92168.pdf

Charged interfaces 20?

and Hunter^193 suggest, however, that the viscoelectric effect was

overestimated by Lyklema and Overbeek and that it is, in fact,

insignificant in most practical situations.

Streaming current and streaming potential

The classical equations relating streaming current or streaming

potential to zeta potential are derived for the case of a single circular

capillary as follows.

Let £s be the potential difference developed between the ends of a

capillary tube of radius a and length / for an applied pressure

difference p. Assuming laminar flow, the liquid velocity vx at a

distance x measured from the surface of shear and along a radius of

the capillary is given by Poiseuille's equation, which can be written in

the form

The volume of liquid with velocity vx can be represented by a hollow

cylinder of radius (a— x) and thickness dx. The rate of flow, d/, in this

cylindrical layer is, therefore, given by

,, 0 ,.. 2-rrp(2ax-x^2 )(a-x)dx

d/ = 2TT(a — x)vKdx = -

477!

The streaming current /s is given by

where p is the bulk charge density. If KO is large, the potential decay
in the double layer and, therefore, the streaming current are located
in a region close to the wall of the capillary tube where x is small
compared with a. Substituting for p (Poisson's equation, d^dr^2 —p/e)
and d/ (neglecting x compared with a) gives

I ™ ...irepa _L^2

T)/ •">i; d/

The solution of this expression (by partial integration), with the

boundary conditions (</> = £ at x = 0 and «/r = 0, di/r/d* = 0 at jc = a)

taken into account, is