Physical Chemistry of Foods

11.2.2 Drainage

It will, of course, take time to obtain a dry foam, since the water has to drain
from it, and drainage is greatly hindered by the narrowness of the films and
channels in the foam. Drainage theory is intricate and not fully worked out,
and we will only consider drainage from a single vertical film due to gravity.
The film has a thicknessd, widthq, and heighth. The gravity force acting on
the film is given by the mass of the film timesg; i.e.,Fgr¼dqhrwg. This gives
a shear stress on each film surface of (1/2)F/hq¼(1/2)gdrw. The surface
can withstand a shear stress due to the formation of an interfacial tension
gradient, resulting from the flow of the liquid; this is discussed in Section
10.7 (see especially Figure 10.28b). The reaction stress due to ag-gradient
would equalDg/h. The maximum value thatDg¼g
g 0 can reach equalsP.
This gives for the maximum height that a film can have to prevent slip, i.e.,
to prevent the film surfaces from moving with the liquid flowing down, is
given by

hmax¼ 2

P

rwgd

ð 11 : 3 Þ

AssumingP¼ 0 :03 N?m
^1 andd¼0.1 mm, we obtainhmax¼6 cm. This
would imply that even for large bubbles and thick films, theg-gradient can
become large enough to prevent slip.
Assuming this to be true, the relation for the volume flow rate
Qðm^3 ?s
^1 Þout of a vertical film is given byQ¼ 2 rwgqd^3 /3Z, whereZis the
viscosity of the continuous phase. Integration yields, for the time it needs for
the film to reach a thicknessd,

tðdÞ& 3

Zh

rgd^2

ð 11 : 4 Þ

Assumingh¼0.5 mm,Z¼2 mPa?s, andd¼ 10 mm, the time would be 3 s;
ford¼20 nm we obtain 9 days. This means that a fairly thin film is rapidly
obtained, but that it would take a long time for a film to become so thin that
colloidal repulsion forces between the two film surfaces become significant
(generally atd&20 nm).
These results should be viewed with caution. In the first place, they
concern a single film, not a foam, and can therefore only give trends.
Secondly, we have assumed the film surfaces to be immobile, which would
imply that the surface dilational properties are completely elastic. This is
rarely the case, especially if it concerns small-molecule surfactants.
Consequently, some slip will occur at the film surfaces, and drainage will
be faster than that of Eq. (11.4). As discussed in Section 10.8.3, several