Interdisciplinary Applied Mathematics

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(7.4)


У2(ф*) = в sinh(^*),

FIGURE 7.1. Schematic diagram of the electric double layer (EDL) next to a negatively charged    solid    surface. Here    ф is    the    electrokinetic    potential,    ф0    is    the


surface electric    potential,    Z    is    the    zeta    potential,    and y is    the distance    measured


from the wall. The Debye length and the EDL thickness are shown by Ad and EDL, respectively.



where ф*(= ф/Z) is the electrokinetic potential normalized with the zeta potential Z, and a is the ionic energy parameter given by



a = ezZ/кв T.



(7.5)



At 20°C, a =1 corresponds to 25.4 mV. The variable в relates the ionic energy parameter a and the characteristic channel length (flow dimension) h to the Debye-Hiickel parameter ш in equation (7.1) as follows:



в



(uih)2


a



We must note    that    for    small    zeta    potential    (Z <    25    mV),    it    is    possible    to


linearize the right-hand side of the Poisson-Boltzmann equation (7.4) via a Taylor series expansion. This results in the Debye-Hiickel approximation



У2(ф*) = фслф*.    (7.6)


In equation (7.4), we presented the Poisson-Boltzmann equation in non dimensional form. Let us consider a two-dimensional channel and assume that the zeta potential Z is known, and that it remains constant along the channel. Under these conditions equation (7.4) can be simplified in the following form:


= в sinh *),    (7.7)



dr/2


where n = y/h and h is the half-channel height. Multiplying both sides of this equation by (2-^-), and integrating with respect to r), the following relation is obtained:



dn


[2 cosh(a’i/>*) — 2cosh(a’i/>*)]5 ,    (7-8)


where both    the    electric    potential    and    its    spatial gradient    at    point    n are


represented as a function of the electric potential at the channel center

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