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The boundary element method (BEM) has become a powerful method for the numerical solution of boundary-value problems (BVPs), due to its ability (at least for problems with constant coefficients) of reducing a BVP for a linear partial differential equation (PDE) defined in a domain to an integral equation defined on the boundary, leading to a simplified discretisation process with boundary elements only. On the other hand, the coefficients in the mathematical model of a physical problem typically correspond to the material parameters of the problem. In many physical problems, the governing equation is likely to involve variable coefficients. The application of the BEM to these equations is hampered by the difficulty of finding a fundamental solution. The first part of this thesis will focus on the derivation of the boundary integral equation (BIE) for the Laplace equation, and numerical results are presented for some examples using constant elements. Then, the formulations of the boundary-domain integral or integro-differential equation (BDIE or BDIDE) for heat conduction problems with variable coefficients are presented using a parametrix (Levi function), which is usually available. The second part of this thesis deals with the extension of the BDIE and BDIDE formulations to the treatment of the two-dimensional Helmholtz equation with variable coefficients. Four possible cases are investigated, first of all when both material parameters and wave number are constant, in which case the zero-order Bessel function of the second kind is used as fundamental solution. Moreover, when the material parameters are variable (with constant or variable wave number), a parametrix is adopted to reduce the Helmholtz equation to a BDIE or a BDIDE. Finally, when material parameters are constant (with variable wave number), the standard fundamental solution for the Laplace equation is used in the formulation. In the third part, the radial integration method (RIM) is introduced and discussed in detail. Modifications are introduced to the RIM, particularly the fact that the radial integral is calculated by using a pure boundary-only integral which relaxes the “star-shaped” requirement of the RIM. Then, the RIM is used to convert the domain integrals appearing in both BDIE and BDIDE for heat conduction and Helmholtz equations to equivalent boundary integrals. For domain integrals consisting of known functions the transformation is straightforward, while for domain integrals that include unknown variables the transformation is accomplished with the use of augmented radial basis functions (RBFs). The most attractive feature of the method is that the transformations are very simple and have similar forms for both 2D and 3D problems. Finally, the application of the RIM is discussed for the diffusion equation, in which the parabolic PDE is initially reformulated as a BDIE or a BDIDE and the RIM is used to convert the resulting domain integrals to equivalent boundary integrals. Three cases have been investigated, for homogenous, non-homogeneous and variable coefficient diffusion problems.

New electrochemical storage and conversion materials hold promise as important additions to the world's energy supply, and the growing ability to control both sequestration and transfer of charge and matter via functionally responsive materials promises to transform the field. Already, new understanding of the role played by nano-scale morphology of materials in transport function has contributed to considerable material improvements, with functional polymers possessing specific chemistry and morphology playing a key role in the future of electrochemical material applications. However, many challenges to optimizing properties still exist due to incomplete descriptions of transport. In this work, fluorescence spectroscopy and single molecule spectroscopy experimental techniques and analysis are developed and employed to reveal details of the mechanisms underpinning ion transport in structurally ordered polyelectrolyte polymer-brush membranes. The studies reveal the existence and nature of heterogeneous transport mechanisms in these polymer films, and provide a description of the dynamic association of molecular ions with the brush. It is also shown that it is possible to tune charged ion transport characteristics in the thin films by controlling the solvent pH, with an effective switching of ion transport rates in these brushes past a threshold pH value. Additionally, Monte Carlo models designed to model molecular scale interactions that give rise to experimental observables are developed to provide additional insight into the physical nature of transport processes in these materials. These models provide additional support for the conclusions of the experimental work.

When a water soluble granular fertilizer, such as potash, is wetted and then dried during storage and transportation processes, clumps or cakes often form in the material even when the maximum moisture content is less than 1% by mass. In order to avoid or decrease these occurred cakes, it is essential to characterize cake strength and to explore the process of cake formulation or caking through theoretical/numerical analysis. In this thesis, both experimental measurements of cake strength and theoretical/numerical simulations for recrystallization near a contact point are used to investigate the relationship between the caking process and the cake strength for important factors such as initial moisture content and drying time. <p>In this research, a centrifugal loading method has been developed to determine cake strength in a caked ring specimen of potash fertilizer where internal tensile stresses dominate. Research on fracture mechanics states that brittle materials, such as caked potash, fail at randomly positioned fracture surfaces in tension so the centrifuge test method is well suited to provide good data. A two-dimensional plane stress analysis was used to determine the area-averaged tensile stress at the speed of the centrifuge when each specimen fractures. Repeated tests and uncertainty calculations give data with a narrow range of uncertainty. <p>The centrifuge test facility was used for a series of tests in which the initial moisture content, drying time, particle size and chemical composition (i.e. magnesium content) of the samples were varied. For particle sizes in the range from 0.85 to 3.35 mm, experimental data show that the cake strength increased linearly with initial moisture content for each drying method and particle size, and decreased with increasing particle size for each initial moisture content and drying method. As well, it was also found that cake strength will increase essentially linearly with magnesium content from 0.02% to 0.1% for samples with the same initial moisture content, particle size and drying method. All data show that potash samples tend to form a stronger cake with a slower drying process. <p>A theoretical/numerical model is presented in this thesis to simulate ion diffusion and crystallization near one contact point between two potash (KCl) particles during a typical drying process. The effects of three independent factors are investigated: initial moisture content; evaporation rate; and degree of supersaturation on the surface surrounding the contact point. The numerical results show that the mass of crystal deposition near the contact point will increase with increased initial moisture content and decreased evaporation rate. These numerical predictions for recrystallization near the contact point are consistent with the experimental data for the cake strength of test samples of particle beds. With variations in the solid crystal surface degree of supersaturation near the contact point, simulations showed up to 5 times the increase in the crystal mass deposition near the contact point. This prediction of increased roughness is consistent with another experimental investigation which showed that the surface roughness of NaCl and KCl surfaces increased by a factor of five after one wetting and drying process.

When a water soluble granular fertilizer, such as potash, is wetted and then dried during storage and transportation processes, clumps or cakes often form in the material even when the maximum moisture content is less than 1% by mass. In order to avoid or decrease these occurred cakes, it is essential to characterize cake strength and to explore the process of cake formulation or caking through theoretical/numerical analysis. In this thesis, both experimental measurements of cake strength and theoretical/numerical simulations for recrystallization near a contact point are used to investigate the relationship between the caking process and the cake strength for important factors such as initial moisture content and drying time. <p>In this research, a centrifugal loading method has been developed to determine cake strength in a caked ring specimen of potash fertilizer where internal tensile stresses dominate. Research on fracture mechanics states that brittle materials, such as caked potash, fail at randomly positioned fracture surfaces in tension so the centrifuge test method is well suited to provide good data. A two-dimensional plane stress analysis was used to determine the area-averaged tensile stress at the speed of the centrifuge when each specimen fractures. Repeated tests and uncertainty calculations give data with a narrow range of uncertainty. <p>The centrifuge test facility was used for a series of tests in which the initial moisture content, drying time, particle size and chemical composition (i.e. magnesium content) of the samples were varied. For particle sizes in the range from 0.85 to 3.35 mm, experimental data show that the cake strength increased linearly with initial moisture content for each drying method and particle size, and decreased with increasing particle size for each initial moisture content and drying method. As well, it was also found that cake strength will increase essentially linearly with magnesium content from 0.02% to 0.1% for samples with the same initial moisture content, particle size and drying method. All data show that potash samples tend to form a stronger cake with a slower drying process. <p>A theoretical/numerical model is presented in this thesis to simulate ion diffusion and crystallization near one contact point between two potash (KCl) particles during a typical drying process. The effects of three independent factors are investigated: initial moisture content; evaporation rate; and degree of supersaturation on the surface surrounding the contact point. The numerical results show that the mass of crystal deposition near the contact point will increase with increased initial moisture content and decreased evaporation rate. These numerical predictions for recrystallization near the contact point are consistent with the experimental data for the cake strength of test samples of particle beds. With variations in the solid crystal surface degree of supersaturation near the contact point, simulations showed up to 5 times the increase in the crystal mass deposition near the contact point. This prediction of increased roughness is consistent with another experimental investigation which showed that the surface roughness of NaCl and KCl surfaces increased by a factor of five after one wetting and drying process.

Le carbure de silicium est un matériau envisagé pour le conditionnement du combustible dans les réacteurs de quatrième génération. Ce travail a pour objectif d'étudier la capacité de confinement des produits de fission et l'évolution de la structure de ce matériau sous les effets combinés de la température et du rayonnement. Les implantations d'ions de basse énergie et l'incorporation d'analogues stables de produits de fission (Cs et I) dans des monocristaux de 6H-SiC ont été réalisées sur l'implanteur ou l'accélérateur du CSNSM. Les irradiations avec des ions lourds de haute énergie ont été effectuées au GANIL. L'évolution du profil des ions implantés et de la structure du cristal a été étudiée par RBS et canalisation. Des informations complémentaires ont été apportées par la spectroscopie d'absorption UV-visible. Les implantations d'ions de basse énergie à température ambiante conduisent à l'endommagement rapide du cristal. Par contre, une élévation de la température d'implantation (600 °C) permet de conserver un faible taux de désordre dans le cristal. Les irradiations avec des ions lourds de haute énergie n'endommagent pas les cristaux de SiC mais au contraire, elles provoquent une guérison du désordre créé préalablement par l'implantation d'ions I de basse énergie. Ces marqueurs d'iode ne diffusent pas lors d'irradiations avec des ions lourds de basse ou de haute énergie à température ambiante ou à 600 °C. Cependant, une diffusion des ions Cs a été observée lors d'un recuit thermique post-implantation à 1300 °C, température à laquelle le cristal qui comportait une couche amorphe étendue commence à retrouver une structure monocristalline.

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Diffusion is a fundamental process which occurs in a wide variety of phases. It plays an important role in chemistry, physics, biology, materials science etc. In recent times, diffusion in confined systems has been widely investigated. Porous aluminosilicates such as zeolites, carbon nano tubes and metal organic frameworks(MOF) provide confined regions within which small molecules can diffuse. Indeed, diffusion within these materials have attracted considerable attention in the past few decades (see for example, “Diffusion in Zeolites and Other Microporous Solids”, J. Ka¨rger and D..M. Ruthven, John Wiley &Sons, NewYork,1992). Diffusion in confined spaces exhibits rich variety. For example, single file diffusion, window effect, levitation effect (LE), super-and sub-diffusive motion have all been observed in confined regions. Levitation effect provides an explanation for the dependence of self-diffusivity on the diameter of the diffusant. Consider a diffusant diffusing within a porous material. The pore network provided by the pore material may be characterized by the void and the neck distribution where the necks are the narrower regions interconnecting larger voids. It has been seen that diffusivity is maximum when the size of the diffusant is large and when it is comparable to the diameter of the bottleneck of the pore network. Recently it has been demonstrated that the levitation effect also exists in dense liquids such as water and dense solids. These developments essentially unify our understanding of diffusion in widely differing condensed matter phases. These results show that there is fundamentally no difference between porous substances and dense media at least with regard to dependence of self-diffusivity on the diameter of the diffusant. Chapter 1 provides a brief introduction to the subject of hydrocarbons confined within zeolites and ionic conductivity in polar solvents. We have given a description of the different applications of zeolites in the area of catalysis, separation etc. Window effect, single file diffusion, levitation effect and enhancement of viscosity of confined fluids are described. A brief review of various computational studies of hydrocarbons confined within zeolites is given. This is followed by a discussion of different experimental techniques and their use in the study of diffusion and adsorption within zeolites by many different groups in the last few decades. In the last section of the chapter we have discussed the anomalous size dependence of ionic conductivity in polar solvents which presumably has its origin in the Levitation Effect(LE). We have explained different theories proposed previously to understand the non-monotonic behavior of ionic conductivity as a function of ionic radius. A molecular dynamics(MD) investigation and quasi-elastic neutron scattering (QENS) study of pentane isomers in zeolite NaY is pre-sented in Chapter 2. QENS provides the first direct experimental evidence for LE. In an earlier study, a maximum in diffusivity as a function of the diameter of the diffusant for monatomic sorbates confined within zeolite NaY was observed by MD simulation. Since LE has been invoked to explain the diffusion in a wide variety of condensed matter phases, an experimental evidence of the levitation effect would be of great value. QENS measurements were carried out by Dr. Herve Jobic. Surprisingly we found that neopentane shows higher diffusivity than n-pentane and isopentane although its cross-sectional diameter perpendicular to the long molecular axis is larger compared to isopentane and n-pentane in agreement with predictions of LE. There is an excellent agreement between QENS results and MD simulation. LE predicts that the isomer with high diffusivity has low activation energy. The activation energies have been calculated from the Arrhenius plots using QENS as well as MD data. These follow the order Ea(n−pentane)>Ea(isopentane)>Ea(neopentane). Various other properties such as potential energy barrier at the bottleneck, velocity auto correlation function, intermediate scattering function, k dependence of the width of the dynamic structure factor have been computed. These provide additional insights into the nature of the motion of these isomers. They suggest that the barrier at the 12-ring window depends on the molecular diameter and levitation parameter of isomer. In Chapter 3, we report molecular dynamics simulation study of n-hexane and 2,2-dimethylbutane(DMB) mixture confined within the pores of zeolite NaY. We have taken an equimolar composition of the mixture consisting of n-hexane and DMB. The total number of hydrocarbon molecules in the system is 128. The simulations were carried out at various temperatures of 170, 200, 250 and 300 K. We have computed the self-diffusivities from the slope of the mean square displacement. It is found that the diffusivity of DMB is 0.82 ×10−9 m2/sec and that of n-hexaneis0.38 ×10−9 m2/sec. All previous studies of linear hydrocarbon and its branched analogue in different zeolites in the literature suggest that it is the linear member which has higher self-diffusivity. The cross-sectional diameter of DMB perpendicular to the long molecular axis is higher than that of n-hexane. Thus, DMB should have lower diffusivity. In order to understand this behavior of diffusivity we have computed the activation energies from the Arrhenius plots. The activation energy of DMB is found to be lower than that of n-hexane. This is inconformity with the levitation effect which states that the molecule with larger diameter comparable to that of the bottleneck diameter has low activation energy. We have also computed the potential energyprofileatthe12-ring window. The potential energy profile shows a barrier for n-hexane and a minimum for DMB at the window. This is in agreement with the previous results on monatomic species. We have computed other properties such as velocity auto correlation function, intermediate scattering function as well as wave number dependence of full width at half maximum of dynamic structure factor. These properties explain in detail the motion of n-hexane and DMB within NaY zeolite. In Chapter 4 molecular dynamics investigation into diffusion of n-decane and 3-methylpentane mixture within zeolite NaY. We have studied an equimolar mixture of n-decane and 3-methylpentane (36 of each) in the supercages of NaY zeolite in such a way that the con-centration is one molecule for every three cages. Simulations were performed at four different temperatures : 300, 350, 400 and 450 K. The distribution and orientation of the molecules inside the cage and at the window plane have been studied. Inside the cage, 3-methylpentane stays more close to the inner surface of the zeolite whereas n-decane prefers to stay close to the center of the cage. Both the species prefer to stay with their long molecular axis parallel to the surface of the zeolite. During passage through the window, 3-methylpentane is closer to the window center than n-decane. The distribution of the angle subtended by the end-to-end vector of the molecule with the normal to the window plane, while the molecular center is in the window plane, shows that 3-methylpentane samples a larger range of orientation than n-decane. This may lead to an entropic barrierfor n-decane. We have computed the diffusivity of both the molecules. Diffusivity of 3-methylpentane is found to be higher than n-decane. This behavior is consistent with the observations made in the last two chapters. The activation energy of 3-methylpentane is found to be 3.17 kJ/mol and forn-decaneitis6.0kJ/mol. This agrees with the prediction of levitation effect. The energy profile a the window shows shallow minimum for both n-decane and 3-methylpentane. Therefore, the energy profile does not describe the nature of motion of the molecules. We have computed the the dihedral angle distribution when the molecule is at the adsorption site and when it is at the window plane. The distributions essentially remain same for 3-methylpentane whereas a considerable change in the distributions is seen for n-decane. The gauche population of n-decane increases at the cost of trans population when it goes from the adsorption site to the window. The lower diffusivity of n-decane can be partly attributed to the change in the dihedral angle. Also, the orientational entropic barrier may be another cause of the slow motion of n-decane. Thus, in the present study the slow motion of n-decane is partly explained by levitation effect and partly by the change in the dihedral angle as well as the entropic barrier. Overall, the results in the last three chapters leads to the main conclusion that the branched isomer will diffuse faster than a linear hydrocarbon in zeolites with 12-ring window such as zeolite NaY. In Chapter 5, diffusion of pentane isomers in zeolites NaX and NaY has been investigated using pulsed field gradient nuclear magnetic resonance(PFG-NMR) and molecular dynamics(MD) techniques respectively. Temperature as well as concentration dependence of diffusivity have been studied. The diffusivities obtained from NMR are roughly an order of magnitude smaller than those obtained from MD. The dependence of diffusivity on loading at high temperatures exhibits a type I behavior according to the classification of K¨arge rand Pfeifer. NMR diffusivities of the isomers exhibit the order D(n−pentane)>D(isopentane)>D(neopentane). The results from MD are in agreement with the QENS results where the diffusivities of the isomers follow the order D(n-pentane)<D(isopentane)<D(neopentane). The activation energies from NMR show Ea(n-pentane)<Ea(isopentane) <Ea(neopentane) whereas those from MD suggest the order Ea(n-pentane) >Ea(isopentane) >Ea(neopentane). The latter follows the predictions of levitation effect whereas those of NMR appears to be due to the presence of defects in the zeolite crystals. The differences between NMR and MD are attributed to the long time and length scales over which NMR samples are probed compared to MD or QENS. Th eresults from these studies suggests that although branched isomer intrinsically have higher diffusivities than linear hydrocarbons in zeolites such as NaY, the presence of defects can effectively annul this higher diffusivity of the branched isomer. Correlation of self-diffusivity and entropy of monatomic sorbates con-fined within zeolite NaY has been investigated in Chapter 6. We have carried out molecular dynamics simulation on monatomic sor-bates within zeolite NaY at 150, 110 and 90 K. As suggested by the Levitation Effect, the self-diffusivity shows a non-monotonic behavior as a function of the diameter of the sorbates. We have computed the entropy of the sorbates of various sizes ranging from 3.07˚ A to 7.0˚ A using the method proposed by Goddard and his co-workers as well as from the radial distribution function. The variation of entropy with the diffusant diameter exhibits a behavior similar to that of the self-diffusivity on diffusant diameter, thereby showing a strong correlation between the entropy and diffusivity. The loss of entropy on adsorption is a minimum for the diffusant with maximum diffu-sivity. This is in agreement with the experimental measurements of Kemball. Thus, entropy follows the prediction of the levitation effect. With decrease in temperature both self-diffusivity as well as entropy show more pronounced maximum as a function of the diameter of the sorbate. The dimensionless diffusivity from three different isotherms follow a Rosenfeld type of excess entropy scaling rule, D∗= Aexp(αSe) where A and α are the scaling coefficients. In Chapter 7 we have investigated the self-diffusivity as well as cor-rected diffusivity of pure methane in faujasite NaY combining quasi elastic neutron scattering experiment and molecular dynamics simu-lation. The QENS experiment carried out at 200 K led to an unex-pected dependence of self-diffusivity on loading for pure methane with the presence of a maximum at 32 CH4/unit cell. This is at variance with previous reports. Typically, diffusivity of a polar species such as methane in a zeolite such as NaY exhibits a monotonic decrease with loading. Molecular dynamics simulation was performed to reproduce this experimentally observed behavior. We could reproduce the diffusivity behavior qualitatively with a maximum at 16 CH4/unit cell. The corrected diffusivities obtained from both experiment as well simulation show similar behavior as the self-diffusivity with maximum at an intermediate loading. The experimental behavior was reproduced only when the interaction of methane with the sodium cation is in-creased suggesting that this interaction may be important. In Chapter8 we have investigated the role of attractive interaction on size dependent diffusivity maximum of ions in water. We have per-formed molecular dynamics simulation of mode lions in water. Earlier study of systems interacting only through van der Waals interaction shows that the size dependent diffusivity maximum or the levitation effect disappears when the attractive term(r−6 term) of the Lennard-Jones potential is put equal to zero. It is not clear whether the absence of the dispersion interaction in a system where there is electrostatic attraction will lead to a size dependent diffusivity maximum. There-fore, two sets of simulations with and without dispersion interaction between the ion and water have been carried out at700Kinorderto understand the influence of the attractive interaction. It is found that the self-diffusivity of the ions indeed exhibits an anomalous maximum as a function of the vanderWaals diameter for both the sets, viz., with dispersion and without dispersion interaction. In fact, the diffusivity maximum is seen to be more pronounced when there is no dispersion interaction. This existence of the maximum in self diffusivity when there is no dispersion interaction between the ion and the water is attributed to the attractive term from electrostatic interactions. De-tailed analysis shows that the solvent shell is more well defined in the presence of dispersion interactions. The velocity auto correlation function shows undulation at short times for the smaller ions indicating rattling motion inside the cage formed by the surrounding water molecules. Smaller ion exhibits a bi-exponential decay while a single exponential decay is seen for the ion with maximum diffusivity in the intermediate scattering function. The solvent structure appears to determine much of the dynamics of the ion. Interesting trends are seen in the activation energies and these can be understood in terms of the Levitation Effect.

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This diploma thesis deals with the issue of lithium-ion accumulators. The thesis focuses on the negative electrode of lithium-ion accumulators made of natural graphite. The first part of this thesis points to the issue of electrochemical cells. In the theoretical part the thesis deals with the SEI layer, advantages, disadvantages, characteristics, operating principle and the use of lithium-ion accumulators. The practical part focuses on the electrochemical properties of negative electrode, especially the determination of the diffusion coefficient. Thesis also deals with electrochemical impedance spectroscopy (EIS) and its use in determining the equivalent replacement circuit and calculating the diffusion coefficient.