ABSTRACT
Before coalescence starts, the value of RFmin is constant throughout the foam and is given by RFmin0, which is an input parameter. Coalescence begins at a given z when the thinnest films at that z (those with RF=RFmin0) become critical (xF=xFc) and rupture. Once this happens, RFmin starts increasing. Because the capillary pressure and the disjoining pressure isotherm are different in different parts of the foam, the degree of coalescence and hence the value of RFmin will be different in different portions of the foam. Thus, RFmin is a function of time t and vertical position z. As long as coalescence occurs, RFmin
increases and the rate of increase is obtained by recognizing that (i.e., xF|ξ,r=0) always corresponds to the critical thickness (xFc), which in turn is a function of the surfactant concentration (cs) at that ξ. Thus, coalescence at a given ξ is characterized by the following conditions:
xF|ξ,r=0=xFc(cs|ξ) (148)
that is,
(149)
Differentiating Eq. (148) with respect to time, we get:
(150)
Combining Eqs. (144) and (150) for r=0, we get:
(151)
which can be rewritten as:
(152)
The following features of Eq. (152) need to be noted. The rate (dRFmin/dt) at which coalescence proceeds at any level depends strongly on the film velocity (Vf) corresponding to the critical thickness (xFc) at that level. Because b1<0, dRFmin/dt increases as the film velocity corresponding to the critical thickness increases. On the other hand, because (dxFc/dcs)<0, the coalescence rate will decrease if the local surfactant concentration increases (i.e., if (∂cs/∂t)>0).
