Roll-up Oil Removal

Imagine a drop of hydrophobic oil on a hydrophilic surface in water. It is not very happy to be in contact with the surface, but it's equally unhappy being in the water. All it takes to tip the balance between surface and water is a surfactant at the surface of the oil so that the head sticks out, offering a more water-compatible surface. There is now a delicate balance between surface and water and with luck the balance allows the drop to separate. A time-lapse video shows the drop gradually rolling-up into a ball and (with or without a last-minute "necking") rising from the surface.

This seems easy-enough to understand. But in reality it is a complex process. What is the required balance of surface energies that will allow roll-up to take place? And why does the roll-up start quickly then become very slow up to the last, quick, moments of release?

A model in a pair of papers ^{1, 2} from Prof Kralchevsky's group in Sofia does a great job of describing these phenomena.

As always, there are three surface energies involved:

1. σ_OW, the interfacial tension between oil and water in the presence of the surfactant
2. σ_OS, the interfacial tension between oil and surface
3. σ_WS, the interfacial tension between water and surface

The roll-up is resisted by a drag coefficient, β, along the line of separation.

At any given time, the drop will have a contact radius, r_c with the surface and this automatically implies a contact angle α with the surface via the basic geometry of the oil drop (with its fixed volume V) as it tries to rise via buoyancy from the surface. Note that α is the actual (geometric) contact angle of the drop, not the equilibrium contact angle.

The key equation is:

`β(dr_c)/(dt) = σ_"OW"cosα + σ_"WS" - σ_"OS"`

Because we want dr_c/dt to be negative (the contact radius decreases with time) clearly we want σ_OS to be much larger than σ_WS - in other words, the surface must be hydrophilic. Roll-up cannot take place on a hydrophobic surface!.

It was said earlier that roll-up can be a slow process. This is readily explained. We know from the Young equation that the equilbrium contact angle α_eq is given by:

`σ_"OW"cosα_"eq" = -(σ_"WS" - σ_"OS")`

In the paper α is defined as the "external angle" while most of us are used to the "internal angle" called here α' which is 180-α. So our key equation becomes:

`β(dr_c)/(dt)= σ_"OW"cosα - σ_"OW"cosα_"eq" = σ_"OW"(cosα - cosα_"eq")`

For a large part of the process, the dynamic, geometry-controlled α will be rather close to the equilibrium angle α_eq so the rate of roll-up will be low. If they ever became equal then roll-up would stop.

The model below doesn't try to take into account all the parameters - that would be too challenging. Instead it takes one of the representative cases from the Kralchevsky papers and lets you move the slider in linear time from t=0 to t=lift-off. Things happen quickly at the start then are very slow most of the way to the end. The values of dr_c/dt, r_c and α are shown. The value for σ_OW in this example is ~4mN/m and the equilibrium angle α_eq is ~ 35°. As the drop volume is large (0.74 mm³ to make it easier to study carefully) the time-scale for the whole process is >2hrs.

The surface in this experiment was glass. Water gradually enters the glass/oil interface via a diffusion process, rendering the glass surface more hydrophilic. If the glass has been pre-treated with acid to make it highly hydrophilic then the roll-up is so fast (σ_WS - σ_OS is so large and negative) that it can't be studied in this careful manner.

The drop shape is calculated via the Young-Laplace algorithm from the more complex Eötvös roll-up app.

There's one remaining problem for roll-up cleaning. What happens to that drop of oil floating around with, perhaps, an indifferent surfactant attached to it? The answer is that with just a little energy it could splat back onto the surface, displacing just a few surfactant molecules as it does. And if the surfactant happens to love attaching its head to the surface then what was previously hydrophilic becomes hydrophobic, positively encouraging the oil to re-deposit. Or the drop could coalesce with another, allowing an even bigger re-deposition back onto the surface.

So although strictly speaking just about any surfactant will do an adequate job at roll-up when studied as a nice scientific problem, most surfactants will be (a) poor stabilisers of the drops, (b) poor emulsifiers of the drops, (c) too attracted to the surface or all of the above. In other words, relying on roll-up is a recipe for disaster. Hence a focus on emulsification via HLD=0 or, for real sophisticates, finding "slow" polymeric surfactants that wrap around the rolled-up drop is going to be vital for effective cleaning.

However, there is one area where, surely, roll-up is highly successful. Given a generally hydophilic kitchen or glass surface just about any simple surfactant sprayed on and wiped with a cloth is going to do a reasonable job of roll-up - with the rolled-up oil safely transferring to the cloth. If the cloth is microfibre its efficacy is much higher because it can reach much more of the surface. That's why the general household cleaning spray is such a success.