Physical Chemistry of Foods

variables are involved and (b) some fundamental problems have not been
fully resolved. Nonetheless, useful general rules can be obtained.

13.4.1 Film Rupture

To arrive at the rate of aggregation of particles, one has to multiply a
frequency factor (f) with a capture efficiency, which is due, in turn, to the
existence of a free energy barrier for contact (DG); see Section 13.2. A
similar situation would exist for film rupture, leading to an equation of the
type

Jrup¼fexp

DG

kBT



& 1031 exp

gd^2

kBT



ð 13 : 29 Þ

giving the number of rupture events to be expected per unit film area per
unit time. It can be derived from theory that the frequency factor would
approximately equal the sound velocity over the molecular volume of the
surfactant at the film interfaces, leading to a value of the order of
1031 m
^2 ?s
^1.

Hole Formation. According to de Vries, the magnitude ofDGwill
be as illustrated in Figure 13.14a. Heat motion of molecules in the film will
occasionally lead to the formation of a small hole, as depicted. If now the
Laplace pressurepLnear position 1 is larger than further away from the hole
(position 2), the hole will spontaneously grow in size and the film will
rupture. Near position 1, pL¼2/d
1/R (see Section 10.5.1), and near
position 2,pL&0; this implies that rupture can occur forR>d/2. To obtain
this situation, the interfacial area of the film has to be increased by an
amount of aboutd^2 , which implies an increase of interfacial free energy by
an amountgd^2 , which then would equalDG.
As an example, assume thatg¼5mN?m
^1 and d¼10 nm. This
results inDG¼ 5? 10
^19 J or 125kBTandJrup¼ 5? 10
^24 m
^2 ?s
^1 ;in
other words, the film would never rupture. Ifdis equal to 3 nm, we arrive at
a rupture rate of 10^26 m
^2 ?s
^1 or 10^14 mm
^2 ?s
^1 ; the film would thus
rupture immediately. This does not agree with experience: films can rupture
at a far greater thickness than 10 nm, and films of 3 nm can be stable.
Moreover, the reasoning given would allow films without surfactant to be
stable, which they never are. The de Vries theory is thus insufficient.

Capillary Waves. At a liquid surface, capillary waves will always
form, due to heat motion and induced by vibration. On a film, symmetrical