Some topics in statistical physics

Abstract

The thesis is in two parts. In Part One, the role of the angle-averaged potential in the theory of non-simple liquids is investigated. Analytic expressions for the angle-averaged potential are obtained and their asymptotic approximations are derived and compared with the exact expressions. The additivity of the angle-averaged potential is discussed. The angle-averaged potential is shown to provide a basis for the expression of the thermodynamic properties of an angle-dependent system. The free energy for such a system is derived as the sum of the free energy of the simple system where the pair interaction is the angle-averaged potential and a set of three body, four body etc. terms involving the triplet and higher correlation functions for the simple system. Some work on the two-dimensional coplanar point dipole system is discussed. A first order perturbation theory is derived where the reference potential is the angle-averaged potential. This theory is compared with the first order perturbation theory of Gubbins and Gray and the Mean Spherical Model of Wertheim for the hard sphere plus imbedded dipole system. The present theory is found to be in best agreement with Monte Carlo studies of the exact dipole system. In Part Two the statistical mechanics of a model adsorbed polymer is developed. The polymer is considered as a string of non-interacting beads connected by freely rotating bonds with an arbitrary bond length distribution. The beads (Monomer units) are free to interact with the flat impenetrable substrate via a one-body potential. Rigorous statistical formulae are derived for the expectation values of the number of beads on the substrate, the spread of the polymer on the substrate, the density of beads off the substrate and the centre of mass of the polymer in terms of an adsorption energy parameter W. These conformational properties are shown to undergo a phase transition at a critical value of W=Wc , where Wc is a functional of the bond length distribution function only. For W>Wc the polymer is adsorbed on the wall and for W<Wc the polymer moves into the bulk solvent and away from the substrate. Some numerical calculations were carried out with real polymer/solvent/substrate systems to discover whether dispersion forces could yield values of W/Wc which could span the complete range of behaviour of the adsorbed polymer. The possibility of temperature and solvent-induced adsorption/ description phase transitions is exhibited. Show more