School of Mathematics and Statistics - Theses

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Type: PhD thesis × Subject: combinatorics × Subject: directed walkers × Subject: polymers ×

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    Generalised directed walker models of adsorption and gelation
    TABBARA, RAMI ( 2015)
    We outline an approach to constructing and solving models of highly interactive systems of single and multiple homopolymers, focusing on adsorption and gelation effects. In particular, solutions of our model allow us describe the critical behaviour of our system, such as the temperature required to graft a polymer onto an attractive surface or the potential energy required to bond a collection of single polymers into a branched, dense gel structure. We begin by outlining a simple counting problem of finding a closed-form expression for the number of walks along a finite lattice that obey some specified step constraints. Of particular relevance is the constraint of self-avoidance which specifies that a given walk is only able to visit a given site once, and we highlight why self-avoiding walks are considered a good approach to modelling polymer conformations. We further highlight how one can incorporate this purely combinatorial problem into the framework of statistical mechanics which allows us to construct a probabilistic model that links the microscopic and macroscopic state of a thermodynamic system. In short, we review the well established approach of transforming the task of modelling a physical system of single or multiple polymers into a combinatorial problem. We then show how self-avoiding directed walkers can be employed to construct idealised models of these polymer systems and review known techniques that allow us to establish exact solutions. While previous work on directed walker models has predominantly focused on, at most, single interaction effects, the applicability of these current techniques to solve more sophisticated models that incorporate multiple polymer conformations and multiple interaction effects was relatively unconsidered. Indeed, we will showcase the versatility of such techniques, central of which is the kernel method, and further unearth extensions of these techniques to solve for a number of highly interactive polymer models, with subsequent analysis revealing a rich set of interesting and unexpected critical behaviour.