This relation only holds for bubbles differing very little in size; in practice,
the conditionESD> 2 gis more realistic. As long as this condition is fulfilled,
no gas will dissolve from the bubble, so it will not shrink; moreover, no
excess gas is available for large bubbles to grow. Even forESD¼g, Ostwald
ripening tends to be quite slow.
Several surfactants cause a large enough modulus, but the problem is
thatESDtends to decrease with time; see, e.g., Figure 10.34. The main reason
is that the decrease in surface tension implies an increase in surface pressure,
and surfactant will thus desorb. For many small-molecule surfactants,ESD
will be virtually zero during bubble shrinkage. For proteins, however, the
rate of decrease in modulus is often quite slow. It can then be useful to
invoke the (apparent)surface dilational viscosityZSD:Dg=ðdlna=dtÞ[Eq.
(10.21)]. Its magnitude can be experimentally determined, and by combining
the observed relation with Eq. (13.33), the shrinkage can be calculated
(although there is no analytical solution). Some experimental results for
single bubbles are given in Figure 13.23, and they agree well with the
prediction. It is seen that a high surface viscosity can considerably retard
Ostwald ripening. Moreover, ‘‘implosion’’ of the bubble then does not
occur: after shrinkage to a certain radius, the shrinkage goes ever more
slowly.
FIGURE13.23 Shrinkage of a single CO 2 bubble. Radius of bubble (a)asa
function of time (t) in three liquids. Curve 1: water,ZSD¼0. Curve 2: beer sample,
ZSD&0.01 (d lnA/dt)0.9. Curve 3: beer sample,ZSD&0.08 (d lnA/dt)0.9(SI
units). (After results by A. D. Ronteltap, A. Prins. Colloids Surf. 47 (1990) 285.)