## ξ (xi), the characteristic length

### Quick Start

The ξ parameter (prounounced "xi") is a key part of HLD-NAC when HLD is close to zero. It's a sort of order parameter; the more the surfactant self-orders, the more it can solubilize oil or water. If we knew how to predict it, then HLD-NAC would become even more powerful.

The problem is that ξ is not a constant for a surfactant, because it depends on the oil. A surfactant has more control (more ordering) over a small oil molecule than a large one.

In the apps, ξ is estimated via the scheme described here, developed by Acosta, who is the first to admit to its limitations. Hopefully, as we gather more systematic HLD-NAC data, we will be able to understand and predict ξ with greater accuracy.

In the meantime, the graphic shows Acosta's view of what ξ really means.

### xi

_{Oil}g/cc

_{Oil}

When we are getting into the Type III zone all thoughts of curvature fail us. In order to understand how much oil can be solubilised by the surfactant we have to know how far the surfactant can extend its influence over the oil. If it is only a short distance then any oil further away than that is "unstabilised" so is not solubilised. It turns out that the characteristic length (called a persistence length by de Gennes), ξ, is what describes this influence. It can be measured via neutron scattering and is some form of correlation distance. But it's important to realise that the same ξ can be measured from the phase volumes of the Type III - a large ξ means a lot of solubilisation power which means a large Type III phase.

So the efficiency of a surfactant in solubilising an oil (or water) depends strongly on ξ. In the early days of HLD-NAC it was convenient to assume that ξ was a constant for each surfactant. And it's certainly the case that some surfactants are much more efficient than others, and the relative ξ values used in the theory made a lot of sense. But it was equally clear that the solubilising power of a surfactant depended strongly on the oil. It is much more difficult to solubilise a large oil molecule than a small one. The fact, therefore, that ξ is oil-dependent is not surprising given the meaning of ξ as being how far the surfactant can extend its influence over the oil. It therefore turns out that ξ depends on both the surfactant and the oil. A convenient measure for this dual dependence is the Overlap Factor, OF:

OF=V_{SurfTail}/ (A_{SurfHead}.V_{Oil}^{1/3})

This makes intuitive sense. The Surfactant terms of Volume/Area gives a length value and the (molecular volume of the oil)^{1/3} also is a length value.

Careful analysis of many surfactant/oil datasets gives two formulae for estimating ξ, one for ionics, one for ethoxylates. Formulae for other classes such as APGs will need more datasets:

ξ_{ionics} = 40.Exp(1.2.OF)

ξ_{ethoxylates} = 0.4.Exp(6.6.OF)

But how do you get the OF from generally known factors? Again, careful analysis reveals a simple correlation for OF that relies on the better-known surfactant Tail length, L, as a stand-in approximation for the V term (clearly the dimensions are not correct, but the correlation works adequately, consider 13 to have the dimensions of distance²) and Head area, A, along with the MWt of the oil and its density ρ:

OF=13(L/A)/[MWt_{Oil}/ρ_{Oil}]^{1/3}

For surfactants such as AOT this approximation works if you say that the tail length is double that of the individual tails.

The overlap factor and it use in estimating the characteristic length has been proposed (and is being evaluated) by the Acosta group. The correlation used here is preliminary, but is already providing key insights into the dual influences of surfactant and oil properties on the value of ξ.

The graphic shows the water (blue) and oil (orange) zones with the surfactant at the interface. The width of the oil phase is 2*L+2*ξ and the width of the water phase is 2*ξ. For convenience the system is shown as a regular repeating structure whilst in reality it is a statistical, fractal structure. The head area is approximated for illustrative purposes.

Clearly this is early days for ξ prediction. The general shape of the theory makes a lot of sense, but it will require much more data (including for different classes of surfactants) to further refine it. For the present the main impact of this thinking on ξ is that the Fish diagrams have had to be updated to take into account the fact that ξ is not a constant.