Model and Experimental Study of Surfactant Solutions and Pure Liquids Contact Angles on Complex Surfaces - Drops and Bubbles in Contact with Solid Surfaces - page 181

only. Mohammadi et al. [77] also studied surfactant solutions of the same and lower
surface tensions as the pure liquids. The surfactant solutions did not show the same
drop in contact angles as the pure liquids did [77], and instead maintained high ad-
vancing CAs (above 90 ◦ in nearly all cases). Mohammadi et al. [77] were the first
to propose that this was due to surfactants inhibiting penetration of the solution into
the roughness; this hypothesis was supported by Milne [78] and will be examined
in the work presented in this chapter.
Overall, it seems that SHS can show high advancing contact angles with surfac-
tant solutions, but may not always do so. Receding contact angles vary between
high and low values between and within studies. Considering the small number of
experimental studies of surfactant wetting on SHS, more investigation is needed to
study what effects (if any) various types of liquid impurities have on the wetting
mode and contact angles of surfactant solution on topographically and chemically
different SHS (i.e., complex surfaces). Theoretical models are also needed. As such,
models based on modified forms of the Cassie and Wenzel equations for wetting of
surfactant solutions on rough/heterogeneous surfaces is presented next in this chap-
ter. The model predictions are then applied to understand and explain experimental
results for wetting of 5 different SHS by various surfactant solutions. The surfactant
results are also put into context by comparing them with wetting of SHS with pure
liquids having similar surface tensions to the surfactant solutions.

C. Contact Angle Model Derivation
The goal in this section is to derive thermodynamic models for the contact angle of
surfactant solutions on complex (i.e., heterogeneous and/or rough) surfaces. This
derivation relies on the use of the Young, Wenzel, and Cassie equations for equi-
librium contact angle on smooth and rough/heterogenous surfaces, leaving aside,
for now, the question of contact angle hysteresis and the consequent advancing and
receding contact angles present. The difference between advancing, receding, and
equilibrium contact angles will be considered when the model is applied later in this
chapter. The Wenzel and Cassie equations are used due to their broad application in
literature.
Gibbs' adsorption equation for the differential change in surface energy with a
differential change in surfactant concentration in an ideal dilute solution is:
d γ xy =− xy RT dln (C S ), (5)

where γ xy is interfacial tension and xy is the surface excess per unit area, or cov-
erage, of surfactants at the interface xy ,where x and y can be any of solid (s),
liquid (l), or vapor (v) phases (e.g., γ lv is liquid-vapor interfacial tension), R is
the universal gas constant, T is the absolute temperature, and C S is surfactant
concentration (throughout this chapter expressed non-dimensionally as the ratio of
concentration over critical micelle concentration (i.e., C/C CMC ).
If a suitable isotherm equation relating xy to ln (C S ) is applied, then Eq. (5) is
integrable for interfacial tension. The general isotherm equation proposed by Zhu