# Modeling and High-Resolution Simulation of Nonlinear Electrophoresis

**Principal Investigators**: J.G. Santiago and Moran Bercovici

Simulation showing the development of an ITP plug consisting of five analytes. The adaptive grid procedure concentrates grid points at regions of high gradients, reduces the total computational time.

We have developed a new approach for spatial discretization of electrokinetic problems which achieves both high accuracy and low computational cost. Precise resolution of interface widths can be crucial for accurate simulation of modern applications. We therefore employed a sixth order compact scheme which is non-dissipative. More important than its high order is the high resolution of the scheme, allowing for the resolution of high wave numbers with fewer grid points. To further reduce the computational cost, we developed an adaptive grid algorithm. The adaptive grid continuously varies the spacing of and clusters grid points in regions of high gradients, thereby reducing the possibility of unstable solutions. We achieve smooth and stable solutions at current densities as high as 5000 using a number of grid points 50 times smaller than equivalent uniform grids which achieve the same resolution. We thereby reduce computational time by a factor of 75.

Schematic illustration and summary of isotachophoresis channel and physical mechanisms included in the current code.

The physical modules implemented in our code include treatment of multi-species, multi-valent equilibrium reactions (including multivalent ampholytes), non-uniform electroosmotic flow and pressure-driven flow. The latter are key to determining ion density gradients in electrokinetic flows with heterogenous electrolytes. For the first time, we include a Taylor-Aris formulation with a multispecies, heterogeneous electrolyte flow solver. The main physical formulations (modules) included in the code are summarized in the above figure. The code also includes a graphical user interface and a database of over 300 analytes, based largely on the tables by Hirokawa. Currently, the code does not account for changes in reactivity and mobility due to ionic strength; but these important mechanisms will be incorporated in near-future improvements. While the code constitutes a rather general tool for CE, FASS, and ITP, among other assays, modern applications of electrophoretic simulations are wide and diverse and no single code can offer an answer to all applications. We will therefore soon offer the code as an open source, written in Matlab, to allow researchers to modify, add, remove, embed, or link part or all of the codes with additional components and other programs. The code will be available for download from this website.

Speed ratio and grid size ratio associated with uniform and adaptive grid simulations of a single interface ITP experiment. LE is 100 mM Hydrochloric acid, TE is 50 mM HEPES, and counter ion is 200 mM TRIS. The channel is a 20 mm long circular capillary with a 50 um diameter. The curves are truncated at the maximum current resolvable by the 300 node adaptive grid.