A FAST AND ACCURATE ISOTACHOPHORESIS SIMULATION CODE

A FAST AND ACCURATE ISOTACHOPHORESIS SIMULATION CODE Moran Bercovici, Sanjiva K. Lele and Juan G. Santiago Stanford University, USA ABSTRACT We developed a numerical code which allows fast and accurate simulation of isotachophoresis (ITP). The multi-species code accounts for equilibrium chemistry, non-uniform electroosmotic flow, and dispersion. Our modeling efforts for the latter are also presented. The goal of our work is to create an efficient, accurate, validated, and uniquely-capable electrophoresis simulation code available for free via the web to the microfluidics community. KEYWORDS: Isotachophoresis, Simulation, High-resolution, Dispersion. INTRODUCTION In ITP, analytes are focused between relatively high mobility leading electrolyte (LE) ions and low mobility trailing electrolyte (TE) ions. The process provides a method for preconcentration due to strong focusing [1]; simultaneous selfsegregation and separation of ions; and unique methods of detection including indirect fluorescence detections methods [2]. New computational tools offer the potential of greatly reducing the amount of time spent on empirical optimization of assays, and can be used to achieve optimal resolution and sensitivity. Successful and efficient optimization of ITP requires accurate representation of coupled advectivediffusion-electromigration physics as well as a fast and accurate simulation. We here present our investigation of dispersion processes in ITP, showing the impact of this mechanism on predictions. We also describe the development of a fast numerical code which includes a unique model for dispersion as well as numerical techniques designed to accurately resolve the sharp concentration interfaces that arise in ITP. NUMERICAL METHOD Several general codes for electrophoresis have been reported [3]-[5]. In particular, SIMUL-5 [1] code developed by a group at Charles University, Czech Republic, has been fairly widely used and is available for free on the web. However, these codes address the challenges of resolving sharp ITP interfaces by either using a large number of grid points, resulting in increased computational time, or by numerical dissipation which leads to nonphysical diffusion. The numerical method we present here combines a high-resolution compact scheme [6] with an adaptive grid to allow more accurate resolution of interfaces while also decreasing computational cost. Figure 1 illustrates the benefits of the method for a simple ITP interface. THEORY Although electromigration, chemical reactions, and electroosmosis have been extensively studied in the past [1],[7], with the exception of the approximate, Twelfth International Conference on Miniaturized Systems for Chemistry and Life Sciences October 12 - 16, 2008, San Diego, California, USA 978-0-9798064-1-4/µTAS2008/$20©2008CBMS

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Figure 1. Predicted concentration profiles showing the effect of spatial discretization and grid adaptation on the resolution of a single ITP interface. N is the number of grid points used. Results obtained after 100 s on a 20 mm long domain, under a current density of 1800 A/m2. LE and TE have respective mobilities of 80⋅10-9 and 40⋅10-9 m2/Vs. (a) Explicit centered second order scheme using an equally spaced grid, which is unable to resolve high wave numbers - resulting in significant oscillations.(b) Sixth-order compact scheme using an equally spaced grid, showing increased resolution and less (yet still appreciable) oscillations. (c),(d) sixth order compact scheme using our adaptive grid procedure results in a smooth interface, even for one third the number of grid points. unvalidated model of [8], there are no efforts dedicated to modeling of dispersion. We study this mechanism by examining a single analyte focused between the leading electrolyte and the trailing electrolyte. To be able to compare to well controlled experiments, we impose an external pressure difference to balance electromigration, resulting in a stationary analyte zone. A scaling analysis of the problem shows that at a steady state axial advection balances radial electromigration, 1 ∂ ⎡ ∂φ ⎤ ∂ ⎡ ∂φ ⎤ r μ c  μ c + u ( r )c ⎥ r ∂r ⎢⎣ ∂r ⎥⎦ ∂x ⎢⎣ ∂x ⎦ where c is species concentration, u is the bulk velocity, μ is the mobility and φ is the electric potential. As the ITP interface curves under the influence of pressure, radial conductivity gradients lead to radial electric fields. The latter move ions perpendicular to the flow’s streamlines, leading to dispersion. To model these effects 2 , where D is we propose an effective diffusivity of the form D = D + K aU eff μ ∂φ / ∂x the diffusivity, U is the dispersion velocity, a is the characteristic size of the cross Figure 2. Experimental visualization of a stationary ITP zone under conditions of increased current and increased dispersion due to pressure. Beyond a certain value, further increase of the current no longer decreases the width of the plug. Experiments were performed in a 20μm by 90μm isotropically etched channel. LE is 100mM and 125mM Histidine. TE is 25mM and 31 mM Histidine. Analyte is a finite 46 mm injection of 1μM Alexa Fluor 488. Electroosmotic flow is suppressed using 0.2% PVP. Twelfth International Conference on Miniaturized Systems for Chemistry and Life Sciences October 12 - 16, 2008, San Diego, California, USA

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section, and K is a constant. This relation is analogous to the effective diffusivity obtained from Taylor dispersion analysis, but is significantly different in that radial electromigration (versus radial diffusion) balances axial advection. RESULTS AND DISCUSSION Figure 2 presents experimental results showing the analyte’s shape for a range of applied currents, while figure 3 compares these results to a non-dispersive model and to the proposed empirical model. The proposed model captures the trends observed in the experiment and shows significant improvement over models where dispersion is ignored, allowing for realistic predictions of isotachopherograms. Clearly, dispersion is a key process in non-linear electrokinetics, and the resolution of ITP in particular. Figure 4 shows an example where our simulation is used to design a buffering system for the focusing, separation, and detection of phenolic compounds. Figure 3. Comparison of experimental measurements and numerical predictions of analyte width (full width at half maximum of area averaged values) versus normalized current. At high currents the proposed effective diffusivity becomes proportional to the electric field leading to a nearly constant interface width. This is in contrast to the non-dispersive model (dashed line) where the width monotonically decreases, leading to more than an order of magnitude error in the width prediction.

Phenol Vanillic acid p-Cresol

Figure 4. Predicted isotachopherogram showing the separation of phenolic compounds in a 50 μm diameter capillary under constant current of 5μA. LE is 100 mM hydrochloric acid and 150 mM Amylamine as counterion (pH 9.2). TE is 22 mM Tetraphenylborate. Computation time here was 83 s using 150 grid points and six reacting electrolytes.

REFERENCES [1] Jung B., et al., Analytical chemistry 78. 7 (2006):2319-27. [2] Khurana, T.K, and Santiago J.G., Analytical chemistry 80. 1 (2008):279-86. [3] Hruska V., Jaros M. and Gas. B., Electrophoresis 27, 5-6 (2006):984-991. [4] Ikuta, N. and Hirokawa T., Journal of chromatography 802, 1 (1998):49-57. [5] Palusinski O.A., et al., AIChE Jounral 32 (1986):215:223. [6] Lele S.K., Journal of computational physics 103, l6-42 (1992):16-42. [7] Mosher, R.A., et al., Journal of chromatography 716, 1-2 (1995):17-26. [8] Saville, D.A., Electrophoresis 11, 11 (1990):899-902. Twelfth International Conference on Miniaturized Systems for Chemistry and Life Sciences October 12 - 16, 2008, San Diego, California, USA

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