On statistical mechanical equations of state for simple fluids : Effective hard spheres and quantum corrections

Nordholm, S., Greberg, H. and Penfold, R., 1991. On statistical mechanical equations of state for simple fluids. Effective hard spheres and quantum corrections. Fluid Phase Equilibria, 90: 307-332.

A comparison is made of the mean field generalised van der Waals theory, based on a variationally determined hard sphere diameter, with the recent equation of state proposed by Song and Mason incorporating a temperature-dependent hard sphere diameter and correlation effects through the second virial coefficient. The simple cell theory ansatz of the former is less accurate but permits a wide range of applications including the estimation of quantum effects on the bulk properties of light fluids at low temperatures. Results for the critical parameters of ³He, ⁴He, H₂, Ne, CH₄ and Ar are examined. The relevance to the corresponding theory of non-uniform fluids is noted.     

Generalized finite element method applied to the calculation of continuum states

A new method for the calculation of continuum energy eigenfunctions in one dimension is presented. It is based on a numerical coupling method first exploited in the so-called finite element method which does not require explicit fitting of boundary conditions. Results from simple test calculations for square well and Morse potentials show the method to be highly accurate and efficient.     

Separable unimolecular reaction rate theory

A new unimolecular reaction rate theory is derived on the basis of a classical model of the reaction. The fundamental assumptions are (i) instantaneous activation-deactivation events represented by a statistical transition probability, (ii) irreversible dissociation, and (iii) separability of internal and external dynamics. The third assumption is found to be satisfied when the statistical transition probability satisfies a generalized strong collision assumption and it leads to a rate theory which depends on internal molecular dynamics only through lifetime information for the isolated molecule. The new theory shows that the rate coefficient is non-Markoffian although the memory effects will not be apparent on the macroscopic time scales of traditional experiments. Thus Slater's “new approach to rate theory” is a special case of the separable rate theory when applied to slow reactions. New experimental techniques are predicted to have the capacity to resolve the memory effects in the rate coefficient and provide lifetime information which would then allow greatly improved accuracy in the prediction of rates for a wide range of unimolecular reactions.     

An atomic diffraction theory of molecular photoionization cross sections

A theory of molecular photoionization cross sections is developed on the basis of locally atomic character of the one-electron final state in the Golden Rule expression for the molecular orbital cross section. Ionization amplitudes from several atomic centres are added and rotationally averaged to produce molecular orbital cross sections displaying a sum of pseudo-atomic cross sections weighted according to the LCAO composition of the orbital and also two-centre products reflecting interference effects. The atomic ionization amplitudes are obtained by use of an atomic central potential constructed by an inversion procedure from the form of the ground state orbital. The theory is of a simple chemical nature but usually of at least semi-quantitative accuracy. In this work we illustrate the nature of the two-centre interference effects in small diatomic molecules (H₂, HF, N₂).     

Effective potential approach to bulk thermodynamic properties and surface tension of molecular fluids I. Single component n-alkane systems

The aim of this work is to develop spherically symmetric effective potentials allowing bulk thermodynamic properties and surface tension of molecular fluids to be predicted semiempirically by the use of statistical mechanical methods. Application is made to the straight chain alkane fluids from methane to decane. An effective Lennard-Jones potential is generated with temperature-dependent parameters fitted to the critical temperature and pressure and to Pitzer's acentric factor. Insertion of this potential into the generalised van der Waals (GvdW) density functional theory yields bulk properties in good agreement with experiments. The surface tension is overestimated for the longer alkane chains. In order to account for the surface tension, an independently adjustable attractive range of interaction is required and obtained through the use of square-well potentials chosen so as to leave the bulk thermodynamics unaltered while the attractive range is fitted to the surface tension at a single temperature. The GvdW theory, which includes binding energy, entropic and profile shape contributions, then generates surface tension estimates that are of good accuracy over the full range of available experimental data. It appears that, given a sufficiently flexible form, effective potentials combined with simple statistical mechanical theory can reproduce both bulk and non-uniform fluid data of great variety in an insighful and practically useful way.     

Numerical study of the effects of constitutive models on plastic buckling of plate elements

Compared with experiments, the J₂ deformation theory of plasticity is known to predict plastic buckling with better accuracy than the more accepted incremental J₂ flow theory. This paradox is commonly known as the ‘plastic buckling paradox’. In an attempt to analyse this discrepancy, the two mentioned constitutive models were implemented in a non-linear finite element code, along with a third non-associative J₂ flow theory. The latter model incorporates a vertex-type plastic flow rule. Using these three constitutive models, the buckling behaviour of plate outstand elements was investigated. Comparisons between the buckling strengths derived are presented. The non-linear static buckling simulations show that the instability introduced by the alternative flow rule of the non-associative model has substantial influence on the buckling behaviour. The acceptance of only small departures from normality was shown to reduce the predicted ultimate capacity of the plates. Furthermore, for plates with small plate slendernesses it was found that the imperfection sensitivity was significantly reduced when using the non-associative flow rule.     

Ergodic collision theory of intermolecular energy transfer

We present a simple theory of collisional energy transfer between molecules based on the assumption of ergodic collisions, i.e., the final state distri     

An efficient solution of weak-collision master equations in thermal unimolecular reaction rate theory

A new finite-basis-set method of solving the weak-collision master equation of thermal unimolecular reactions in the general pressure regime is presented. Consecutive sections of the equilibrium probability density are used as basis functions. Significant advantages in terms of efficiency and applicability are obtained. Representative calculations are performed to illustrate the method's convergence properties and storage requirements. Calculations of collisional efficiency factors in the low-pressure limit β₀ and weak-collision broadening factors F^WC(ω) are performed to offer a simple concise representation of weak-collision effects. It is noted that weak-collision effects can be incorporated into fall-off curves with an accuracy of within an order of magnitude by simple scaling of the strong-collision fall-off curves. However, for accurate representation of weak-collision effects the weak-collision broadening must be taken into account.     

Strong collision rate coefficients in unimolecular reaction rate theory

The traditional and a recently proposed renormalized approximation to the rate coefficient obtained from a strong collision master equation are derived and compared. It is shown that they deviate significantly when applied to the unimolecular reactions of large molecules with low activation energies at elevated temperatures. The derivation is extended to yield a method whereby the exact rate coefficient can be calculated without increasing the level of numerical effort. An analysis of the exact rate coefficient in the high and low collision frequency limits is found to verify the validity of the traditional approximate rate coefficient in these limits. The renormalized rate coefficient fails in the low frequency limit. Both approximations can be significantly in error at intermediate collision frequencies and the use of the exact expression is recommended.