Phase behaviour of a generalised penetrable sphere model: two and three body interaction potentials

Abstract

There has been considerable interest in recent years in the properties on systems of particles where the pairwise interaction in bounded. Such interactions allow particles to overlap, imposing only a finite energy cost. This feature is particularly pertinent to systems such as star polymers, dendrimers and micelles, where the effective interactions are softened due to the internal degrees of freedom of the particles. The main focus has been on continuous pair potentials, such as the generalised Gaussian potential, leading to enormous progress. Computer simulation studies have mapped out many phase diagrams, with a particular emphasis on the curious, multiple-occupancy crystal structures that form, where more than one particle occupy each lattice site. Relatively simple theoretical approaches, based on a mean field theory, yield very good agreement with simulations in both the fluid and solid phases, suggesting these systems are very well understood.   A model that is less straightforward, however, is the penetrable sphere model (PSM). This potential is discontinuous, being a positive constant when the particles overlap and zero otherwise. The phase behaviour of PSM is less well explored by simulation than that of the continuous potential models and the fairly simple theories, noted above, work nothing like so well. Part of the research presented here involves simulating some of the neglected regions of the phase diagram and comparing the results with theory.   Most simulation and theoretical research in statistical mechanics focuses on the systems with pairwise additive potentials, while the analysis of three body interactions is considerably less well studied. In addition, both face-centred cubic (FCC) and body-centred cubic (BCC) structures are found in the Gaussian core model (GCM) and GEM-4 with pure two body interaction. However, in the PSM, which represents the infinite limit of the generalised Gaussian potential, only FCC structure has been observed so far. This raises the question of whether including three-body interactions might result in a richer phase diagram, similar to those seen in GCM and GEM-4. Practically, three-body interactions contribute the thermodynamical properties of the models of star polymers and the highly size-asymmetrical colloid-polymer mixtures.1 Given the relatively simplicity of systems with bounded potentials compared to more realistic interactions, we carry out simulations and some theoretical analysis on a simple three body version of the PSM, with the aim of improving our understanding of the effects of three body interactions. In our model there is an additional energy contribution if three particles overlap simultaneously, but zero contribution otherwise.   The analysis of the standard, two-body PSM focuses on the fluid phase at relatively high temperatures and densities. We calculate the radial distributions functions under these conditions and invert them to obtain the direct correlation function and the bridge function. These quantities play crucial theoretical roles and our data helps to identify where the assumptions of simple mean field theory become invalid. This analysis will also, we hope, provide a testing bed for new theoretical approaches to the fluid state of the PSM. A good account of the fluid state direct correlation function will hopefully also lead, in the future, to better theoretical descriptions of the solid phase.   The research on the three-body PSM is, to our knowledge, novel. We use computer simulation to calculate the thermodynamic and structural properties of the fluid phase and provide a provisional analysis of the crystalline phase formed at high density. If the two-body potential is chosen to be attractive and the three-body potential is repulsive, then we locate vapour-liquid transitions. We present a comparison of these data with the predictions of a mean-field theory and, at least for the vapour-liquid transition, the theory works poorly. At higher de

Date of Award	1 Aug 2025
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Andrew Masters (Supervisor) & Carlos Avendano (Supervisor)