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.
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 ,
γwc+ γclcosθc= γwl
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<
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)  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 . Moreover, its use is justified only
for systems where the interfacial tension equals the inter-
facial free energy . 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” , this method is invalid. For
the same reason, the PA technique cannot be used to de-
termine γwc. 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  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  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
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  developed a TI method by
the use of “cleaving potentials”, to obtain γwland γwcfor
hard sphere systems at coexistence. In another work ,
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 , 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
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.  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 
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.  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
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  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”  or “phantom walls ” , 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
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)  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-
tion, under conditions of liquid-vapor coexistence. Later,
Grzelak and Errington (GE)  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
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  potential. If two particles i and j of types α and β
are separated by a distance rij, the interaction potential
is written as
φαβ(rij) − φαβ(rc) − φ′
for 0 < rij≤ rc,
0 for rij> rc,
αβ(rij= rc)[rij− rc]
where the prime in φ′
spect to r and
αβdenotes the derivative with re-
φαβ(rij) = 4ǫαβ
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
pp, respectively. Time
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-
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
for 0 < z ≤ zcw,
for z > zcw
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
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  and
Grzelak and Errington , we also consider a truncated
and shifted LJ potential for the particle-particle (pp) and
particle-structured wall (pw) interactions,
φαβ(rij) − φαβ(rc)
for rij< rc,
for rij≥ rc
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 , 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
III. CALCULATION OF INTERFACIAL FREE
The Hamiltonian of our model, corresponding to the
LJ system interacting with a solid wall, can be written
upp(rij) + Uwall
with Npthe total number of LJ particles and Uwallthe
wall-particle potential. For interactions of the LJ system
with a flat wall, Uwall=?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 =
j=1upw(rij) (with Nw the total number of wall
particles). In Eq. (7), there is no kinetic energy term
? ? ?
−H(r,p) + PNALz
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:
This thermodynamic definition of the interfacial tension
is equivalent to the mechanical definition :
[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 
γ′= γ + A∂γ
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
γ =Gsystem− Gbulk
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.
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
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 . 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
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
N(z) = ρ(z)kBT
?z − zi
?z − zi
T(z) =ρ(z)kBT −
?z − zi
?z − zi
where θ is the Heavyside step function, zij= zj−zi, and
ρ(z) is the local density given by
(A/2) × ∆z.
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
Ffw(zi− zb)Θ(zi− z)
Ffw(zt− zi)Θ(z − zi)
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.
γ from TI
In a TI, the free energy of a state of interest is com-
puted with respect to a reference state . 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 
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
z + (1 − λ)zcw
z + (1 − λ)zcw
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.
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
ufw(λ,z = zi− Z)
and thus the partition function can be written as
? ? ?
−H(r,p,λ) + PzALz
The derivative of the Gibbs free energy with respect to
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.
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) =
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
?(λi+1− λi) .
The partial derivative of H(λ) with respect to λ is given
z + (1 − λ)zcwufw(z,λ) .
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)
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
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,
pp(rij) + (1 − µ)u(2)
ufw(λ = 1,z = zi− Z) .
The thermodynamic integrand in this step is
∂H/∂µ = ∂upp(r,µ)/∂µ = −u(2)
and thus the free energy difference can be expressed as
∆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
showed that the contribution of
γ =Gfw− Gbulk
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?
ufw(z = zi−Z)+λ2?
Now, the λ-dependent Hamiltonian is
(1 − λ2)
ufw(z = zi− Z,λ) + λ2
and the derivative of the Hamiltonian with respect to λ
(λ − 1)
ufw(z = zi− Z) + λ
So, finally the interfacial free energy of the LJ system in
contact with a structured wall (sw) is given by
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  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 , 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-
To calculate γwlwe consider systems of 4000 particles.
The structured walls contain between 200-1200 particles,
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 . At small undercool-
ings, the hard sphere fluid freezes into the (111) crystal
near the wall . 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  and Grzelak-Errington , 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  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  and Grzelak-
Errington , but in the NV T ensemble at a fixed liquid
0 10 203040 5060
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
0 0.2 0.40.60.81
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
?∂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.
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
0.0 2.0 4.0 6.0 8.0 10.0
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).
010 20304050 60
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
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
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
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
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
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
The interfacial free energy of the crystal-melt interface
is influenced by the orientation of the crystal in contact
with the melt . 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.  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
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
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 1.0 2.0 3.04.0
εpw = 0.5
εpw = 1.0
εpw = 0.1
ρw = 4
ρw = 2.048
ρw = 1.372
ρw = 0.863
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).
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
, while diamonds and filled right triangles are the data
obtained by Grzelak-Errington  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
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
Harris using a PA technique .
Errington  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
Later Grzelak and
In Fig. 9, our re-
0.40.40.6 0.60.8 0.8111.2 126.96.36.199
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
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 . 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. 
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-
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 . 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
0 0.25 0.5 0.75
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,
is commensurate. However, a stacking of the planes in
an incommensurate manner such as
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.
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
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
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