Density functional theory for molecular orientation of hard rod fluids in hard slits

A density functional theory (DFT) is used to investigate molecular orientation of hard rod fluids in a hard slit. The DFT approach combines a modified fundamental measure theory (MFMT) for excluded-volume effect with the first order thermodynamics perturbation theory for chain connectivity. In the DFT approach, the intra-molecular bonding orientation function is introduced. We consider the effects of molecular length (i.e. aspect ratio of rod) and packing fraction on the orientations of hard rod fluids and flexible chains. For the flexible chains, the chain length has no significant effect while the packing fraction shows slight effect on the molecular orientation distribution. In contrast, for the hard rod fluids, the chain length determines the molecular orientation distribution, while the packing fraction has no significant effect on the molecular orientation distribution. By making a comparison between molecular orientations of the flexible chain and the hard rod fluid, we find that the molecular stiffness distinctly affects the molecular orientation. In addition, partitioning coefficient indicates that the longer rodlike molecule is more difficult to enter the confined phase, especially at low bulk packing fractions.     

SAFT-D theory for hard sphere and hard disc chain fluids

SAFT-dimer (SAFT-D) theory is reformulated to yield an improved equation of state for the hard sphere chain fluid. Two sets of the equation of state are proposed by employing Chiew's expressions for the contact values of the m hard sphere site-site correlation function g(σ). Comparison with molecular simulation data shows that the improved SAFT-D equation of state predicts the compressibility factor more accurately than Ghonasgi and Chapman's equation of state. It has been shown that SAFT-dimer theory can be applied readily to fused hard sphere chain fluids by considering the correct value of the effective chain length (m*). SAFT-dimer theory is also extended to the 2-dimensional tangent and fused hard disc chain fluids. For the fused hard disc dimer fluid, the SAFT equation of state is found to be more accurate than the Boublik hard disc dimer equation of state. For tangent hard disc chain fluids, the results obtained from SAFT-dimer theory are compared with Monte Carlo results for 5-mers and with GFD theory for 4-mers, 8-mers and 16-mers.     

An Ellipsoidal Model for Small Multilayer Particles

This paper presents an ellipsoidal model that is constructed for small layered nonspherical particles and methods for constructing “effective” multilayer ellipsoids, the light-scattering properties of which would be close to the corresponding properties of original particles. In the case of axisymmetric particles, prolate or oblate spheroids (ellipsoids of revolution) are implied. Numerical calculations of the polarizability and scattering cross sections of small layered nonspherical particles, including nonconfocal (similar) spheroids, Chebyshev particles, and pseudospheroids, are performed by different approximate and rigorous methods. Approximate approaches involve the use of an ellipsoidal model, in which the polarizability of a layered particle is determined in two ways. In the first case, the polarizability is calculated in the approximation of confocal spheroids, while, in the second case, it is sought as a linear combination of the polarizabilities of embedded spheroids proportionally to the volumes of layers. Among rigorous methods, the extended boundary conditions method and the generalized separation of variables method are applied. On the basis of a comparison of the results obtained with rigorous and approximate approaches, their drawbacks and advantages are discussed.     

Principle of corresponding states for hard polyhedron fluids

ABSTRACT

We introduce an equation of state for hard polyhedra and validate our new theory with simulations performed for 22 different shapes spanning a range of particle anisotropy. The derived expression not only shows excellent agreement with simulations but also exhibits ‘corresponding states’-like behaviour across shape space, allowing for a systematic reduction of thermodynamic properties onto a single master curve. Additionally, we propose a scaling-type argument for predicting the order–disorder transition packing fraction for hard polyhedron fluids that accurately captures the observed transitions in simulations. Our works suggest that hard-polyhedron systems can be thought of as perturbations about a hard-sphere where corrections in the principle axes of inertia and excess volume account for differences in relative orientation between neighbouring particles and each polyhedron's intrinsic asphericity, respectively. Additionally, our theory greatly benefits from requiring only knowledge of the inherent geometry for a polyhedron of interest with no necessary fitting parameters and thus provides a good heuristic rule of thumb for targeting relevant regions of simulation interest for novel systems of hard polyhedra.     

Equations of state for the hard disk fluids

By using the published accurate new virial coefficients B₁₁～B₁₄ for the hard disk fluids [C. Zhang and B.M. Pettitt, Mol. Phys., 2014, 112, 1427], we here propose a new updated version of Tian-Gui-Mulero equation of state [J.X. Tian, Y.X. Gui and A. Mulero, Phys. Chem. Chem. Phys., 2010, 12, 13597]. Compared with other proposals, the new version stands strongly to be the only which can reproduce the known virial coefficients B₂～B₁₃ at the same time that can describe the relation of the compressibility factor versus the packing fraction for the hard disk fluids with high accuracy.     

Thermodynamic inequalities for molecular fluids

We consider the application of the classical Gibbs inequality for the Helmholtz free energy to a fluid whose potential energy depends on positions and orientations of the constituent molecules. Properties of the given fluid are related to the properties of a corresponding ‘reference’ fluid whose potential is the unweighted orientational average of the given potential. It is shown that Aˇ-A ₀, where A and A ₀ are the free energies respectively of the given and corresponding reference fluids, and that the high temperature limit of A - A ₀ may be either zero or negative infinite, depending on the nature of the potential. As consequences, it is shown that no general inequality exists relating the entropies of these systems, and that the internal energies U and U ₀ are plausibly expected to satisfy Uˇ-U ₀ at sufficiently high temperature. The case for a general inequality relating U and U ₀ is not decided.     

Transport coefficients of hard sphere fluids

New calculations have been made of the self-diffusion coefficient D, the shear viscosity η_s, the bulk viscosity η_b and thermal conductivity λ of the hard sphere fluid, using molecular dynamics (MD) computer simulation. A newly developed hard sphere MD scheme was used to model the hard sphere fluid over a wide range up to the glass transition (～0.57 packing fraction). System sizes of up to 32 000 hard spheres were considered. This set of transport coefficient data was combined with others taken from the literature to test a number of previously proposed analytical formulae for these quantities together with some new ones given here. Only the self-diffusion coefficient showed any substantial N dependence for N < 500 at equilibrium fluid densities (ε 0.494). D increased with N, especially at intermediate densities in the range ε ～ 0.3–0.35. The expression for the packing fraction dependence of D proposed by Speedy, R. J., 1987, Molec. Phys., 62, 509 was shown to fit these data well for N ～ 500 particle systems. We found that the packing fraction ε dependence of the two viscosities and thermal conductivity, generically denoted by X, were represented well by the simple formula X/X ₀ = 1/[1 ? (ε/ε₁)]^m within the equilibrium fluid range 0 > ε > 0.493. This formula has two disposable parameters, ε and m, and X ₀ is the value of the property X in the limit of zero density. This expression has the same form as the Krieger-Dougherty formula (Kreiger, I. M., 1972, Adv. Colloid. Interface Sci., 3, 111) which is used widely in the colloid literature to represent the packing fraction dependence of the Newtonian shear viscosity of monodisperse colloidal near-hard spheres. Of course, in the present case, X _o was the dilute gas transport coefficient of the pure liquid rather than the solvent viscosity. It was not possible to fit the transport coefficient normalized by their Enskog values with such a simple expression because these ratios are typically of order unity until quite high packing fractions and then diverge rapidly at higher values over a relatively narrow density range. At the maximum equilibrium fluid packing fraction ε = 0.494 for both the hard sphere fluid and the corresponding colloidal case a very similar value was found for η_s/η_o ?30–40, suggesting that the ‘crowding’ effects and their consequences for the dynamics in this region of the phase diagram in the two types of liquid have much in common. For the hard sphere by MD, D_o/D ～ 11 at the same packing fraction, possibly indicating the contribution from ‘hydrodynamic enhancement’ of this transport coefficient, which is largely absent for the shear viscosity. Interestingly the comparable ratio for hard sphere colloids is the same.     

Equilibrium properties of hard non-sphere fluids

Derivation of the thermodynamic properties of fluids of hard non-spherical molecules of arbitrary symmetry is based on the decoupling approximation. Theoretical expressions are given and calculations made for the equation of state and virial coefficients for hard ellipsoids. These results are compared with Monte Carlo values and show fair agreement in all cases. The theoretical predictions for the equation of state for binary mixtures are compared with the Monte Carlo results for hard spheres and hard prolate spherocylinders. Theoretical expressions for the first order quantum correction to the free energy, pressure and virial coefficients are also given. The quantum effects increase with increase of density and with increase of anisotropy parameter.     

Perturbation theory for non-axial molecular fluids

The thermodynamic perturbation theory in which all angle-dependent interactions are considered as a perturbation of the central potential is applied to study the equilibrium properties of a fluid composed of non-axial molecules. The influence of a large number of anisotropic pair and three-body non-additive interactions have been taken into account. Using the same set of force parameters the calculation is made for gaseous pressure second and third virial coefficients and liquid phase thermodynamic properties (Helmholtz free-energy, configurational energy, pressure and entropy). It is shown that the non-axial approximation is an improvement over the axial one. Excellent agreement between theory and experiment is obtained for ethylene.     

Fortran codes for the correlation functions of hard sphere fluids

Our Fortran codes for hard sphere fluids and their mixtures for the correlation functions that arise from the Percus–Yevick theory and the Verlet–Weis semi-empirical correction have proven useful during a period of nearly four decades and continue to be useful. In order to make these codes even more widely available, a brief summary is presented here and listings of these codes are given in the electronically accessible Supplementary Material to this paper.     

Classical and quantum hard sphere fluids: Theory and experiment

A comprehensive picture is presented of what can be reconstructed of the equations of state, at all densities, of both classical and quantum hard sphere fluids (the latter in their ground states) just from the first few coefficients which are known of the low-density expansions. Extrapolation techniques, mostly Padé and generalizations thereof, are employed and comparisons with computer simulation studies are made wherever these are available.     

Integral equation approximations for fluids of hard spheres with linear quadrupoles

In this paper we solve numerically several integral equation theories for the dense quadrupolar hard-sphere fluid. Closure approximations obtained by expanding the hypernetted-chain equation are shown to give pair-correlation functions and internal energies in good agreement with Monte Carlo calculations. The mean spherical approximation, however, is found to be extremely poor.     

Modified Parsons-Lee theory for fluids of linear fused hard sphere chains

The Parsons-Lee theory has been modified to study the liquid-crystalline phase behaviour of the linear fused hard sphere chain fluids. The modification of the Parsons-Lee theory is based on the application of the so-called effective molecular volume instead of the real molecular volume. This alteration does not mean any change for the Parsons-Lee treatment of the hard convex bodies but does for the non-convex ones. The results of the modified Parsons-Lee theory show very good agreement with simulations not only for the location of the isotropic-nematic phase transition but for the equation of state.     

Local size segregation in polydisperse hard sphere fluids

The structure of polydisperse hard sphere fluids, in the presence of a wall, is studied by the Rosenfeld density functional theory. Within this approach, the local excess free energy depends on only four combinations of the full set of density fields. The case of continuous polydispersity thereby becomes tractable. We predict, generically, an oscillatory size segregation close to the wall, and connect this, by a perturbation theory for narrow distributions, with the reversible work for changing the size of one particle in a monodisperse reference fluid.     

Perturbation theory for equilibrium properties of molecular fluids

Perturbation theory for the angular pair correlation function g(r₁₂ ω ₁ ω ₂), using a fluid with isotropic intermolecular forces as the reference system, is applied to the calculation of a variety of macroscopic properties. Comparisons with experiment are made for methane, oxygen and nitrogen (and carbon monoxide for infra-red and Raman band moments) in the dense fluid and liquid states. Theoretical expressions are given and calculations made for thermodynamic properties (isothermal compressibility, pressure, configurational energy, entropy and specific heat) both along and away from the vapour-liquid co-existence curve, for infra-red and Raman band moments, and for neutron scattering cross sections. Excellent agreement with experiment is obtained for all properties, except for the infra-red and Raman band moments; this latter comparison is inconclusive because of large experimental uncertainties. The anisotropic intermolecular forces are found to have very little effect on the liquid isothermal compressibility, in agreement with the first-order theory. Molecular anisotropy has a relatively small effect on the configurational energy and on the Helmholtz free energy, but the effect is large for pressure and specific heat. The pressure is more sensitive to short-range anisotropic forces than the other properties, whereas the specific heat is particularly sensitive to the long-range anisotropic forces. Mean squared torques (derived from infra-red and Raman band moments) are very sensitive to the strengths of the anisotropic forces, and are more sensitive to higher terms in the multipole series than are the other properties. The structure factors for oxygen and nitrogen are found to be little affected by the anisotropic forces.     

Integral equation approximations for fluids of hard spheres with dipoles and quadrupoles

The LHNC, QHNC and mean spherical approximations are solved for fluids of hard spheres with dipoles and linear quadrupoles. The theories are evaluated by comparison with new Monte Carlo results. The dielectric constant is found to decrease rapidly with increasing quadrupole moment.     

On the compressibility equation of state for multi-component adhesive hard sphere fluids

The compressibility equation of state (EOS) for a multi-component sticky hard sphere model alternative to Baxter's one is investigated within the mean spherical approximation (MSA). For this model and this closure, as well as for a more general class of models and closures leading to Baxter functions q_ij(r) with density-independent stickiness coefficients, no compressibility EOS can exist for mixtures, unlike the one-component case (in view of this, an EOS recently reported in the literature turns out to be incorrect). The reason is the failure of the Euler reciprocity relation for the mixed second-order partial derivatives of the pressure with respect to the partial densities. This is in turn related to the inadequacy of the approximate closure (in particular, the MSA). A way out to overcome this drawback is presented in a particular example, leading to a consistent compressibility pressure, and a possible generalization of this result is discussed.     

Perturbation theory based direct correlation function for fluids with hard core potentials

Using second-order Barker–Henderson perturbation theory we are able to derive an explicit expression for the direct correlation function of fluids with hard core potentials. Using the obtained direct correlation function, one can explicitly calculate all thermodynamic properties of simple fluids with hard core potentials. Comparisons with computer simulation data show good agreement for both thermodynamic properties and the static structure factor of the hard core double Yukawa potential.     

Liquid-vapour coexistence of molecular fluids

We derive, for a liquid-vapour system with pairwise interactions, an infinite set of sum rules relating the pressure and temperature to integrals over the density profile and pair correlation function. For approximate forms for the variation of the pair correlation function through the liquid-vapour interface, these sum rules reduce to equations describing the coexistence curve of the fluid. Good agreement with experiment is obtained for all fluids considered.     

The fifth virial coefficients of fused hard sphere fluids

The virial coefficients up toB ₅ of 34 fused hard sphere models have been computed using about (4–10)×10⁵ trial configurations of a modified Monte Carlo integration method. The principle of the conformal behaviour of hard particle systems is tested against these data. It is found that while for the fourth coefficient the principle is satisfied in all the cases, the fifth coefficients exhibit more complicated behaviour and do not conform, in general, to this principle.