## Microemulsion Curvature

### Quick Start

Curvature is important for understanding HLD-NAC, but what is it? Having rejected the naive ideas of Critical Packing Parameter, here is a more powerful view, based on Tchakalova's CIT ideas, that describe how oils and additives get into the interface domain and change the curvature in either direction.

This is an expert-level app so you will have to read the text in detail to understand what is going on.

### Microemulsion Curvature

_{s}Å³

_{o}Å³

_{a}Å³

_{s}Å²

_{o}Å²

_{a}Å²

_{tot}Å³

_{tot}Å²

It is helpful to look at curvature in other ways. The interactive demonstration below puts curvature effects into a wider perspective.

A paper from the Kunz group in Regensburg^{1} shows that curvature, HLD (with some nuances necessary for NAC) and PP, packing parameter, are each a different way of looking at the same phenomenon. Packing parameter is different from the well-known CPP (Critical Packing Parameter - discussed in the next section) which is assumed to be a constant for any given surfactant. This has given the impression that some surfactants with a CPP<1 can only give o/w emulsions and those with CPP>1 can only give w/o emulsions, whereas any surfactant can be made (given the right oil, salinity and temperature) to give o/w or w/o. So PP is a variable that depends on formulation conditions.

The PP is brought to life by the CIT (Constant Interface Thickness) paper from Tchakalova's group at Firmenich^{2}. The app uses their approach to think through what is happening with different species in all relevant phases. In practice, the measurements required to generate the necessary data require too much effort for most practical formulators and in a later paper^{3} Tchakalova's team used HLD theory for analysis of a large number of fragrance/surfactant interactions (and introduced the PIT-Shift EACN measurement technique described in the Measurements section). But because HLD, curvature and PP are inter-related one can use one approach experimentally to gain insights into what is going on fundamentally.

There are two key aspects to the CIT calculations:

First: It is assumed that there are 4 key regions where an extra molecular species (additive) might reside. In the paper these are fragrance molecules such as eugenol but they can also be "linkers", "co-surfactants" or, indeed, "hydrotropes" in the meaning of molecules which alter the behaviour of the interfacial layer of a surfactant.

- O - Oil
- W - Water
- C - Micellar core
- I - Interface

Second: For any given set of conditions one can calculate a PP based on three variables:

- V, the effective volume of the surfactant tail
- l, the effective length of that tail (typically 80% of the extended length of an alkyl chain)
- A, the effective area of the surfactant head.

with `PP=V/(A.l)`

Note that the word "effective" is included in the definition of each parameter. A stand-alone surfactant molecule has a V, l and A that can all be estimated from molecular models and from which the CPP can be calculated. But that is irrelevant to a real surfactant system where the V depends on how much (if any) oil and additive creep into the tail region, where l is averaged over a range of conformations and A depends on salinity (for ionics, where salts decrease head repulsion), temperature (for ethoxylates) and the extent to which oil and additive are inserted into the interfacial region.

To implement the model, the two key assumptions are:

- All the surfactant is at the interface
- The interfacial layer thickness is constant (hence CIT).

This model is distinct from the more common "wedge" model which assumes that there is no oil in the tail region and that any additive goes only into the tail, affecting V but leaving A unchanged. There are, of course, circumstances where this applies and in the app you can adjust sliders to emulate this mode (set τ and α to zero as discussed below). But CIT is surely more powerful because it is more general.

The master equations is:

`PP=V/(Al) =[V_s+(τ+αλ)V_"o"+λV_a]/ ([A_s+(τ+αλ)A_"o"+λA_a>]l)`

The top line of the equation shows the intrinsic volumes of the surfactant tail, V_{s}, the oil V_{o} and additive V_{a} and they are combined depending on:

- τ, the mole ratio, oil/surfactant in the interfacial region,
- α is the mole ratio, oil/additive, of oil molecules attracted to the interface by the additive and
- λ is the mole ratio, additive/surfactant.

The bottom line is similar (apart from the extra factor l), where we have Areas instead of Volumes.

The Radius of curvature, R, can be calculated from PP knowing that:

`PP=1-l/R+l^2/(3R^2)`

So now all that is required is to enter the key values for each of the factors. These values can be measured experimentally from macroscopic measurements, but for the app it is easier to enter them directly. The default values when you load the app for the first time are those for eugenol given in the paper.

The outputs are the total tail volume, V_{tot}, the total head area, A_{tot}, the packing parameter, PP and radius of curvature, R, where a negative R means curvature in the w/o direction. When R is very large it is artificially limited to avoid the complex issues near zero curvature. The graphic shows the head and tail regions changing shape as the different sliders are adjusted and shows the overall curvature that results. The graphic is illustrative only - a 2D representation of a 3D phenomenon.

One aspect of the theory at first puzzled me - the effect of the additive is symmetrical on V and A, yet our intuition is that a hydrophobic additive should relatively increase V_{tot} and a hydrophilic additive should relatively increase A_{tot}. But of course, the balance of V_{a} and A_{a} shifts between the two cases, and α will be small for the hydrophilic case, giving only a small amount of extra oil (which has a large V/A ratio) in the interface. Therefore not only are our intuitions confirmed but the theory gives a clear way to think about the multiple effects involved in a shift from a hydrophobic to hydrophilic additive.

^{1}Werner Kunz, Fabienne Testard, and Thomas Zemb
*Correspondence between Curvature, Packing Parameter, and Hydrophilic-Lipophilic Deviation Scales around the Phase-Inversion Temperature*. Langmuir 2009, 25, 112-115

^{2}Vera Tchakalova, et.al. *Solubilization and interfacial curvature in microemulsions I. Interfacial expansion and co-extraction of oil* Colloids and Surfaces A: Physicochem. Eng. Aspects 331 (2008) 31–39

^{3}Vera Tchakalova and Wolfgang Fieber, *Classification of Fragrances and Fragrance Mixtures Based on Interfacial Solubilization* J Surfact Deterg (2012) 15:167–177