Direct numerical simulation of electrokinetic transport phenomena: variational multi-scale stabilization and octree-based mesh refinement

Finite element modeling of charged species transport has enabled the analysis, design, and optimization of a diverse array of electrochemical and electrokinetic devices. These systems are represented by the Poisson-Nernst-Planck (PNP) equations coupled with the Navier-Stokes (NS) equation. Direct numerical simulation (DNS) to accurately capture the spatio-temporal variation of ion concentration and current flux remains challenging due to the (a) small critical dimension of the electric double layer (EDL), (b) stiff coupling, large advective effects, and steep gradients close to boundaries, and (c) complex geometries exhibited by electrochemical devices. In the current study, we address these challenges by presenting a direct numerical simulation framework that incorporates: (a) a variational multiscale (VMS) treatment, (b) a block-iterative strategy in conjunction with semi-implicit (for NS) and implicit (for PNP) time integrators, and (c) octree based adaptive mesh refinement. The VMS formulation provides numerical stabilization critical for capturing the electro-convective instabilities often observed in engineered devices. The block-iterative strategy decouples the difficulty of non-linear coupling between the NS and PNP equations and allows using tailored numerical schemes separately for NS and PNP equations. The carefully designed second-order, hybrid implicit methods circumvent the harsh timestep requirements of explicit time steppers, thus enabling simulations over longer time horizons. Finally, the octree-based meshing allows efficient and targeted spatial resolution of the EDL. These features are incorporated into a massively parallel computational framework, enabling the simulation of realistic engineering electrochemical devices. The numerical framework is illustrated using several challenging canonical examples

Nonlinear Electrokinetic Flow near Permselective Membrane

Ph.DDOCTOR OF PHILOSOPH

New porous medium Poisson-Nernst-Planck equations for strongly oscillating electric potentials

We consider the Poisson-Nernst-Planck system which is well-accepted for describing dilute electrolytes as well as transport of charged species in homogeneous environments. Here, we study these equations in porous media whose electric permittivities show a contrast compared to the electric permittivity of the electrolyte phase. Our main result is the derivation of convenient low-dimensional equations, that is, of effective macroscopic porous media Poisson-Nernst-Planck equations, which reliably describe ionic transport. The contrast in the electric permittivities between liquid and solid phase and the heterogeneity of the porous medium induce strongly oscillating electric potentials (fields). In order to account for this special physical scenario, we introduce a modified asymptotic multiple-scale expansion which takes advantage of the nonlinearly coupled structure of the ionic transport equations. This allows for a systematic upscaling resulting in a new effective porous medium formulation which shows a new transport term on the macroscale. Solvability of all arising equations is rigorously verified. This emergence of a new transport term indicates promising physical insights into the influence of the microscale material properties on the macroscale. Hence, systematic upscaling strategies provide a source and a prospective tool to capitalize intrinsic scale effects for scientific, engineering, and industrial applications

Electrokinetically forced turbulence in microfluidic flow.

While laminar flow heat transfer and mixing in microfluidic geometries has been investigated experimentally, as has the effect of geometry-induced turbulence in microfluidic flow (it is well documented that turbulence increases convective heat transfer in macrofluidic flow), little literature exists investigating the effect of electrokinetically-induced turbulence on heat transfer at the micro scale. Successful research in this area could be invaluable in creating more efficient heat exchangers for emerging microscale electronics as well as to fields requiring greater control of mixing in microfluidic devices

Charge relaxation dynamics of an electrolytic nanocapacitor

Understanding ion relaxation dynamics in overlapping electric double layers (EDLs) is critical for the development of efficient nanotechnology based electrochemical energy storage, electrochemomechanical energy conversion and bioelectrochemical sensing devices as well as controlled synthesis of nanostructured materials. Here, a Lattice Boltzmann (LB) method is employed to simulate an electrolytic nanocapacitor subjected to a step potential at t = 0 for various degrees of EDL overlap, solvent viscosities, ratios of cation to anion diffusivity and electrode separations. The use of a novel, continuously varying and Galilean invariant, molecular speed dependent relaxation time (MSDRT) with the LB equation recovers a correct microscopic description of the molecular collision phenomena and enhances the stability of the LB algorithm. Results for large EDL overlaps indicated oscillatory behavior for the ionic current density in contrast to monotonic relaxation to equilibrium for low EDL overlaps. Further, at low solvent viscosities and large EDL overlaps, anomalous plasma-like spatial oscillations of the electric field were observed that appeared to be purely an effect of nanoscale confinement. Employing MSDRT in our simulations enabled a modeling of the fundamental physics of the transient charge relaxation dynamics in electrochemical systems operating away from equilibrium wherein Nernst-Einstein relation is known to be violated.Comment: Accepted for publication in the Journal of Physical Chemistry C on October 30 2014. Supplementary info available free of charge via the Internet at http://pubs.acs.org. Revised version includes more details on the computation of the molecular speed dependent relaxation time (MSDRT) and emphasizes the Galilean invariance of the computed MSDR

Augmented saddle point formulation of the steady-state Stefan--Maxwell diffusion problem

We investigate structure-preserving finite element discretizations of the steady-state Stefan--Maxwell diffusion problem which governs diffusion within a phase consisting of multiple species. An approach inspired by augmented Lagrangian methods allows us to construct a symmetric positive definite augmented Onsager transport matrix, which in turn leads to an effective numerical algorithm. We prove inf-sup conditions for the continuous and discrete linearized systems and obtain error estimates for a phase consisting of an arbitrary number of species. The discretization preserves the thermodynamically fundamental Gibbs--Duhem equation to machine precision independent of mesh size. The results are illustrated with numerical examples, including an application to modelling the diffusion of oxygen, carbon dioxide, water vapour and nitrogen in the lungs.Comment: 27 pages, 5 figure

Nonlinear Dynamic Modeling, Simulation And Characterization Of The Mesoscale Neuron-electrode Interface

Extracellular neuroelectronic interfacing has important applications in the fields of neural prosthetics, biological computation and whole-cell biosensing for drug screening and toxin detection. While the field of neuroelectronic interfacing holds great promise, the recording of high-fidelity signals from extracellular devices has long suffered from the problem of low signal-to-noise ratios and changes in signal shapes due to the presence of highly dispersive dielectric medium in the neuron-microelectrode cleft. This has made it difficult to correlate the extracellularly recorded signals with the intracellular signals recorded using conventional patch-clamp electrophysiology. For bringing about an improvement in the signalto-noise ratio of the signals recorded on the extracellular microelectrodes and to explore strategies for engineering the neuron-electrode interface there exists a need to model, simulate and characterize the cell-sensor interface to better understand the mechanism of signal transduction across the interface. Efforts to date for modeling the neuron-electrode interface have primarily focused on the use of point or area contact linear equivalent circuit models for a description of the interface with an assumption of passive linearity for the dynamics of the interfacial medium in the cell-electrode cleft. In this dissertation, results are presented from a nonlinear dynamic characterization of the neuroelectronic junction based on Volterra-Wiener modeling which showed that the process of signal transduction at the interface may have nonlinear contributions from the interfacial medium. An optimization based study of linear equivalent circuit models for representing signals recorded at the neuron-electrode interface subsequently iv proved conclusively that the process of signal transduction across the interface is indeed nonlinear. Following this a theoretical framework for the extraction of the complex nonlinear material parameters of the interfacial medium like the dielectric permittivity, conductivity and diffusivity tensors based on dynamic nonlinear Volterra-Wiener modeling was developed. Within this framework, the use of Gaussian bandlimited white noise for nonlinear impedance spectroscopy was shown to offer considerable advantages over the use of sinusoidal inputs for nonlinear harmonic analysis currently employed in impedance characterization of nonlinear electrochemical systems. Signal transduction at the neuron-microelectrode interface is mediated by the interfacial medium confined to a thin cleft with thickness on the scale of 20-110 nm giving rise to Knudsen numbers (ratio of mean free path to characteristic system length) in the range of 0.015 and 0.003 for ionic electrodiffusion. At these Knudsen numbers, the continuum assumptions made in the use of Poisson-Nernst-Planck system of equations for modeling ionic electrodiffusion are not valid. Therefore, a lattice Boltzmann method (LBM) based multiphysics solver suitable for modeling ionic electrodiffusion at the mesoscale neuron-microelectrode interface was developed. Additionally, a molecular speed dependent relaxation time was proposed for use in the lattice Boltzmann equation. Such a relaxation time holds promise for enhancing the numerical stability of lattice Boltzmann algorithms as it helped recover a physically correct description of microscopic phenomena related to particle collisions governed by their local density on the lattice. Next, using this multiphysics solver simulations were carried out for the charge relaxation dynamics of an electrolytic nanocapacitor with the intention of ultimately employing it for a simulation of the capacitive coupling between the neuron and the v planar microelectrode on a microelectrode array (MEA). Simulations of the charge relaxation dynamics for a step potential applied at t = 0 to the capacitor electrodes were carried out for varying conditions of electric double layer (EDL) overlap, solvent viscosity, electrode spacing and ratio of cation to anion diffusivity. For a large EDL overlap, an anomalous plasma-like collective behavior of oscillating ions at a frequency much lower than the plasma frequency of the electrolyte was observed and as such it appears to be purely an effect of nanoscale confinement. Results from these simulations are then discussed in the context of the dynamics of the interfacial medium in the neuron-microelectrode cleft. In conclusion, a synergistic approach to engineering the neuron-microelectrode interface is outlined through a use of the nonlinear dynamic modeling, simulation and characterization tools developed as part of this dissertation research