Model and Experimental Study of Surfactant Solutions and Pure Liquids Contact Angles on Complex Surfaces - Drops and Bubbles in Contact with Solid Surfaces - page 183

where θ y is the Young contact angle for pure liquid on a clean smooth surface.
Following the same procedure but starting with Eq. (2) one arrives at:

cos − 1 cos θ w γ lv + r sl RT

n sl

K sl C n s S )

θ w (C S ) = ln ( 1 +

)

(10b)

r sv RT
n sv

K sv C n sv

− +

ln ( 1

S

γ lv −

K lv C n l S )

lv RT
n lv

ln ( 1 +

which describe the change of contact angles with surfactant concentration for rough
surfaces wet in the Wenzel mode. Equation (10b) will be referred to as the modified
Wenzel equation throughout this chapter. θ w is the Wenzel contact angle for a pure
liquid on a clean surface.
Again following the same procedure but starting with Eq. (3), one arrives at:

cos − 1 cos θ c γ lv −

f m sv RT
n sv

K sv C n sv θ c (C S ) = ln ( 1 + )

S

∀ m

K sl C n s S ) m

(10c)

sl RT
n sl

− ln ( 1 +

γ lv −

+ K lv C n l S ) ,

lv RT
n lv

ln ( 1

where θ c is the Cassie contact angle for a pure liquid on a clean surface. Equation
(10c) is for Cassie mode wetting of an arbitrary surface. As stated in the introduc-
tion, it is valid for any surface (smooth or rough, homogeneous or heterogeneous,
with or without vapor remaining under the drop). Equation (10c) will be referred to
as the modified full Cassie equation throughout this chapter. It can be simplified to
the Eq. (10a), (10b), or (10d) by suitable choice of m and f m . Considering a SHS
with air remaining under the drop, one can simplify Eq. (10c) to:

cos − 1 cos θ c γ lv + f 1 sl RT

n sl

K sl C n s S )

θ c (C S ) = ln ( 1 +

K lv C n l S )

(10d)

f 2 lv RT
n lv

+ ln ( 1 +

γ lv −

+ K lv C n l S ) .

lv RT
n lv

ln ( 1

Equation (10d) is the modified version of the Cassie equation expressed for SHS
(i.e., Eq. (3b)), and will be referred to as the modified Cassie equation throughout
this chapter. The common (but not always allowable [31]) simplification of f 1 = f