Transport in an Electrokinetic Valve

This tutorial presents an example of pressure-driven flow and electrophoresis in a microchannel system.

Researchers often use a device similar to the one in this example as an electrokinetic sample injector in biochips to obtain well-defined sample volumes of dissociated acids and salts and to transport these volumes. The model presents a study of a pinched injection cross valve during the focusing, injection, and separation stages. Inspiration for the model comes from a study by Ermakov and others (Ref. 1). Focusing is obtained through pressure-driven flow of the sample and buffer solution, which confines the sample in the focusing channel. When the system reaches steady state, the pressure-driven flow is turned off and an electric field is applied along the channels. This field drives the dissociated sample ions in the focusing zone at right angles to the focusing channel and through the injection channel. A clean separation of the sample ions is important, so the model examines the effect on ion separation of different configurations of the electric field.

This specific case does not account for electroosmosis because the channel surfaces are subjected to a treatment that minimizes the extension of the electric double layer.

This model requires either the CFD Module, the Microfluidics Module, or the Subsurface Flow Module

Model Definition

Figure 1 shows a 2D cross section of the geometry in the xz-plane and points out the different channels and boundaries. The horizontal channel serves as the focusing channel, while the vertical channel is the injection channel. The actual model is in 3D with rectangular pipes whose corners are rounded. For geometry dimensions refer to Table 1 below.

Table 1: Model Dimensions.
	Horizontal channel	Vertical channel	crossing area
Dimensions (μm)
- x	340	20	28
- y	20	20	20
- z	20	340	28
Position (μm)
- x	-100	0	-4
- y	0	0	0
- z	0	-200	-4
Rounding (μm)
- radius	4	4	4
- direction	in	in	out

Figure 1: The focusing stage involves pressure-driven flow of both the sample and the buffering solution. The device applies an electric field over the focusing channel.

The device operation and hence the modeling procedure takes place in two stages: focusing and injection.

In the focusing stage, the device injects a buffering solution through pressure-driven convection into the vertical channels from the top and bottom. At the same time, it forces the sample solution through the horizontal focusing channel (see Figure 1). The buffering solution neutralizes the acids contained in the sample except for a very thin region confined to the crossing between the horizontal and vertical channels. This means that the dissociated ions are only in a needle-shaped region in the focusing zone.

Next, in the injection stage the device turns off the convective flow and then applies a vertical field to migrate the sample from the focusing channel to the injection point at the lower end of the vertical channel. The sample ions are negatively charged and migrate in opposite direction to the electric field. This model studies two different configurations (See Table 2) for the applied electric field. In the first configuration (Injection stage, Mode A) electric field is only applied in the vertical direction. In the second configuration (Injection stage, Mode B) the electric field is applied in both the horizontal and vertical directions (Figure 2).The horizontal field focuses the sample during the initial part of the injection stage in order to obtain a well-separated sample.

Figure 2: During the injection stage, the device turns off convective flow and applies an electric field. The horizontal field avoids the broadening of the sample, while the vertical field injects the sample into the vertical channel in the direction opposite to the electric field.

Table 2: Applied Electric field configuration.
Inlet	Focusing stage	injection stage, Mode a	injection stage, Mode B
Sample inlet	Electric potential, V = -1V	Electric insulation	Electric potential, V = -1V
Outlet	Ground	Electric insulation	Ground
Upper buffer inlet	Electric insulation	Electric potential, V = -3.2V	Electric potential, V = -3.2V
Lower buffer inlet	Electric insulation	Ground	Ground

The model assumes that the charged sample concentration is very low compared to other ions dissolved in the solution. This implies that the sample concentration does not influence the solution’s conductivity and that you can neglect the concentration gradients of the charge-carrying species, which are present in a much higher concentration than the sample ions. Such an electrolyte is known as a supporting electrolyte.

Several equations describe the model: the Stokes flow equations, the equation for current balance, and a mass balance using the Nernst–Planck equation. This model uses the steady-state solution for the focusing stage as the initial condition for the injection stages.

Now consider the formulation of the model equations.

The Focusing Stage

The Stokes flow equations give the global mass a momentum balance in the focusing stage:

In these equations, h denotes the dynamic viscosity (SI unit: kg/(m·s)), u is the velocity (SI unit: m/s), p is the pressure (SI unit: Pa).

The total balance of charges for a supporting electrolyte comes from the divergence of the current-density vector, which in a supporting electrolyte is given by Ohm’s law:

Here κ is the electrolyte’s conductivity (SI unit: S/m) and V is the potential (SI unit: V). The balance of current at steady state then becomes

which gives

The flux vector for the sample ions comes from the Nernst-Planck equation

which leads to the following mass balance equation at steady state for species i:

Here ci is the concentration (SI unit: mol/m3), Di represents the diffusivity (SI unit: m2/s), zi equals the charge number (which equals 1 for this model), umi is the mobility (SI unit: s·mol/kg), and F is Faraday’s constant (SI unit: C/mol).

For the pressure-driven flow, assume that the flow has fully developed laminar form in all inlets, that all sides have no slip conditions, and that the fluid flows freely out from the end of the focusing channel.

The boundary conditions for the charge balance determine the potential at the respective inlet and outlet boundary

where i denotes the index for each boundary. This model also assumes that all wall boundaries are insulating:

The boundary conditions for the mass balance of the sample during the focusing stage appear below. The equation

gives the concentration at the inlet of the sample, while the equation

gives the concentration of the buffer at both boundaries of the vertical channel. At the outlet boundary, convection and migration are the dominating transport mechanisms (that is, diffusion is negligible), so that

The Injection Stage

In the injection and separation stages, the device turns the flow off and changes the configuration of the electric field. You again solve the charge-balance equations but with new boundary conditions:

The mass balance for the dilute species comes from a time-dependent mass balance:

The model assumes that the convective contribution is zero.

The boundary conditions for the current-balance equation imply that the potential is locked at all boundaries except for the walls,

Further assume the walls are electrically insulated, which yields

As opposed to the focusing state, the boundary conditions for the mass balance are changed. In the injection stage, set the concentration at the inlet boundary:

For all other boundaries, assume that migration is the dominating transport mechanism, so that:

The time-dependent solution requires an initial condition for the mass balance, which you obtain from the steady-state solution of the focusing stage:

Results and Discussion

This example analyzes the focusing stage and two configurations for the injection stages. Recall that the first injection-stage configuration (Mode A) applies the electric field only over the injection channel while the inlet and outlet boundaries of the focusing channel are insulated; the second injection-stage configuration (Mode B) applies the electric field over both channels.

Figure 3 shows the steady-state concentration distribution during the focusing stage along with the distribution at the beginning of the injection stage. Note that the vertical flows from the upper and lower injection channels focus the concentration on a very narrow region near the crossing area of the channels. Further away from the crossing area, however, the concentration spreads again more equally over the channel.

Figure 3: The steady-state concentration distribution during the focusing stage and prior to the injection stage.

Figure 4 and Figure 5 compare the concentration distribution for the two configurations at two times, specifically 0.06 s and 0.12 s after the beginning of the injection stage. The figures on the left show that for Mode A the concentration boundary is practically stationary in the horizontal direction. Consequently, the vertical electric field can continuously draw ions from the focusing channel, which results in poor separation and a poorly defined sample volume of the substance. For Mode B the situation is very different. The horizontal electric field draws the concentration boundary to the left, and the channels separate rapidly. Consequently, this scheme draws a well-defined sample volume of the substance into the injection channel.

Figure 4: The concentration distribution at a time 0.06 s after starting the injection stage for the Mode A configuration (left) and Mode B configuration (right).

Figure 5: The concentration distribution at a time 0.12 s after starting the injection stage for the Mode A configuration (left) and Mode B configuration (right).

It is also possible to observe the difference between the two configurations if you look at the concentration along a line through the middle of the injection channel, examining it at several times after the start of the injection stage (Figure 6). The maximum concentration moves down the injection channel with time. The peaks are higher in the upper axis corresponding to Mode A, but they are much wider than for Mode B. A considerable amount of concentration appears at the left of the peak, and the sample remains attached to the focusing area — resulting in an unwanted distortion of the sample package. The narrow peaks of Mode B, on the other hand, form nice bell curves throughout the downward transport in the injection channel, resulting in a well-defined sample package.

Figure 6: Concentration profile for Mode A (top) and Mode B (bottom) along the injection channel at various time steps: 0 s, 0.06 s, 0.12 s, 0.18 s, 0.24 s, 0.30 s, 0.36 s, 0.42 s, 0.48 s, 0.54 s, and 0.6 s after initialization of the injection stage. The origin of the x-axis marks the centerline of the focusing channel.

This study illustrates that modeling is extremely valuable in the investigation of electrophoretic transport. You can vary the configuration of the potential to obtain even better focusing and injection stages for the valve under study.

Notes About the COMSOL Implementation

Interfaces

In COMSOL Multiphysics you define the model with the following physics interfaces:

•

The Creeping Flow interface solves the fluid flow in the channels governed by Stokes equations.

•

The Electric Currents interface solves the equation for current balance.

•

The Transport of Diluted Species interface solves the Nernst-Planck equation.

Computing the solution

The operation of the actual device proceeds in two stages, the focusing stage and the injection stage. This model simulates two settings of the injection stage so in total it works in three phases.

The first phase defines the domain settings and boundary conditions for the focusing phase. Then the model solves the interfaces sequentially with a nonlinear solver in the following sequence:

Creeping Flow interface

Electric Currents interface

Transport of Diluted Species interface

Each step uses the solution from the previous one. The model stores the last solution for use as the initial value for the consequent modeling.

In the second phase you change the domain settings and boundary conditions to handle the injection stage Mode A. In a real device you would turn off the convective flow; in the model you simulate this by setting the velocity parameters of the Electrokinetic Flow interface to zero. Thus it uses no information from the Laminar Flow interface.

Solving the second phase starts from the stored solution of the first phase, and the model solves the Electric Currents interface with a nonlinear solver. Then you select a time-dependent solver and solve the Transport of Diluted Species interface. This solution is the result for the injection stage Mode A.

In the third phase you again change domain settings and boundary conditions but this time for the injection stage Mode B; you then solve for the final solution the same way as in the second phase.

Reference

1. S.V. Ermakov, S.C. Jacobson, and J.M. Ramsey, Technical Proc 1999 Int. Conf. on Modeling and Simulation of Microsystems, Computational Publications, 1999.

Chemical_Reaction_Engineering_Module/Electrokinetic_Effects/electrokinetic_valve

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Property	Variable	Value	Unit	Property group
Electrical conductivity	sigma_iso ; sigmaii = sigma_iso, sigmaij = 0	1[S/m]	S/m	Basic
Relative permittivity	epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0	1	1	Basic
Density	rho	1e3[kg/m^3]	kg/m³	Basic
Dynamic viscosity	mu	1e-3[Pa*s]	Pa·s	Basic