Optimizing Band Dispersion in an Electroosmotic Flow Through a Curved Microchannel

This model uses the Optimization Module to minimize the turn induced dispersion of a chemical species in an electroosmotic flow through a curved channel (Ref. 1, Ref. 2, and Ref. 3). The dispersion is mainly caused by two factors: the difference in path length between the inner and outer boundaries of the curved channel, and the difference in electric field magnitude along these two boundaries. The curve induced dispersion may impede the ability to detect the species. The downstream concentration profile produced by the optimized geometry has a maximum concentration almost three times larger than that produced by the original geometry.

Model Definition

The original geometry of the device is shown in Figure 1. The width of the channel is 100 μm and the outer radius ro for the curved channel is 150 μm. The length L for the straight section of the microchannel is 1 mm. The chemical species band is introduced at x = −600 μm and modeled as the Gaussian pulse with a peak concentration of 1 mol/m3.

The solute band is carried by electroosmosis through the curved geometry. The steady state electroosmotic flow is modeled by solving the Creeping-flow equations with the Helmholtz-Smoluchowski slip boundary condition applied at the channel walls:

where ueo is the electroosmotic slip velocity, ε0 is the permittivity of free space, εr is the relative permittivity of the solution, μ is the dynamic viscosity of the fluid, ζ is the zeta potential and Et is the tangential component of the electric field as the wall. The electric field required to estimate the slip velocity is computed by solving Laplace’s equation.

Figure 1: The initial model geometry is a constant width microchannel with a 180-degree turn.

The solute transport through the curved channel is modeled using the transient convection-diffusion equation. The diffusivity, D, of the species is taken as 1·10−11 m2/s, which results in a Peclet number of roughly 6100, based on the slip velocity and width, W, of the channel (Pe = ueoW/D).

The first study models the dispersion in the solute band through the constant radius geometry with a 180-degree turn. In the second study the Optimization Module is used to optimize the geometry for minimal solute band dispersion. The inner channel curve is represented by a Bézier curve (Ref. 4). Five parameters are chosen to optimize the geometry including the inner radii, control points for the Bézier curve and their corresponding weights. A more complicated geometric discretization could be achieved by employing multiple Bézier curves. The Bézier curve parameters are used as the optimization variables with the objective of minimizing the solute dispersion. The solute dispersion for the given Peclet number can be related to the difference in time taken by the solute molecules to traverse along inner and outer edges of microchannel. Hence the following objective function is utilized to arrive at the optimal design:

where, tin and tout are the times taken by the solute to move around the curve along the inner and outer walls respectively. Using this simple objective function means that only the steady state solution of the velocity field is required for the optimization step.

Results and Discussion

The velocity field magnitude (surface plot) and the electric potential (contours) are plotted for the original geometry in Figure 2. The equipotential lines are closer along the inner curve which results in a non-uniform electric field and velocity field in the curved section. The solute band traverses inner edge more quickly and consequently there is substantial dispersion (see Figure 3).

The velocity magnitude is almost uniform across the cross section of the microchannel in the optimized geometry shown in Figure 4. Additionally the tapered section in the optimized geometry increases the path length along the inner microchannel edge. The concentration surface plot in Figure 5 shows the reduction in solute dispersion in the optimized geometry.

Figure 2: Velocity magnitude (surface) and electric potential (contour) for the original geometry.

Figure 3: Concentration surface plot in the original geometry showing dispersion due to curved channel.

Figure 4: Velocity magnitude (surface plot) and electric potential (contour plot) in the optimized geometry.

Figure 5: Concentration surface plot illustrating the optimized design for minimized dispersion.

Figure 6 compares the average concentration passing through the detector (domain 3) for original as well as optimized geometry. The obtained average concentration intensity peak is almost three times higher for the optimized geometry compared to the original geometry for the given Peclet number.

Figure 6: The average concentration through the detector for the original and optimized geometries.

References

1. S.K. Griffiths and R.H. Nilson, “Low-Dispersion Turns and Junctions for Microchannel Systems,” Analytical Chemistry, vol. 73, no. 2, pp. 272–278, 2001.

2. S.K. Griffiths and R.H. Nilson, “Band Spreading in Two-Dimensional Microchannel Turns for Electrokinetic Species Transport,” Analytical Chemistry, vol. 72, no. 21, pp. 5473–5482, 2000.

3. J.L. Molho, A.E. Herr, B.P. Mosier, J.G. Santiago, T.W. Kenny, R.A. Brennen, G.B. Gordon, and B. Mohammadi, “Optimization of Turn Geometries for Microchip Electrophoresis,” Analytical Chemistry, vol. 73, no. 6, pp. 1350–1360, 2001.

4. M. Jain, A. Rao, and K. Nandakumar, “Study on Groove Shape Optimization for Micromixers”, presented at COMSOL Conference 2012, Boston.

Microfluidics_Module/Fluid_Flow/microchannel_dispersion_optimization

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Global Definitions

Geometry parameters

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Model parameters

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Geometry 1

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Materials

Water

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Property	Variable	Value	Unit	Property group
Electrical conductivity	sigma_iso ; sigmaii = sigma_iso, sigmaij = 0	1e-3	S/m	Basic
Relative permittivity	epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0	er_w	1	Basic
Density	rho	rho0	kg/m³	Basic
Dynamic viscosity	mu	mu0	Pa·s	Basic

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Wall 1

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Initial Values 1

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Variables 1

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Name	Expression	Unit	Description
cini	c0gp1(x[1/m])(y>0)	mol/m³	Concentration Gaussian Pulse
t_in	intop1(1/spf.U)	s	Time taken along inner curve
t_out	intop2(1/spf.U)	s	Time taken along outer curve
c_avg	aveop1(c)	mol/m³	Average concentration at detector surface