## Coalescence

### Quick Start

When two emulsion drops collide, do they come together, coalesce, to form a larger one? It depends on the energy barrier, E, around the drop, along with the drop radius r, the % oil and the bulk viscosity, which affects how fast the drops collide. The app lets you see what happens either to the number of drops or the typical radius of the drops.

The big question is: "What is E?" No one knows, but a well-packed surfactant is going to have a higher value than a loose one.

### Coalescence

_{max}hr

Although Creaming and Ostwald Ripening are both important for emulsion stability, the biggest problem when creating an emulsion is coalescence. If two drops bump into each other with an energy greater than the barrier between them then they join to form a larger drop.

The process is hugely complex and models that are sophisticated enough to capture the complexity require too many unknown parameters to be useful to the general surfactant community. The version here (usually known as the Davies and Rideal model) is a great simplification. Nevertheless, it captures a lot of the key ideas.

The starting point is that the diffusion coefficient, D, of a drop of radius r in a medium of viscosity η and with Boltzmann thermal energy kT is given by:

D= kT/6πηr

The collision rate between particles that meet at distance R=2r is given by:

CollisionRate= 4πDRn²

Of course, every collision does not result in a coalescence event. The assumption is that if there is an energy barrier E the coalescence rate is given by:

CoalescenceRate= 4πDRn²exp(-E/kT)

Combining the diffusion equation with the coalescence equation gives us a rate that is independent of particle radius, i.e. R and r cancel out:

CoalescenceRate= (4kT/3η)n²exp(-E/kT)

Integrating this tells us the density of particles (number/cc) at time t, n_{t} starting from an original density n_{0}:

1/n_{t} = 1/n_{0} + t(4kT/3η)exp(-E/kT)

The app lets you specify the initial drop radius, the volume% oil, the viscosity and the barrier, E, in units of kT. Because T is absolute temperature, modest changes of temperature around 25°C have little effect so T is not an input. The n-mode option plots the droplet density as % of the original. The r-mode option plots the droplet radius with time, using a simplistic conversion of n_{t} into r given the fixed volume of oil.

The conclusion from exploring the app is obvious: the drop-to-drop barrier, E, is of huge importance. Because of its exponential dependence, a relatively modest reduction from the default of 20 down to 17 has a massive effect on coalescence.

Getting a high E with a good surfactant is not all that hard - given enough time. The problem when we're making emulsions is that it takes time for the surfactant to form a good barrier around the drop, so most of the small drops we make coalesce and have to be re-formed to have a chance to end up as a nice, fine, emulsion drop.

How to get a high value of E is discussed in the DLVO app.