Evaluation of second and third dielectric virial coefficients for non-polarisable molecular models

<p>The dielectric constant, ε, of a dilute vapour can be estimated from the dielectric virial equation of state (VEOS), but the long-ranged nature of the electrostatic interactions complicates the evaluation of coefficients of this series. We propose a formulation of the second and third dielectric coefficients of a general non-polarisable molecular model that permits their reliable calculation using Mayer sampling Monte Carlo. We demonstrate for three models: dipolar hard spheres, dipolar Lennard–Jones, and TIP4P water. The coefficients are used to compute ε for each model as a function of density, which are compared to molecular-simulation data. The form of the VEOS relating ε to density depends on the dielectric constant ε′ of the embedding medium. Three choices are examined: vacuum (ε′ = 1), self-consistent (ε′ = ε) and tin foil (ε′ = ∞). The vacuum-boundary form is found to be unreliable, losing accuracy at low density and yielding divergent results for ε at moderate densities. In contrast, the series formulated using the tin-foil boundary produces accurate and stable values of ε for almost all conditions and models examined here, even when truncated at second order (which itself is shown to be a large improvement over the first-order Clausius–Mossotti–Debye formula).</p> <p></p

Quantum virial coefficients of molecular nitrogen

<p>We report virial coefficients up to third order in density for molecular nitrogen, investigating 103 temperatures in the range (15 K, 3000 K). All calculations are based on an <i>ab initio</i>-based potential taken from the literature. Path-integral Monte Carlo (PIMC) is applied to account for nuclear quantum effects, and these results are compared to a more approximate but faster semiclassical treatment. Additionally, we examine a PIMC approach that employs semiclassical beads for the path-integral images, but find that it offers marginal advantage. A recently developed orientation sampling algorithm is used in conjunction with Mayer sampling to compute precise virial coefficients. We find that, within the precision of our calculations of the second-order coefficient (<i>B</i>₂), semiclassical methods are adequate for temperatures greater than 250 K, and are needed to correct classical behaviour for temperatures as high as 800 K. For the third-order coefficient (<i>B</i>₃), the semiclassical methods are adequate above 150 K, and are required up to the highest temperature examined (3000 K) in order to correct the classical treatment within the precision of the calculations. However, three-body contributions to the potential are much more significant than nuclear quantum effects for the evaluation of <i>B</i>₃.</p

Thermodynamic Properties of Supercritical CO₂/CH₄ Mixtures from the Virial Equation of State

Mixture virial coefficients to seventh order are presented for the system CO₂/CH₄ at four supercritical temperatures: 323.15, 373.15, 473.15, and 573.15 K. Values are evaluated via the Mayer sampling Monte Carlo method using a three-site TraPPE model for CO₂ and a one-site model for CH₄. The coefficients are used to compute seven thermodynamic properties (viz., compressibility factor, isothermal compressibility, volume expansivity, isochoric and isobaric heat capacities, Joule–Thomson coefficient, and speed of sound) as a function of mole fraction and density for these temperatures. Comparison is made with corresponding data in the literature as obtained by molecular dynamics simulation, covering densities up to about twice the critical density. Key conclusions are as follows, noting that some exceptions are observed in each case: (a) The virial equation of state (VEOS) to fourth or fifth order describes all properties to within the simulation uncertainty for densities up to at least the critical density, and the addition of terms up to seventh order extends this range considerably. (b) The accuracy of the VEOS is severely diminished for conditions approaching the critical point (the present work extends down to a reduced temperature of 1.06 for CO₂), and the study of the pure component behavior suggests the critical singularity blocks convergence for conditions at considerably higher temperatures, albeit at correspondingly higher pressures. (c) Comparison of the VEOS at different orders provides a reliable guide to its accuracy at a given order, so the VEOS can provide a self-assessment of its accuracy when independent data for comparison are unavailable. (d) The VEOS provides a good description of the Joule–Thomson coefficient, including the inversion point in particular. The third-order series is needed to obtain behavior that is qualitatively correct, and the addition of higher-order terms steadily improves the accuracy quantitatively. (e) Under conditions where the seventh-order series is converged, properties can be computed to a given precision with VEOS using much less computational effort in comparison to molecular simulation

Effects of Finite Size and Proton Disorder on Lattice-Dynamics Estimates of the Free Energy of Clathrate Hydrates

We consider lattice dynamics methods for calculation of the free energy of clathrate hydrate phases, specifically the cubic structure I (sI), cubic structure II (sII), and hexagonal structure H (sH) phases, in the absence of guest molecules; water molecules are modeled with the TIP4P potential. We examine in particular the effects of finite size and proton disorder on the calculated free energies and consider these in the context of the free-energy differences between the phases. We find that at 300 K, the finite-size, proton-disorder, and quantum effects between phases are, respectively, on the order of 0.4, 0.03, and 0.1 kJ/mol. Details of the calculations are provided, with emphasis on efficiencies developed to handle various aspects related to rigid-molecule lattice dynamics with electrostatic interactions in large crystalline systems

Virial Coefficients and Equations of State for Hard Polyhedron Fluids

Hard polyhedra are a natural extension of the hard sphere model for simple fluids, but there is no general scheme for predicting the effect of shape on thermodynamic properties, even in moderate-density fluids. Only the second virial coefficient is known analytically for general convex shapes, so higher-order equations of state have been elusive. Here we investigate high-precision state functions in the fluid phase of 14 representative polyhedra with different assembly behaviors. We discuss historic efforts in analytically approximating virial coefficients up to <i>B</i>₄ and numerically evaluating them to <i>B</i>₈. Using virial coefficients as inputs, we show the convergence properties for four equations of state for hard convex bodies. In particular, the exponential approximant of Barlow et al. (<i>J. Chem. Phys</i>. <b>2012</b>, <i>137</i>, 204102) is found to be useful up to the first ordering transition for most polyhedra. The convergence behavior we explore can guide choices in expending additional resources for improved estimates. Fluids of arbitrary hard convex bodies are too complicated to be described in a general way at high densities, so the high-precision state data we provide can serve as a reference for future work in calculating state data or as a basis for thermodynamic integration

Free energy and concentration of crystalline vacancies by molecular simulation

<p>We present an approach for evaluating the concentration of vacancy defects in crystalline materials by molecular simulation. The proposed method circumvents the problem of equilibration of the number of ‘lattice sites’ <i>M</i>, which characterises the trade-off between more, smaller lattice cells (with some vacant), versus fewer, larger cells. Working in a grand-canonical framework, we instead fix <i>M</i> and solve for the chemical potential <i>μ</i> that ensures thermodynamic consistency of the ensemble-averaged pressure and the grand potential. Having determined <i>μ</i> this way for the given <i>M</i>, the equilibrium vacancy concentration and free energy are easily determined. Methods are demonstrated for the classical Lennard–Jones fcc crystal, examining all states where the crystal is stable. We find for this system that the effect of equilibrating <i>M</i> is negligible at all conditions. Also, although the vacancy fraction varies by many orders of magnitude with temperature and density, we find that the value at melting is nearly independent of density, equal to about . Results further show that a lattice-energy approximation (ignoring entropic effects) underestimates the correct concentration by four orders of magnitude at almost all conditions; ignoring only anharmonic contributions underestimates the vacancy concentration at melting by nearly one order of magnitude.</p