Foam Ostwald Ripening

A foam changes both by drainage and by Ostwald ripening. When both processes are happening simultaneously it is hard to disentangle what is going on. In this section we assume that, for whatever reason (perhaps because the foam is already well-drained), drainage is insignificant. The drainage section makes the opposite assumption, assuming (e.g. because the gas is C2F6), that there is no Ostwald ripening.

The separation of effects is for more than just convenience. There is a key difference between the two effects Ostwald ripening depends strongly on the surfactant species and on the walls of the foam; the connections (Plateau Borders) ? between bubbles are of no significance. Drainage has little dependence on the surfactant species and the walls are of no significance - only the channels along the connections between bubbles are of significance. Of course how you get your bubbles in the first place depends on a big mixture of factors, but in this app we are only concerned with what happens once the foam in a certain state has been produced.

Smaller bubbles (higher internal pressure) lose out to bigger bubbles (lower internal pressure) via gas diffusion throught the walls. This is a complex phenomenon because the pressures created by the crowding of the foam are an additional factor and the changes in bubble size cause changes in the crowding pressures etc. The analysis of Slavka Tcholakova and her colleagues at U. Sofia¹ has enabled a powerful model to be created, which is implemented in this app, with some simplifications and therefore some loss of rigour compared to the original.

The starting point is an initial distribution of bubbles, assumed to be a Gaussian volumetric distribution with a given Peak diameter (D[3,2]) and G(aussian)-Width (skewed if necessary to approach zero at zero radius). From this volumetric distribution all the other distributions (number, area...) at t=0 are also calculated, so in addition to the Volume-Surface average, D[3,2], you can also plot the Number average N, the Surface-Length average D[2,1] and the Mean Volume average D[4,3]. In addition to the timescale, tMax, of the simulation (typically 3600s as the foam films are, by definition, stable if they have reached the Ostwald state), the thickness of the film, h and the air volume fraction Φ in the foam (typically 90% for these stable foams that are well-drained, with a minimum of 75% because below that they are not close-packed), only 2 other parameters are required - k, the permeability of the cell walls to the gas and h, the thickness of the cell walls which is assumed to be constant over the modest range of bubble pressures.

k, however,is made up of four components: D=Diffusion coefficient through the aqueous phase, H, the Henry constant of the gas in the aqueous phase and k_ml, the permeability of the monolayer surfactant film and h the wall thickness: k=HD/(h+2D/k_ml). Although the calculation is based on k, you can create your own value of k (which you can then manually update) by varying D, H and k_ml and h. You will see the dramatic effect of changing from a gas such as CO2 (H=0.832, so fizzy drink foams ripen quickly) to N2 (H=0.015, which is why Guinness uses ~70% N2 in its foam) or to C2F6 (H=0.0014, which gives very Ostwald-stable foams for experimentalists). Note that in this Ostwald app we use Φ, the air fraction. In the Drainage app we use ε, the liquid fraction. Given that Φ=1-ε it is easy to swap between conventions.

Default values for these parameters are:
k=50E^-5 m/s; h=30 nm; Ψ=0.9 (D=2E^-9 m²/s; H=0.02 (dimensionless); k_ml=0.05 m/s;) A typical starting distribution might be a diameter of 200µm with a Gaussian width of 100µm. Of course the surface tension γ is also required.

In terms of the output, you can plot the diameter average of your choice: Number (N), Surface-Length (D[2,1]), Volume-Surface (D[3,2]) or Mean Volume (D[4,3]). If the Scale option is checked then the output distribution is shown scaled with respect to the original (i.e. broader and therefore a lower maximum peak height). If large k values are used (>300) then the calculations slow down as the numerics have to take smaller steps to cope with the bigger changes.

It is interesting to take data from the original paper¹ to see how changes in the formulation alter the final size distribution. The baseline is an SLES/Betaine surfactant blend which is a good foamer with N2 as the gas. The starting size is assumed to be 300µm with a 100µm G-Width, γ=22mN/m and k=75E^-5 and the D[3,2] values are calculated after 1200s, i.e. 20min.