Research Papers: Fundamental Issues and Canonical Flows

Effects of Convection on Isotachophoresis of Electrolytes

[+] Author and Article Information
Partha P. Gopmandal

Department of Mechanical and
Materials Engineering,
Washington State University,
Pullman, WA 99164-2920
Department of Mathematics,
National Institute of Technology Patna,
Patna, Bihar 800005, India

S. Bhattacharyya

Department of Mathematics,
Indian Institute of Technology Kharagpur,
Kharagpur, West Bengal 721302, India
e-mail: somnath@maths.iitkgp.ernet.in

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received May 13, 2014; final manuscript received January 19, 2015; published online March 30, 2015. Assoc. Editor: Prashanta Dutta.

J. Fluids Eng 137(8), 081202 (Aug 01, 2015) (12 pages) Paper No: FE-14-1248; doi: 10.1115/1.4029888 History: Received May 13, 2014; Revised January 19, 2015; Online March 30, 2015

In this study, the form of the analytes distribution in isotachophoresis (ITP) in the presence of a convective flow is analyzed in a wide rectangular microchannel. The imposed convection is considered due to a mismatch of electroosmotic (EO) slip velocity of electrolytes of different electrophoretic mobilities. We compute the two-dimensional (2D) Nernst–Planck equations coupled with the Navier–Stokes equations for fluid flow and an equation for electric field. We use a control volume method along with a higher-order upwind scheme to capture the sharp variation of variables in the transition zones. The convection of electrolytes produces a smearing effect on the steepness of the electric field and ion distribution in the interface between two adjacent electrolytes in the ITP process. The dispersion of the interface in plateau-mode and the sample in peak-mode is analyzed through the second- and third-order moments. The dispersion due to nonuniform EO flow (EOF) of electrolytes is found to be different from the case when the dispersion is considered only due to an external pressure driven Poiseuille flow. The nonuniform EOF of electrolytes produces less dispersion and skewness in the sample distribution when the molecular diffusivity of the sample ionic species is close to the harmonic mean of the diffusivity of adjacent electrolytes. We find that the EOF may become advantageous in separating two analytes of close diffusivity. Our results show that the one-dimensional (1D) Taylor–Aris model is suitable to predict the dispersed ITP when the average convection speed of electrolytes is in the order of the ITP speed.

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Grahic Jump Location
Fig. 1

Distribution of sample ions in the microchannel having height H = 25 μm for Dt = Dl∕3, Ds = Dl∕2, and Dl = 7×10-10 m2/s when initial molar mass (a) Ms = 40 μm × HCl∞; (b) Ms = 10 μm × HCl∞. The bulk concentration of LE is taken as Cl∞ = 0.001 M and current density is 270.2 A∕m2.

Grahic Jump Location
Fig. 2

Comparison of computed solution for the width of the transition zone in ideal ITP with the analytical results when H = 25 μm and Dl = D0 = 7.0 × 10-10 m2/s. We have shown the variation of width of the transition zone with E0 for fixed Dr = 0.7.

Grahic Jump Location
Fig. 3

Normalized axial velocity contours and slip velocity along the wall when EOF of LE and TE is considered with u¯ = 3 × UITP. Here Dr = 0.7,Dl = D0 = 7 × 10-10 m2/s, j0 = 445.03 A/ m2, and H = 25 μm. (a) μtEOF = 2.84 × 10-8 m2/(Vs) and μlEOF = 1.22 × 10-7 m2/(Vs), which leads to vr = utEOF/ulEOF = 1/3; (b) μtEOF = 8.53 × 10-8 m2/(Vs),μlEOF = 4.06 × 10-8 m2/(Vs) corresponds to vr = 3.

Grahic Jump Location
Fig. 8

Variation of standard deviation (σ) for interaction of EOF on ITP. Here Dl = D0 = 7.0 × 10-10 m2/s,E0=105 V/m and channel height H = 25 μm. (a) Variation of σ with EO slip velocity ratio vr. Here Dr = 0.7,j0 = 445.03 A/ m2, and UITP = 2.23 × 10-3 m/s. The EO mobility of LE is taken as μlEOF = 3.54 × 10-8 m2/(Vs) and μtEOF is adjusted by changing the value of vr. (b) Variation of σ with diffusivity ratio Dr. The EO mobility of LE and TE is taken to be same as μtEOF = μlEOF = 3.54 × 10-8 m2/(Vs).

Grahic Jump Location
Fig. 9

Variation of the standard deviation when ITP is interacted with (a) Poiseuille flow EO mobility of LE, i.e., μlEOF,u¯ = αUITP as well as (b) EOF by varying the EO mobility of LE, i.e., αμlEOF with vr = 3. Here the channel height H = 25 μm and UITP = 2.23 × 10-3 m/s,Dr = 0.7,j0 = 445.03 A/ m2; the diffusivity of LE and common ions is taken as Dl = D0 = 7 × 10-10 m2/s. For EOF dispersion case, EO slip velocity ratio is considered to be vr = 3, the EO mobility of TE is taken as μtEOF = 3.54 × 10-8 m2/(Vs).

Grahic Jump Location
Fig. 10

Comparison of area averaged concentration profile of LE and TE with 1D Taylor–Aris dispersion model within the transition zone in dispersed ITP for different values of the average convection speed u¯ = αUITP,α = 1,3, and 5. (a) Dispersion due to Poiseuille flow. Solid line, computed results; dashed lines with symbols, Taylor–Aris dispersion model and (b) dispersion due to mismatch of EOF. Solid line, computed results when vr = 3; dashed lines, computed results when vr = 0.33; dashed lines with symbols, Taylor–Aris dispersion model.

Grahic Jump Location
Fig. 11

Comparison of the area averaged sample concentration of our 2D model with the 1D Taylor–Aris model and the analytical solution proposed by Garcia-Schwarz et al. [20] when Ds = 0.5Dl,Dt = 1/3Dl, and Dl = 7.0 × 10-10 m2/s. The EO mobility of TE and LE is taken to be same as μtEOF = μlEOF = 3.59 × 10-8 m2/(Vs). We consider three different values of the EO mobility of sample electrolyte. (a) μsEOF = 0.898 × 10-8 m2/(Vs); (b) μsEOF = 3.59 × 10-8 m2/(Vs); and (c) μsEOF = 7.19 × 10-8 m2/(Vs).

Grahic Jump Location
Fig. 12

Comparison of the area averaged sample concentration of our 2D model with the 1D Taylor–Aris model and the analytical solution proposed by Garcia-Schwarz et al. [20] when Dt = 1/3Dl,Dl = 7.0 × 10-10 m2/s and Ds is either close to Dt or Dl. (a) Ds = 0.4Dl; (b) Ds = 0.8Dl. The EO mobility of TE, sample electrolyte, and LE is taken to be same as μtEOF = μsEOF = μlEOF = 3.59 × 10-8 m2/(Vs).

Grahic Jump Location
Fig. 14

Variation of the (a) standard deviation and (b) skewness of the sample distribution with (k2 − 1)/(k1 − 1). Here the parameter (k2 − 1)/(k1 − 1) is changed by changing k2 for fixed k1. The diffusivity of LE is taken to be Dl = 7.0 × 10-10 m2/s and k1 = Dl/Dt and k2 = Dl/Ds. The EO mobility of TE, sample electrolyte, and LE is taken to be same as μtEOF = μsEOF = μlEOF = 3.59 × 10-8 m2/(Vs).

Grahic Jump Location
Fig. 7

Variation of area averaged axial electric field Ex¯ within transition zone for ideal ITP and dispersed ITP cases with Dr = 0.7,j0 = 445.03 A/ m2 and H = 25 μm. We have considered the dispersion due to Poiseuille and mismatch of EOF of TE and LE with average convection speed u¯ = 3 × UITP = 6.69 × 10-3 m/s. For ITP with EOF case we considered μ-EOF = 2.84 × 10-8 m2/(Vs),μlEOF = 1.22 × 10-7 m2/(Vs) which yields vr = 1∕3 and μtEOF=8.53 × 10-8 m2/(Vs),μlEOF = 4.06× 10-8 m2/(Vs) yields vr = 3.

Grahic Jump Location
Fig. 6

Variation of the scaled concentration of LE (cl) within the transition zone for ITP without imposed fluid convection and dispersed ITP when Dr = 0.7,j0 = 445.03 A/ m2, and H = 25 μm. (a) ITP without imposed convection (u¯ = 0); (b) dispersion due to Poiseuille flow with α = 3; (c) Dispersion due to EOF with μtEOF = 2.84 × 10-8 m2/(Vs),μlEOF = 1.22 × 10-7 m2/(Vs) which yields u¯ = 3 × UITP(α = 3) and vr = 1∕3; (d) dispersion due to EOF with μtEOF = 8.53 × 10-8 m2/(Vs), μlEOF = 4.06 × 10-8 m2/(Vs) which yields u¯ = 3 × UITP(α = 3) and vr = 3. For (b)–(d); the average convection speed is u¯ = 3 × UITP = 6.69 × 10-3 m/s.

Grahic Jump Location
Fig. 5

Time evolution of standard deviation (σ, in μm) for interaction of EOF on ITP. Here Dr = 0.7,Dl = D0 = 7.0 × 10-10 m2/s,j0 = 445.03A/ m2; channel height H = 25 μm, and UITP = 2.23×10-3m/s with average slip velocity u¯ = 3UITP when EO slip velocity ratio between two adjacent electrolytes is vr = 3 and 1/3.

Grahic Jump Location
Fig. 4

Comparison of wall velocity is with Baier et al. [12]. EO mobility of TE and LE is μtEOF = 9.912 × 10-8 m2/(Vs) and μtEOF = 3.54 × 10-8 m2/(Vs),j0 = 0.75 kA/ m2, and αtot = 1/2(ulEOF-utEOF).

Grahic Jump Location
Fig. 13

Variation of the (a) standard deviation and (b) skewness of the sample distribution with Dt = 1/3Dl,Dl = 7.0 × 10-10 m2/s by changing the diffusivity of sample species. The EO mobility of TE, sample electrolyte, and LE is taken to be same as μtEOF = μsEOF = μlEOF = 3.59 × 10-8 m2/(Vs).



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