Extended version of the van der Waals capillarity theory

An extended version of the van der Waals capillarity theory describing the liquid-vapor interface in the temperature range from the triple to the critical point is suggested. A model functional of thermodynamic potential for a two-phase Lennard-Jones system taking into account the effect of the highest degree terms of gradient expansion has been constructed. The identity of the thermodynamic and the mechanical definition of Tolman's length has been proved in the framework of the adopted form of functional. The properties of nuclei of the liquid and the vapor phase are described. The paper determines: the work of formation of a nucleus, density profiles, size dependences of the surface tension, and the parameter delta in the Gibbs-Tolman-Koenig-Buff equation.     

Metastable extension of the sublimation curve and the critical contact point

Molecular dynamics methods have been employed in order to calculate the (p,rho,T)-properties and the internal energy of gas and crystal phases in stable and metastable equilibrium coexistence states for a model system consisting of 2048 Lennard-Jones particles. Thermal and caloric equations of state and the spinodal curves of the vapor and crystal phases have been determined. A new algorithm for the computation of phase equilibrium curves is suggested. Employing this method, the sublimation curve and its metastable extension to temperatures above the triple point have been calculated. It is found that the crystal-gas phase equilibrium terminates on the spinodal of the superheated crystal. The point of contact of the sublimation line and the spinodal is a singular point of the thermodynamic surface of states of a simple system considered.     

Lateral capillary forces between solid bodies on liquid surface: a lattice Boltzmann study

When two solid bodies are placed on the surface of a dense liquid under gravitation, they deform the liquid surface to experience a lateral capillary force between themselves that can be attractive and repulsive, depending on the wettabilities and weights of the bodies. In the present study, the lateral capillary force between two square bodies at a liquid-vapor interface has been examined using numerical simulations based on a two-dimensional two-phase lattice Boltzmann (LB) method. The particular situations were simulated, where every body was vertically constrained and had the fixed triple points at its upper or lower corners. Here, the triple point indicates the place at which vapor, liquid, and solid phases meet. The interaction force between these two bodies was calculated as a function of the separation distance, the interfacial tension, and the gravitational acceleration. The simulation results agree well with the analytical expression of the lateral capillary interaction, indicating that our LB method can reproduce the interaction force between two bodies of various wettabilities at a liquid-vapor interface in mechanical equilibrium.     

Monte Carlo simulation methods for computing the wetting and drying properties of model systems

We introduce general Monte Carlo simulation methods for determining the wetting and drying properties of model systems. We employ an interface-potential-based approach in which the interfacial properties of a system are related to the surface excess free energy of a thin fluid film in contact with a surface. Two versions of this approach are explored: a "spreading" method focused on the growth of a thin liquid film from a surface in a mother vapor and a "drying" method focused on the growth of a thin vapor film from a surface in a mother liquid. The former provides a direct measure of the spreading coefficient while the latter provides an analogous drying coefficient. When coupled with an independent measure of the liquid-vapor surface tension, these coefficients enable one to compute the contact angle. We also show how one can combine information gathered from application of the spreading and drying methods at a common state point to obtain direct measures of the contact angle and liquid-vapor surface tension. The computational strategies introduced here are applied to two model systems. One includes a monatomic Lennard-Jones fluid that interacts with a structureless substrate via a long-ranged substrate potential. The second model contains a monatomic Lennard-Jones fluid that interacts with an atomistically detailed substrate via a short-ranged potential. Expanded ensemble techniques are coupled with the interface potential approach to compile the temperature- and substrate strength-dependence of various interfacial properties for these systems. Overall, we find that the approach pursued here provides an efficient and precise means to calculate the wetting and drying properties of model systems.     

Phase equilibria and surface tension of pure fluids using a molecular layer structure theory (MLST) model

This paper presents a new model based on thermodynamic and molecular interaction between molecules to describe the vapour–liquid phase equilibria and surface tension of pure component. The model assumes that the bulk fluid can be characterised as set of parallel layers. Because of this molecular structure, we coin the model as the molecular layer structure theory (MLST). Each layer has two energetic components. One is the interaction energy of one molecule of that layer with all surrounding layers. The other component is the intra-layer Helmholtz free energy, which accounts for the internal energy and the entropy of that layer. The equilibrium between two separating phases is derived from the minimum of the grand potential, and the surface tension is calculated as the excess of the Helmholtz energy of the system. We test this model with a number of components, argon, krypton, ethane, n-butane, iso-butane, ethylene and sulphur hexafluoride, and the results are very satisfactory.     

Computation of surface tensions using expanded ensemble simulations

A method for the direct simulation of the surface tension is examined. The technique is based on the thermodynamic route to the interfacial tension and makes use of the expanded ensemble simulation method for the calculation of the free energy difference between two inhomogeneous systems with the same number of particles, temperature, and volume, but different interfacial area. The method is completely general and suitable for systems with either continuous or discontinuous interactions. The adequacy of the expanded ensemble method is assessed by computing the interfacial tension of the planar vapor-liquid interface of Lennard-Jones, Lennard-Jones dimers, Gay-Berne, and square-well model fluids; in the latter, the interactions are discontinuous and the present method does not exhibit the asymmetry of other related methods, such as the test area. The expanded ensemble simulation results are compared with simulation data obtained from other techniques (mechanical and test area) with overall good agreement.     

Surface tension of water droplets upon homogeneous droplet nucleation in water vapor

A method has been proposed for determining interfacial free energy from the data of molecular dynamics simulation. The method is based on the thermodynamic integration procedure and is distinguished by applicability to both planar interfaces and those characterized by a high curvature. The workability of the method has been demonstrated by the example of determining the surface tension for critical nuclei of water droplets upon condensation of water vapor. The calculation has been performed at temperatures of 273–373 K and a pressure of 1 atm, thus making it possible to determine the temperature dependence of the surface tension for water droplets and compare the results obtained with experimental data and the simulation results for a “planar” vapor–liquid interface.     

Density functional theory of size-dependent surface tension of Lennard-Jones fluid droplets using a double well type Helmholtz free energy functional

A double well type Helmholtz free energy density functional and a model density profile for a two phase vapor-liquid system are used to obtain the size-dependent interfacial properties of the vapor-liquid interface at coexistence condition along the lines of van der Waals and Cahn and Hilliard density functional formalism of the interface. The surface tension, temperature-density curve, density profile, and thickness of the interface of Lennard-Jones fluid droplet-vapor equilibrium, as predicted in this work are reported. The planar interfacial properties, obtained from consideration of large radius of the liquid drop, are in good agreement with the results of other earlier theories and experiments. The same free energy model has been tested by solving the equations numerically, and the results compare well with those from the use of model density profile.     

Phase equilibria and interfacial tension of fluids confined in narrow pores

Correlation between phase behaviors of a Lennard-Jones fluid in and outside a pore is examined over wide thermodynamic conditions by grand canonical Monte Carlo simulations. A pressure tensor component of the confined fluid, a variable controllable in simulation but usually uncontrollable in experiment, is related with the pressure of a bulk homogeneous system in equilibrium with the confined system. Effects of the pore dimensionality, size, and attractive potential on the correlations between thermodynamic properties of the confined and bulk systems are clarified. A fluid-wall interfacial tension defined as an excess grand potential is evaluated as a function of the pore size. It is found that the tension decreases linearly with the inverse of the pore diameter or width.     

A novel mesophase formed by top-shaped molecules in the bulk and unsupported thin films: a molecular dynamics study

We have used molecular dynamics simulations to investigate the ordering of top-shaped molecules in bulk phases and in unsupported thin films. Each rigid anisotropic molecule was composed of 11 Lennard-Jones interaction centers (beads). In an attempt to enhance the nematic stability in preference to smectic, the three central beads were assigned a larger Lennard-Jones diameter than the tail beads, giving the molecule a shape resembling a top. The molecular model was found to exhibit an unusual bulk mesophase with long-range orientational order and with molecular center-of-mass positions arranged in parallel interdigitated layers, with layer spacing smaller than half the length of the long axis of a molecule. However, despite the toplike molecular shape, no nematic phase was observed in the pressure range studied. Unsupported films of the isotropic liquid were cooled in order to locate a triple point between the novel mesophase, vapor, and isotropic liquid. At temperatures slightly above the triple point, enhanced surface ordering of molecules was found to occur in the unsupported film. At temperatures slightly below the triple point, the preferred molecular alignment in the unsupported film was parallel to the interface, in violation of arguments that have been proposed based on the relative enthalpies of various cleavage planes for close-packed structures.     

Effect of long-range interactions on the surface tension

The effect of long-range interactions on the surface tension at a liquid-gas interface was considered. An analytical expression for the correction to the surface tension for the cutoff of the particle interaction potential at the distance r _c was derived based on a step density profile. For the Lennard-Jones fluid, this correction was calculated numerically from the results of computer simulations of the density profiles. It was established that, in the vicinity of the triple point, the correction is as great as ～6% at the potential cutoff radius r _c=6.78 molecular diameters, a quantity insensitive to the form of the density profile in the interfacial layer.     

Wall-liquid and wall-crystal interfacial free energies via thermodynamic integration: A molecular dynamics simulation study

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.     

Simultaneous prediction of interfacial tension and phase equilibria in binary mixtures: An approach based on cubic equations of state with improved mixing rules

Vapor–liquid interfacial tensions of miscible mixtures have been predicted by applying the gradient theory to an improved Peng–Robinson equation of state. The modified Huron–Vidal mixing rule model has been considered for fitting vapor–liquid equilibrium data of miscible polar and non-polar mixtures and, then, for predicting the interfacial tension of these mixtures. According to results, an accurate and globally stable fitting of the vapor–liquid equilibrium data results on a physically coherent prediction of interfacial tensions in the full concentration range. In addition, we present a criteria based on the geometry of the grand potential function along the interface for assessing the predictive value of the GT. Calculations for subcritical binary mixtures are presented and compared to experimental data and the Parachor method for demonstrating the potential of the unified approach suggested in this work.     

Modeling of phase equilibrium and vapor adsorption on carbon black based on a combination of a lattice theory and equation of state

A thermodynamic approach is developed in this paper to describe the behavior of a subcritical fluid in the neighborhood of vapor-liquid interface and close to a graphite surface. The fluid is modeled as a system of parallel molecular layers. The Helmholtz free energy of the fluid is expressed as the sum of the intrinsic Helmholtz free energies of separate layers and the potential energy of their mutual interactions calculated by the 10-4 potential. This Helmholtz free energy is described by an equation of state (such as the Bender or Peng-Robinson equation), which allows us a convenient means to obtain the intrinsic Helmholtz free energy of each molecular layer as a function of its two-dimensional density. All molecular layers of the bulk fluid are in mechanical equilibrium corresponding to the minimum of the total potential energy. In the case of adsorption the external potential exerted by the graphite layers is added to the free energy. The state of the interface zone between the liquid and the vapor phases or the state of the adsorbed phase is determined by the minimum of the grand potential. In the case of phase equilibrium the approach leads to the distribution of density and pressure over the transition zone. The interrelation between the collision diameter and the potential well depth was determined by the surface tension. It was shown that the distance between neighboring molecular layers substantially changes in the vapor-liquid transition zone and in the adsorbed phase with loading. The approach is considered in this paper for the case of adsorption of argon and nitrogen on carbon black. In both cases an excellent agreement with the experimental data was achieved without additional assumptions and fitting parameters, except for the fluid-solid potential well depth. The approach has far-reaching consequences and can be readily extended to the model of adsorption in slit pores of carbonaceous materials and to the analysis of multicomponent adsorption systems.     

First correction to surface tension for the curvature of an interface

Tolman parameter δ_T, which determines the first correction to surface tension for the interface curvature, is calculated in molecular-dynamic experiments performed for Lennard-Jones fluid within the temperature range from the critical to the triple point (and slightly below). It is shown that parameter δ_T is positive and slightly depends on temperature; its absolute magnitude is no larger than 0.1–0.2 molecular diameters and does not coincide with the distance between the equimolecular dividing surface and the surface of tension in a flat interfacial layer, as calculated through the first moment of the pressure tensor. The results of the moleculardynamic experiments are compared with the δ_T values calculated in terms of the extended version of the van der Waals theory of capillarity. It is established that taking into account terms of higher orders than the squared density gradient in the expansion of the free energy of an inhomogeneous system does not reverse the negative sign at δ_T and slightly affects its value.     

Test-area simulation method for the direct determination of the interfacial tension of systems with continuous or discontinuous potentials

A novel test-area (TA) technique for the direct simulation of the interfacial tension of systems interacting through arbitrary intermolecular potentials is presented in this paper. The most commonly used method invokes the mechanical relation for the interfacial tension in terms of the tangential and normal components of the pressure tensor relative to the interface (the relation of Kirkwood and Buff [J. Chem. Phys. 17, 338 (1949)]). For particles interacting through discontinuous intermolecular potentials (e.g., hard-core fluids) this involves the determination of delta functions which are impractical to evaluate, particularly in the case of nonspherical molecules. By contrast we employ a thermodynamic route to determine the surface tension from a free-energy perturbation due to a test change in the surface area. There are important distinctions between our test-area approach and the computation of a free-energy difference of two (or more) systems with different interfacial areas (the method of Bennett [J. Comput. Phys. 22, 245 (1976)]), which can also be used to determine the surface tension. In order to demonstrate the adequacy of the method, the surface tension computed from test-area Monte Carlo (TAMC) simulations are compared with the data obtained with other techniques (e.g., mechanical and free-energy differences) for the vapor-liquid interface of Lennard-Jones and square-well fluids; the latter corresponds to a discontinuous potential which is difficult to treat with standard methods. Our thermodynamic test-area approach offers advantages over existing techniques of computational efficiency, ease of implementation, and generality. The TA method can easily be implemented within either Monte Carlo (TAMC) or molecular-dynamics (TAMD) algorithms for different types of interfaces (vapor-liquid, liquid-liquid, fluid-solid, etc.) of pure systems and mixtures consisting of complex polyatomic molecules.     

The surface tension of a solid at the solid-vacuum interface, an evaluation from adsorption and wall potential calculations

A method for the evaluation of quantities that are experimentally inaccessible such as the surface tension at the solid-vacuum interface and the superficial tension of the fluid in contact with the solid is presented. The approach is based on consideration of an equilibrium of a fluid in solid capillary wherein a balance between surface and capillary forces has been replaced by conceptual alternative interfacial and centrifugal forces. This approach involves the simultaneous numerical solution one the special forms of the Gibbs equation for solid-fluid interface and a generalized Kelvin equation derived earlier. The latter equation takes into account interactions between the solid thick cylindrical wall and confined fluid, this body-body interaction potential has been primarily calculated using the Lennard-Jones (6-12) expression for the atom-atom pair potentials and expressed by hypergeometrical functions having good convergences. All numerical calculations shown here have been performed for the model graphite-argon system at 90 K. Finally, an analysis of the accuracy of the proposed method is considered.     

New approach to defining thermodynamic surface tension of solids

Using either the chemical potential of the immobile component of a solid dissolved in a fluid phase or the corresponding component of the tensor of chemical potential in solid phase, a new concept of the grand thermodynamic potential of solid-fluid two-phase system is proposed. For a planar interfacial surface, this makes it possible to generalize the notion of thermodynamic surface tension σ introduced by Gibbs that has the meaning of the formation work of a unit surface. This tension is determined as the specific surface excess of the grand thermodynamic potential. This definition of the thermodynamic surface tension does not depend on the position of the dividing surface and is common for fluids and solids. It is shown that, at the arbitrary position of dividing surface, the difference between thermodynamic σ and mechanical @[gamma] surface tensions for solid surface is determined by the nonuniformity of the tensor of chemical potential in a solid, as well as by its anisotropy in the bulk of solid phase.     

Verification of Onsager's reciprocal relations for evaporation and condensation using non-equilibrium molecular dynamics

Non-equilibrium molecular dynamic (NEMD) simulations have been used to study heat and mass transfer across a vapor-liquid interface for a one-component system using a Lennard-Jones spline potential. It was confirmed that the relation between the surface tension and the surface temperature in the non-equilibrium system was the same as in equilibrium (local equilibrium). Interfacial transfer coefficients were evaluated for the surface, which expressed the heat and mass fluxes in temperature and chemical potential differences across the interfacial region (film). In this analysis it was assumed that the Onsager reciprocal relations were valid. In this paper we extend the number of simulations such that we can calculate all four interface film transfer coefficients along the whole liquid-vapor coexistence curve. We do this analysis both for the case where we use the measurable heat flux on the vapor side and for the case where we use the measurable heat flux on the liquid side. The most important result we found is that the coupling coefficients within the accuracy of the calculation are equal. This is the first verification of the validity of the Onsager relations for transport through a surface using molecular dynamics. The interfacial film transfer coefficients are found to be a function of the surface temperature alone. New expressions are given for the kinetic theory values of these coefficients which only depend on the surface temperature. The NEMD values were found to be in good agreement with these expressions.