Modular simulation of electrode array action

PI John McCaskill and Uwe Tangen RUB

The ECCell project depends on the use of digital electrode arrays to control the transport and concentration of charged molecules. The electrokinetic theory of electrolyte solutions manipulated by electrode arrays with non-blocking electrodes (supporting a DC field through ongoing Faradaic reactions at the electrode surface) and time dependent (digital) voltages is still complex because of a number of factors:

(i) screening by ions in solution introduces thin boundary layers (double layers) which would make a multiscale treatment essential in direct numerical simulation

(ii) the voltages used are not small compared with thermal voltages kT/e (25mV) making a direct linear treatment untenable. Strongly nonlinear effects such as charge saturation and reversal are well documented, depend on details of the electrolyte composition (e.g. multivalent cations) and qualitatively change the response.

(iii) there are multiple timescales of relaxation to an external voltage change, including the build up of the Stern layer, double layer and bulk ion distribution 

(iv) in dilute electrolytes, concentration of ions by the fields is limited by finite ion size effects and ignoring these can lead to unphysical concentrations of ions near surfaces. The packing needs to be described by an equation of state such as Carnaghan-Starling that takes into account the hard ion core repulsion.

(v) not only do ions migrate in the presence of the electric fields, but the entire fluid can be accelerated as electroosmotic flow

(vi) the nonlinear interaction between ion redistribution, surface reactions and electrostatic field buildup must be computed including all of the problems above

The recent review by Bazant [1] delineates some of these problems. The general framework proposed there is the modified Poisson-Nenrnst-Planck (mPNP) equations for the non-reactive part and the modified Butler-Volmer-Frumkin theory for relating the faradaic currents at the electrodes to the half-reactions there. 

Our approach is to try to separate the thin boundary layer effects from the geometric effects as far as possible, using analytic theory derived from the above framework for the double layer in a standard geometry such as a parallel plate capacitor) and then deriving time-dependent boundary conditions for the electric potential in the bulk region (i.e. outside the double layer) for our more complex electrode arrays.

Under a wide range of conditions, the complex potential drop in the double layers allows the potential in the bulk region to be treated linearly by solving Laplace's equation, and so we can superpose the effects of separate electrodes linearly. In order to allow the rapid simulation of switching-electrode-controlled migration of molecules like DNA, it is useful to also have analytic solutions for the electric field obtained from solving Laplace's equations with the appropriate boundary conditions. We have invested some effort to derive appropriate solutions, and the simplest approximate forms are already being evaluated in transport control scenarios.

One such scenario of central importance is that of traveling wave electrophoresis (TWE). TWE allows the rapid  separation of different charged oligomers with digital voltages in closed loop lanes, and hence is the most promising route for the integration of separation processes into chemical process control.

We have constructed analytic solutions to the electric field induced by electrodes in a planar array both with Neumann and Dirichilet boundary conditions, and optionally with a thick or thin layer bounded by a conducting (ITO) or insulating (glass) lid. The treatment of the insulating boundary conditions on the SiO2 floor between electrodes (Neumann) poses some problems for an analytic treatment in 3D (conformal methods are available in 2D), with many authors assuming linear or bilinear forms for the surface potential there. In 3D, the effective surface charge distribution (and hence potential boundary condition) can be obtained by solving a linear integral equation, or approximated as described above. The former is more efficient than using the infinite series solutions resulting from standard techniques such as Fourier transform.

[1]  Bazant et al. Towards an understanding of nonlinear electrokinetics at large applied voltages in concentrated solutions. Arxiv preprint arXiv:0903.4790 (2009)