Thermodynamic properties of molecular fluid mixtures of hard ellipsoids

Thermodynamic properties of molecular fluid mixtures of hard ellipsoids are calculated. Numerical results are given for equation of state and excess-free energy of the binary mixture of both additive and non-additive hard ellipsoids. It is found that the equation of state and free energy of mixtures increase with increase of anisotropy parameterx ₀.     

Thermodynamic and structural properties of liquids modelled by ‘2-Lennard-Jones centres’ pair potentials

Molecular dynamics calculations have been carried out for model liquid systems of N (=108 or 256) molecules interacting through two Lennard-Jones (12–6) centres coinciding with the positions of the atomic masses (the ‘atom-atom’ pair potential). The objectives were (a) to study the dependence of the properties on the molecular anisotropy defined by the reduced distance l*=l/σ between the centres in the range 0·5–0·8; and (b) to compare the computed quantities with those of real liquids (F₂, Cl₂, Br₂, CO₂). This paper deals with thermodynamic and structural features. Time-dependent correlations will be treated in a future communication.

In the liquid region not too far from the triple point the energy and pressure isochores are well represented by straight lines, the slopes of which increase with density and anisotropy. Thermodynamically consistent expressions for the energy and pressure as functions of density and temperature have been obtained for each system.

With Lennard-Jones parameters adjusted so as to secure the best overall fit, the agreement between experimental and computed thermodynamic properties is very satisfactory for F₂ (l*=0·505), quite good for Cl₂ and Br₂ (l*=0·608–0·63), but rather poor for CO₂ (l*=0·793). The ‘interatomic distances’ are close to the experimental values.

The static structural correlations are discussed in terms of the pair-correlation functions (pcf) g _A(r*) for the separation between ‘atoms’, the first few functions g_ll'm (R*) which arise from the expansion of the g(R*, θ₁, θ₂, φ₁₂) in spherical harmonics, and the pcf's for certain special near-neighbour configurations. The computed atom-atom structure factor is compared with the experimental data for liquid Br₂.

Mean square forces and torques have been evaluated and are related to some experimental results.     

Entropy driven demixing in binary mixtures of hard convex bodies: influence of molecular geometry

The recently introduced approach (Bosetti, H., and Perera, A., 2001, Phys. Rev. E, 63, 021206) for studying entropy driven demixing in binary mixtures of hard convex bodies is applied here to analyse the influence of molecular geometrical factors such as shape and size ratio on isotropic fluid-fluid phase separation. The theory is shown to be self-consistent, in the sense that it allows the calculation of both the binodal and the spinodal of the demixing, and both curves merge at the same lower consolute point. In the case of 3-dimensional fluids, demixing is usually allowed for sufficiently thick solutes, provided no orientational ordering destabilizes the mixture. Fluid—fluid demixing is explicitly forbidden in cases of equal breadth of the solute and the solvent molecules, regardless of the aspect ratio of both particles. In particular, mixtures of prolate and oblate particles will demix more easily. The influence of shape is more importantly seen at moderate elongations. Finally, within the present approximation, demixing is forbidden for binary mixtures of 2-dimensional fluids, irrespective of the shape of the bidimensional molecules.     

Spherical reference systems for hard dumbells

Spherical reference systems for use in perturbation theories of hard dumbell molecules are investigated. In particular the reference potential obtained by angle averaging the Boltzmann factor of the full molecular potential (ε_eff) is shown to lead to reasonable pair distribution functions, but to pressures much lower than those of the full hard dumbell system, whereas an effective hard sphere reference potential can be designed to give pressures in close agreement to those of the full system, though it fails to give reasonable pair distribution functions.

The ε_eff reference system properties are calculated using a new iterative solution of the Percus-Yevick equation for soft cores. These results are compared with a Monte Carlo simulation of the same system.     

Hund's rules reformulated as inequalities and some related inequalities for exchange integrals,Slater-Condon parameters energy differences and correlation energies

Measurements have been made of the neutron scattering structure factors of liquid nitrogen and liquid oxygen at 77 K and 84 K respectively in the Q-value range of 3 to 36 Å^-1. ‘White’ incident thermal neutrons were produced in the wavelength range of 0·3 to 3·0 Å by a pulsed electron linac and detected in a total-scattering time-of-flight spectrometer. Qualitative agreement has been obtained between these present data and a simple molecular form factor in the Q-value range of 12–36 Å^-1.

Using reactor data [1], structure factors over a total Q-value range of 0·3 to 36 Å^-1 were obtained. Fourier transforms of the structure factor curves yield pair distribution functions which show a distinct separation of the inter and intra-molecular distances in the liquid. A comparison is made with results for the solid phase.     

The thermodynamics of symmetric two centre Lennard-Jones liquids

We have carried out molecular dynamic simulations for the thermodynamic properties of two centre Lennard-Jones fluids at lower densities and higher temperatures than have been studied previously, and have also made simulations for one additional shape. The results, together with results already given in the literature, are presented in a parameterized form which is very convenient to use in testing the simulation results against data for real liquids. Tables of second virial coefficients are also provided. We illustrate the use of our results by an analysis of pure liquid ethane, which is found to be well represented by such a model with L* = 0·67, ε/k = 137·5 K and σ = 3·506 Å. We also suggest that the experimental thermodynamic properties of suitable liquid mixtures can, with the aid of a theory for the equivalent pure liquid parameters (L _x *, ε _x , σ _x ), be satisfactorily interpreted using the general results given in this paper.     

Neutron diffraction studies of liquid carbon suboxide

The structure factor S _m(Q) for liquid carbon suboxide has been determined for a Q-value range of 0·4 to 60 Å^-1 by neutron diffraction measurements using a steady-state (reactor) and a pulsed (linac) neutron source. The bond lengths of the molecule have been determined from the data and give good agreement with the results of electron diffraction measurements on the vapour phase after application of a molecular recoil correction term. The quasi-linear nature of the molecule is confirmed but the shape of the form factor indicates that large amplitude bending motion probably occurs in the liquid phase.

Oscillations in the intermolecular pair correlation function are observed to have a regular periodicity extending to 12 Å but details of orientational effects cannot be established from a single diffraction measurement.     

Electron spin resonance study of radiation-produced radical pairs in single crystals of 1-methyluracil

Five different types of radical pairs produced in single crystals of 1-methyl-uracil by irradiation with X-rays were identified. Partial differentiation between pairs was achieved by utilizing different stabilities versus ultraviolet illumination or annealing, and variation of relative yields with temperature. The composition of pairs was derived from hyperfine couplings, the orientation from the maxima of dipolar couplings. These were found to agree with the directional cosines obtained from the crystal structure. Further evidence was provided by half-field spectra (Δm _S=2). The zero-field splitting constant D was determined for five pairs, the value of E for one pair. Second-order effects observed at 9·5 GHz and at 35 GHz have been fully analysed for two pairs by application of the appropriate theory which permitted to determine the absolute sign of D. Two pairs are homogeneously composed of hydrogen abstraction radicals 1-methyleneuracil. These are located two layers apart for one pair with the link normal to the molecular planes (D/gβ = -107·5 G), and within one plane for the other (-224·0 G). The three remaining pairs consist heterogeneously of an abstraction radical and a hydrogen addition radical 1-methyl-5,6-dihydrouracil-6-yl. Two of these are intraplanar (-180·0 G; -129·0 G), and the other one located in neighbouring layers (-183·0 G).     

Calculation of thermodynamic properties of dense fluid neon using statistical-mechanical perturbation theory

The theory and potentials described in the preceding paper are used for the calculation of thermodynamic properties of fluid neon between 73 and 348 K and up to the melting line. A polynomial equation of state for correlating the densities between 73 and 323 K and between 30 MPa and 1 GPa is presented. The calculations have been extended up to 6 GPa and the performance of the EXP-6 effective pair potential and the HFD-C2, HFD-B and XC3 pure pair potentials is compared. The effect of the Axilrod-Teller many-body correction term on the pure pair potentials is studied. In the ranges 98-348 K and 0·6-6 GPa the density data are correlated by a MBWR equation of state, and a polynomial expression is given for the sound velocity in these ranges. The pressure and temperature dependences of the specific heat at constant volume are shown. A comparison is made between the experimental pVT and soundvelocity data of Kimura et al. at 295 K and up to 3·5 GPa and our calculations.     

The structure factor for liquid bromine by neutron diffraction

Thermal neutrons from steady-state (reactor) and pulsed (linac) neutron facilities have been used to study the structure factor for liquid bromine at 20°C, covering a wide range of values (0·6-35 Å^-1) of the momentum transfer (?Q). The diffraction pattern at high Q-values (>10 Å^-1) gives information on the structural properties of the individual molecules but detailed interpretation is complicated by the vibrational motion which causes a systematic variation in the periodic oscillations of the molecular form-factor. An internuclear distance (bond length) of 2·28 ± 0·01 Å is found to be suitable for data at lower Q-values and has been used in the analysis of the liquid structure. The results show that some form of orientational correlation between molecules must be present and the nuclear (atom) pair correlation function is split into two peaks for the coordination shell corresponding to nearest-neighbour molecules. The results are compared with other studies of liquid bromine and similar neutron experiments for liquid nitrogen and liquid oxygen.     

Quantum corrections to thermodynamics of polar hard sphere fluids

The quantum corrections to the thermodynamic properties of polar hard sphere fluids and fluid mixtures are estimated taking into account the influence of dipole and quadrupole moments. Expressions are given for the second virial coefficient, free energy and pressure and results are given for different values ofμ* andϑ*. The first order quantum correction arises due to the translational contribution only. The quantum effect increases with density,μ* andϑ*. Numerical results are also estimated for binary mixtures of (i) hard spheres and dipole hard spheres and (ii) hard spheres and quadrupole hard spheres. The ‘excess’ free energy for dipole hard sphere binary mixture is also reported. It is found that the ‘excess’ quantum effect depends on the concentration and the particle diameter ratio and increases with increase ofμ* andϑ*.     

On the critical non-additivity driving segregation of asymmetric binary hard sphere fluids

A previously proposed version of thermodynamic perturbation theory, appropriate for singular pair interactions between particles, is applied to binary mixtures of hard spheres with non-additive diameters. The critical non-additivity Δ_C required to drive fluid–fluid phase separation is determined as a function of the ratio ξ ≤ 1 of the diameters of the two species. Δ_C(ξ) is found to decrease with ξ and to go through a minimum for ξ ? 0.015 before increasing sharply as ξ → 0, irrespective of the total packing-fraction η of the mixture. These results are the basis of an estimate of the range of size ratios for which a binary mixture of additive hard spheres exhibit a fluid–fluid miscibility gap. This range is conjectured to be 0.01 ? ξ ? 0.1.     

Virial coefficients and equations of state for mixtures of hard discs,hard spheres and hard hyperspheres

The composition-independent virial coefficients of a d-dimensional binary mixture of (additive) hard hyperspheres following from a recent proposal for the equation of state of the mixture (Santos, A., Yuste, S. B., and López de Haro, M., 1999, Molec. Phys., 96, 1) are examined. Good agreement between theoretical estimates and available exact or numerical results is found for d = 2, 3, 4 and 5, except for mixtures whose components are very disparate in size. A slight modification that remedies this deficiency is introduced and the resummation of the associated virial series is carried out, leading to a new proposal for the equation of state. The case of binary hard sphere mixtures (d = 3) is analysed in some detail.     

Perturbation calculation of the distribution functions of mixtures of hard spheres

The distribution functions for a mixture of additive hard spheres of diameter σ_ii are calculated by means of an expansion in powers of σ _ij ⁿ - dⁿ , using an unperturbed single-component hard-sphere fluid of diameter d. Although the expansion converges only when the hard spheres in the mixture are nearly equal in size, it is useful because it is the only practical scheme available for obtaining the distribution functions of multicomponent hard-sphere mixtures.     

The direct determination of pair potential energy functions for mixed interactions: Ar-Kr

Measurements of the viscosity coefficients for argon-krypton gas mixtures over the temperature range 120–1600 K are inverted to give the pair potential energy function for Ar-Kr. The inversion procedure requires an estimate of the well-depth parameter ε/k and a method for its determination using second virial coefficient data is described. The resulting Ar-Kr potential function has characteristic parameters ε/k=165 K, σ=0·348 nm and r _m=0·3902 nm. The conformality of this potential with those for the like-molecule interactions, and the value of parametric combining rules for this system are also investigated.     

Dielectric properties of a mixture of dipolar hard spheres

A theory for the dielectric constant, ε, of a fluid mixture of dipolar hard spheres is formulated by generalizing the methods developed by Ramshaw and Wertheim for the pure fluid case. The resulting expression for ε depends on the pair distribution functions, g _αβ(r ₁, θ₁, r ₂, θ₂) for a dipolar mixture. Due to the unavailability of exact representations for these dipolar pair distribution functions, the results of the mean spherical approximation are employed in the formalism developed. Numerical results are given for ε as calculated from the pair distribution functions for a spherical volume of macroscopic dimensions. The compositional dependence of the ε obtained in this way for a specific mixture is compared with the corresponding properties of the well established theories of Clausius-Mossotti-Debye and Onsager. In addition, the relative importance of the dipole moment and size of the hard sphere parameters in determining ε for a dipolar mixture (the correlative behaviour of which is described by the mean spherical approximation) is evaluated. It is found that the differences in hard core diameters can be largely ignored, in that ε for an ‘effective’ single component fluid can be given to within 2–5 per cent relative error (at worst) of the mean spherical approximation's result. Such an ‘effective pure fluid’ is described as having the same polarization content as the actual mixture being considered. Thereby, the properties of the effective fluid are determined by the quantity y = 4πβ(m ₁ ² ρ₁ + m ₂ ² ρ₂)/9 where m_i and ρ _i are the dipole moment and number density of component i in the binary mixture, with β = (kT)^-1.     

The virial coefficients of hard hypersphere binary mixtures

The third, fourth and fifth virial coefficients of hard hypersphere binary mixtures with dimensionality d = 4, 5 have been calculated for size ratios R ≥ 0.1, R = ≡ σ₂₂/σ₁₁, where σ _ii is the diameter of component i. The composition independent partial virial coefficients have been evaluated by Monte Carlo integration of the corresponding Mayer modified star diagrams. The results are compared with the predictions of Santos, S., Yuste, S. B., and Lopez de Haro, M., 1999, Molec. Phys., 96, 1 of the equation of state of a multicomponent mixture of hard hyperspheres, and the good agreement gives strong support to the validity of that recipe.     

Methyl group tunnelling and torsion in 2,4-hexadiyne

The tunnelling splitting of the ground torsional level of solid 2,4-hexadiyne and transitions to excited torsional states have been measured at low temperatures using neutron inelastic scattering. At 4 K the tunnelling splitting is 1·060 μeV (0·0086 cm^-1). It decreases as the temperature is raised, to 0·834 μeV (0·0067 cm^-1) at 35 K, and to less than 0·6 μeV at 50 K. A V-2←V=0 transition in the torsional vibration has been observed at 222 cm^-1 which shifts to 160 cm^-1 in the fully deuterated compound.

The values of the torsional frequencies, tunnelling frequency, and the change of tunnelling splitting with temperature have been fitted exactly to a potential energy for rotation of a methyl group given by

with a barrier to rotation of 432 cm^-1.

Changes in the tunnelling transitions as the temperature increases are compared with existing theories of the mechanism.