Fluid Surfaces and Interfaces - Surfactant Science and Technology - page 86

If the idealized concept of a mathematical dividing plane between phases is
employed to represent the interface, the adsorption of a solution component can
conveniently be pictured as the existence at the interface of a concentration of
the adsorbed material, n s , that differs from its concentration, n i b , in one or both
of the bulk phases, where ''s'' denotes the surface phase and ''b,'' the bulk. The
amount of component i in the surface phase in excess of that amount would have
been present had each phase extended up to the dividing plane S-S without
changing composition is referred to as the ''surface excess concentration'' of i,
specifically,

i . Formally, it is given by

n i
A

i ¼ ð 3 : 1 Þ

where A is the interfacial area. In principle, i may be either positive or negative,

and its value will be determined by the somewhat arbitrary choice of the location of
the dividing surface S-S.
When interfacial adsorption occurs, the energy of the interface changes. To
understand and predict the role of surfactant adsorption, it is necessary to know
the amount of material adsorbed at the interface of interest. The Gibbs equation,
which relates changes in the interfacial energy of a system to the degree of adsorp-
tion of a species at the interface and the compositions of the bulk phases, forms the
basis for understanding the thermodynamics of the adsorption process. Under con-
ditions of constant temperature and pressure, the basic equation is given as

ds i ¼ 1 dm 1 2 dm 2 3 dm 3 ð 3 : 2 Þ

where s i is the interfacial energy,

i is the surface excess of component i at the
interface, and m i is its chemical potential in each bulk phase.
The change in the free energy G of a system may be given by

X m i dn i

dG ¼ SdT þ VdP þ s dA þ ð 3 : 3 Þ

where G, S, P, V, and T have their usual thermodynamic meanings and A and n i
are as defined above. At equilibrium and under constant conditions of T, P, and
n i , Eq. (3.3) reduces to

¼ s dA ð : Þ

dG 3 4

If the surface excess of component i is allowed to vary by adsorption, then
ds ¼ P n i

X

Ad m i ¼

i dm i ð 3 : 5 Þ

As pointed out above, the value of i is defined by the choice of the location of the

dividing surface. To simplify the mathematics, it is convenient to define S-S so that
the surface excess of one component, usually one of the bulk solvent phases, will