# Interdisciplinary Applied Mathematics

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When designing integrated microfluidic systems of the type shown in Figure 18.1, some important objectives are to:

1.    Increase the throughput.

2.    Improve the homogeneity of the mixture.

3.    Obtain higher separation efficiency.

4.    Perform detection faster.

However,    it    may    not    be possible    to    attain    all    these    objectives,    and there

can be a trade-off leading to an optimized design. In this section, using the techniques discussed in Chapter 17, “easy-to-use” circuit and device models are presented, which can be used to explore the design space and select an optimal design for integrated microfluidic systems to perform various functions. The model development is illustrated using the example shown in Figure 18.1. The models are, however, general enough that they can be applied or extended to other microfluidic systems. The development of a compact/circuit model for fluid flow due to a combined pressure and electrical potential gradient is first discussed. The compact model is described in two parts, namely, the electrical model and the fluidic model.

###### 18.1.1 Electrical Model

For microfluidic devices that rely on the electrokinetic force as the driving force, the electric field must be computed first. In the case of electroosmotic flow (see Chapter 7 for details; here we restate only the essential equations to derive the circuit models), the potential field due to an applied potential can be computed by solving the Laplace equation:

У2ф = 0,    (18.1)

where ф is the electrical potential. Since equation (18.1) predicts a linear potential drop for simple straight channels, the potential variation can be represented by linear electrical resistances. In order to develop a complete circuit    that    takes    into    account    the    charge    stored in    the    electrical    double

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