Effects of the repulsive and attractive forces on phase equilibrium and critical properties of two-dimensional non-conformal simple fluids

https://doi.org/10.1016/j.molliq.2020.115234Get rights and content

Highlights

  • Non conformal interactionpotentials with tunable repulsive and attractive softness are proposed.

  • Thermophysical properties for two dimensional fluids were predicted with computer simulations.

  • The interactions proposed can be applied for modelling quasi-two dimensional systems occuring in nature.

Abstract

Molecular simulations in the canonical and isothermal-isobaric ensembles were performed for a two-dimensional simple fluid using a set of non conformal molecular interactions. The specific characteristics of this interaction model give it some advantages over other well known interaction potentials employed traditionally in the study of thermodynamic properties of soft matter systems. The functional form employed for modelling the effective potentials is the one proposed by the so-called ANC theory (Approximate Non-Conformal Theory), which allows to modify separately the contributions of the repulsive and attractive terms by means of a molecular parameter called softness. The effect of both, repulsive and attractive softness, on the liquid-vapour coexistence, liquid-solid transition and vaporization curve as well as on the value of the critical properties were explored. In fact, the linear dependence on the softness parameters exhibited by the critical properties, allows to extrapolate the linear correlations presented in this work, to reproduce the critical point of a two-dimensional Lennard-Jones fluid. With the aim of showing how versatile is the interaction model presented in this work, adequate values of the softness parameters were selected to reproduce qualitatively the liquid-vapour coexistence and critical points for both, a two-dimensional model fluid with a double exponential interaction potential, widely employed in molecular simulations of simple fluids; and a thin monolayer of methane molecules adsorbed on a planar graphite substrate, a system that can be considered in good approximation as two-dimensional and that has been studied in a lot of work for four decades. Finally, the effects of the dimensionality of the system was evaluated by quantifying the relative percentage differences exhibited by the thermodynamic properties explored in this paper with respect to its three-dimensional counterpart. The results presented in this work show not only the predictive capacity of the linear correlations mentioned, but the robustness of the ANC interactions to reproduce thermophysical properties and phase equilibria for model and real two-dimensional fluids. Discriminating the influence of repulsive and attractive contributions in molecular interaction models is undoubtedly of great relevance to have a better understanding of thermodynamic properties of simple and complex fluids.

Introduction

Since 1873, when van der Waals introduced corrections to the equation of state of gases, it has been clear the relevance that the appropriate modelling of molecular interactions has in the description of the phase equilibrium of fluids. In his work, van der Waals included the effects of both, the finite volume of the molecules and the attractive forces that should exist among them. A great deal of effort has been devoted in recent decades to the development of analytical functions that, in the same spirit of van der Waals' original idea, appropriately describe the repulsive and attractive contributions of molecular interactions. Examples of these functions that are worth mentioning, due to their simplicity and their ease to be applied to theoretical and computer simulation studies, are the Square-Well (SW), Sutherland [1,2], Kihara [3], Yukawa [4], and the well-known Lennard-Jones (LJ) [5] potentials. Part of their simplicity is based on the spherical symmetry of all of them. The accuracy of these models in the prediction of thermodynamic properties of both, two-dimensional (2D) and three-dimensional (3D) systems depends strongly on the complexity of the molecules under study which are not spherical at all. Besides that, even in the study of simple fluids, there are thermodynamic inconsistencies that can not be ignored. For instance, it has been hard to propose an effective potential capable to reproduce simultaneously the pair distribution function and the virial coefficients. Hence, while it is true that there are a lot of work in this direction, our knowledge in this field is far to be complete and the contribution of experimental, theoretical and numerical results in this field is valuable in the development of models that, starting from the description of molecular interactions be capable of predicting macroscopic (thermodynamic) properties, which is the central aim of statistical physics. In this sense, from a theoretical point of view, it is always relevant to have a more profound understanding of the dimensionality effects on the thermodynamic properties, which justify the study of 2D systems but, on the other hand, this relevance is not only theoretical: there are many systems in nature that can be considered quasi-two dimensional, for this, to have interaction potentials that allow to model these systems is valuable also from a practical perspective.

For instance, the knowledge of structural and dynamical properties of membranes in biological systems has been traditionally studied at the microscopic level. This is due to the fluid character of the membranes in physiological conditions, and in part to the lack of experimental data that be directly interpretable in terms of microscopic attributes. In particular, of special biological relevance is the understanding of local variations in the pressure as well as in the stress in the study of membrane bending [[6], [7], [8], [9]]. However, since this information is inaccessible experimentally, interaction potentials as engine of molecular simulations, play a central role in the quantitative prediction of pressure profiles in membranes. Other examples of systems that can be considered in good approximation as two-dimensional, and where the variation of the pressure is important to understand the inhomogenities exhibited, include lipid bilayers and membrane proteins [10,11]. A detailed discussion of the role that plays the two-dimensional pressure in molecular dynamics simulations (MD) of these systems can be found in the Sumith and Maroo work [12]. Attributes such as the area density, chemical potentials and surface pressure of a surfactant monolayer at a 2D vapour-liquid interface have been determined by Moghimikheirabadi et al. [13]. Starting from a three-dimensional (3D) model system of amphiphilic surfactants, they developed an effective two-dimensional interaction potential that was used in Monte Carlo simulations in the Gibbs Ensemble (GEMC) to determine these thermophysical properties. On the other hand, monolayers adsorbed on a substrate can be considered to a large extent as two-dimensional, as it has been showed by different experimental works [[14], [15], [16]]. Neutron scattering experiments has been performed to study the physical properties of argon adsorbed on graphite. It has been observed that these films form a triangular lattice in two dimensions at low temperatures [14]. Additionaly, X ray diffraction techniques have been used to describe the structure of alkali-metals layers adsorbed also on graphite substrates [15]. These experiments allowed to determine, from the static structure factor, whether the alkali-metal is in an amorphous or in a liquid phase.

Theoretical efforts have also been developed to study the role that molecular interactions play on the calculation of different thermodynamic properties of 2D systems. Cuadros et al. [17,18] based on the Weeks, Chandler and Andersen (WCA) perturbation theory [[19], [20], [21]], studied the individual influence of repulsive and attractive forces on thermophysical properties such as the potential and internal energies, pressure, Helmholtz free energy and chemical potential. Also in the context of 2D systems, starting from the framework of the Barker-Henderson theory, Trejos et al. [22], developed an analytical expression for the Helmholtz free energy for a 2D-SW fluid in a recent work. Then, from this equation of state, they predicted the liquid-vapour coexistence curves for LJ and Yukawa 2D fluids. Finally, it is worth mentioning the work of Cuadros and collaborators [[23], [24], [25]] where they proposed semi-theoretical equations of state based on the WCA theory for 2D and 3D-LJ fluids.

On the other hand, molecular simulations have proven to be a central tool to scrutinize the effect of molecular interactions on critical, structural and phase equilibrium properties, allowing exhaustive computer experiments under different conditions. Of special interest to the study developed in this paper, there are several works devoted to analyze the effect of the potential range on the phase equilibrium properties. Méndez-Maldonado and collaborators [26] performed molecular dynamics simulations for 2D soft-Yukawa fluids and discussed the effect of the range of interaction on the orthobaric densities and on the line tension. A systematic study on the behaviour of the pressure and internal energy for 2D repulsive-Yukawa systems was done by Kryuchkov et al. [27] using MD simulations. Other interaction models that have been subject of many numerical simulations in order to study the phase equilibrium in 2D systems are Lennard Jones and Square Well (SW) potentials [[28], [29], [30], [31], [32], [33], [34]]. For the SW model, Vörtler et al. [35] studied the simulation cell size dependence of chemical potential isotherms in subcritical square-well fluids, while Chapela and collaborators [36] analyzed the phase diagram with special emphasis in the region of low temperatures and/or high densities. Both works were developed using 2D Monte Carlo simulations.

Special emphasis should be placed on LJ interactions which have been subject of numerous analysis in the thermodynamic study of simple fluids. In a meticulous work, Frenkel and Smit [29,37] studied the effect of the cutoff radius in a 2D-LJ system on the LV coexistence curve and its critical coordinates. They compared their results with those of the equation of state proposed by Reddy and O'Shea [38] and numerical simulation results of Singh [28], Nicolaides [39] and Rovere et al. [40,41], and discussed the influence of the potential tail on the phase diagram. Different from Sikkenk observations [42], Frenkel and Smit distinguished the thermodynamic predictions of the simulations when the LJ potential is truncated or considered in its original form. More research on two-dimensional LJ fluids can be found in the literature. Panagiotopoulos [43,44] studied phase behaviour near the critical point for 3D and 2D. Feng [45] presented and successfully applied analytical expressions that include corrections to the truncated 2D-LJ potential. In a more recent work, Ouyang and collaborators [46], using the Maxwell construction method and MD results, analyzed the effects of the radius cutoff on the LV coexistence in 2D-LJ systems. In particular, Jiang and Gubbins, performed Monte Carlo simulations in the Gibbs ensemble employing the LJ potential as effective interaction to study liquid-vapour (LV) equilibrium of methane monolayers adsorbed in graphite substrates [16].

As can be seen in most of these works, it is the attractive contribution that is systematically tested, leaving aside the analysis of the repulsive contribution. This is due to the fact that in many of this models it is not easy to discriminate the influence of this contribution on the thermophysical properties of interest. Potoff and Benard-Brunel [47] in an attempt to address this question, developed a force field of atom-united inspired by the known Mie potentials [48]. They took into account the influence of the repulsion to adequately describe the vapour pressure without losing precision in the calculation of liquid saturated densities. Another effort was done by Martínez-Valencia et al. [49] and Ibarra-Tandi et al. [50] who studied the effect of repulsive forces on the surface tension and the physical adsorption phenomena, respectively. Basically, they used the Morse potential plus an extra repulsive term. With the aim of analyzing separately the influence of both, repulsive and attractive contributions, on the LV coexistence curve and the critical properties behaviour for 3D fluids, Okumura and Yonezawa [51] used the Morse, Stillinger-Weber and the Mie family potentials. However, in the case of non-conformal potentials, which are understood as non-scalable because of their profile differences, it is dificult to perform a systematic analysis of these contributions on the thermodynamic properties mentioned. This difficulty relies on the fact that uncoupling the repulsive and attractive contribution in order to perform a comprehensive study of their effects on thermodynamic properties, as the one presented in this work, is a non trivial task when considering many of the interaction models reported in literature.

An alternative option to avoid this difficulty was proposed by del Río and Ramos in the so called Approximmate Non Conformal Theory (ANC) [52,53]. This theory proposes a family of intermolecular potentials, in which a molecular parameter -the softness- allows to systematically module the shape of their repulsive and attractive parts. In a more refined version of this theory, referred to as ANC2s, a softness parameter is defined separately for the repulsive and attractive part of the interaction. Besides that, in this version of the ANC theory, this parameter also acquires well-defined physical attributes: the hardness/softness of the molecules in the fluid under study and the range of molecular interaction, respectively [54]. Motivated by the versatility and flexibility of ANC2s interactions, in this work we present a new set of molecular interaction potentials with the features described above, along with a systematic study of the influence of every contribution (repulsive and attractive) on different thermodynamic properties for 2D fluids. The dependence of the critical properties, the liquid-vapour (LV) and liquid-solid (LS) equlilibrium, as well as the vaporization curve in a 2D simple fluid is studied. In addition, once a set of numerical simulations have been performed for different values of the softness parameters, after an extrapolation of a linear fit to the trend exhibited by the critical points, a prediction of the critical coordinates in terms of the softness for a 2D-LJ fluid is presented.

On the other hand, and with the aim of showing the versatility of ANC2s potentials to model different interactions, adequate values of the repulsive and attractive softness allowed to estimate the LV coexistence curve and critical points for two systems: (i) a 2D simple fluid whose interaction is represented by a double-exponential model (DE) proposed by Wu and Brooks [55] and (ii) a physisorbed system of methane molecules on a planar graphite substrate [16]. In addition, in a suplementary section a quantitative analysis of the dimensionality effects when ANC2s interactions are considered is included. LV coexistence, LS transition and vapour pressure are compared for both 3D and 2D systems. In fact, in the present work it is exhibited another important feature of the ANC2s potentials. It was found that the LV coexistence curves and vapour pressure for different values of the repulsive and attractive softness, collapse in the same curve when they are rescaled with respect to their corresponding critical properties. These master curves were shown in the research of Okumura and Yonezawa [51] and in a later work published by Orea et al. [56] for some combinations of repulsive and attractive exponents of the interactions considered in those works. It is important to highlight that the conformality with respect to the critical properties found in the referred works, as well as in the present research is not a general result. For instance, for the SW potentials family, whose range is variable, such a master curve was not observed [57].

The contribution of this research is to present the family of ANC2s potentials as a set of effective interactions whose flexibility not only allows to discriminate the repulsive and attractive contribution to thermophysical properties separately, but also to show it as a versatile set of interactions with which it is possible to reproduce results of other 2D potential models, specifically, LJ and DE simple fluids in two dimensions. In principle, it would be possible to choose the values for the softness parameter to make ANC2s potentials conformal to other interaction models. Then, according to the theorem of corresponding states, ANC2s interactions should be capable to reproduce at least with the same degree of accuracy thermodynamic properties as well as other potentials do.

Section snippets

Intermolecular non-conformal potentials

The ANC2s potentials used in this work depend on four molecular parameters, the position and depth of the minimum (δ and ε) plus two non-dimensional parameters called the repulsive and attractive softness (sR and sA) which are responsible of tuning the shape of the potential. The ANC2s interactions are represented by the functionuANC2srδεsRsA=εuANCr/δsRifrδεuANCr/δsAifr>δwhere uANC is a modified LJ potential,uANCzs=11+z21/s1/212211+z21/s1/26and z = r/δ is the intermolecular distance in

Simulation methodology

Numerical simulations were performed in LAMMPS to analyze the effects of repulsive and attractive softness on phase equilibrium, vaporization curve and critical properties of a 2D simple fluid. ANC2s interactions were implemented in LAMMPS as a table, and periodic boundary conditions, minimum image convention and neighbor list were included. The integrator and thermostat used were velocity-Verlet and Nose-Hoover, respectively. All results presented here were obtained from an initial

Liquid-vapour and liquid-solid coexistence

Fig. 2(a) shows the effect on the orthobaric densities of liquid-vapour coexistence when the repulsive softness is varied in the range 0.3 ≤ sR ≤ 0.9, with the attractive softness fixed at sA = 1.0. As the numerical value of repulsive softness is increased, the critical point is displaced to higher temperatures and densities. As can be expected, the latter means that, for a fixed temperature, the difference between the saturated vapour and liquid densities becomes larger as sR is increased.

Conclusions

In the present work, a systematic analysis to study separately the effects of the repulsive and attractive contributions of the molecular interaction on the phase diagram of 2D simple fluids was performed. The LV and LS coexistence curves and the critical coordinates, as well as the vaporization curve for a 2D simple fluid was studied. Additionally, a quantitative analysis on the dimensionality effects for the same coexistence curves was done. Small numerical values of sR and sA that, from the

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

Computational resources were provided by Yoltla at UAM-I and Miztli at UNAM by grant No. LANCAD-UNAM-DGTIC-276. JLL gratefully thanks for computer resources, technical advice and support provided by Laboratorio Nacional de Supercómputo del Sureste de México (LNS), a member of the CONACYT national laboratories, by grant No.201903091N.

References (93)

  • J. Unguris et al.

    Ar and Kr adsorption on Ag(111)

    Surf. Sci.

    (1981)
  • S.K. Singh et al.

    Effect of pore morphology on vapor–liquid phase transition and crossover behavior of critical properties from 3D to 2D

    Fluid Phase Equilib.

    (2011)
  • H. Graben et al.

    Third virial coefficient for the Sutherland (∞, ν) potential

    Rev. Mod. Phys.

    (1964)
  • W. Sutherland

    LII. The viscosity of gases and molecular force, the London, Edinburgh, and Dublin philosophical magazine

    J. Sci.

    (1893)
  • T. Kihara

    Virial coefficients and models of molecules in gases

    Rev. Mod. Phys.

    (1953)
  • J.E. Jones

    On the determination of molecular fields II. From the equation of state of a gas

  • E. Lindahl et al.

    Spatial and energetic-entropic decomposition of surface tension in lipid bilayers from molecular dynamics simulations

    J. Chem. Phys.

    (2000)
  • S.E. Feller et al.

    Constant surface tension simulations of lipid bilayers: the sensitivity of surface areas and compressibilities

    J. Chem. Phys.

    (1999)
  • S.C. Maroo

    Surface-heating algorithm for water at nanoscale

    J. Phys. Chem. Lett.

    (2015)
  • S.C. Maroo

    Origin of surface-driven passive liquid flows

    Langmuir

    (2016)
  • Y. Sumith et al.

    A direct two-dimensional pressure formulation in molecular dynamics

    J. Mol. Graph. Model.

    (2018)
  • A. Moghimikheirabadi et al.

    Gas–liquid phase equilibrium of a model Langmuir monolayer captured by a multiscale approach

    Phys. Chem. Chem. Phys.

    (2019)
  • H. Taub et al.

    Neutron-scattering studies of the structure and dynamics of Ar 36 monolayer films adsorbed on basal-plane-oriented graphite

    Phys. Rev. Lett.

    (1975)
  • H. Zabel et al.

    Planar diffusive motion of alkali-metal intercalant atoms in graphite

    Phys. Rev. Lett.

    (1983)
  • S. Jiang et al.

    Vapour-liquid equilibria in two-dimensional Lennard-Jones fluids: unperturbed and substrate-mediated films

    Mol. Phys.

    (1995)
  • F. Cuadros et al.

    The role of attractive forces in determining the thermodynamic properties of two-dimensional Lennard-Jones fluids

    Mol. Phys.

    (1995)
  • H.C. Andersen et al.

    Relationship between the hard-sphere fluid and fluids with realistic repulsive forces

    Phys. Rev. A

    (1971)
  • J.D. Weeks et al.

    Role of repulsive forces in determining the equilibrium structure of simple liquids

    J. Chem. Phys.

    (1971)
  • J.D. Weeks et al.

    Perturbation theory of the thermodynamic properties of simple liquids

    J. Chem. Phys.

    (1971)
  • V.M. Trejos et al.

    Vapor-liquid equilibrium and equation of state of two-dimensional fluids from a discrete perturbation theory

    J. Chem. Phys.

    (2018)
  • A. Mulero et al.

    Vapour–liquid equilibrium properties for two-and three-dimensional Lennard-Jones fluids from equations of state

    Aust. J. Phys.

    (1999)
  • G. Méndez-Maldonado, M. González-Melchor, J. Alejandre, Phase equilibria and interfacial properties of two-dimensional...
  • N.P. Kryuchkov et al.

    Thermodynamics of two-dimensional Yukawa systems across coupling regimes

    J. Chem. Phys.

    (2017)
  • R.R. Singh et al.

    Monte Carlo simulation of phase equilibria for the two-dimensional Lennard-Jones fluid in the Gibbs ensemble

    J. Chem. Phys.

    (1990)
  • B. Smit et al.

    Vapor–liquid equilibria of the two-dimensional Lennard-Jones fluid(s)

    J. Chem. Phys.

    (1991)
  • K. Mon et al.

    Finite size effects for the simulation of phase coexistence in the Gibbs ensemble near the critical point

    J. Chem. Phys.

    (1992)
  • N. Wilding et al.

    Density fluctuations and field mixing in the critical fluid

    J. Phys. Condens. Matter

    (1992)
  • S. Jiang

    Computer simulation, theoretical and experimental investigation of fluids in micropores

    (1993)
  • S. Jiang et al.

    Computer simulation study of adsorption, isosteric heat and phase transitions of methane on graphite

    MRS Proc.

    (1992)
  • J. Recht et al.

    Finite-size effects and approach to criticality in Gibbs ensemble simulations

    Mol. Phys.

    (1993)
  • H.L. Vörtler et al.

    Simulation of chemical potentials and phase equilibria in two-and three-dimensional square-well fluids: finite size effects

    J. Phys. Chem. B

    (2008)
  • J.C. Armas-Pérez et al.

    Liquid-vapor equilibrium and interfacial properties of square wells in two dimensions

    J. Chem. Phys.

    (2013)
  • B. Smit

    Phase diagrams of Lennard-Jones fluids

    J. Chem. Phys.

    (1992)
  • M.R. Reddy et al.

    The equation of state of the two-dimensional Lennard–Jones fluid

    Can. J. Phys.

    (1986)
  • D. Nicolaides et al.

    Priv. Comm.

    (1992)
  • M. Rovere et al.

    Block density distribution function analysis of two-dimensional Lennard-Jones fluids

    EPL

    (1988)
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