^{1}, Partha P. Gopmandal

^{1}, Tobias Baier

^{2,a)}and Steffen Hardt

^{2}

### Abstract

A particular mode of isotachophoresis (ITP) employs a pressure-driven flow opposite to the sample electromigration direction in order to anchor a sample zone at a specific position along a channel or capillary. We investigate this situation using a two-dimensional finite-volume model based on the Nernst-Planck equation. The imposed Poiseuille flow profile leads to a significant dispersion of the sample zone. This effect is detrimental for the resolution in analytical applications of ITP. We investigate the impact of convective dispersion, characterized by the area-averaged width of a sample zone, for various values of the sample Péclet-number, as well as the relative mobilities of the sample and the adjacent electrolytes. A one-dimensional model for the area-averaged concentrations based on a Taylor-Aris-type effective axial diffusivity is shown to yield good agreement with the finite-volume calculations. This justifies the use of such simple models and opens the door for the rapid simulation of ITP protocols with Poiseuille counterflow.

T. Baier and S. Hardt kindly acknowledge support by the German Research Foundation (DFG) through the Cluster of Excellence 259. S. Hardt acknowledges helpful discussions with Juan Santiago, Stanford University.

I. INTRODUCTION

II. GOVERNING EQUATIONS

III. NUMERICAL METHODS

IV. TAYLOR-ARIS DISPERSION MODEL

V. RESULTS AND DISCUSSION

VI. CONCLUSIONS

### Key Topics

- Carrier mobility
- 19.0
- Electrolytes
- 19.0
- Electric fields
- 17.0
- Diffusion
- 13.0
- Poiseuille flow
- 12.0

## Figures

Sketch of the situation considered. A single sample zone is sandwiched between a leading (LE) and trailing (TE) electrolyte. The sample ions migrate from left to right at velocity U ITP in an applied electric field. A pressure driven flow from right to left is applied that exactly counters the electromigration, such that the sample plug is stationary. The channel length, L = L t + L s + L l , is the sum of lengths occupied by the leading and trailing electrolytes as well as the sample. We will consider L s ≪ L t , L l .

Sketch of the situation considered. A single sample zone is sandwiched between a leading (LE) and trailing (TE) electrolyte. The sample ions migrate from left to right at velocity U ITP in an applied electric field. A pressure driven flow from right to left is applied that exactly counters the electromigration, such that the sample plug is stationary. The channel length, L = L t + L s + L l , is the sum of lengths occupied by the leading and trailing electrolytes as well as the sample. We will consider L s ≪ L t , L l .

Sample concentration obtained in the 2D model for k 1( = μ l /μ t = D l /D t ) = 3, k 2( = μ l /μ s = D l /D s ) = 2 with channel depths of (a) H = 10 μm (Pe = 40) and (b) H = 30 μm (Pe = 120). The amount of sample is . Both coordinates axes show the length in units of 10−5 m, but were rescaled differently for better visualization of the sample distribution.

Sample concentration obtained in the 2D model for k 1( = μ l /μ t = D l /D t ) = 3, k 2( = μ l /μ s = D l /D s ) = 2 with channel depths of (a) H = 10 μm (Pe = 40) and (b) H = 30 μm (Pe = 120). The amount of sample is . Both coordinates axes show the length in units of 10−5 m, but were rescaled differently for better visualization of the sample distribution.

Effect of the channel height on the area-averaged sample concentration profile and comparison with the corresponding profile obtained in the Taylor-Aris dispersion model for fixed values of the diffusivity ratios, i.e., k 1( = μ l /μ t = D l /D t ) = 3 and k 2( = μ l /μ s = D l /D s ) = 2 when the depth of the channel is chosen as H = 10 μm and H = 30 μm, respectively. The amount of sample is . Dashed lines represent the results obtained from the 2D model and solid lines corresponding results from the 1D model.

Effect of the channel height on the area-averaged sample concentration profile and comparison with the corresponding profile obtained in the Taylor-Aris dispersion model for fixed values of the diffusivity ratios, i.e., k 1( = μ l /μ t = D l /D t ) = 3 and k 2( = μ l /μ s = D l /D s ) = 2 when the depth of the channel is chosen as H = 10 μm and H = 30 μm, respectively. The amount of sample is . Dashed lines represent the results obtained from the 2D model and solid lines corresponding results from the 1D model.

Area-averaged sample concentration profile obtained in the 2D simulation for different values of the diffusivity of sample electrolyte, i.e., k 2( = μ l /μ s = D l /D s ) = 1.1 (Pe = 55), k 2 = 2 (Pe = 100), k 2 = 2.7 (Pe = 135). Here, the LE to TE diffusivity ratio, i.e, k 1( = μ l /μ t = D l /D t ) = 3 and channel depth is H = 25 μm. The amount of sample is (a) ; (b) . A comparison with the corresponding profile obtained in the Taylor-Aris dispersion model is also made. Dashed lines represent the results obtained from the 2D model and the solid line represents corresponding results from the 1D model.

Area-averaged sample concentration profile obtained in the 2D simulation for different values of the diffusivity of sample electrolyte, i.e., k 2( = μ l /μ s = D l /D s ) = 1.1 (Pe = 55), k 2 = 2 (Pe = 100), k 2 = 2.7 (Pe = 135). Here, the LE to TE diffusivity ratio, i.e, k 1( = μ l /μ t = D l /D t ) = 3 and channel depth is H = 25 μm. The amount of sample is (a) ; (b) . A comparison with the corresponding profile obtained in the Taylor-Aris dispersion model is also made. Dashed lines represent the results obtained from the 2D model and the solid line represents corresponding results from the 1D model.

Sample concentration obtained in the 2D model for k 1( = μ l /μ t = D l /D t ) = 3 and k 2( = μ l /μ s = D l /D s ) = 2 when (a) , (b) , (c) , and (d) with . The channel depth is H = 25 μm. Note that both the coordinate axes are multiplied by a factor 10−5 m.

Sample concentration obtained in the 2D model for k 1( = μ l /μ t = D l /D t ) = 3 and k 2( = μ l /μ s = D l /D s ) = 2 when (a) , (b) , (c) , and (d) with . The channel depth is H = 25 μm. Note that both the coordinate axes are multiplied by a factor 10−5 m.

Comparison of the area-averaged sample concentration profiles obtained in the 2D simulation with the corresponding profile obtained in the Taylor-Aris dispersion model for fixed values of k 1( = μ l /μ t = D l /D t ) = 2 and k 2( = μ l /μ s = D l /D s ) = 1.5 when (a) , (b) , (c) , and (d) with . The depth of the channel is H = 25 μm. The solid line represents the results obtained from 1D model and dashed lines from 2D model.

Comparison of the area-averaged sample concentration profiles obtained in the 2D simulation with the corresponding profile obtained in the Taylor-Aris dispersion model for fixed values of k 1( = μ l /μ t = D l /D t ) = 2 and k 2( = μ l /μ s = D l /D s ) = 1.5 when (a) , (b) , (c) , and (d) with . The depth of the channel is H = 25 μm. The solid line represents the results obtained from 1D model and dashed lines from 2D model.

Variation of the (a) standard deviation and (b) skewness of the sample distribution with k 2( = μ l /μ s = D l /D s ) for a fixed value of k 1( = μ l /μ t = D l /D t ) = 2, 3, 4, and 5. The amount of sample is . Lines with unfilled symbols represent the results from the 1D model (Taylor-Aris model), lines with filled symbols those from the 2D model.

Variation of the (a) standard deviation and (b) skewness of the sample distribution with k 2( = μ l /μ s = D l /D s ) for a fixed value of k 1( = μ l /μ t = D l /D t ) = 2, 3, 4, and 5. The amount of sample is . Lines with unfilled symbols represent the results from the 1D model (Taylor-Aris model), lines with filled symbols those from the 2D model.

Variation of the (a) standard deviation and (b) skewness of the sample distribution with k 2( = μ l /μ s = D l /D s ) for a fixed value of k 1( = μ l /μ t = D l /D t ) = 3. The Péclet number is Pe = 50k 2, the depth of the channel H = 25 μm. The amount of sample is and 4 μm. Lines with unfilled symbols represent the results from the 1D model (Taylor-Aris model), lines with filled symbols those from the 2D model.

Variation of the (a) standard deviation and (b) skewness of the sample distribution with k 2( = μ l /μ s = D l /D s ) for a fixed value of k 1( = μ l /μ t = D l /D t ) = 3. The Péclet number is Pe = 50k 2, the depth of the channel H = 25 μm. The amount of sample is and 4 μm. Lines with unfilled symbols represent the results from the 1D model (Taylor-Aris model), lines with filled symbols those from the 2D model.

Variation of the (a) standard deviation and (b) skewness of the sample distribution with the channel depth, where k 1(= μ l /μ t = D l /D t ) = 3, k 2( = μ l /μ s = D l /D s ) = 2, and . The Péclet number varies with the channel depth as Pe = 100· H/(25 μm).

Variation of the (a) standard deviation and (b) skewness of the sample distribution with the channel depth, where k 1(= μ l /μ t = D l /D t ) = 3, k 2( = μ l /μ s = D l /D s ) = 2, and . The Péclet number varies with the channel depth as Pe = 100· H/(25 μm).

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