# Interdisciplinary Applied Mathematics

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r, _ eAs,i

(yst ,i —    i

xH,i

where e is the permittivity of the fluid in the channel, Asy is the inner surface area of the ith channel, and xHy is the thickness of the stern layer. The capacitance of the diffuse layer, Cdlji, is given by the expression (Davies and Rideal, 1966)

Cdl,

JT

iAS

where jTy is the intrinsic surface charge density on the channel wall, kB is Boltzman’s constant, T is the temperature, z is the valence of the counterion, e is the charge of an electron, c is the concentration of the counterion in the bulk solution, and R is the universal gas constant. The capacitance of the wall for a cylindrical channel, Cwall i, is given by the expression

Cwall,i

cAs у о!п(^)’

where ri    is    the    inner radius    of the    channel    and ro is    the    outer    radius of the

channel.

When no potential difference is applied across the channel wall, no charge is induced in the channel wall. As a result, the capacitance of the channel wall can be neglected in the computation of the effective capacitance.

FIGURE 18.2. (a) A typical cross-shaped channel segment of a microfluidic system. The electrical    potentials,    ф1-4,    are    given.    Vi    and    V2    are    the transverse

applied potentials. (b) The electrical network representation for the cross-shaped channel. Rch,i-4    are    the electrical    resistances,    гф1-4    are    the    surface    potentials    of

the channel walls, and Ceff,i-4 are the capacitances of the EDLs.

For example,    there is    no    wall    capacitance    for    i = 2,    3, 4, since    there is    no

applied voltage across the channel, as shown in Figure 18.2(a). Typically, the capacitance of the stern layer is much higher than the capacitance of the diffuse layer (Oldham and Myland, 1994). Also, when capacitances are connected in series (as in this case) , the capacitance with the smaller value dominates. Therefore, in most cases the effective capacitance, Ceff, can be approximated by the diffuse layer capacitance, Cdl. The effective capacitance can be related to the surface potential by the expression

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