In this thesis several classes of directed paths are considered as models
of linear polymers in a dilute solution. We obtain the generating functions for
each model by considering factorization arguments.
Information about the polymer behaviour can be extracted from the
singularity structure of the associated generating functions.
By using modified versions of these models we study the adsorption and
localization of polymer molecules, the behaviour of polymers subject to
a tensile force, the effects of stiffness, as well as the behaviour of polymers
in confined geometries.
In each of these situations the resulting generating functions
contain at least two physical singularities. We identify the phase transitions
in these systems by a changeover in the dominant singularity of the generating
function.
In the study of localization and polymers subject to a force, we utilize both
homopolymer and random copolymer models. For copolymers, the physically
relevant properties are obtained by considering a quenched average of the
free energy over all possible monomer sequences. This procedure is intractable
even for the simplest models. By considering the Morita approximation for several
walk models we obtain results which give a bound on the corresponding features
of the quenched system.
We use a mapping between a simple model of duplex DNA and an adsorbing Motzkin
path in order to study the mechanical unzipping of duplex DNA. From this
model, we obtain force-temperature diagrams which show re-entrant
behaviour of the force. We also develop a simple low temperature theory to
describe the behaviour of the force close to T=0 and find that the shape
of the force-temperature curve is associated with entropy in the ground state
of the system.
We consider the effect of stiffness on polymer adsorption and find that
the phase transition is second order for all finite stiffness parameters.
For systems of polymers in confined geometries, we find that the behaviour of
the polymer depends on the distance between the confining surfaces and
the associated interactions with each surface. In this problem, there exist
regimes where the polymer exerts a force on the surfaces which can be
attractive, repulsive or zero.
| Identifer | oai:union.ndltd.org:TORONTO/oai:tspace.library.utoronto.ca:1807/16728 |
| Date | 19 January 2009 |
| Creators | Iliev, Gerasim K. |
| Contributors | Whittington, Stuart |
| Source Sets | University of Toronto |
| Language | en_ca |
| Detected Language | English |
| Type | Thesis |
| Format | 1616791 bytes, application/pdf |
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