92168.pdf

Charged interfaces 179

p is related to $ by Poisson's equation, which for a flat double layer

takes the form

(7,2)

where e is the permittivity*.

Combination of equations (7.1) and (7.2) gives

ck^2 ~ € KT (73)

The solution^87 of this expression, with the boundary conditions (^

= t^o when x = 0; and ^ = 0, d^/dx = 0 when jc = °°) taken into

account, can be written in the form

(7,4)

..,,

(7'6)

where NA is Avogadro's constant and c is the concentration of
electrolyte.
If ze^J2kT « 1 (kT/e = 25.6 mV at 25°C), the Debye-Huckel
approximation,

2kT

1+

2kT

can be made and equations (7.4) and (7.5) simplify to

*The permittivity of a material is the constant e in the rationalised expression, F =
(Q,Q 2 X(47rer^2 ), where F is the force between charges d and Q 2 separated by a
distance r. The permittivity of a vacuum Co according to this definition is equal to 8.854
x HT^12 kg ' m~^3 s^4 A^2. The dielectric constant of a material is equal to the ratio
between its permittivity and the permittivity of a vacuum, and is a dimensionless
quantity.