## Interdroplet Pair Potential

Polymeric steric interactions arise when emulsion droplets get so close together that the emulsifier layers overlap (Figure 3.15). This type of interaction can be conveniently divided into two contributions (Hiemenz 1986, Hunter 1986):

The elastic contribution is due to the compression of the interfacial membrane, whereas the mixing contribution is due to the intermingling of the polymer chains (Figure 3.15).

### 3.5.2.1. Mixing Contribution

If it is assumed that the polymer molecules in the layers interpenetrate each other without the layers being compressed (Figure 3.15a), then the interaction is entirely due to mixing of the

Compression

FIGURE 3.15 Steric interactions between emulsion droplets can be divided into an elastic contribution which involves compression of the polymer layers and a mixing contribution which involves interpenetration of the polymer chains.

polymers. The theories describing polymeric steric interactions are much less well developed than those describing electrostatic or van der Waals interactions. The major reason for this is that polymeric steric interactions are particularly sensitive to the precise structure, orientation, packing, and interactions of the polymer molecules at the interface (Hunter 1986, Claesson et al. 1995). These parameters vary from system to system and are difficult to account for theoretically or to measure experimentally. Mathematical theories have been developed for a number of simple well-defined systems, and it is informative to examine these because they provide some useful insights into more complex systems (Hunter 1986). For example, the following equation has been derived to account for the mixing contribution when the polymer molecules are permanently attached to the droplet surface and there is a constant number of polymer chains per unit surface area (Hunter 1986):

where m is the mass of polymer chains per unit area, 8 is the thickness of the adsorbed layer, Na is Avogadro's number, % is the Flory-Huggins parameter, vp is the partial specific volume of the polymer chains, and Vs is the molar volume of the solvent. The Flory-Huggins parameter depends on the relative magnitude of the solvent-solvent, solvent-segment, and segment-segment interactions and is a measure of the quality of a solvent. It is related to the effective interaction parameter (w) which was introduced in Chapter 2 to characterize the compatibility of molecules in mixtures: % = w/RT. In a good solvent (% < 0.5), the polymer molecules prefer to be surrounded by solvent molecules. In a poor solvent (% > 0.5), the polymer molecules prefer to be surrounded by each other. In an indifferent (theta) solvent (% = 0.5), the polymer molecules have no preference for either solvent or polymer molecules. In the original Flory-Huggins theory, it was assumed that % was entirely due to enthalpic contributions associated with the molecular interactions. In practice, it is more convenient to assume that % also contains entropic contributions since interactions involving changes in the structural organization of the solvent can then be accounted for (e.g., hydrophobic interactions) (Evans and Wennerstrom 1994). Whether the mixing contribution is attractive or repulsive depends on the quality of the solvent. In a good solvent, the increase in concentration of polymer molecules in the interpenetration zone is thermodynamically unfavorable (wmix positive) because it reduces the number of polymer-solvent contacts and therefore leads to a repulsive interaction between the droplets. Conversely, in a poor solvent, it is thermo-dynamically favorable (wmix negative) because it increases the number of polymer-polymer contacts and therefore leads to an attractive interaction between the droplets. In an indifferent solvent, the polymer molecules have no preference as to whether they are surrounded by solvent or by other polymer molecules, and therefore the mixing contribution is zero. Thus, by altering solvent quality, it is possible to change the mixing contribution from attractive to repulsive or vice versa. In food emulsions, this could be done by varying temperature or by adding alcohol or electrolyte to the aqueous phase.

### 3.5.2.2. Elastic Contribution

If it is assumed that the polymer layers surrounding the emulsion droplets are compressed without any interpenetration of the polymer molecules (Figure 3.15b), then the interaction is entirely elastic. When the layers are compressed, a smaller volume is available to the polymer molecules and therefore their configurational entropy is reduced, which is energetically unfavorable, and so this type of interaction is always repulsive (welastic positive).

The magnitude of the elastic contribution can be calculated from a statistical analysis of the number of configurations the polymer chains can adopt before and after the layers are compressed (Hiemenz 1986, Dickinson 1992): 