92168.pdf

Charged interfaces 201

E is applied parallel to the surface. Each layer of liquid will rapidly

attain a uniform velocity relative and parallel to the surface, with

electrical and viscous forces balanced. Equating the electrical and

viscous forces on a liquid layer of unit area, thickness dx and distance

x from the surface, and having a bulk charge density p,

Eodx = ( —) -( ~]

v dj,+<u TdJ,,

d ( dv\TJ \AX

dx\ dx J

Inserting the Poisson equation in the form p = _

L dxl dx

gives

d ( d\jA_^_( c

dvl dx) dxl c

Integrating,

ditf dv

—Ee — = 17 — + constant

dx dx

The integration constant is zero, since at x = °°, d&ldx = 0 and dvldx

= 0. Integrating again (assuming that e and 77 are constant throughout

the mobile part of the double layer),

-Ee\l/ = 77V + constant

If electrophoresis is being considered, the boundary conditions are
$ = 0, v = OatJt = °° and fy — £, v = — VE at the surface of shear,
where VE is the electrophoretic velocity - i.e. the velocity of the
surface relative to stationary liquid. Therefore,

or

_"E_<re (7.24)

~ ~