5.5 Equivalent circuits for flow and current in electrolyte-filled microchannels [electrodynamics top]

While a complete description of fluid and charge transport in a microchannel typically involves a relatively complicated analysis, 1D models often provide an approximate solution that guides intuition or system design. We treat microchannels as 1D elements and their intersections as nodes, and model both the electrical properties and the fluid flow properties with circuit-like elements.

This approach is well suited for analyzing microfluidic devices for several reasons. First, microchannels have lengths much longer than their diameters, so the fully-developed flow approximation is accurate. Second, complicated microchannel networks are ubiquitous in microsystems.

5.5.1 Electrical circuit equivalents of hydraulic components

To describe the current and voltage in a microsystem using 1D models, we describe microchannels in terms of equivalent circuit elements and replace the microchannel system with an electrical circuit. We model microchannels as resistors, junctions as nodes, and electrodes ascapacitors. The governing equation isOhm’s law (ΔV = IR or Δ = ) and the boundary conditions are given byKirchoff’s current law at each node.

Electrical circuit equivalent of microchannels Our 1D model of a microchannel is that of aresistor with resistance R, specified both by the geometry and the fluid conductance σ:

Where L is the length of the channel, σ is the bulkconductivity of the fluid, and A is the cross-sectional area. The SI units for conductivity are ohms per meter Ω^-1m^-1, but it is more commonly reported in terms of μS∕cm or mS∕m, where a Siemens is an inverse Ohm (1 S = 1Ω^-1).

For microchannels with discrete changes in cross-sectional area, we can describe them with several discrete resistors. For systems with cross-sectional areas or conductivities that vary continuously and slowly, we can use a 1D integral relation:

(5.77)

where x₁ and x₂ denote the locations of the beginning and ending of the microchannel (L = x₂ -x₁). Here, σ and A are assumed to be functions of x.

Electrical circuit equivalent of microchannel intersections In a 1D model, we treat the intersection between channels as a node, and apply Kirchoff’s law:

where I denotes current into the node.

Electrical circuit equivalent of electrodes Electrodes denote the interface between a solid conductor and an electrolyte solution. Fields are applied to our microchannel system by connective voltage sources to metal wires inserted into reservoirs or micropatterned metal electrodes on the surfaces of the microdevices. The interface (described in more detail in Chapter 16) has acapacitance, which can be roughly approximated by

Here C is the capacitance, A_electrode is the area of the electrode surface, ε is the electrical permittivity of the fluid, and λ_D is the Debye length of the electrolyte solution, which is typically between 1–100 nm (see Chapter 9). The reactions that take place at the electrode—consuming or creating electrons in the electrode through chemical reactions in the electrolyte that consume or create ions—lead to an effective resistance, as well, which can be predicted with electrode kinetics models and is a function of the catalytic behavior of the electrode (Section 5.2.2). For microfluidic systems, which tend to have high resistances, the resistance of the electrode can often be neglected.

From the above relations, we see that potentials applied to microchannels can be modeled (if the microchannel is well approximated by a 1D model) as if they were being applied to resistor/capacitor systems. An example of this for a four-port microfluidic device is shown in Figures 5.18 and 5.19.

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