Wetting Dynamics of Aqueous Surfactant Solutions on Polymer Surfaces - Drops and Bubbles in Contact with Solid Surfaces - page 81

Figure 11. Contact angle and drop base of a water drop, containing a non-ionic surfactant on a hy-
drophobic surface: ( ! ) contact angle; ( 1 ) drop diameter.

A general form of the spreading rate law in such case is given by Eyring's bi-
exponential form [25]

d r
d t =

A A e − b A cos θ A R e b R cos θ , − (3)

where θ is dynamic contact angle, i.e., the contact angle as a function of time
θ

θ(t) . The indices A and R describe the parameters of advancing and reced-
ing movement of the TPC line. For θ> , the TPC-line is advancing, e.g., in the
case of spreading of aqueous surfactant solutions; for θ< , the TPC-line is reced-
ing, e.g. during evaporation of a water drop. The spreading law (3) reflects the
barrier character of the TPC line movement [26], which was confirmed for the
adsorption-desorption mechanism of the TPC movement [37, 38]. Most of wet-
ting characteristics such as mobility and immobility of the TPC line—pinning ef-
fect [39], quasi-static advancing and receding contact angles, characteristic spread-
ing velocity—can be interpreted with the help of equation (3). The parameters
A A , R and b A , R depend on a particular mechanism, but in the case of equilibrium
d r/ d t

=

= 0, the following relationship is valid

1
b A + b R ln A A

=

cos

A R .

(4)

Thecosineof is defined by the Young equation (1) and does not depend on any
mechanisms. From a simple geometrical consideration [40, 41], the base radius r
for a small drop can be expressed as

cos θ) 3 / 2 3

π

( 1 +

r 3 =

cos θ) V,

(5)

( 1 − cos θ) 1 / 2 ( 2 +

where V is the drop volume, which is supposed to remain constant, and θ is the
contact angle changing with time. In Fig. 11, the contact angle and base radius of
an evaporating drop (water) are illustrated.