Page 218 - Introduction to Colloid and Surface Chemistry

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 v x 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 v x 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
2
,, 0 , . . 2-rrp(2ax-x )(a-x)dx
d/ = 2TT(a — x)v Kdx = -
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
2
compared with a. Substituting for p (Poisson's equation, d^dr —p/e)
and d/ (neglecting x compared with a) gives
irepa 2 i;
I ™ ... _L
T)/ •"> 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