# Interdisciplinary Applied Mathematics

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layer (EDL), capacitive elements also need to be included in modeling the electrical domain. The EDL can be decomposed into the stern layer and the diffuse    layer.    As    the    stern    layer    and    the    diffuse    layer    store charge,

the capacitance associated with these layers is important. In addition, the capacitance of the channel wall, which arises due to a potential difference across the channel wall, needs to be taken into account. The electrical resistance of the EDL can be safely neglected, since the effective resistance of the EDL is much higher than the resistance of the channel filled with buffer (Hayes and Ewing, 2000). Figure 18.2(a) and Figure 18.2(b) illustrate a typical cross-shaped channel segment (this is similar to the cross shapes formed by S1, S2, M1, S3 or B1, B2, B3, Ml or B4, B5, B6, M2 or S4, S5, S6, M2 in Figure 18.1) in a microfluidic system and its circuit representation, respectively.

The electrical resistance of a solution-filled simple straight channel is given by the expression

psol,iLi

AcA

R

ch,i

where psol,i is    the    electrical    resistivity of    the    solution    in    the    ith    channel,

i = 1, 2,… 4    (see    Figure    18.2(b)),    Li    is the    length of the    ith    channel,    Acy

is the cross-sectional area of the ith channel, and Rch,i is the electrical resistance of the ith channel.

The expression for the effective capacitance, shown in Figure 18.2(b), is given by

(Ceff ,i)-1

(Cst,i)-1 + (Cdl,i)

1

(Cwall,i)

1

where Cst,i is the capacitance of the stern layer of the ith channel, Cdly is the capacitance    of    the    diffuse layer    of the    ith    channel,    and    Cwalti    is the

capacitance of the ith channel wall; Cstii is given by the expression (Oldham and Myland, 1994)

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