Model and Experimental Study of Surfactant Solutions and Pure Liquids Contact Angles on Complex Surfaces - Drops and Bubbles in Contact with Solid Surfaces - page 182

and Gu [79] is used here since it allows for the expression of several 'types' of
adsorption (e.g., Langmuir, S-type or 2 plateau). The isotherm of Eq. (6) can also
be applied to all three interfaces.

K xy C n xy

xy

S

xy = , (6)

K xy C n xy

1 +

S

where xy is the limiting surface coverage (i.e., the maximum possible concen-
tration of surfactant on an interface) and K xy is the product of the equilibrium
constants for the first and second adsorption steps. The exponent, n xy ,isusedasan
empirical fitting parameter [80].
Combining Eqs (5) and (6) and integrating gives an expression for interfacial
tension as a function of surfactant concentration.

xy RT
n xy

K xy C n xy γ xy −

= +

γ xy (C S ) ln ( 1 ), (7)

S

where γ xy is the interfacial tension for the pure liquid, i.e., for C S = 0.

Proceeding, one can consider the Young, Wenzel and Cassie equations (Eqs (1)-
(3)). Taking the derivative of Eq. (1) for a given material, m , with respect to ln (C S ) ,
and applying Eq. (5), one obtains Eq. (8) describing the relationship between con-
tact angle and surfactant concentration for a smooth surface.
γ lv
RT

dcos θ y | m
dln (C S )

× − lv cos θ y | m + ( sv − sl ) | m = 0 . (8)

Equation (8) is identical to that obtained by El Ghzaoui [47]. Substituting the
isotherm model of Eq. (6) into Eq. (8) yields:

cos θ y | m

K lv C S n lv
lv

lv RT
γ lv

dcos θ y | m

dln (C S ) − 1 + K lv C S n lv

lv

(9)

m
with the γ lv term in Eq. (9) given by Eq. (7) expressed at the liquid-vapor inter-
face. In this chapter, solid-vapor adsorption of surfactant on hydrophilic surfaces
is considered to model the autophobic effect. It is not needed (setting sv

sl RT
γ lv

K sl C S n sl
sl

sv RT
γ lv

K sv C S n sv

sv

= −

1 + K sl C S n sl

sl

1 + K sv C S n sl

sv

= 0) for

hydrophobic surfaces [27, 70].
Equation (9) is a linear, inhomogeneous, ordinary differential equation with
variable coefficients. Applying the technique of variation of parameters results in
Eq. (10a), a modified Young's equation relating changes in smooth surface contact
angle with surfactant concentration:
θ y | m (C S )

cos − 1 ⎛

⎞ m

(10a)

sl RT
n sl

sv RT
n sv

K sl C n s S ) K sv C n sv cos θ y γ lv + ln ( 1 + − ln ( 1 + )

S

⎝ ⎠

= ,

lv RT
n lv

K lv C n l S )

γ lv − ln ( 1 +