Bachelor / Master Thesis
Transient behavior of electrical double layers in electrokinetic flows
by Poisson-Nernst-Planck equation
In lab-on-a-chip (LoC) systems, samples consisting of fluid and particles are transported,
manipulated and analyzed in structures of length scales from 100 µm to several nm.
The mechanism of choice for microfluidic manipulation and actuation is electrokinetics.
Electrokinetics includes electro-osmosis and electrophoresis.
Electro-osmosis is the flow of a dilute electrolyte solution caused by an applied electric field.
Electrophoresis is the movement of charged particles relative to a fluid in an electric field.
Typical applications are electrokinetic pumps, traps or filters for driving, fixing or
separating bioparticles depending on their charge.
Electrokinetic phenomena can be simulated by coupling a fluid simulation (considering fluid-particle interaction)
to a solver for a partial differential equation (PDE) that describes the electric potential.
Such a PDE is Poisson-Boltzmann equation (PBE) or the more advanced Poisson-Nernst-Planck equation (PNPE).
The PNPE describes the transient behavior of electrical double layers (EDLs), considering the ion distribution in the fluid .
EDLs result from ions that are attracted towards charged surfaces or particles (e.g. cells) in electrolyte solutions.
The PNPE becomes relevant when transient EDL behavior is of interest or when EDLs are overlapping, as in nano-channels.
(widely applicable Lattice Boltzmann from Erlangen) software framework provides a powerful
simulation environment for multi-physics applications.
As illustrated in the upper image above, we can simulate the electro-osmotic flow formation in a microchannel by means of the Poisson-Boltzmann equation (when EDL forms much faster than the flow develops).
In the movie above, we show a simulation of uncharged particles in electro-osmotic flow.
The Poisson-Boltzmann equation that was used for the simulations, models the steady-state of the EDL.
It can not model overlapping EDLs or the transient behavior of the EDL.
In contrast, the Poisson-Nernst-Planck equation can model those effects.
The goal of this thesis is to implement a solver for the Poisson-Nernst-Planck equation, based on the existing solver for the Poisson-Boltzmann equation.
This solver for elliptic PDEs (e.g. discretized by means of the finite differnence method) is encapsulated in a waLBerla module.
The PNPE can be described as a parabolic PDE, coupled with an elliptic PDE.
Thus the existing solver should be extended to be able to solve parabolic PDEs by means of a trapezoidal rule time integration scheme (e.g. Crank-Nicholson).
In order to solve the PNPE, both equations should finally be coupled and solved.
The physical correctness of the simulations should then be verified.
This includes the comparison of numerical results for the steady-state solution of the PNPE and the solution of the PBE.
It can also be done by comparing numerical results to analytical solutions.
Finally, the effect of an external electric field on charged particles in a fluid and the surrounding EDL could be simulated and visualized.
Here, the solver for the PNPE is coupled to the existing LBM solver.
- Interest in simulation and modelling of physical phenomena
- C++ programming skills
- Background in numerics
 M. Wang and Q. Kang, "Modeling electrokinetic flows in microchannels using coupled lattice Boltzmann methods",
Journal of Computational Physics, vol. 229, no. 3, pp. 728-744 (2010)
 D. Hlushkou et.al, "Coupled lattice-Boltzmann and fnite-difference simulation of electroosmosis in microfluidic channels"
Comput. Math. Appl., vol. 55, no. 7, pp. 1601-1610, (2008)
 M. Kaltenbacher, "Numerical simulation of mechatronic sensors and actuators", Springer, (2010)