Spread Lipid Films on Liquid Surfaces (Langmuir–Blodgett Films) - Surface and Colloid Chemistry: Principles and Applications - page 86

σ = −∫ ρ d x (4.24)

= [D/4 π][d ψ/d x] (4.25)

when integration is zero and infinity. At x = 0, the magnitude of ψ reaches ψ o :

σ = [2 D R T c/1000 π] 1/2 [sinh (ε ψo/2 k T)] (4.26)

The average thickness of the double layer, 1/ k , that is, the Debye-Huckel length, is
given as (Chattoraj and Birdi, 1984):

1/ k = [1000 D R T/8 π N5 ε5 c] 0.5 (4.27)

At 25°C, for uni-univalent electrolytes, one gets

k = 3.282 10 7 c −1 [cm] (4.28)

For small values of ψ, one gets the following relationships:

σ = [ D R T k /2 π N ε] sinh[ε ψ o /2 k T ] (4.29)

This relates the potential charge of a plane plate condenser to the thickness 1/ k . The
expression based upon the Gouy model is derived as

σ = 0.3514 10 5 sinh [0.0194 ψ o ] (4.30)

= Γ z N ε (4.31)

where the magnitude of Γ can be experimentally determined, and thus the magni-
tude of ψ o can be estimated. The free energy change due to the electrostatic work
involved in charging the double layer is (Adamson and Gast, 1997; Chattoraj and
Birdi, 1984; Birdi, 1989)

Fe = I ψo σ d ψ (4.32)

By combining these equations, the expression for Π el can be written (Chattoraj and
Birdi, 1984):

Π el = 6.1 c 1/2 [cosh sinh −1 (134/A el C 1/2 )] −1 (4.33)

The quantity [ k T] is approximately 4 10 −14 erg at ordinary room temperature (25°C),
and [ k T/ε] = 25 mV. The magnitude of Π el can be estimated from monolayer studies
at varying pH. At the isoelectric pH, the magnitude of Π el will be zero (Birdi, 1989).
These Π versus A isotherms data at varying pH subphase have been used to estimate
Π el in different monolayers.