Errington [36] utilized GCTMMC simulations to obtain

free energy profiles of the same system over a wide range

of densities, with the fluid confined by a structured wall

on one side and a hard wall at the other side or the fluid

confined between two identical structured walls. Both of

their approaches with one or two structured walls lead

to the same results within the statistical errors. Using

TI in the NVT ensemble, we computed the interfacial

free energy of the same system, with the liquid confined

by two identical structured walls.

sults are reported along with data from the two previous

works [16, 36]. Data obtained by Tang-Harris systemat-

ically deviate from our estimates of γwlas ǫpwincreases.

Their data also has a large statistical error. However, our

predictions are in good agreement with those of Grzelak

and Errington.

Later Grzelak and

In Fig. 9, our re-

0.40.40.6 0.60.80.8111.21.2 1.41.4

T

3.2

3.2

3.4

3.4

3.6

3.6

γ

3.1

3.3

3.5

3.7

0.51 1.52

εpw

3.2

3.4

3.6

γ

FIG. 10: (Color online) Crystal-flat wall interfacial free en-

ergy γwl(diamonds) and liquid-flat wall interfacial excess free

energy γwc (squares) as a function of temperature. NPNAT

simulations were carried out at PN = 3 and the wall-liquid

interaction strength ǫw = 1.

crystal was considered in determining γwc. Inset corresponds

to γwl and γwc as a function of the interaction strength ǫw at

coexistence: PN = 3.0 and T = 0.8. Symbols are same as in

the main graph.

The (111) orientation of the

B.

γwc

We will compute the crystal-wall interfacial energy by

the TI method only, since the crystal can support stress

and hence the interfacial tension and interfacial free en-

ergy are not the same, thus invalidating the applicabil-

ity of the PA method [23]. In performing simulations of

crystal in contact with walls on both sides, the number

of particles N must be chosen such that it is compatible

with the long range order of the crystal. This is in con-

trast to a liquid, where choosing a large enough N yields

a sufficiently large system and the two walls on either

side of the liquid do not influence each other. However,

the crystal has a long-range order and merely choosing a

large N may not necessarily be commensurate with this

order. Such an incommensurate N is associated with

long range elastic distortion, that propagates from one

wall to the other leading to an inaccurate value for γwc.

This had already been pointed out by D. Deb et al. [32]

who studied the interfacial free energy of a hard-sphere

crystal confined between softly repulsive walls described

by the WCA potential.

As specified earlier, we restrict our attention to the

close-packed (111) orientation of the crystal in contact

with a flat wall along the z axis. To evaluate γwc, a

bulk fcc crystal with the (111) orientation along z axis

is simulated in the NPT ensemble. Periodic boundary

conditions are employed in all directions to determine

the average equilibrium lattice constant and hence the

density of the crystal. A fcc crystal with this density

was chosen as the initial configuration for the TI sim-

ulations to compute γwc. The length of the simulation

box was chosen such that an integer number of unit cells

along the x, y and z directions adapted exactly into the

simulation box. Then, independent simulations in the

NPNAT ensemble were carried out at each value of the

λ parameter during the two-step TI scheme. In Fig. 10,

we plot γwc for crystal in contact with a flat wall, as a

function of temperature up to the coexistence tempera-

ture at PN = 3.0. For comparison, γwl is also plotted

at the same pressure. Similar to γwl, γwcdecreases as a

function of temperature.

To predict the wetting behavior of the crystal in con-

tact with the wall at crystal-liquid coexistence, one needs

γclin addition to γwland γwc. However, without knowl-

edge of γcl, it is still possible to predict whether the crys-

tal will completely wet the wall (θc= 0◦) or do so only

partially (0 < θc< 180◦). To this end, simulations were

carried out at coexistence (PN = 3.0,T = 0.8) for the

bulk liquid and crystal in contact with a flat wall at var-

ious interaction strengths between the wall and the bulk

liquid or crystal. The data is reported in the inset of

Fig. 10. No wetting layer was observed near the walls

during wall-liquid simulations, allowing for a determina-

tion of the wall-liquid interfacial free energy directly at

coexistence. We find that γwc> γwl, showing that there

is incomplete wetting of a flat wall by the (111) orienta-

tion of the LJ crystal and that the contact angle can be

varied by changing ǫpw.

While γclhas not been determined in this work, David-

chack and Laird [44, 45], obtained the crystal-liquid in-

terfacial free energy at coexistence for a similar LJ model.

The bulk liquid and crystal densities at the coexistence

temperature T = 0.809 for their model is same as for our

system at PN = 3.0 and T = 0.8. At T = 0.809, they

obtained γcl= 0.428 ± 0.004. Since the LJ model used

in this work is not very different from their potential, it

is safe to assume that the crystal-liquid interfacial free

energies will not be far apart. Using γcl= 0.428 ± 0.004

and the values of γwland γwcused in this work, we ob-

tain contact angles of 97.4◦, 103.8◦and 113.6◦respec-

Page 14

14

tively, signifying partial wetting of the flat wall. This

is in contrast to the hard sphere case, where the (111)

orientation of the crystal led to complete wetting of the

wall [32]. Such a situation of incomplete wetting will fa-

cilitate the study of heterogeneous nucleation of a crystal

droplet at a wall-liquid boundary, and enable us to test

the predictions of classical nucleation theory.

0 0.51 1.5

ρw

2 2.53

0.5

1

1.5

2

2.5

3

γwc

0 0.25 0.5 0.75

εpw

1

0.5

1

1.5

2

2.5

3

γwc

FIG. 11: (Color online) Crystal-structured wall interfacial free

energy, γwc, as a function of the structured wall density, for

the (111) orientation of the crystal in contact with the (111)

orientation of the structured wall. NPNAT simulations were

carried out at a normal pressure and temperature PN = 3

and T = 0.5 respectively, with the crystal-wall interaction

strength ǫwc = 0.5. The inset shows γwc as a function of ǫpw

for a structured wall density ρw = 0.647, other parameters

remaining the same.

Having obtained the flat wall-crystal interfacial free

energy, we can now compute the structured wall-crystal

interfacial excess free energy γwc. We choose to investi-

gate the (111) orientation of the crystal in contact with

the (111) orientation of the structured wall. To obtain

γwc, commensurate surfaces of the wall must be in con-

tact with the crystal on both sides. We know that in the

fcc structure, there is an ABCABCABC... stacking of

the lattice planes along the (111) orientation. The same

order of the planes must be kept for the crystal plane

in contact with the wall. For example, the the following

stacking of the planes,

AwBwCwAwBwCwAcBcCcAcBcCc...

......AcBcCcAwBwCwAwBwCw.

(37)

is commensurate. However, a stacking of the planes in

an incommensurate manner such as

AwBwCwAwBwCwAcBcCcAcBcCc...

......AcBcCcCwBwAwCwBwAw

(38)

will lead to long range deformation of the crystal.

In Fig. 11 and its inset, we plot γwc as a function of

the structured wall density and, in the inset, as a func-

tion of the wall-crystal interaction strength ǫpw. Similar

to the liquid case we find that the interfacial free en-

ergy decreases with ǫpw due to the stronger attraction

between the crystal and the wall. Unlike the liquid case,

Fig. 11 shows that while the main trend for γwc is to

increase with decreasing density of the structured wall,

there is a sharp dip when the density of the wall equals

the density of the crystal. This is easy to understand,

since less energy will be needed to create an interface,

when the structured wall has the same structure as the

crystal than when there is a mismatch between the wall

and crystal structures leading to a relatively unfavorable

interaction between them.

V.CONCLUSION

We propose a thermodynamic integration (TI) scheme

to compute interfacial free energies of liquids or crystals

in contact with flat or structured walls from molecular

dynamics simulation. In this work, this scheme has been

applied to Lennard-Jones systems, but it can be easily

generalized to other interaction models. The implemen-

tation of our method is simple, and, as demonstrated

above, our method provides reliable and accurate esti-

mates of γwland γwcthat enter in Young’s equation (1).

In particular for structured walls (substrates), to the best

of our knowledge, there are no simulation studies cal-

culating the substrate-crystal interfacial free energy γwc.

Most of the previous simulation works on structured walls

[15–18, 26, 27] have been limited to the calculation of the

interfacial free energy γwlusing the integration over the

pressure anisotropy (PA). The PA method, however, does

not give reliable results in general, and, in contrast to

our TI scheme, it is not applicable to substrates that can

support stress (such as structured walls where the wall

particles are allowed to move and are thus not fixed to

their ideal lattice positions, see discussion above). There-

fore, the TI scheme proposed in this work can be consid-

ered as a novel approach to obtain accurate values for

substrate-liquid or substrate-crystal interfacial free ener-

gies and thus it will be useful in studies of wetting and

nucleation problems.

Acknowledgments

One of the authors (R. B.) thanks the DLR-DAAD

fellowship program for financial support. The authors

acknowledge financial support by the German DFG SPP

1296. Computer time at the NIC J¨ ulich is gratefully

acknowledged.

Page 15

15

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