This thesis contains four articles related to mathematical aspects of Density Functional Theory. In Paper A, the theoretical justification of density methods formulated with current densities is addressed. It is shown that the set of ground-states is determined by the ensemble-representable particle and paramagnetic current density. Furthermore, it is demonstrated that the Schrödinger equation with a magnetic field is not uniquely determined by its ground-state solution. Thus, a wavefunction may be the ground-state of two different Hamiltonians, where the Hamiltonians differ by more than a gauge transformation. This implies that the particle and paramagnetic current density do not determine the potentials of the system and, consequently, no Hohenberg-Kohn theorem exists for Current Density Functional Theory formulated with the paramagnetic current density. On the other hand, by instead using the particle density as data, we show that the scalar potential in the system's Hamiltonian is determined for a fixed magnetic field. This means that the Hohenberg-Kohn theorem continues to hold in the presence of a magnetic field, if the magnetic field has been fixed. Paper B deals with N-representable density functionals that also depend on the paramagnetic current density. Here the Levy-Lieb density functional is generalized to include the paramagnetic current density. It is shown that a wavefunction exists that minimizes the "free" Hamiltonian subject to the constraints that the particle and paramagnetic current density are held fixed. Furthermore, a convex and universal current density functional is introduced and shown to equal the convex envelope of the generalized Levy-Lieb density functional. Since this functional is convex, the problem of finding the particle and paramagnetic current density that minimize the energy is related to a set of Euler-Lagrange equations. In Paper C, an N-representable Kohn-Sham approach is developed that also include the paramagnetic current density. It is demonstrated that a wavefunction exists that minimizes the kinetic energy subject to the constraint that only determinant wavefunctions are considered, as well as that the particle and paramagnetic current density are held fixed. Using this result, it is then shown that the ground-state energy can be obtained by minimizing an energy functional over all determinant wavefunctions that have finite kinetic energy. Moreover, the minimum is achieved and this determinant wavefunction gives the ground-state particle and paramagnetic current density. Lastly, Paper D addresses the issue of a Hohenberg-Kohn variational principle for Current Density Functional Theory formulated with the total current density. Under the assumption that a Hohenberg-Kohn theorem exists formulated with the total current density, it is shown that the map from particle and total current density to the vector potential enters explicitly in the energy functional to be minimized. Thus, no variational principle as that of Hohenberg and Kohn exists for density methods formulated with the total current density. / <p>QC 20140523</p>

Doctor of Philosophy / Department of Electrical and Computer Engineering / Kenneth H. Carpenter / A computer simulation of transient current density distributions in thin conductors was developed using a time-stepped implementation of the integral equation method on a finite element mesh. A study of current distributions in thin conductors was carried out using AC analysis methods. The study of the AC current density distributions was used to develop a circuit theory model for the thin conductor which was then used to determine the nature of its transient response. This model was used to support the design and evaluation of the transient current density solver. A circuit model for strip lines was made using the Partial Inductance Method to allow for simulations with the SPICE circuit solver. Magnetic probes were designed and tested that allow for physical measurements of voltages induced by the magnetic field generated by the current distributions in the strip line. A comparison of the measured voltages to simulated values from SPICE was done to validate the SPICE model. This model was used to validate the finite-integration model for the same strip line. Formulation of the transient current density distribution problem is accomplished by the superposition of a source current and an eddy current distribution on the same space. The mathematical derivation and implementation of the time-stepping algorithm to the finite element model is explicitly shown for a surface mesh with triangular elements. A C++ computer program was written to solve for the total current density in a thin conductor by implementing the time-stepping integral formulation. Evaluation of the finite element implementation was made regarding mesh size. Finite element meshes of increasing node density were simulated for the same structure until a smooth current density distribution profile was observed. The transient current density solver was validated by comparing simulations with AC conduction and transient response simulations of the SPICE model. Transient responses are compared for inputs at different frequencies and for varying time steps. This program is applied to thin conductors of irregular shape.

Understanding the characteristics of special nuclear materials in LiCl-KCl eutectic salt is extremely important in terms of effective system operation and material accountability for safeguarding pyroprocessing technology. By considering that uranium (U) is the most abundant and important element in the used nuclear fuel, measurements and analyses of U properties were performed in LiCl-KCl eutectic salt. Therefore, the electrochemical techniques such as cyclic voltammetry (CV), open circuit potential (OCP), Tafel, linear polarization (LP), and electrochemical impedance spectroscopy (EIS) were conducted under different experimental conditions to explore the electrochemical, thermodynamic, and kinetic properties of U in LiCl-KCl eutectic. The ultimate goal of this study was to develop proper methodologies for measuring and analyzing the exchange current density (i0) of U3+/U reaction, which has not been fully studied and understood in literature. In the preliminary study, cerium (Ce) was selected as a surrogate material for uranium and its behavior was being explored with the developments of experimental methods. CV was performed to evaluate Ce properties such as the diffusion coefficients (D), apparent standard reduction potential (E0*), Gibbs free energy (DG), and activity coefficient (g). In addition, EIS methods were adapted and specific experimental procedures were established for the proper i0 measurements providing repeatable and reproducible data sets. The i0 values for Ce3+/Ce pair were ranging from 0.0076 A cm-2 to 0.016 A cm-2, depending on the experimental conditions. These preliminary results give insight in developing the experimental setups and methods to evaluate the properties of U in LiCl-KCl. Plus, Ce is one of the lanthanide (Ln) fission products in electrorefiner (ER) system; therefore, the resulting data values yield useful information of the fundamental behaviors of Ln elements in the system. Based on these developed methodologies, the experimental designs and routines were established to explore the main properties (e.g., D, E0*, etc.) of UCl3 in LiCl-KCl eutectic salt under different concentrations (0.5 wt% to 4 wt% UCl3) and temperatures (723 K to 798 K). Specially, the i0 values of U3+/U were evaluated via EIS, LP, Tafel, and CV methods. All i0 values had linear trends with the change of concentration and temperature; however, these values measured by LP, Tafel, and CV methods were greatly influenced by the change in electrode surface area. Overall, the i0 values agreed within 33% relative error range with the EIS method being the most consistent and accurate in comparison to reported literature values. The measured values of i0 were ranging from 0.0054 A cm-2 to 0.102 A cm-2. Therefore, an extremely reliable database for i0 was provided and it is feasible to anticipate the i0 kinetics in other experimental conditions by using the provided equation models. Furthermore, GdCl3 was added to the LiCl-KCl-UCl3 system to explore the effects of other elements on the U properties such as the diffusion coefficients, thermodynamic properties, and i0 kinetics. The diffusion coefficient was generally decreased by 12 ~ 35% with addition of GdCl3 in LiCl-KCl-UCl3; however, the apparent standard potentials and exchange current density follow the same trends with data obtained without GdCl3 additions. Hence, the results indicate that the thermodynamic and kinetic values for U3+/U reaction in LiCl-KCl eutectic salt are not greatly influenced by the presence of GdCl3.

<p>In this thesis we report on calculations for open quantum mechanical and certain microwave systems. The models refer to a quantum point contact and an electron cavity. We model this open system with an imaginary potential as source and drain, and use the finite difference method to make our calculations. We report on general features of the model we have found, and compare our calculations with measurements made on microwave cavities.</p>

Many modeling studies have been carried out to simulate the current distribution across the channel and shoulder direction in a proton exchange membrane (PEM) fuel cell. However the modeling results do not show agreement on the current density distribution. At the same time, no experimental measurement result of current density distribution across the channel and the shoulder direction is available to testify the modeling studies. Hence in this work, an experiment was conducted to separately measure the current densities under the channel and the shoulder in a PEM fuel cell by using the specially designed membrane electrode assemblies. The experimental results show that the current density under the channel is lower than that under the shoulder except when the fuel cell load is high. Afterwards two more experiments were carried out to find out the reason causing the higher current density under the shoulder. The effects of the electric resistance of gas diffusion layer (GDL) in the lateral and through-plane directions on the current density distribution were studied respectively. The experimental results show that it is the through-plane electric resistance that leads to the higher current density under the shoulder. Moreover, a three-dimensional fuel cell model is developed using FORTRAN. A new method of combining the thin-film model and homogeneous model is utilized to model the catalyst layer. The model is validated by the experimental data. The distribution of current density, oxygen concentration, membrane phase potential, solid phase potential and overpotential in a PEM fuel cell have been studied by the model. The modeling results show that the new modeling method provides better simulations to the actual transport processes and chemical reaction in the catalyst layer of a PEM fuel cell.

The production of chlorate, chlorine and sodium hydroxiderelies heavily on electrical energy, so that savings in thisarea are always a pertinent issue. This can be brought aboutthrough increased mass transfer of reacting species to therespective electrodes, and through increased catalytic activityand uniformity of current density distribution at theseelectrodes. This thesis will present studies involvingmathematical modelling and experimental investigations of theseprocesses. They will show the effect that hydrodynamicbehaviour has on the total current density and cell voltages,along with the effects on current density distributions andindividual overpotentials atthe respective electrodes. Primary, secondary and psuedo-tertiary current densitydistribution models of a chlor-alkali anode are presented anddiscussed. It is shown that the secondary model presentsresults rather similar to the pseudo-tertiary model, when thecurrent density distribution is investigated, although thepotential distribution differs rather markedly. Furthermore, itis seen that an adequate description of the hydrodynamicsaround the anode is required if the potential distribution, andthereby the prevalence of side-reactions, is to be reasonablepredicted. A rigorous tertiary current density distribution model ofthe chlorate cell is also presented, which takes into accountthe developing hydrodynamic behaviour along the height of thecell. This shows that an increased flowrate gives more uniformcurrent density distributions. This is due to the fact that theincreased vertical flowrate of electrolyte replenishes ioncontent at the electrode surfaces, thus reducing concentrationoverpotentials. Furthermore, results from the model lead to theconclusion that it is the hypochlorite ion that partakes in themajor oxygen producing side-reaction. A real-scale cross-section of a segmented anode-cathode pairfrom a chlorate cell was designed and built in order to studythe current density distribution in industrial conditions.These experiments showed that increased flowrate brought aboutmore even current density distributions, reduced cell voltageand increased the total current density. An investigation ofthe hydrodynamic effects on the respective electrodeoverpotentials shows the anode reactions being more favoured byincreased flowrate. This leads to the conclusion that theuniform current density distribution, caused by increasedflowrate, occurs primarily through decreasing the concentrationoverpotential at the anode rather than by decreasing thebubble-induced ohmic drop at the cathode. Finally, results from experiments investigating thebubble-induced free convection from a small electrochemicalcell are presented. These experiments show that Laser DopplerVelocimetry is the most effective instrument for investigatingthe velocity profiles in bubble-containing electrochemicalsystems. The results also show that the flow can transform fromlaminar to turbulent behaviour on both the vertical andhorizontal planes, in electrochemical systems where bubbles areevolved.

Biological tissues are generally conductive and knowing the current distribution in these tissues is of great importance in many biomedical applications. Radio frequency current density imaging (RF-CDI) is a technology that measures current density distributions at the Larmor frequency utilizing magnetic resonance imaging (MRI). RF-CDI computes the applied current density, J, from the non-invasively measured magnetic field, H, produced by J. The previously implemented RF-CDI techniques could only image a single slice at a time. The previous method for RF current density reconstruction only computed one component of J. Moreover, this reconstruction required an assumption about H, which may be easily violated. These technical constraints have limited the potential biomedical applications of RF-CDI. In this thesis, we address the limitations of RF-CDI mentioned above. First, we implement a multi-slice RF-CDI sequence with a clinical MRI system and characterize its sensitivity to MRI random noise. Second, we present a novel method to fully reconstruct all three components of J without relying on any assumption of H. The central idea of our approach is to rotate the sample by 180 degrees in the horizontal plane to collect adequate MR data from two opposite sample orientations to compute one component of J. Furthermore, this approach can be extended to reconstruct the other two components of J by adding one additional sample orientation in the horizontal plane. This method has been verified by simulations and electrolytic phantom experiments. We have therefore demonstrated for the first time the feasibility of imaging the magnitude and phase of all components of the RF current density vector. The work presented in this thesis is expected to significantly enhance RF-CDI to image biological subjects. The current density vector J or any component of J can be measured over multiple slices without the compromise of motions of organs and tissues caused by gravitational force, thanks to the method of horizontal rotations. In addition, the reconstruction of the complex conductivity of biological tissues becomes possible due to the current advance in RF-CDI presented here.

In this thesis we report on calculations for open quantum mechanical and certain microwave systems. The models refer to a quantum point contact and an electron cavity. We model this open system with an imaginary potential as source and drain, and use the finite difference method to make our calculations. We report on general features of the model we have found, and compare our calculations with measurements made on microwave cavities.

Biological tissues are generally conductive and knowing the current distribution in these tissues is of great importance in many biomedical applications. Radio frequency current density imaging (RF-CDI) is a technology that measures current density distributions at the Larmor frequency utilizing magnetic resonance imaging (MRI). RF-CDI computes the applied current density, J, from the non-invasively measured magnetic field, H, produced by J. The previously implemented RF-CDI techniques could only image a single slice at a time. The previous method for RF current density reconstruction only computed one component of J. Moreover, this reconstruction required an assumption about H, which may be easily violated. These technical constraints have limited the potential biomedical applications of RF-CDI. In this thesis, we address the limitations of RF-CDI mentioned above. First, we implement a multi-slice RF-CDI sequence with a clinical MRI system and characterize its sensitivity to MRI random noise. Second, we present a novel method to fully reconstruct all three components of J without relying on any assumption of H. The central idea of our approach is to rotate the sample by 180 degrees in the horizontal plane to collect adequate MR data from two opposite sample orientations to compute one component of J. Furthermore, this approach can be extended to reconstruct the other two components of J by adding one additional sample orientation in the horizontal plane. This method has been verified by simulations and electrolytic phantom experiments. We have therefore demonstrated for the first time the feasibility of imaging the magnitude and phase of all components of the RF current density vector. The work presented in this thesis is expected to significantly enhance RF-CDI to image biological subjects. The current density vector J or any component of J can be measured over multiple slices without the compromise of motions of organs and tissues caused by gravitational force, thanks to the method of horizontal rotations. In addition, the reconstruction of the complex conductivity of biological tissues becomes possible due to the current advance in RF-CDI presented here.

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