Highly accurate simulation of electrokinetic transport

Tinh Vo

Computational Multiphase Flow, TU Darmstadt

08 Jul, 2025

Persons

Tinh Vo, M.Sc.

  • Education
    • Case Western Reserve University – Mechanical Engineering B.S.E.
    • Technical University of Munich (TU München) – Computational Mechanics M.Sc.
  • Research interests
    • Coupled multiphysics
    • Optimization
    • High Performance Computing

Advisor: Dr.Ing. Holger Marschall

  • Research group leader: Computational Multiphase Flow (CMF)

  • Chief Product & Innovation Officer @ IANUS Simulation

Project overview

Physical phenomenon and application

Electrically driven flows

  • Focuses
    • Near walls
    • At high wall change

  • Resolving the Stern layer numerically is the main challenge
    • Understood as being exactly one atom thick
    • Previous models describes unbounded, exponential value increase in species concentration even at moderate voltages
    • Breakdown of continuum assumptions

  • Contributions to understanding of mechanics:
    • Charging time
    • Permselectivity
    • Species transport

Numerical Methods

Governing equations

Navier-Stokes Equations

\[ \frac{\partial}{\partial t}[\rho\mathbf{v}] + \nabla \cdot \{\rho\mathbf{v}\mathbf{v}\} = -\nabla p + \nabla \cdot \left\{\mu\left[\nabla\mathbf{v} + (\nabla\mathbf{v})^{\text{T}}\right]\right\} + \mathbf{f}_b \] \[ \frac{\partial\rho}{\partial t} +\nabla\cdot (\rho\mathbf{u}_{}) = 0 \]

Species Transport &

Gauss law

\[ \frac{ \partial c_{i} }{ \partial t } + \mathbf{u} \cdot \nabla c_{i} = \nabla \cdot(D_{i}\nabla c_{i}) + \nabla \cdot [( \underbrace{ D_{i} \dfrac{ez_{i}}{kT} } _{ \text{Stokes-Einstein relation} }\nabla \Psi)c_{i} ] \] \[ \nabla^2 \Psi _{E} =\frac{-\rho_E}{\epsilon\epsilon_{0}} = \frac{-F \sum_{i} z_{i} c_{i}}{ \epsilon\epsilon_{0}} \]

Proposed Benchmarks

Case setup¹

Internal electric potential results for thin double layer and small applied potential 1

Described leading order solutions of pore charging and electric potential field of a close ended pore,

replicating transmission line model solutions

Governing equation discrepancy

Does not contain convective/advective term

Improvement Proposals

Implicitly(Block) Coupled solver

  • Motivation: Numerical
    • Segregated solvers are numerically not robust
  • Goals:
    • Numerically robust
    • Stable

Coupled Navier-Stokes and Poisson-Nernst-Planck solution system 1

Sorption-enabled Stern layer

Description

  • Motivation: Numerical/Physical
    • Standard linear discretization does not properly represent exponential (concentration) profiles near walls
  • Goals:
    • Subgrid-scale correction
    • Include sorption processes

➡ Increase accuracy near walls without larger mesh

Ion concentration profiles resulting from a classical Nernst-Planck formulation solved numerically (left) 1

Ionic sorption processes near walls 2

Sorption-enabled Stern layer

Governing equations

Part of the boundary condition of a ionic system: \[ D_{i} \partial_{n} c_{i} + c_{i} \mu _{i} \partial_{n } \Psi =0 \]

Analytical solution: \[ c_{i|b} = c_{i|P}e^{-\mu _{i}/D_{i} \left( \Psi _{|b} -\Psi _{|P} \right) } \]

Sorption source term: \[ D_{i} \partial_{n} c_{i} + c_{i} \mu _{i} \partial_{n } \Psi = -s_i \]

Solution treatment:

unknown

Thermodynamically Consistent Flux

Formulation

Change the construction of the electromigration flux

\[ \frac{ \partial c_{i} }{ \partial t } + \mathbf{u} \cdot \nabla c_{i} = \underbrace{ \nabla \cdot(D_{i}\nabla c_{i}) + \nabla \cdot (D_{i} \dfrac{ez_{i}}{kT}\nabla \Psi c_{i}) } _{ \nabla \cdot \mathbf{J_{i}} } \] \[ \frac{\partial c_{i}}{\partial t} + \nabla \cdot ( \underbrace{c_{i}\color{red}{\mathbf{v}_{}} } _\text{advection}+ \mathbf{J_{i}}) =0 \]

Existing formulation:

\[ \mathbf{J}_{i} = \underbrace{D_{i}\nabla c_{i} } _\text{diffusion} + \underbrace{\left[ \frac{D_{e,i} z_{i}e}{k_{b} T} \right]c_{i}(-\nabla \Psi )} _\text{electromigration} \]

  • No guarantee of mass conservation

Improved Nernst-Planck Proposal 1

Barycentric velocity:

\[ \mathbf{v } = \frac{1}{\rho} \sum_{i} \rho_{i} \mathcal{v }_{i} \]

Flux:

\[ \begin{align} \mathbf{J}_{i} = -\sum _{j =1} ^{N-1} M_{i j} &\left( \nabla \frac{\mu _{j} -\mu _{N}}{T} n_{i} \right. \\ & \quad \left. + \frac{1}{T} \left( \frac{z_{j}}{m_{j}} - \frac{z_{N}}{m_{N}} \right)\nabla \Psi \right) \\ \end{align} \]

  • Conservation requirement:

\[ \sum_{\alpha{}}^{N} \mathbf{J}_{\alpha{}} =0 \]

Thermodynamically Consistent Flux

Goals

  • Quasi-incompressible formulation
    • Pressure-velocity coupling
    • Density- ion concentration coupling
  • Implementation and validation
    • Ion mass conservation

Atomistic - continuum coupling: Candidates

Grid-based Mesoscale Simulations

Representative setup for mesoscale simulations 1

Candidate framework

Nanoscale simulations

Represent molecular ion crowding

Previous work simulating nucleate boiling with molecular simulation 2

Candidate framework

Atomistic - continuum coupling: Target Framework

Schematics of coupling overlap layer 1

Previous work velocity coupling between molecular and mesoscale simulations using an overlap layer 2

High Performance computing:

  • Motivation: Numerical
    • Distributed computing enables concurrent computations
  • Goal:
    • Dynamic, adaptive mesh refinement
    • Load Balancing

Computation domain being distributed across multiple processes 1

Adaptive mesh refinement

Bonus slides