Flocculation

The Creaming app just has single droplets rising to the surface (and a simple adjustment for volume fraction). This is unrealistic when it comes to real surfactant-based emulsions. Instead the individual droplets have a chance to combine into bigger clusters of the original particles and these clusters, with their larger volume, cream faster. This clustering is flocculation. One can debate whether these flocculated clusters will coalesce, but that is a discussion for the Coalescence app. Here it is simply assumed that an emulsion with a volume fraction φ₀ will flocculate and cream, giving a packed surface of φ_max which would typically be 0.64 for random packed spheres or 0.74 for close-packed spheres.

The question is, how quickly does that process happen and how does it depend on φ₀, viscosity, η, particle radius, a, density difference Δρ, the height of the sample, H₀ and on a dimensionless "pressure" p₀ which represents the effective pressure the particles exert on each other just by being near each other. Leave this at 1.3x10^-3 for a "typical" emulsion then play with the value when you want to more closely model your own emulsion.

The calculations here are all based on the work¹ of a U. Sofia team under Prof Theodor Gurkov. The complex methodology is solved using code directly translated from the C original kindly provided by Prof Gurkov. The paper is based on the classic Buscall and White theory. As this was pointed out to me only recently by an expert in the area, I will see if a separate Buscall and White app would be useful.

The idea seems simple. At each time step the particles (individual or flocs) rise with their characteristic velocity. By stepping through the entire column from top to bottom during each time step the new positions of the particles can be calculated. But it isn't so simple. The velocity depends not just on the local friction via Stokes law but on the local "pressure" which in turn depends on the volume fraction of the emulsion drops. The Sofia team used considerable sophistication to arrive at a numerically tractable system.

The graph on the left plots, at time t_show, the volume fraction of oil φ against the % distance down the tube, with 100% being the bottom of the tube. At short times the whole tube is at φ₀ apart from a numerically necessary φ_max point at the start. At long times the system becomes fully packed so that no more oil moves. The graph on the left plots the height H(t) of the boundary between oil and clear phase at time t up to the time t_max. The two views together give you a clear picture of what is going on in the system, with t_show giving the local picture in a tube at the given time and the plot up to t_max showing the overall trend of what you would see in a tube. Because the flocculation process slows down with time, the t_max plot uses a log scale.

It is interesting to note that most of the input values do not feature in the actual calculation! The calculations are done using dimensionless parameters which are then translated into the real values for output. In other words, for a given φ₀ and φ_max the creaming process is identical in dimensionless terms. The key translation is the term H₀/U₀, i.e. a correction for the height of the tube and the initial particle velocity U₀. This in turn is given by:

`U_0=(2Δρga^2)/(9η)`

i.e. the Stokes Law velocity we saw in the Creaming app.

The calculations are surprisingly fast for such complexity. They slow down (because of the numerical dependence on U₀) for low viscosities, large particles and long times, so reduce t_max in these cases because the particles cream quickly and nothing much happens at longer times. Conversely, the app shows that to avoid creaming, create sub-micron emulsion particles - but hopefully you knew that already.