Physical Chemistry of Foods

solutioncand a final surface excessG?, the amount of surfactant needed to
give an adsorbed monolayer could be provided by a layer of solution
adjacent to the interface of thicknessd¼G?/c. AssumingG?to be about
3mg?m
^2 and a surfactant concentrationcof 3? 106 mg?m
^3 (0.3%), we
obtaind¼ 1 mm, i.e., a very thin layer would suffice. Applying Eq. (5.21), we
would arrive at a halving time for adsorptiont0.5¼d^2 /D, whereDis the
diffusion coefficient of the surfactant in the solution. The conditions
underlying Eq. (5.21) are not fully met, however, and a more elaborate
analysis (accounting for the development of a concentration gradient and
the decreasing possibility for adsorption with increasing values ofG/G?)
leads to

tads& 10

G^2?

c^2 D

ð 10 : 6 Þ

where tads is the time needed to obtain almost complete adsorption
(G/G?& 0 :97).
For small-molecule surfactants, D would be of the order of
3? 10
^10 m^2 ?s
^1. For the above-mentioned quantities, Eq. (10.6) then
predicts an adsorption time of about 30 ms, a time too short to measureG
(or even g with common methods). The surfactant concentration can,
however, be far smaller. Figure 10.6 shows that Na-stearate, a very surface
active amphiphile, gives for c ¼0.13 mol?m
^3 a surface excess G¼
9? 10
^6 mol?m
^2. This would lead totads&10 (9? 10
^6 /0.13)^2 /3? 10
^10 ¼
160 s. For a still lower concentration, adsorption would also occur, butG?
would then be smaller, and the calculated adsorption times are of
comparable magnitude. In other words, for small-molecule surfactants,
diffusion times are always short, from milliseconds to a few minutes.
This need not be true for macromolecular surfactants, since these are
much more surface active than amphiphiles (see Figure 10.13), and the
diffusion coefficients are also smaller, say, by a factor of 4. From Figure
10.13 we derive thatb-casein at a concentration as small as 900 mg?m
^3 can
provide a surface excess of 2.5 mg?m
^2 at the A–W interface. This then
would lead totads&10(2.5/900)^2 /7? 10
^11 & 106 s or about 12 days. Very
long times can indeed be observed.
If the surface equation of state is known for the surfactant adsorbing,
its combination with a diffusion equation would yield the dependence ofg
on time. It is, however, often very difficult to predict the evolution ofg(t).
Following are somecomplications:

1. Convection. Transport of surfactant to the interface may be by
   convection rather than diffusion. Convection is likely to occur if the distance
   over which the surfactant has to be transported (about equal toG?/c) is not