Multicomponent charge transport in electrolyte solutions

The work presented in this thesis investigates the mathematical modelling of charge transport in electrolyte solutions, within the nanoporous structures of electrochemical devices. We compare two approaches found in the literature, by developing onedimensional transport models based on the Nernst-Planck and Maxwell-Stefan equations. The development of the Nernst-Planck equations relies on the assumption that the solution is infinitely dilute. However, this is typically not the case for the electrolyte solutions found within electrochemical devices. Furthermore, ionic concentrations much higher than those of the bulk concentrations can be obtained near the electrode/electrolyte interfaces due to the development of an electric double layer. Hence, multicomponent interactions which are neglected by the Nernst-Planck equations may become important. The Maxwell-Stefan equations account for these multicomponent interactions, and thus they should provide a more accurate representation of transport in electrolyte solutions. To allow for the effects of the electric double layer in both the Nernst-Planck and Maxwell-Stefan equations, we do not assume local electroneutrality in the solution. Instead, we model the electrostatic potential as a continuously varying function, by way of Poisson’s equation. Importantly, we show that for a ternary electrolyte solution at high interfacial concentrations, the Maxwell-Stefan equations predict behaviour that is not recovered from the Nernst-Planck equations. The main difficulty in the application of the Maxwell-Stefan equations to charge transport in electrolyte solutions is knowledge of the transport parameters. In this work, we apply molecular dynamics simulations to obtain the required diffusivities, and thus we are able to incorporate microscopic behaviour into a continuum scale model. This is important due to the small size scales we are concerned with, as we are still able to retain the computational efficiency of continuum modelling. This approach provides an avenue by which the microscopic behaviour may ultimately be incorporated into a full device-scale model. The one-dimensional Maxwell-Stefan model is extended to two dimensions, representing an important first step for developing a fully-coupled interfacial charge transport model for electrochemical devices. It allows us to begin investigation into ambipolar diffusion effects, where the motion of the ions in the electrolyte is affected by the transport of electrons in the electrode. As we do not consider modelling in the solid phase in this work, this is simulated by applying a time-varying potential to one interface of our two-dimensional computational domain, thus allowing a flow field to develop in the electrolyte. Our model facilitates the observation of the transport of ions near the electrode/electrolyte interface. For the simulations considered in this work, we show that while there is some motion in the direction parallel to the interface, the interfacial coupling is not sufficient for the ions in solution to be "dragged" along the interface for long distances.

Detailed analysis of a conservative difference approximation for the time fractional diffusion equation

Diffusion equations that use time fractional derivatives are attractive because they describe a wealth of problems involving non-Markovian Random walks. The time fractional diffusion equation (TFDE) is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α ∈ (0, 1). Developing numerical methods for solving fractional partial differential equations is a new research field and the theoretical analysis of the numerical methods associated with them is not fully developed. In this paper an explicit conservative difference approximation (ECDA) for TFDE is proposed. We give a detailed analysis for this ECDA and generate discrete models of random walk suitable for simulating random variables whose spatial probability density evolves in time according to this fractional diffusion equation. The stability and convergence of the ECDA for TFDE in a bounded domain are discussed. Finally, some numerical examples are presented to show the application of the present technique.

A second order control-volume finite-element least-squares strategy for simulating diffusion in strongly anisotropic media

An unstructured mesh �nite volume discretisation method for simulating di�usion in anisotropic media in two-dimensional space is discussed. This technique is considered as an extension of the fully implicit hybrid control-volume �nite-element method and it retains the local continuity of the ux at the control volume faces. A least squares function recon- struction technique together with a new ux decomposition strategy is used to obtain an accurate ux approximation at the control volume face, ensuring that the overall accuracy of the spatial discretisation maintains second order. This paper highlights that the new technique coincides with the traditional shape function technique when the correction term is neglected and that it signi�cantly increases the accuracy of the previous linear scheme on coarse meshes when applied to media that exhibit very strong to extreme anisotropy ratios. It is concluded that the method can be used on both regular and irregular meshes, and appears independent of the mesh quality.

Implicit difference approximation for the time fractional diffusion equation

In this paper, we consider a time fractional diffusion equation on a finite domain. The equation is obtained from the standard diffusion equation by replacing the first-order time derivative by a fractional derivative (of order $0<\alpha<1$ ). We propose a computationally effective implicit difference approximation to solve the time fractional diffusion equation. Stability and convergence of the method are discussed. We prove that the implicit difference approximation (IDA) is unconditionally stable, and the IDA is convergent with $O(\tau+h^2)$, where $\tau$ and $h$ are time and space steps, respectively. Some numerical examples are presented to show the application of the present technique.

Numerical approximation of a fractional-in-space diffusion equation (II) - with nonhomogeneous boundary conditions

In this paper, a space fractional di®usion equation (SFDE) with non- homogeneous boundary conditions on a bounded domain is considered. A new matrix transfer technique (MTT) for solving the SFDE is proposed. The method is based on a matrix representation of the fractional-in-space operator and the novelty of this approach is that a standard discretisation of the operator leads to a system of linear ODEs with the matrix raised to the same fractional power. Analytic solutions of the SFDE are derived. Finally, some numerical results are given to demonstrate that the MTT is a computationally e±cient and accurate method for solving SFDE.