Foam Drainage

The assumptions here are that you have a foam that has stable walls (the DLVO parameters are strong enough to stop the walls from breaking) and that there is no Ostwald ripening (e.g. because the foam is made with C2F6 gas). The reason for simplification is that things are complicated enough without trying to couple ripening with drainage. The theory here is based on a key paper by Koehler¹ and the work of Saint-Jalmes².

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.

The standard assumption is that the walls of the foam contain virtually no liquid and that no drainage takes place along them. Anyone familiar with Poiseuille will realise that with a thickness, h, of ~30nm and a 1/h³ dependency there can be no significant bulk flow. Instead the flow is along the Plateau Borders (PB) between bubbles, and the PB contain most of the liquid. For bubbles of diameter D the length L of the PB is given by L=D/2.7. The cross-section of a PB is a sort of triangle with radius r (there is some deliberate fudging of "r" here; experts will know what is meant) and a cross-sectional area A=0.161r². r depends on the liquid fraction ε (our main factor of interest for drainage) and is given by r=L√(ε/0.17). As ε decreases, r decreases and the ability to flow down the PB also decreases. The nodes of the foam are assumed to contain relatively little of the liquid. Their importance in the overall flow varies in terms of the types of flow discussed in the next paragraph.

Life would be relatively simple if flow through the PB was always of one type. For a very rigid surfactant surface the flow is the same as the flow down any fixed surface - the flow goes as 1/r³ because the "no slip" condition at the walls slows down the overall flow. This is Poiseuille flow. For a surfactant that is very mobile at the surface, the walls play a much more limited role (in effect the wall moves with the flow) so the flow is unimpeded ("plug flow") and drainage is faster. In this case the nodes take on a new significance as they now cause the main resistance to flow. It is very important to distinguish between these two extremes:

• Flow that is limited by the "no slip" condition down the PB is called "Channel-dominated" flow
• Flow down slippery PB is called "Node-dominated" flow

Sadly in many cases the system can be a mixture of both modes and that mixture can change during drainage so it is not easy to know what is going on. Because we have the luxury of running an app rather than a real experiment, you can explore what happens when the system stays constant at either extreme. It's then up to you to decide whether your surfactant is relatively mobile (e.g. SLES) or rigid (e.g. foams stiffened with long-chain fatty acids or alcohols). For those who know the surface shear viscosity, µ_s of their surfactant system then a value M can be calculated, M=0.9µ√ε/µ_s. When M<<1 you are in Channel-dominated mode, when M>1 you are in Node-dominated mode. But as most of us don't know where our surface shear viscosity is within the typical range of 10^-4 to 10^-2 g/s this calculation is of little help.

A further choice of equations is included. Stevenson, with his focus on foam flotation and fractionation has derived a pragmatic equation based on two parameters, m and n which is Channel-dominated. The "n" value describes the dependency on ε. His own work has found that n=2 works well for some foams. In this app an n=1 option is also included to mimic the other Channel-dominated formula.

To simplify things to some representative equations we need equations for just three things:

1. The speed, v_f at which the front containing the original ε moves down the foam column over time
2. The average speed v_m at which the mid-point ε value moves down the foam column over time
3. The final profile ε(z) once drainage is complete

Point #2 starts to make sense once you get used to seeing the ε/z plot changing over time, where z is the axis down the column, with 0 being the top of the column. The equations are (with some simplifications):

The Stevenson equivalent of the Channel-dominated Front Velocity is `v_f=(ρgr^2mε^n)/μ` where r is the bubble radius, here (with some liberties) assumed to be D/2 rather than the harmonic mean radius. Stevenson's default values of m=0.016 and n=2 are used.

So let's do some calculations. The key inputs are foam height, H, bubble diameter D, solution viscosity µ and density ρ and the starting liquid fraction ε₀. In many experiments it's not entirely clear where the foam column begins, so H can be ill-defined. So don't worry too much about the details of what happens when z approaches H. Rather than specify a time, the time t required for the main front at ε₀ to travel H is calculated and the curves remain the same for different values of H etc. with only the time labels changing. It seems odd that t is calculated trivially via t=H/v_f but the front moves at a velocity dependent on ε₀; what happens as the front moves on does not affect its speed, and the draining liquid does not build up an ε value higher than ε₀, so the progress of the draining front is constant. The straight-line graphs in Channel mode are not far from reality observed in many experiments; the gentle curves in Node mode can also be observed. The fact that all the curves sweep to the equilibrium value at z=0 without apparently moving down the ε scale at z=0 is at odds with some data but is seen in other datasets. If a more satisfactory equation becomes available, the app will be updated to resolve these issues.

Note that, for a given bubble diameter (which no doubt depends on the surfactant but is a given in this app), the properties of the surfactant play no role in the drainage time other than the choice between Channel- and Node-dominated. However, the surface tension, γ, is one of the inputs as it helps control the equilibrium drainage curve, the one at infinite time, shown as Equil. in the graph. If the calculated equilibrium curve looks odd that's because the set-up is outside its range of applicability. The factor z₀ in the equilibrium equation has to be estimated from H which itself is not so well defined. Note that in this Drainage app we use ε, the liquid fraction. In the Ostwald app we use Φ, the air fraction. Given that Φ=1-ε it is easy to swap between conventions.

What is the main conclusion from all this? That big, wet bubbles with low viscosity solutions (no added glycerol!) drain faster than small, dry ones with high viscosity. That is, of course, obvious. But it's good to have some numerics to be able to explore what happens and to have some notion of the difference between mobile and immobile surfactants.