An Asymptotic Analysis of Space Charge Layers in a Mathematical Model of a Solid Electrolyte

We review a model for a solid electrolyte derived under thermodynamics principles. We non-dimensionalise and scale the model to identify small parameters, where we identify a scaling that controls the width of the space-charge layer in the electrolyte. We present asymptotic analyses and numerical solutions for the one dimensional zero charge flux equilibrium problem. We introduce an auxiliary variable to remove singularities from the domain in order to facilitate robust numerical simulations. From the asymptotics we identify three distinct regions: the bulk, boundary layers, and intermediate layers. The boundary and intermediate layers form the space charge layer of the solid electrolyte, which we can further distinguish as strong and weak space-charge-layers respectively. The weak space-charge-layer is characterised by a length, $\lambda$, which is equivalent to the Debye length of a standard liquid electrolyte. The strong space-charge-layer is characterised by a scaled Debye length, which is larger than $\lambda$. We find that both layers exhibit distinct behaviour, we see quadratic behaviour in the strong space-charge-layer and exponential behaviour in the weak space-charge-layer. We find that matching between these two asymptotic regimes is not standard and we implement a pseudo-matching approach to facilitate the transition between the quadratic and exponential behaviours. We demonstrate excellent agreement between asymptotics and simulation.Comment: 24 pages plus 14 page supplementary materials, 19 figures total (counting subfigures) Submitted to SIAM Journal on Applied Mathematic

Asymptotic reduction of a porous electrode model for lithium-ion batteries

We present a porous electrode model for lithium-ion batteries using Butler--Volmer reaction kinetics. We model lithium concentration in both the solid and fluid phase along with solid and liquid electric potential. Through asymptotic reduction, we show that the electric potentials are spatially homogeneous which decouples the problem into a series of time-dependent problems. These problems can be solved on three distinguished time scales, an early time scale where capacitance effects in the electrode dominate, a mid-range time scale where a spatial concentration gradient forms in the electrolyte, and a long-time scale where each of the electrodes saturate and deplete with lithium respectively. The solid-phase concentration profiles are linear functions of time and the electrolyte potential is everywhere zero, which allows the model to be reduced to a system of two uncoupled ordinary differential equations. Analytic and numerical results are compared with full numerical simulations and experimental discharge curves demonstrating excellent agreement.Comment: Accepted in SIAM Journal on Applied Mathematic

From exam to education: the math exam/educational resources

peer-reviewedThe Math Exam/Education Resources (MER) is an open online learning resource hosted at The University of British Columbia (UBC), aimed at providing mathematics education resources for students and instructors at UBC. In this paper, there will be a discussion of the motivation for creating this resource on the MediaWiki platform, key features of the implementation that support student learning (including the evolution of the MER wiki from an exam database to more general learning resource), data on student use and response, potential for future development, and a brief description of how the project was implemented. Preliminary correlation data between wiki usage and exam performance are shared along with some preliminary data from an ongoing impact study.peer-reviewe

A numerical framework for singular limits of a class of reaction diffusion problems

We present a numerical framework for solving localized pattern structures of reaction-diffusion type far from the Turing regime. We exploit asymptotic structure in a set of well established pattern formation problems to analyze a singular limit model that avoids time and space adaptation typically associated to full numerical simulations of the same problems. The singular model involves the motion of a curve on which one of the chemical species is concentrated. The curve motion is non-local with an integral equation that has a logarithmic singularity. We generalize our scheme for various reaction terms and show its robustness to other models with logarithmic singularity structures. One such model is the 2D Mullins-Sekerka flow which we implement as a test case of the method. We then analyze a specific model problem, the saturated Gierer-Meinhardt problem, where we demonstrate dynamic patterns for a variety of parameters and curve geometries. (C) 2015 Elsevier Inc. All rights reserved

Existence, stability, and dynamics of ring and near-ring solutions to the saturated Gierer--Meinhardt model in the semistrong regime

We analyze a singularly perturbed reaction-diffusion system in the semi-strong diffusion regime in two spatial dimensions where an activator species is localized to a closed curve, while the inhibitor species exhibits long range behavior over the domain. In the limit of small activator diffusivity we derive a new moving boundary problem characterizing the slow time evolution of the curve, which is defined in terms of a quasi steady-state inhibitor diffusion field and its properties on the curve. Numerical results from this curve evolution problem are illustrated for the Gierer-Meinhardt model (GMS) with saturation in the activator kinetics. A detailed analysis of the existence, stability, and dynamics of ring and near-ring solutions for the GMS model is given, whereby the activator concentrates on a thin ring concentric within a circular domain. A key new result for this ring geometry is that by including activator saturation there is a qualitative change in the phase portrait of ring equilibria, in that there is an S-shaped bifurcation diagram for ring equilibria, which allows for hysteresis behavior. In contrast, without saturation, it is well-known that there is a saddle-node bifurcation for the ring equilibria. For a near-circular ring, we develop an asymptotic expansion up to quadratic order to fully characterize the normal velocity perturbations from our curve-evolution problem. In addition, we also analyze the linear stability of the ring solution to both breakup instabilities, leading to the disintegration of a ring into localized spots, and zig-zag instabilities, leading to the slow shape deformation of the ring. We show from a nonlocal eigenvalue problem that activator saturation can stabilize breakup patterns that otherwise would be unstable. Through a detailed matched asymptotic analysis, we derive a new explicit formula for the small eigenvalues associated with zig-zag instabilities, and we show that they are equivalent to the velocity perturbations induced by the near-circular ring geometry. Finally, we present full numerical simulations from the GMS PDE system that confirm the predictions of the analysis

Production of nitrate spikes in a model of ammonium biodegradation

Nitrification at the site of a contaminant ammonium plume from a former coal carbonisation plant can be modelled with three competing bacterial populations of Nitrosomonas, Nitrobacter, and Brocadia anammoxidans. Oscillations of chemical species at the site can be explained by a reduced model of ammonium competition between Nitrosomonas and B. anammoxidans which effectively acts as an activator-inhibitor system. Stable oscillations occur in conditions of low nutrient (ammonium) supply and this causes a spatial travelling wave in a borehole profile when diffusion is introduced

An improved approximation for hydraulic conductivity for pipes of triangular cross-section by asymptotic means

In this paper, we explore single-phase flow in pores with triangular cross-sections at the pore-scale level. We use analytic and asymptotic methods to calculate the hydraulic conductivity in triangular pores, a typical geometry used in network models of porous media flow. We present an analytical formula for hydraulic conductivity based on Poiseuille flow that can be used in network models contrasting the typical geometric approach leading to many different estimations of the hydraulic conductivity. We consider perturbations to an equilateral triangle by changing the length of one of the triangle sides. We look at both small and large triangles in order to capture triangles that are near and far from equilateral. In each case, the calculations are compared with numerical solutions and the corresponding network approximations. We show that the analytical solution reduces to a quantitatively justifiable formula and agrees well with the numerical solutions in both the near and far from equilateral cases.Science Foundation Ireland under Grant No. SFI/13/IA/1923 The Mathematics Applications Consortium for Science and Industry under Grant No. 12/IA/1683 The Carswell Family Foundation NSERC Vanier Canada Graduate scholarship Grant No. 434051. The Natural Sciences and Engineering Research Council of Canada Discovery Grant 2019-06337