Mathematical models for polymer-nematic interactions

This dissertation considers a mathematical model that consists of a nematic liquid crystal layer sandwiched between two parallel bounding plates, across which an external field may be applied. Particular attention is paid to the effect of an applied field on the layer as well as the interaction between the liquid crystal molecules and the molecules of the substrate. The system studied may be considered as a simple model of a Liquid Crystal Display (LCD) device, and the results obtained are discussed and interpreted within this context. The first part of this dissertation considers a study that investigates how the number and type of solutions for the director orientation within the layer change as the field strength, anchoring conditions and material properties of the nematic liquid crystal layer vary. During this investigation, particular attention is paid to how the inclusion of flexoelectric effects alters the Freedericksz and saturation thresholds. In the second part of the dissertation, the interaction between nematic liquid crystal (NLC) and polymer coated substrates with and without an external applied field is considered. Under certain conditions, such polymeric substrates can interact with the NLC molecules, exhibiting a phenomenon known as director gliding or easy axis gliding. Mathematical models for gliding, inspired by the physics and chemistry of the interaction between the NLC and polymer substrate are presented. These models, though simple, lead to non-trivial results, including loss of bistability under gliding. Perhaps surprisingly, it is observed that externally imposed switching between the steady states of a bistable system may reverse the effect of gliding, preventing loss of bistability if switching is sufficiently frequent. These findings may be of relevance to a variety of technological applications involving liquid crystal devices, and particularly to a new generation of flexible Liquid Crystal Displays (LCDs) that implement polymeric substrates. Finally, this dissertation considers how well the proposed models fit published experimental data. The results of two experimental papers are discussed, and a quantitative fit of the mathematical model to the data is made

A model for Chagas disease with oral and congenital transmission.

This work presents a new mathematical model for the domestic transmission of Chagas disease, a parasitic disease affecting humans and other mammals throughout Central and South America. The model takes into account congenital transmission in both humans and domestic mammals as well as oral transmission in domestic mammals. The model has time-dependent coefficients to account for seasonality and consists of four nonlinear differential equations, one of which has a delay, for the populations of vectors, infected vectors, infected humans, and infected mammals in the domestic setting. Computer simulations show that congenital transmission has a modest effect on infection while oral transmission in domestic mammals substantially contributes to the spread of the disease. In particular, oral transmission provides an alternative to vector biting as an infection route for the domestic mammals, who are key to the infection cycle. This may lead to high infection rates in domestic mammals even when the vectors have a low preference for biting them, and ultimately results in high infection levels in humans

Populations at year 30 as functions of with .

<p>The number of infected humans, infected dogs, vectors, and infected vectors, all at year 30, as functions of . Here while all other parameters are set to the baseline values.</p

Infected humans at year 30 as a function of , , and .

<p>The number of infected humans at year 30 as a function of the vector consumption rate and congenital transmission probabilities (where ). All other parameters are the baseline values.</p

Higher initial conditions.

<p>Simulation results of the model with baseline parameters and higher initial conditions.</p

Populations at year 30 as functions of with .

<p>The number of infected humans, infected dogs, vectors, and infected vectors, all at year 30, as functions of . Here while all other parameters are set to the baseline values.</p

Infected humans at year 30 as a function of and .

<p>The number of infected humans at year 30 as a function of and , where and all other parameters are set to the baseline values.</p

Infected humans and infected dog populations at year 30 in different scenarios.

<p>The infected human and infected dog levels after 30 years in different scenarios where dogs can be infected through vector biting only, oral consumption only, or both biting and consumption. Here and all other parameters are set to the baseline values.</p