# Interdisciplinary Applied Mathematics

Скачать в pdf «Interdisciplinary Applied Mathematics»

(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

Скачать в pdf «Interdisciplinary Applied Mathematics»