Emulsification Drop Size Model

It would be wonderful to have a single equation that predicted how the drop size in an emulsion depended on the key input parameters.

While it might be the case that sophisticated multiscale modelling can do a reasonable job, most of us don't have the ability to access such models. At the other extremes are simple models (such as the one used in the HLD-NAC EStability) that predict drop size from Hinze-Kolmogorov theory. In these, the drop radius (r_d) from the Sauter diameter (d₃₂) can be calculated in terms of the density of the continuous phase (ρ), the diameter of the impeller of the mixer (D), the interfacial energy at the water/surfactant/oil boundary, γ_OW, and the speed of agitation (rpm):

`r_d=0.063.D[ρ("rpm"/60)^2D^3/γ_"OW"]^-0.6`

In general such simple equations are not too helpful. A paper¹ by Tcholakova and colleagues at U. Sofia provides four equations that seem to do an excellent predictive job over a super-wide range of emulsification conditions. These are:

1. A Davies variant of the Hinze-Kolmogorov equation that incorporates the effects of the viscosity of the disperse phase, η_D.
2. An equation that defines a transitional volume fraction, φ_TR, above which a very different behaviour is observed.
3. The rather simple equation describing drop size at high(er) volume fractions above φ_TR.
4. The Yaron, Gal-Or equation describing the viscosity of the continuous phase, η_C in terms of φ.

One of the key issues with Hinze-Kolmogorov equations is that they are defined in terms of the energy dissipation per unit volume, ε, which depends on the specific disperser. This can be calculated from rotation speed, R and rotor diameter D via ε=b₁(rpm)³D². It is the formulator's job to estimate b₁ for their own system. In addition the variant of Hinze-Kolmogorov used here has two further parameters, A₁ and A₂. The default values are a reasonable match to those in the original paper (which used two different homogenizers); you will have to find your own via fitting to a few representitive cases.

The equation for φ_TR is a bit too unpleasant to show here. The Yaron, Gal-Or equation is described in the Emulsion Viscosity app. Finally this gives the two required equations for the droplet diameter, d. In this app the exact definition of d (d₃₂ or d_V95) is left blurred. Below φ_TR:

`d=A_1(1+A_2[η_D d^0.333b_1^0.333 ND^0.667]/γ)^0.6 γ^0.6/(ρ^0.6 b_1^0.4 N^1.2 D^0.8)`

Above φ_TR the equation is much simpler and is mostly based on the shear rate, SR, in the homogenizer. This in turn is calculated as SR=πDN/h where h is the rotor-stator gap:

`d=0.001A_3SR^-0.5(η_D/η_"EM")^0.167`

For those who are observant, the first equation is transcendental, it has the desired value, d, on both sides of the equation. The φ_TR equation is equally transcendentally inconvenient. That is why one has apps. To avoid a sharp and unrealistic jump between the two modes at φ_TR a transition zone is included using a weighted average. This is a purely cosmetic addition and is not part of the paper.

The equations represent the summary of a lot of work by the Tcholakova team. The use of the Davies version of Hinze-Kolmogorov required a lot of experimental validation to distinguish it from alternatives. When drop viscosities and φ are low the equation becomes equivalent to simpler Hinze-Kolmogorov versions. The calculation of φ_TR is connected to the change between inertial break-up of drops, which is the mode at lower viscosities and φ, to a viscous break-up regime. The difference is that in the inertial case the drops are larger than the turbulent eddies and deform from the "outside" pressures from the flow, while in the viscous domain the drops are smaller than the eddies and deform by viscous dissipation from inside and between the eddies.

Although the idea of φ_TR has been around for a long time (connected to the Kolmogorov scale, λ), it needed the combination of a number of factors, including the Yaron, Gal-Or viscosity term before the team could match the transition to their experimental data.

The equation above φ_TR is an excellent fit to their data - and nothing more. Elegant alternative methodologies were tested but they showed no match to the experimental data and no justification for (or alternative to) the fitted formula has yet been found.

One striking outcome of the theory is that it shows how and why some relatively useless surfactants in terms of interfacial energy can give emulsions just as good as those which produce lower interfacial energies - something which goes against years of intuition. It is all about the balance of viscosities, φ and interfacial energy, and sometimes the latter factor is of no importance. It is incredibly important to know this. Some "bad" surfactants, such as many food surfactants, give high interfacial tensions while also (e.g. because they are large proteins) providing robust defence against coalescence or ripening. The lack of need for low interfacial tension is especially clear in the regime above φ_TR which is why, perhaps, so many food emulsions with high concentrations of fats and oils are stable.

At present this set of equations is the best available for describing the whole range of emulsification possibilities with just a few adjustable constants which can be determined experimentally with relatively little effort.