Partition Coefficient

One key property of surfactants is often overlooked - the partition coefficient, or how much a given surfactant prefers to be in the oil phase rather than the water phase. If at equilibrium the concentration of surfactant in the oil is C_oil and in water is C_water then the partition coefficient is given by K_p = C_oil/C_water

Why does the partition coefficient matter? If you believe in Bancroft's rule (“The phase in which an emulsifier is more soluble constitutes the continuous phase”) then it's useful information, even though there are plenty of exceptions to Bancroft. More generally, if the partition coefficient is wildly in favour of one phase rather than the other then the chances are that it won't be such a good interfacial molecule because any surfactant molecule deep inside one phase is doing nothing useful at the interface.

The key problem with partition coefficients is that they say nothing about absolute solubility. If one surfactant is soluble to a level of 0.001% in both phases and another is soluble to 10% in both phases, they both have a partition coefficient of 1, but will possess totally different practical properties. This problem becomes more severe when you try to understand trends in partition coefficients as the surfactant tail is systematically changed from short to long. If this results in an increase of the partition coefficient it is "obvious" that the longer tail means that the surfactant is more soluble in the oil. However, this is often wrong - the surfactant with the longer chain may be less soluble in the oil than the one with the shorter chain. The increase in the partition coefficient is therefore not due to increased solubility but to a smaller decrease in solubility in the oil relative to the decrease in solubility in the water.

Another issue is that partition coefficients are meaningfully measured at concentrations below the CMC so that the effects of micelles are more-or-less eliminated. If the concentration of surfactant used in a given application is much higher than CMC then the sub-CMC partition coefficient might not be directly applicable.

Despite these complications, it is still the case that partition coefficients should be the sort of numbers we routinely know for our surfactants as they tell us a lot about the inherent nature of the molecules. Sadly this is not the case. An excellent paper¹ from the U. Stuttgart's Colloid & Interface Science group provides the reasons why measurement should be routine, why the trends they observe (changing surfactant head group, surfactant tail length and oil) are happening (the oil partition example above is taken from the paper) and, importantly, why the partition coefficient is relatively easy to measure.

Measuring partition coefficients

Take a 50:50 oil water mix, add surfactant to the water at a known concentration somewhat below its CMC, shake and allow to stand for a day or so. Commercial surfactants (as the paper shows) can take some time to equilibrate while chemically pure ones will equilibrate relatively quickly. Then measure the surfactant concentration in the water phase.

But even with sophisticated analytical equipment it is relatively hard to measure surfactant concentrations. Fortunately there is an elegant trick. Simply take a small sample of the aqueous phase and measure its surface tension γ_equil. [You may want to get the starting concentration in the water by measuring the original surface tension γ_orig - that's what's used in the app.] From the ST v Concentration graph that is routinely used to identify CMC and to find important properties such as Γ_m it is trivial to go from the measured ST to the actual concentration. (If you are measuring at concentrations above CMC you first have to dilute the samples by a known amount to bring them into the sub-CMC range.) After that, calculating the partition coefficient is merely arithmetic. Knowing the original concentration C_orig and the equilibrium concentraton C_equil:

`K_p=(C_"orig"-C_(equil))/C_(equil`

The app lets you try out the idea using the simplified isotherm from the CMC app.

What does the paper find for the three types of surfactant studied: ethoxylates, phosphine oxides and APGs? The answer is that "it's complicated". There is no obvious correlation with other properties such as HLD (and certainly not HLB!) or CMC, and dependencies on surfactant chain lengths are small for the relatively hydrophilic (low K_p) APGs, large for the DMPOs (from ~1 at C10 and ~55 for C14), with a relatively small oil dependency for the specific APG and DMPO surfactants and relatively large (60 down to 30) when the oil changed from hexane to dodecane for C12EO6.