In this thesis, motivated by modelling polymers, the topological entanglement complexity of systems of two self-avoiding polygons (2SAPs), stretched polygons and systems of self-avoiding walks (SSAWs) in a tubular sublattice of Z3 are investigated. In particular, knotting and linking probabilities are used to measure a polygonfs selfentanglement and its entanglement with other polygons respectively. For the case of 2SAPs, it is established that the homological linking probability goes to one at least as fast as 1-O(n^(-1/2)) and that the topological linking probability goes to one exponentially rapidly as n, the size of the 2SAP, goes to infinity. For the case of stretched polygons, used to model ring polymers under the influence of an external
force f, it is shown that, no matter the strength or direction of the external force, the knotting probability goes to one exponentially as n, the size of the polygon, goes to infinity. Associating a two-component link to each stretched polygon, it is also proved that the topological linking probability goes to unity exponentially fast as n→∞. Furthermore, a set of entangled chains confined to a tube is modelled by a system of self- and mutually avoiding walks (SSAW). It is shown that there exists a positive number γ such that the probability that an SSAW of size n has entanglement complexity (EC), as defined in this thesis, greater than γn approaches one exponentially as n→∞. It is also established that EC of an SSAW is bounded above by a linear function of its size. Using a transfer-matrix approach, the asymptotic form of
the free energy for the SSAW model is also obtained and the average edge-density for span m SSAWs is proved to approach a constant as m→∞. Hence, it is shown that EC is a ggoodh measure of entanglement complexity for dense polymer systems modelled by SSAWs, in particular, because EC increases linearly with system size, as the size of the system goes to infinity.
| Identifer | oai:union.ndltd.org:USASK/oai:usask.ca:etd-09082008-102610 |
| Date | 09 September 2008 |
| Creators | Atapour, Mahshid |
| Contributors | Soteros, Chris, Kusalik, Anthony J. (Tony), Diao, Yuanan, Bremner, Murray R., Bickis, Mikelis G., Srinivasan, Raj |
| Publisher | University of Saskatchewan |
| Source Sets | University of Saskatchewan Library |
| Language | English |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | http://library.usask.ca/theses/available/etd-09082008-102610/ |
| Rights | restricted, I hereby certify that, if appropriate, I have obtained and attached hereto a written permission statement from the owner(s) of each third party copyrighted matter to be included in my thesis, dissertation, or project report, allowing distribution as specified below. I certify that the version I submitted is the same as that approved by my advisory committee. I hereby grant to University of Saskatchewan or its agents the non-exclusive license to archive and make accessible, under the conditions specified below, my thesis, dissertation, or project report in whole or in part in all forms of media, now or hereafter known. I retain all other ownership rights to the copyright of the thesis, dissertation or project report. I also retain the right to use in future works (such as articles or books) all or part of this thesis, dissertation, or project report. |
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