1. Introduction
Using knowledge of the molecular structure of liquids to predict their macroscopic behavior is important for several applications [
1,
2,
3,
4,
5]. One of the most rigorous solution theories is the Kirkwood–Buff (KB) theory, where a sound connection between macroscopic and microscopic properties for isotropic multicomponent fluids is established [
6,
7]. Kirkwood and Buff derived a relation between several thermodynamic properties and integrals of radial distribution functions (RDFs) over infinite and open volumes
in the grand-canonical ensemble [
6]:
where
r is the particle distance, and
is the RDF, of the infinitely large system, for species
and
. KB integrals can also be expressed in terms of density fluctuations in open systems [
6,
7,
8,
9]. While KB integrals were derived for open and infinite systems, many studies use molecular simulation to estimate KB integrals, where only finite systems can be studied. In Reference [
10], a review of the methods available in literature for computing KB integrals from molecular simulations is presented.
To accurately estimate
, it is possible to use KB integrals of finite and open subvolumes
V embedded in larger reservoirs. In this way, the grand-canonical ensemble, in which KB integrals in the thermodynamic limit were derived, is mimicked. This approach is referred to as the small system method (SSM) [
5,
11,
12]. According to the SSM, properties of small subvolumes, that can be of the order of a few molecular diameters, are treated in terms of thermodynamics of small systems rather than classical thermodynamics. According to Hill’s thermodynamics of small systems, properties of open embedded subvolumes scale with the inverse size of the subvolumes [
13,
14]. This also applies to KB integrals of finite subvolumes,
[
10,
12]. For a specific system,
computed with subvolumes of different sizes, scales linearly with the inverse size of the subvolume [
10,
12,
15,
16]. For spherical subvolumes
V inside a simulation box, we have
where
is the characteristic length of the subvolume
V with surface area A. For a sphere,
L is the diameter (
). In Equation (
2),
is related to surface effects of the subvolume. Using Hill’s formulation of small-system thermodynamics [
14], it was shown that properties of small systems can be written in terms of volume and surface contributions [
17]. In Reference [
17], Hill’s thermodynamics were applied to several properties, including pressure. From the volume contribution of pressure, the homogeneous pressure is obtained, while the Gibbs surface relation was obtained from the surface contribution [
17]. This last contribution is proportional to the surface tension. In the case of KB integrals, the surface term, or contribution,
, can also be defined from Gibbs surface equation [
17]. From a microscopic point of view, it originates from interactions between molecules inside the subvolume and molecules across the boundary of the subvolume [
12,
15]. These surface effects vanish in the thermodynamic limit, but for systems used in molecular dynamics (MD) simulations these effects cannot be neglected [
18]. As a result, the quantitative and qualitative study of surface contributions is essential for estimating
from integrals of finite subvolumes
.
KB integrals of finite and open subvolumes
are defined in terms of fluctuations in the number of particles, which relate to double integrals of RDFs over the subvolume
V [
12],
where
and
are the number of molecules of type
and
, in volume
V. The brackets
denote an ensemble average in an open system. Equation (
3) is applicable to isotropic molecular fluids where the orientations of molecules are already integrated out. While it is possible to compute KB integrals
from fluctuations in the number of particles (i.e., the right hand side of Equation (
3)), it is more practical to use RDFs. RDFs are readily computed by most molecular simulation software packages. The double integrals in the left hand side of Equation (
3) can be transformed to a single integral using a weight function
[
12],
The function
depends on the geometry of
V. For spherical and cubic subvolumes, theoretically derived functions are available in References [
12,
16], respectively. It is possible to numerically obtain the function
for an arbitrary shape as shown in Reference [
19]. In this work, spherical subvolumes will be used, for which,
where
x is the dimensionless distance
[
12,
15,
16]. The scaling of finite integrals
with the size of the subvolumes
L is used to compute KB integrals in the thermodynamic limit
(for convenience, indicies
and
will be dropped from this point onwards). Specifically,
is computed from extrapolating the linear part of the scaling of
with
to the limit
[
12,
15,
17]. A disadvantage of this approach is that a linear regime is not always easily identified [
15].
To avoid extrapolating
, Krüger and Vlugt [
16] proposed a direct estimation of KB integrals in the thermodynamic limit:
The accuracy of the estimation depends on the function
[
20]. Krüger and Vlugt [
16] considered three different estimations and found that integrals computed using the function
provided the best estimation of
,
KB integrals computed using Equations (
6) and (
7) will be denoted by
. To derive the expression for
, the starting point was the scaling of KB integrals with
. First, an explicit estimation of
in Equation (
2) was derived. In the work of Krüger and Vlugt [
16],
has the following form
It is important to note that the structure of Equation (
8) is similar to KB integrals in the thermodynamic limit (Equation (
1)). So, analogous to Equation (
2),
can be defined as,
where
C is a constant. For finite systems,
can be computed using
where the function
is given in Equation (
5). The similarity between the expression for KB integrals (Equation (
4)) and surface term (Equation (
8)) in the thermodynamic limit allows for deriving an estimation for surface effects as in Equation (
6). Using Equation (
6), and Equation (
8) an explicit expression for surface effects in the thermodynamic limit, denoted here by
, is obtained from
with
in Equation (
7).
An alternative method to extrapolate KB integrals
to the thermodynamic limit is to use the scaling of
with
L, rather than the scaling of
with
. The scaling of
in Equation (
2) can be rewritten as
By fitting the linear part of the scaling of
with
L, it is possible to obtain
and
. Finding the slope and intercepts of a straight line is easier than extrapolating the linear regime of the scaling of
with
. Another advantage of this approach is that an estimation of the surface effects is automatically computed. This estimation can be compared to other available methods for computing
. So far, it is shown that three methods are available for estimating
from integrals of finite subvolumes:
Using the scaling of
(Equation (
4)) with
. To estimate
, the linear regime of the scaling is extrapolated to the limit
.
Using the direct extrapolation formula
(Equation (
6)) combined with the function
(Equation (
7)). This will converge to
for large
L.
Computing
from fitting the linear regime of the scaling of
with
L (Equation (
12)). The values of the integrals
are computed using Equation (
4).
To simplify the estimation of KB integrals, it would be useful to evaluate the performance of these methods in terms of accuracy and practicality. Similarly, different methods are available to compute the surface term in the thermodynamic limit :
Using the expression in Equation (
11).
From the scaling of
with
L (Equation (
9)).
is computed using Equation (
10). The value of
is obtained from the slope of the scaling; as
, in which
C is a constant.
From the scaling of
with
L (Equation (
9)). The value of
is obtained from the intercept of the scaling.
The objective of this work is to test the estimation of KB integrals
and the surface effects
using the approaches discussed earlier. For both
and
, the effect of the size of the system is studied. These effects are investigated for both Lennard–Jones (LJ) and Weeks–Chandler–Andersen (WCA) fluids [
21] at different densities. Finally, this work aims at quantifying the contributions of the surface term when computing KB integrals of LJ fluid at various densities.
This paper is organized as follows: In
Section 2, the methods used to compute RDFs, KB integrals, and the surface term of KB integrals of LJ and WCA fluids are presented.
Section 2 includes the details of the MD simulations. In
Section 3, the results are presented, which include KB integrals and the surface term for WCA and LJ systems at different sizes and densities.
Section 4 summarises the main findings of this work.
2. Methods
RDFs of systems of particles interacting via the LJ potential are computed using MD simulations in the
ensemble. Systems with different densities and number of particles are studied. Also, systems of particles interacting via the Weeks–Chandler–Andersen (WCA) potential [
22], where only the repulsive part of the LJ potential is included, are considered. The common approach of particles counting was implemented to compute RDFs. While this is not carried out in this work, it is possible to investigate other methods. It would be interesting to see if force-based computations of RDFs improve the convergence of computed KB integrals [
23,
24,
25]. For each system, the computed RDF is used to compute KB integrals
and the surface term
in the thermodynamic limit. For both quantities, the methods discussed in
Section 1 are used. In this section, the numerical details of computing RDFs and the required integrals are briefly discussed.
According to Kirkwood–Buff theory, KB integrals are defined for open and infinite systems [
6]. When computing KB integrals using molecular simulations of closed systems, it is essential to correct RDFs for finite-size effects [
10,
12,
15]. Recently, a number of corrections for the RDFs have been proposed [
12,
26,
27]. In Reference [
15] it was demonstrated that the accuracy of computing KB integrals improves when the Ganguly and van der Vegt correction [
26] is applied. Applying the Ganguly and van der Vegt correction results in RDFs that are consistent with the physical behavior of fluids. For example, Equation (
13) converges to
for a single-component ideal gas, which is the correct value in the thermodynamic limit. The Ganguly and van der Vegt [
26] correction is based on the excess (or depletion) of the density of the system beyond a distance
L from a central molecule
. The corrected RDF is
is obtained from a simulation in a finite system with total volume
.
is the excess number of particles of type
in a sphere of radius
r around a particle of type
, which is computed by
For all systems studied in this work, RDFs are corrected using the Ganguly and van der Vegt corrections. The corrected RDFs are numerically integrated to obtain
,
,
, and
. Once these quantities are obtained, various methods are implemented to estimate KB integrals
and the surface terms
in the thermodynamic limit.
Table 1 provides the relations and description of the methods considered to estimate
. Similarly,
Table 2 presents information regarding the methods used to estimate
.
Simulation Details
RDFs of LJ and WCA fluids were computed using MD simulations and then used to estimate KB integrals and surface effects. The simulations were carried out using an in-house FORTRAN code. All RDFs were computed from simulations in the ensemble. The systems were simulated at a dimensionless temperature , dimensionless densities ranging from 0.2 to 0.8 and using number of particles N equals to 100, 500, 1000, 5000, 10,000, 30,000, and 50,000. For each size, the length of the simulation box L was set according to the required density.
All MD simulations started from a randomly generated configuration for which an energy minimization was used to eliminate particle overlaps. A sufficient number of time steps was used to initialize the system. After initialization, RDFs were sampled every 100 time steps. For both initialization and production, a dimensionless time step equal to 0.001 was used. The simulation length was chosen depending on the size of the system and the available computational resources. For instance, for systems with , 1 × production time steps were carried out, while for the maximum size N = 50,000, 7 × steps were used. Multiple independent simulations were performed for each point (, N). The resulting RDFs were then averaged and used to compute and . At high densities (), RDFs from at least 10 runs are used. At lower densities, at least 20 runs are performed to enhance statistics.