Atom transfer radical polymerization is a new and important living polymerization mechanism because it can produce many different polymers with controlled microstructures and novel properties. The commercialization of these new polymers will require detailed polymer reaction engineering investigations. Mathematical models are essential in this stage because they can summarize our knowledge on polymers made by ATRP and help us to find the optimum conditions for their synthesis. This thesis studies the polymerization kinetics of ATRP with mathematical models based on our own experimental work and experimental data published by other researchers. ATRP with both monofunctional and bifunctional initiators are considered. This is one of very few studies combining detailed mathematical models for polymerization kinetics and polymer microstructure and experimental results in the area of ATRP. Fundamental mathematical models were used to study the main features of ATRP. Population balances and the method of moments were used to predict polymer average properties, while Monte Carlo models were used to predict the complete microstructural distributions. This type of comparison between different modeling techniques is seldom done in the literature, even for other polymerization techniques, and can lead to a better understanding of polymerization mechanisms and mathematical modeling techniques. Since the discovery of ATRP, approximately ten years ago, little attention has been given to bifunctional initiators. This thesis tries to extend our knowledge on this important class of initiators. Comparison between monofunctional and bifunctional initiators, both through mathematical modeling and experimentally, showed that bifunctional initiators have some advantages over monofunctional initiators for ATRP. Polymers made with bifunctional initiators have narrow molecular weight distributions, higher molecular weight averages, and higher monomer conversion for the same polymerization time. In addition to homopolymerization studies, this thesis presents mathematical models for copolymerization with ATRP and for processes combining ATRP and coordination polymerization. These models describe the detailed microstructures of these copolymers and permit a better understanding of ATRP with its advantages and pitfalls. An interesting conclusion from these modeling studies in atom transfer radical copolymerization is that the Mayo-Lewis terminal model is applicable to ATRP and that the copolymer composition in ATRP is independent of the equilibrium constants (activation and deactivation). In order to develop and validate these mathematical models, we collected experimental data in our own laboratories and also used experimental data available in the literature. Our experimental work focused on the homopolymerization and copolymerization of styrene, because of the commercial importance of this monomer and also due to the relative simplicity of its polymerization. Experimental data collected from the literature covered the following systems: bulk homopolymerization of styrene, solution polymerization of styrene, solution polymerization of methyl methacrylate, bulk polymerization of n-butyl acrylate, bulk copolymerization of styrene and n-butyl acrylate. Different characterization techniques were used to determine polymer properties. Molecular weight and molecular weight distribution were measured using gel permeation chromatography (GPC); copolymer chemical composition was determined with nuclear magnetic resonance (NMR) and Fourier-transform infrared (FTIR). We have also done copolymerization with styrene and acrylonitrile (SAN) because it is one of the least understood ATRP system and also because its potential industrial importance. The ability to synthesize polymers with novel molecular architectures is one of the advantages of living polymerization techniques. In this thesis, we used ATRP to produce amphiphilic copolymers composed of polystyrene and polyethylene glycol methacrylate macromonomers. We have shown that ATRP can produce these very interesting polymers with two different types of macroinitiators.

Understanding flow processes of sea and fresh water through complex coastal regions is of utmost importance for a number of applications of interest to the scientific and engineering community, including wetland health and restoration, inland flooding due to tropical storms and hurricanes, and navigation through coastal waters. In such regions, the existence of vegetation increases flow resistance, which is a major factor in determining velocity and water level distribution in wetlands and inland. Commonly, the momentum loss due to vegetation is included in a bottom friction term in the model equations; however, such models may oversimplify the complex resistance characteristics of such a system. With recent increases in computational capabilities, it is now feasible to develop and implement more intricate resistance models that more accurately capture these characteristics. We present two methods for modeling flow through vegetated regions. With the first method, we employ mathematical and computational upscaling techniques from the study of subsurface flow to parametrize drag in a complex heterogeneous region. These parameterizations vary greatly depending on Reynolds number. For the coastal flows in which we are interested the Reynolds number at different locations in the domain may vary from order 1 to order 1000, so we must consider laminar and fully turbulent flows. Large eddy simulation (LES) is used to model the effects of turbulence. The geometry of a periodic cell of vegetative obstacles is completely resolved in the fluid mesh with a standard no-slip boundary condition imposed on the fluid-vegetation boundaries. The corresponding drag coefficient is calculated and upscaling laws from the study of inertial flow through porous media are used to parametrize the drag coefficient over a large range of Reynolds numbers. Simulations are performed using a locally conservative, stabilized continuous Galerkin finite element method on highly-resolved, unstructured 2D and 3D meshes. The second method we present is an immersed structure approach. In this method, separate meshes are used for the fluid domain and vegetative obstacles. Taking techniques from immersed boundary finite element methods, the effects of the fluid on the vegetative structures and vice versa are calculated using integral transforms. This method allows us to model flow over much larger scales and containing much more complicated obstacle geometry. Using a simple elastic structure model we can incorporate bending and moving obstacles which would be extremely computationally expensive for the first method. We model flexible vegetation as thin, elastic, inextensible cantilever beams. We present two numerical methods for modeling the beam motion and analyze their computational expense, stability, and accuracy. Using the immersed structure approach, a fully coupled steady-state fluid-vegetation interaction model is developed as well as a dynamic interaction model assuming dynamic fluid flow and quasi-static beam bending. This method is verified using channel flow and wave tank test problems. We calculate the bulk drag coefficient in these flow scenarios and analyze their trends with changing model parameters including stem population density and flow Reynolds number. These results are compared to well-respected experimental results. We model real-life beds of Spartina alterniflora grass with representative beds of flexible beams and perform similar comparisons. / text

The objective of this dissertation was to advance the knowledge about mathematics instruction regarding the use of exploratory graphical embodiments in Pre-K to College levels. More specifically, the study sought to find out which graphical representations generate the highest learning effect sizes as well as which teaching method is the most supportive when graphical representations are applied. The dissertation is organized into three coherent research studies that correspond to different schooling levels. The primary method of data analysis in this study was meta-analysis supported by synthesis of qualitative and comparative studies. A total of 73 primary studies (N = 9055) from 22 countries conducted over the past 13 years met the inclusion criteria. Out of this pool, 45 studies (N = 7293) were meta-analyzed. The remaining 28 studies (N = 1762) of qualitative or mixed method designs where scrutinized for common themes. The results support the proposed hypothesis that visualization aids mathematics learning. At the primary level, the mean effect size for using exploratory environment was ES = 0.53 (SE = 0.05, 95% CI: 0.42-0.63), the mean effect size for using computerized programs at the grade levels 1-8 was ES = 0.60 (SE = 0.03, 95% CI: 0.53-0.66), and the results of applying congruent research techniques at the high school and college levels revealed an effect size of ES = 0.69 (SE = 0.05, 95% CI: 0.59–0.79). At each of the teaching level, implementing an exploratory environment generated a moderate effect size when compared to traditional teaching methods. These findings support a need for a broader implementation of exploratory learning media to mathematics school practice and provide evidence to formulate a theoretical instructional framework.

A model is developed to describe the bulk thermally-initiated free-radical polymerization of styrene between 100 °C and 170 °C. This model incorporates a comprehensive thermal initiation mechanism including generation and consumption of a Diels-Alder adduct intermediate species. A semi-empirical break-point treatment of diffusion control on reaction kinetics is used to account for autoacceleration behaviour. Using recently-developed statistical techniques, parameters are ranked based on their influence on model predictions, uncertainty in their initial values and correlation between their effects. The four top-ranked parameters (of the 40 total model parameters) are chosen for estimation to improve the fit between model predictions and literature data. After estimation of these four parameters, and hand-tuning of two additional autoacceleration parameters, the model predicts conversion data with a standard error of 5 %. The model also provides an excellent fit to a single MWD curve obtained from a literature experiment performed at 100 °C. Simulations are used to show that chain-end degradation reactions are not important in the temperature range of interest. The model is then extended to include industrially-relevant dicumyl peroxide and biphenyl peroxide chemical initiation. Additional peroxide-induced mid-chain scission reactions are considered as they may have an important influence on the molecular weight of polystyrene. To improve trends in predictions of Mn and Mw, the stationary-state hypothesis is applied to the initial adduct concentration. Parameters are then ranked, and selected for estimation using recently-developed statistical techniques. While significant improvements in predictions of conversion data are obtained, it is necessary to manually tune several parameters and scrutinize the reaction scheme in detail. To improve trends in predictions of Mn and Mw, mid-chain scission reactions are turned off and chain-transfer to monomer is implemented. Nine of the 48 total parameters are selected for estimation, resulting in a 73 % decrease in the objective function value compared with predictions using literature values. The final step of this work will be to estimate parameters using a large, proprietary industrial data set. Using this data set, it may be possible to estimate additional parameters which may lead to improved model predictions. / Thesis (Master, Chemical Engineering) -- Queen's University, 2012-02-22 12:43:35.531

Adaptive evolution is the process by which natural selection, acting on variation within a population, promotes the survival of individuals that are more successful at reproducing and contributing to future generations. Evolutionary processes in microbes occur at the intersection of population genetics, natural selection, and underlying mechanistic constraints, to give rise to the repertoire of adaptation observed in nature. Understanding microbial adaptive evolution is of critical importance for human health for example, through the emergence of pathogenicity and antibiotic resistance. Moreover, the stability and function of natural and artificial ecosystems is contingent on the evolving interactions between microbes, and between microbes and the environment. We present a modelling framework, based on the theory of adaptive dynamics, to investigate how cellular resource allocation trade-offs affect the adaptation process. We used resource-consumer theory, which explicitly models the interactions between cells and their environment, together with matrix models of structured populations, to implement phenotype-determined cellular strategies of resource allocation between mutually exclusive processes. We then analyse the outcome of competitions between different phenotypes across environmental and competitive conditions. We applied our methods to the evolution of strategies (phenotypes) for resource allocation between two competing cellular process in microbial populations growing in chemostat-like environments. We calculated the adaptively stable strategies for several models and showed how state-structured population models can be mapped to simpler chemostat models on invariant manifolds. We then extended our analysis to the case where a limiting nutrient may be utilized using two alternative metabolic pathways. We described how the total population fitness of a metabolic strategy can be constructed from the individual decisions of its constituent members. We developed numerical methods to simulate and analyse general models of adaptive dynamics using principles from graph theory and discrete Markov processes. The methods were used to explore the evolution of nutrient use strategies for microbial populations growing on two and three substitutable nutrients. We highlight the importance of the ancestral phenotype in channelling the adaptation process, which, together with the choice of the mutational kernel, influences the adaptively stable strategies and modes of co-existence. In a related finding, we show how some phenotypes are adaptively stable only in the presence of a competitor lineage that modifies the environment in a manner that permits another phenotype to invade. Our methods also reveal instances where historical contingency and chance have an important effect on determining the stable nutrient use strategies. Finally, we demonstrate the existence of adaptively stable periodic solutions whereby, under some conditions, phenotype successions are cyclical. Our work builds on the foundation of adaptive dynamics theory to provide a general framework for analysing models of microbial adaptation. We focused on understanding the implications of underlying constraints and cellular resource allocation trade-offs in the context of adaptive evolution.

Microbial biofilms are defined as clusters of microbial cells living in self-produced extracellular polymeric substances (EPS), which always attached to various kinds of surfaces. In this thesis, we studied several mathematical models of biofilm in the human body and marble environment. Some related background of biofilm growth and some basic existing numerical models were introduced in the first chapter. In the first project, we introduced how biofilm affects the local oxygen concentration near the neutrophil cells in the human body with three one-dimensional reaction-diffusion models from different geometries. In nature, microbial biofilm development can be observed on almost all kinds of stone monuments and can also be associated with the problem of monument conservation. In the second part of my research, we built the deliquescence models for biofilm growth environment in the first model and added biomass into consideration in the second one. Also, we analyzed the stability of the equilibria. In the third part, we applied the weather data collected from the weather station on the roof of the Jefferson Memorial to the deliquescence model with biofilm. Furthermore, compared the simulation result for biofilm growth in cold and warm weathers. In the last part of this thesis, we analyzed the biofilm activity with support vector regression. The machine learning model we obtained can be used to find the growth trends of biofilm for any pair of temperature and relative humidity data. / Mathematics

Recruitment and derecruitment (R/D) of airways is known to significantly influence mechanical properties of the respiratory system during artificial ventilation, particularly in states of lung injury. The prevailing view of this phenomenon treats airway R/D as a static function of pressure. Recent experimental and clinical data suggests that this is not the case, but rather that R/D is an inherently dynamic process. In order to quantitatively assess the dynamics of lung recruitment during mechanical ventilation we extended a mathematical model by Bates and Irvin (9) for the purpose of fitting experimental data. The model of the lung consists of a parallel network of flow pathways with identical resistive and elastic elements. Each pathway is allowed to be either open, whereby it accumulates flow and decreases overall lung stiffness, or closed, increasing lung elastance and not participating in ventilation. The pathways are characterized by unique critical closing and opening pressures, and opening and closing velocities, each chosen from probability distribution functions. The rate of transition between an open and closed state depends on the magnitude difference between the pressure in the respiratory system and each unit’s critical pressure times the airway’s opening or closing velocity constant. Since the exact form of the pressure dependence governing recruitment and derecruitment remains unknown we explored four model variants to predict how opening or closing behavior is altered in injury. The lung model was coupled with a computational model of a mechanical ventilator in order to simulate elastance changes following deep inflation (DI) at three levels of Positive End Expiratory Pressure (PEEP). Elastance measurements came from healthy or lung injured mice at 4, 14, 24 or 48 hours following intratracheal instillation of saline (control) or hydrochloric acid (injury). The Nelder and Mead simplex optimization method was used to minimize error between model variants and average experimental elastance for each condition. By comparing the residual error of the fits for each model, we have demonstrated that only one variant was able to recreate both the transient response to deep inflations and the response to static PEEP. In fitting the best model to data from individual mice we obtained estimates for parameters governing opening and closing behavior. Statistics and model sensitivity were determined for each parameter in every experimental condition. Comparison of parameter values between groups revealed a significant increase in closing and opening pressures from health to injury, which worsened with increasing injury severity. The progressive increase in critical pressures as injury worsens implicates surfactant deactivation as the likely cause of increased propensity for airway closing during acute lung injury.

A critical step in mounting an immune response is antigen recognition by T cells. This step proceeds by productive interactions between T cell receptors (TCR) on the surface of T cells and foreign antigen, in the form of peptide-major-histocompatibility-complexes (pMHC), on the surface of antigen-presenting-cells (APC). Antigen recognition is exceedingly difficult to understand because the vast majority of pMHC on APCs are derived from self-proteins. Nevertheless, T cells have been shown to be exquisitely sensitive, responding to as few as 10 antigenic pMHC in an ocean of tens of thousands of self pMHC. In addition, T cells are extremely specific and respond only to a small subset of pMHC by virtue of their specific TCR. To explain the sensitivity of T cells to pMHC it has been proposed that a single pMHC may serially bind multiple TCRs. Integrating present knowledge on the spatial-temporal dynamics of TCR/pMHC in the T cell-APC contact interface, we have constructed mathematical models to investigate the degree of TCR serial engagements by pMHC. In addition to reactions within clusters, the models capture the formation and mobility of TCR clusters. We find that a single pMHC serially binds a substantial number of TCRs in a TCR cluster only if the TCR/pMHC bond is stabilized by coreceptors and/or pMHC dimerization. In a separate study we propose that serial engagements can explain T cell specificity. Using Monte Carlo simulations, we show that the stochastic nature of TCR/pMHC interactions means that multiple binding events are needed for accurate detection of foreign pMHC. Critical to our studies are estimates of TCR/pMHC reaction rates and mobilities. In the second half of the thesis, we show that Fluorescence Recovery After Photobleaching (FRAP) experiments can reveal effective diffusion coefficients. We then show, using asymptotic analysis and model fitting, that FRAP experiments can be used to estimate reaction rates between cell surface proteins, like TCR/pMHC. Lastly, we use FRAP experiments to investigate how the actin cytoskeleton modulates TCR mobility and report effective reaction rates between TCR and the cytoskeleton.

Atom transfer radical polymerization is a new and important living polymerization mechanism because it can produce many different polymers with controlled microstructures and novel properties. The commercialization of these new polymers will require detailed polymer reaction engineering investigations. Mathematical models are essential in this stage because they can summarize our knowledge on polymers made by ATRP and help us to find the optimum conditions for their synthesis. This thesis studies the polymerization kinetics of ATRP with mathematical models based on our own experimental work and experimental data published by other researchers. ATRP with both monofunctional and bifunctional initiators are considered. This is one of very few studies combining detailed mathematical models for polymerization kinetics and polymer microstructure and experimental results in the area of ATRP. Fundamental mathematical models were used to study the main features of ATRP. Population balances and the method of moments were used to predict polymer average properties, while Monte Carlo models were used to predict the complete microstructural distributions. This type of comparison between different modeling techniques is seldom done in the literature, even for other polymerization techniques, and can lead to a better understanding of polymerization mechanisms and mathematical modeling techniques. Since the discovery of ATRP, approximately ten years ago, little attention has been given to bifunctional initiators. This thesis tries to extend our knowledge on this important class of initiators. Comparison between monofunctional and bifunctional initiators, both through mathematical modeling and experimentally, showed that bifunctional initiators have some advantages over monofunctional initiators for ATRP. Polymers made with bifunctional initiators have narrow molecular weight distributions, higher molecular weight averages, and higher monomer conversion for the same polymerization time. In addition to homopolymerization studies, this thesis presents mathematical models for copolymerization with ATRP and for processes combining ATRP and coordination polymerization. These models describe the detailed microstructures of these copolymers and permit a better understanding of ATRP with its advantages and pitfalls. An interesting conclusion from these modeling studies in atom transfer radical copolymerization is that the Mayo-Lewis terminal model is applicable to ATRP and that the copolymer composition in ATRP is independent of the equilibrium constants (activation and deactivation). In order to develop and validate these mathematical models, we collected experimental data in our own laboratories and also used experimental data available in the literature. Our experimental work focused on the homopolymerization and copolymerization of styrene, because of the commercial importance of this monomer and also due to the relative simplicity of its polymerization. Experimental data collected from the literature covered the following systems: bulk homopolymerization of styrene, solution polymerization of styrene, solution polymerization of methyl methacrylate, bulk polymerization of n-butyl acrylate, bulk copolymerization of styrene and n-butyl acrylate. Different characterization techniques were used to determine polymer properties. Molecular weight and molecular weight distribution were measured using gel permeation chromatography (GPC); copolymer chemical composition was determined with nuclear magnetic resonance (NMR) and Fourier-transform infrared (FTIR). We have also done copolymerization with styrene and acrylonitrile (SAN) because it is one of the least understood ATRP system and also because its potential industrial importance. The ability to synthesize polymers with novel molecular architectures is one of the advantages of living polymerization techniques. In this thesis, we used ATRP to produce amphiphilic copolymers composed of polystyrene and polyethylene glycol methacrylate macromonomers. We have shown that ATRP can produce these very interesting polymers with two different types of macroinitiators.

A critical step in mounting an immune response is antigen recognition by T cells. This step proceeds by productive interactions between T cell receptors (TCR) on the surface of T cells and foreign antigen, in the form of peptide-major-histocompatibility-complexes (pMHC), on the surface of antigen-presenting-cells (APC). Antigen recognition is exceedingly difficult to understand because the vast majority of pMHC on APCs are derived from self-proteins. Nevertheless, T cells have been shown to be exquisitely sensitive, responding to as few as 10 antigenic pMHC in an ocean of tens of thousands of self pMHC. In addition, T cells are extremely specific and respond only to a small subset of pMHC by virtue of their specific TCR. To explain the sensitivity of T cells to pMHC it has been proposed that a single pMHC may serially bind multiple TCRs. Integrating present knowledge on the spatial-temporal dynamics of TCR/pMHC in the T cell-APC contact interface, we have constructed mathematical models to investigate the degree of TCR serial engagements by pMHC. In addition to reactions within clusters, the models capture the formation and mobility of TCR clusters. We find that a single pMHC serially binds a substantial number of TCRs in a TCR cluster only if the TCR/pMHC bond is stabilized by coreceptors and/or pMHC dimerization. In a separate study we propose that serial engagements can explain T cell specificity. Using Monte Carlo simulations, we show that the stochastic nature of TCR/pMHC interactions means that multiple binding events are needed for accurate detection of foreign pMHC. Critical to our studies are estimates of TCR/pMHC reaction rates and mobilities. In the second half of the thesis, we show that Fluorescence Recovery After Photobleaching (FRAP) experiments can reveal effective diffusion coefficients. We then show, using asymptotic analysis and model fitting, that FRAP experiments can be used to estimate reaction rates between cell surface proteins, like TCR/pMHC. Lastly, we use FRAP experiments to investigate how the actin cytoskeleton modulates TCR mobility and report effective reaction rates between TCR and the cytoskeleton.