Virial Pressure Research Articles

Empirical expressions of the pair correlation functions at contact and equation of state of the liquid hard-sphere mixtures at large size ratio

The virial–pressure equation of state of hard-sphere mixtures in the colloidal limit obtained from ad hoc expressions of the contact values of the pair correlation functions and the related question of the fluid–fluid phase separation are revisited. The sensitivity of the location of the binodal to the exponential behaviour of the contact value of the big spheres pair correlation function and its size ratio dependence are analyzed by considering a model for the mean force potential that is used to parameterise the contact value. Agreement with simulation data or accurate integral equations theories is improved when compared with previous approaches based on the virial–pressure equation of state.

Control of density fluctuations in atomistic-continuum simulations of dense liquids

We present a control algorithm to eliminate spurious density fluctuations associated with the coupling of atomistic and continuum descriptions for dense liquids. A Schwartz domain decomposition algorithm is employed to couple molecular dynamics for the simulation of the atomistic system with a continuum solver for the simulation of the Navier-Stokes equations. The lack of periodic boundary conditions in the molecular dynamics simulations hinders the proper accounting for the virial pressure leading to spurious density fluctuations at the continuum-atomistic interface. An ad hoc boundary force is usually employed to remedy this situation. We propose the calculation of this boundary force using a control algorithm that explicitly cancels the density fluctuations. The results demonstrate that the present approach outperforms state-of-the-art algorithms. The conceptual and algorithmic simplicity of the method makes it suitable for any type of coupling between atomistic and continuum descriptions of dense fluids.

Scattering problem with nonlocal separable potential of rank-two: Application to statistical mechanics

In this paper, we present a formulation of statistical mechanics of a thermodynamic system consisting of free particles and independent correlated pairs interacting via nonlocal potential in terms of the scattering properties. Some quantum statistical properties such as energy, heat capacity, second virial coefficient, virial pressure and quantum correction of kinetic energy are described analytically. The difference between the resolvents of the interacting and free Hamiltonians, represented as R ^ ( z ) , that is associated with particle correlations is used for the evaluation of the properties. The statistical properties are related to correlated states, when making a pole expansion of the analytically continued momentum matrix element of R ^ ( z ) . The present work illustrates these relations for a three-dimensional nonlocal separable potential of rank-two.

Representability problems for coarse-grained water potentials

The use of an effective intermolecular potential often involves a compromise between more accurate, complex functional forms and more tractable simple representations. To study this choice in detail, we systematically derive coarse-grained isotropic pair potentials that accurately reproduce the oxygen-oxygen radial distribution function of the TIP4P-Ew water model at state points over density ranges from 0.88 to 1.30 g/cm3 and temperature ranges from 235 to 310 K. Although by construction these effective potentials correctly represent the isothermal compressibility of TIP4P-Ew water, they do not accurately resolve other thermodynamic properties such as the virial pressure, the internal energy, or thermodynamic anomalies. Because at a given state point the pair potential that reproduces the pair structure is unique, we have therefore explicitly demonstrated that it is impossible to simultaneously represent the pair structure and several key equilibrium thermodynamic properties of water with state-point dependent radially symmetric pair potentials. We argue that such representability problems are related to, but different from, more widely acknowledged transferability problems and discuss in detail the implications this has for the modeling of water and other liquids by coarse-grained potentials. Nevertheless, regardless of thermodynamic inconsistencies, the state-point dependent effective potentials for water do generate structural and dynamical anomalies.

‘Exact’ integral equation theory and local formulation for excess thermodynamic properties of hard spheres

A hybrid hard sphere bridge function is proposed, which, in combination with the standard Ornstein–Zernike integral equation, can predict extremely accurately hard sphere compressibility, virial pressure, and correlation function. Second, a local formulation for determination of excess chemical potential is derived out, which, in combination with the present hybrid hard sphere bridge function and OZ integral equation, can predict the excess chemical potential also extremely accurately. The resultant excess entropy is in excellent agreement with that from the Carnahan–Starling equation of state. The present formalism performs excellently over the whole density range, i.e. from zero to freezing density, and is largely superior to a formalism available in the literature.

Beware of density dependent pair potentials

Density (or state) dependent pair potentials arise naturally from coarse-grainingprocedures in many areas of condensed matter science. However, correctly usingthem to calculate physical properties of interest is subtle and cannot be uncoupledfrom the route by which they were derived. Furthermore, there is usually nounique way to coarse-grain to an effective pair potential. Even for simple systemslike liquid argon, the pair potential that correctly reproduces the pair structurewill not generate the right virial pressure. Ignoring these issues in naiveapplications of density dependent pair potentials can lead to an apparentdependence of thermodynamic properties on the ensemble within whichthey are calculated, as well as other inconsistencies. These concepts areillustrated by several pedagogical examples, including effective pair potentialsfor systems with many-body interactions, and the mapping of charged(Debye–Hückel) and uncharged (Asakura–Oosawa) two-component systems ontoeffective one-component ones. The differences between the problems oftransferability and representability for effective potentials are also discussed.

Self-contained equilibrium theory for freely jointed Lennard-Jones chain fluids and similar models

On the basis of the van der Waals picture and an extension of the Weeks-Chandler-Andersen prescription, it is assumed that the average monomer-monomer radial distribution function of the freely jointed Lennard-Jones (LJ) chain fluid can be approximated by that of a hypothetical reference chain fluid in which the interatomic forces are entirely repulsive and identical to the repulsive forces generated by the LJ potential. The structural information for the reference chain system is approximated using the cavity correlation function of freely jointed hard sphere rings with an appropriately chosen hard sphere diameter. It is found that both the virial pressure and the excess internal energy obtained from the energy equation are quantitatively accurate at high densities and temperature. The method and analytical expressions used here employ no simulation input and are all relevant not only for an LJ potential but also any intermonomer potential with steep short range repulsion and longer range attraction.

Virial pressure of periodic systems with long range forces

The virial expression of the pressure of systems with periodic boundary conditions is discussed for long range Coulomb interactions. It is shown, that the mechanical definition of the pressure leading to the virial pressure agrees with the thermodynamical definition for such systems.

Statistical mechanical models with effective potentials: Definitions, applications, and thermodynamic consequences

Realistic interactions that operate in condensed matter systems can exhibit complicated many-particle characteristics. However, it is often useful to seek a more economical description using at most singlet and pair effective interactions that are density dependent, to take advantage of the theoretical and computational simplifications that result. This paper analyzes the statistical mechanical formalism required to describe thermal equilibrium in that kind of approach. Two distinct interpretations are available for the role of density dependence. Either one can be treated with internal consistency, but generally they lead to differing thermodynamic predictions. One regards the density dependence of effective interactions as merely a passive index for the state at which the optimal choice of those effective interactions was determined (Case I). The other treats the density as an active variable on the same footing as particle coordinates (Case II). Virial pressure and isothermal compressibility expressions in terms of particle distribution functions are displayed for both cases. Under special circumstances it is possible for the two interpretations to yield the same pressure isotherms; the conditions producing this coincidental concordancy have been explored by means of density expansions.

Precollisional velocity correlations in a hard-disk fluid with dissipative collisions.

Velocity correlations are studied in granular fluids, modeled by the inelastic hard sphere gas. Making a density expansion of the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy for the evolution of the reduced distributions, we predict the presence of precollisional velocity correlations. They are created by the propagation through correlated sequences of collisions (ring events) of the velocity correlations generated after dissipative collisions. The correlations have their origin in the dissipative character of collisions, being always present in granular fluids. The correlations, that manifest microscopically as an alignment of the velocities of a colliding pair produce modifications of collisional averages, in particular, the virial pressure. The pressure shows a reduction with respect to the elastic case as a consequence of the velocity alignment. Good qualitative agreement is obtained for the comparison of the numerical evaluations of the obtained analytical expressions and molecular dynamics results that showed evidence of precollisional velocity correlations [R. Soto and M. Mareschal, Phys. Rev. E 63, 041303 (2001)].

Equation of State for Liquid Tin Tetraiodide under Hydrostatic Pressure

Isothermal-isobaric molecular dynamics simulations were carried out for SnI 4 in its liquid state under hydrostatic pressure up to about 5 GPa. The equation of state was obtained from the virial pressure, which was found to be in good agreement with that derived by Boublík regarding a system consisting of hard-convex bodies as a reference fluid. The virial coefficients determined from the equation of state were presented so as to facilitate experimental confirmation.

Density functional theory for pair correlation functions in polymeric liquids

A density functional theory is presented for the pair correlation functions in polymeric liquids. The theory uses the Yethiraj–Woodward free-energy functional for the polymeric liquid, where the ideal gas free-energy functional is treated exactly and the excess free-energy functional is obtained using a weighted density approximation with the simplest choice of the weighting function. Pair correlation functions are obtained using the Percus trick, where the external field is taken to be a single polymer molecule. The minimization of the free energy in the theory requires a two molecule simulation at each iteration. The theory is very accurate for the pair correlation functions in freely jointed tangent-hard-sphere chains and freely rotating fused-hard-sphere chains, especially at low densities and for long chains. In addition, the theory allows the calculation of the virial pressure in these systems and shows a remarkable degree of consistency between the virial and compressibility pressure.

Virial theorem and pressure calculations in the GGA

It is shown that using a generalized gradient approximation (GGA) functionaldescribing exchange and correlation (xc) effects in electronic structurecalculations implies additional xc terms in the virial theorem and thepressure tensor. These new expressions generalize the forms well known in thelocal density approximation and are useful for checking the numericalprecision of self-consistent field calculations for atoms and solids, and foraccurate pressure computations in heavy elements. Numerical examples are given.

A study of associating Lennard–Jones chains by a new reference radial distribution function

A new radial distribution function (RDF) for the Lennard–Jones (LJ) fluid is derived around the LJ potential size ( σ). The theoretically based RDF is completely analytical and real. Comparisons with computer simulation data at various conditions indicate that the RDF is very accurate at r<1.2 σ, and the accuracy is consistent from low to high densities. The new RDF is utilized to study associating LJ chains by combining with Wertheim's first-order perturbation theory. The resulting equation of state is tested against computer simulation data for second virial coefficient and pressure of LJ chains, and for pressure and the fraction of non-associating molecules in associating LJ chains. The calculated results indicate that the new equation of state is very satisfactory and its performance is comparable to several empirical equations reported in the literature. Another feature of the new equation of state is that both chain bonding and association sites are adjustable for practical purposes.

Monte Carlo simulation study of the high-temperature phase diagram of model C60 molecules

The constant-NVT Monte Carlo simulation is performed for model C60 molecules interacting via the Girifalco potential and a full free-energy analysis is made to predict the high-temperature phase diagram. The repulsive part of the C60 potential is very steep and the attractive part is relatively short-ranged. For such a system accurate computations of the virial pressure are difficult in simulations and it is argued that the discrepancies among the previous results for the phase diagram of C60 can partly be attributed to the uncertainties of the virial pressure involved in simulations. To avoid this difficulty we take the energy route to calculate equation of state (EOS), in which the absolute (Helmholtz) free energy is obtained by performing isochoric integration of the excess internal energy. A difficulty of the energy route in the high-temperature limit is resolved by the aid of an analytic method. The exact second and third virial coefficients are also used in the analysis of the fluid EOS. The pressure route is taken to calculate the EOS of the solid phase, in which the virial pressure is numerically more stable than in the fluid phase. The resulting high-temperature phase diagram of C60 is quite systematic and free from uncertainties, and the liquid–vapor critical point is found at Tc=1980 K and ρc=0.44 nm−3, whereas the triple point at Tt=1880 and ρt=0.74 nm−3, confirming the existence of a stable liquid phase over the range of ∼100 K.

Pressure calculation in polar and charged systems using Ewald summation: Results for the extended simple point charge model of water

Ewald summation and physically equivalent methods such as particle-mesh Ewald, kubic-harmonic expansions, or Lekner sums are commonly used to calculate long-range electrostatic interactions in computer simulations of polar and charged substances. The calculation of pressures in such systems is investigated. We find that the virial and thermodynamic pressures differ because of the explicit volume dependence of the effective, resummed Ewald potential. The thermodynamic pressure, obtained from the volume derivative of the Helmholtz free energy, can be expressed easily for both ionic and rigid molecular systems. For a system of rigid molecules, the electrostatic energy and the forces at the atom positions are required, both of which are readily available in molecular dynamics codes. We then calculate the virial and thermodynamic pressures for the extended simple point charge (SPC/E) water model at standard conditions. We find that the thermodynamic pressure exhibits considerably less system size dependence than the virial pressure. From an analysis of the cross correlation between the virial and thermodynamic pressure, we conclude that the thermodynamic pressure should be used to drive volume fluctuations in constant-pressure simulations.

Self-consistent density functional theory and the equation of state for simple fluids

The accuracy of the self-consistent density functional theory, based on the idea of universality of the so-called 'bridge functional', is tested by comparison with the extensive simulations for the Lennard-Jones bulk fluid (Johnson et al. 1993, Molec. Phys., 78, 591). In this benchmark test very good agreement with the simulation results is obtained, and thermodynamic consistency is accurately obeyed between the 'Energy', 'Virial (Pressure)', and 'Compressibility' independent routes to the equation of state.

Phase stability of binary non-additive hard-sphere mixtures: A self-consistent integral equation study

We have tested the capabilities of a new self-consistent integral equation, closely connected with Verlet’s modified closure, for the study of fluid-fluid phase separation in symmetric non-additive hard-sphere mixtures. New expressions to evaluate the chemical potential of mixtures are presented and play a key role in the construction of the phase diagram. The new integral equation, which implements consistency between virial and fluctuation theorem routes to the isothermal compressibility, together with chemical potential and virial pressure consistency via the Gibbs-Duhem relation, yields a phase diagram which especially at high densities agrees remarkably well with the new semi-Grand Ensemble Monte Carlo simulation data also presented in this work. Deviations close to the critical point can be understood as a consequence of the inability to enforce virial-fluctuation consistency in the neighborhood of the spinodal decomposition curve.

A Born–Green–Yvon equation for flexible chain-molecule fluids. II. Applications to hard-sphere polymers

Using the method of Born, Green, and Yvon we have recently derived a coupled set of integral equations for the intermolecular site–site distribution functions which describe the microscopic equilibrium structure of a fluid of flexible chain molecules. Here, we carry out an average over this set of equations to arrive at a single equation for the average intermolecular distribution function. Intramolecular distribution functions appearing in this equation are approximated by results for isolated chain molecules. Numerical calculations have been performed over a range of fluid densities for chains composed of 8, 16, 20, and 50 tangent hard spheres. The resulting intermolecular distribution functions are in good to fair agreement with simulation data. This structural information is used to compute second virial coefficients and pressure equations of state via the virial route to thermodynamics. Although the theory consistently underestimates the pressure, it gives very accurate results for the second virial coefficient for all chain lengths studied.

A Born–Green–Yvon equation for flexible chain-molecule fluids. I. General formalism and numerical results for short hard-sphere chains

The Born–Green–Yvon approach is used to construct an integral equation for the site–site distribution functions describing the microscopic equilibrium structure of a fluid of flexible chain molecules. The multisite distribution functions appearing in this formally exact equation are related to the site–site functions via a set of site–site superposition approximations. Intramolecular distribution functions are approximated using exact results for an isolated chain molecule. Numerical results are presented for fluids of tangent hard-sphere 3-mers and 4-mers. These results, which include the site–site distribution functions over a range of densities, second virial coefficients, and the virial pressure equations of state, compare well with Monte Carlo data. In particular, the contact value of the average site–site distribution function is given quite accurately.