Wall-liquid and wall-crystal interfacial free energies via thermodynamic integration: a molecular dynamics simulation study.
ABSTRACT 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.
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ABSTRACT: Different computational techniques in combination with molecular dynamics computer simulation are used to determine the wall-liquid and the wall-crystal interfacial free energies of a modified Lennard-Jones (LJ) system in contact with a solid wall. Two different kinds of solid walls are considered: a flat structureless wall and a structured wall consisting of an ideal crystal with the particles rigidly attached to fcc lattice sites. Interfacial free energies are determined by a thermodynamic integration scheme, the anisotropy of the pressure tensor, the non-equilibrium work method based on Bennett acceptance criteria, and a method using Cahn's adsorption equations based on the interfacial thermodynamics of Gibbs. For the flat wall, interfacial free energies as a function of different densities of the LJ liquid and as a function of temperature along the coexistence curve are calculated. In the case of a structured wall, the interaction strength between the wall and the LJ system and the lattice constant of the structured wall are varied. Using the values of the wall-liquid and wall-crystal interfacial energies along with the value for the crystal-liquid interfacial free energy determined previously for the same system by the "cleaving potential method," we obtain the contact angle as a function of various parameters; in particular, the conditions are found under which partial wetting occurs.The Journal of Chemical Physics 08/2013; 139(8):084705. · 3.12 Impact Factor
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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
Page 2
2
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-
Page 3
3
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
Page 4
4
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
Page 5
5
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.