illustrated in the Question just discussed. Information about Gis also
needed when the stability of a dispersion has to be studied or explained,
since several forms of instability greatly depend on the Gvalue of the
adsorbed surfactant.
For adsorption at asolid surface, it is often envisaged that the surface
contains a finite number of identical specific adsorption sites, i.e., atomic
groups that can each bind an adsorbate molecule. Further assuming (a) the
binding to be reversible (which generally implies that it is not due to
formation of a covalent bond) and (b) that the occupation of binding sites
does not affect the affinity of the surfactant for neighboring sites, the
following simple adsorption isotherm was derived by Langmuir:
Y¼
G
G?
¼
c=c 0 : 5
1 þc=c 0 : 5¼
c^0
1 þc^0ð 10 : 3 Þwhere Yis the proportion of adsorption sites occupied andc0.5is the
concentration in solution at whichY¼0.5; 1/c0.5is thus a measure of the
surface activity of the adsorbate. The normalized Langmuir isotherm is
shown in Figure 10.7. At very smallc, the adsorbed amount is proportional
toc; the concentration in solution has to be relatively large to achieve a high
saturation of the adsorption sites, e.g.,c^0 &100 forY¼0.99. However, the
assumptions on which the Langmuir equation is based are not nearly always
met, especially not at highY.
Forfluid surfacesand one (pure) surfactant, we can apply the Gibbs
equation to construct an adsorption isotherm, provided that the relation
betweengand surfactant activity is precisely known. In Figure 10.6b such
results are shown. Strikingly, a constant surface excess is reached at
concentrations markedly below the CMC. This is due to the (virtually)
constant slope ofgversus logcin this region. Strictly speaking,Gcannot be
constant for increasing surfactant activity. However, surfactant concen-
tration rather than activity has been used in making Figure 10.6b, and it
is very likely that the activity coefficient of the surfactant will decrease with
increasing c near the CMC (see, e.g., Figure 2.7c). In other words,
dg/d lnawould actually still increase, leading to a (slightly) increasingG
value.
The Langmuir equation may be applicable for adsorption on fluid
surfaces under ‘‘ideal’’ conditions. The parameter Y should then be
interpreted as being equal toG/G?, whereG?is the maximum surface
excess attainable. Comparison with an actual adsorption isotherm (taken
from Figure 10.6) in Figure 10.7 shows marked differences. These must be
due to deviations from ideality. The most important causes ofnonideality
are (a) a difference in molecular size between surfactant and solvent, and (b)