Your oil emulsion drops will rise to the surface at a velocity that depends on the difference in densities, ρ, of the oil and water, the radius r of the drop and the viscosity η of the fluid. However, if you increase the volume fraction φ of the emulsion, the drops slow each other down.
This is an idealised calculation. In practice you will tend to get Flocculation which follows different rules.
The speed with which emulsion particles will move to the surface is described via the Stokes equation which says that the velocity, v, is given by:
- r - the radius of the emulsion particle
- η - the viscosity of the emulsion
- ρO - the density of the oil
- ρW - the density of the aqueous phase
- g - gravity
The drops are slowed down by crowding and Richardson and Zaki tells us that the velocity is reduced by a factor of (1-φ)5.65 where φ is the volume fraction. (As pointed out by Dr Paul Stevenson, the power is normally quoted as 4.65 but that's the Lagrangian view, for our Eulerian view it's 5.65).
Clearly the bigger the emulsion drop, the larger the density difference between oil and water and the lower the viscosity, the faster it rises. In addition to the velocity, the approximate time taken for the emulsion to cream (i.e. a simple calculation of Hours=Height/Velocity) is shown.
The calculation on creaming tells you what will happen with drops of a given size, it also suggests that particles of less than, say, 1000nm radius would not cream significantly. It doesn't tell you why the drops are that size and how that size might change. There are 3 issues on drop size:
- The size of the drops immediately after dispersion. In HLD terms, dropsize after dispersion is likely to be smallest when an efficient surfactant is used, one where HLD is somewhat lower than 0 (for w/o) or somewhat higher than 0 (for o/w).
- Their tendency to grow via coalescence. In HLD terms, coalescence (two drops bumping into each other and forming a bigger one) is easy when the interfacial energy is low, i.e. when HLD~0. Of course there are other factors that affect coalescence. DLVO theory tells us that charged surfactants can be stabilised by charge repulsion and surfactants with long chains sticking out (such as EO50 groups) can be stabilised by steric effects. These matters are discussed in all standard textbooks on colloidal stability.
- Their tendency to grow via Ostwald ripening. The subtle effects of HLD on Ostwald ripening are discussed in the next section, though, not surprisingly, the message is the same - being very far from HLD=0 gives drop growth and rapid creaming.
The above effects are explored in the Emulsion Stability section under NAC
There is one other surfactant effect of note. It is common in cosmetics to use such large amounts of surfactant that they form viscous liquid-crystalline phases. And, of course, higher viscosity means lower creaming rates. Sometimes these phases are encouraged by adding NaCl (an effect partly explicable by HLD), the cheapest possible viscosity enhancer. If these large amounts of surfactant are required for other reasons (e.g. this is a shampoo formulation) such an approach is OK, otherwise it seems rather inelegant.