At the air/liquid interface: And the solid/liquid interface: Lowers the surface tension. Stabilizes dispersions.

## Culinary applications: Sauces

• Mayonnaise is a classic example of an emulsion of an oil in water. Howard McGee gives an extensive discussion of how to prepare this well-known condiment:
• The surface tension of water makes it highly-favorable for the water and oil to exist in distinct phases. Energy, in the form of vigorous mixing, needs to be added to the mixture to create a dispersion of oil droplets in water. As an order of magnitude estimate, 15 ml of oil can separate into 30 billion drops in the final product! Enthusiastic mixing by hand can achieve droplets on the order of 3 micron, but industrial-grade homogenizers can produce drops less than one micron in size.
• As described in the previous section, this process of dispersing the droplets can be made easier with the presence of surfactants, also known as emulsifiers. In mayonnaise, the phospholipid lecithin in the eggs serves this purpose. The proteins in the egg yolks contain separate hydrophobic and hydrophillic regions, which is also effective. Warm, raw eggs yolks are traditionally used since they are more flexible and can flow more easily than their refrigerated or cooked counterparts. The casein in milk and cream are also sometimes used in emulsions.
• However, it is not enough to simply create the droplets: something is needed to keep them from coalescing into larger drops. In mayonnaise, the polymers in mustard seeds do the job.

A derivation by Gibbs gives a relation between the chemical potential of a solute in solution, the surface tension of an interface and the excess concentration the solute at that interface. The interface is considered to be wide compared to the concentration gradients; the excess number of moles associated with that interface is calculated and is expressed as a surface concentration, moles per area:

Morrison. Fig. 15.1
 The differential of the total energy: $dU=TdS-pdV+\sigma dA+\sum{\mu _{i}dn_{i}}$ Integrating to get the total energy: $U=TS-pV+\sigma A+\sum{\mu _{i}n_{i}}$ Taking the differential gives the Gibbs-Duhem relation $SdT-Vdp+Ad\sigma +\sum{n_{i}d\mu _{i}}=0$ Defining that relation for both bulk phases: $S^{\alpha }dT-V^{\alpha }dp+\sum{n_{i}^{\alpha }}d\mu _{i}^{\alpha }=0$ $S^{\beta }dT-V^{\beta }dp+\sum{n_{i}^{\beta }}d\mu _{i}^{\beta }=0$ Chemical potentials are constant: $d\mu _{i}=d\mu _{i}^{\alpha }=d\mu _{i}^{\beta }$ Subtracting the phases from the total: $\left( S-S^{\alpha }-S^{\beta } \right)dT-\left( V-V^{\alpha }-V^{\beta } \right)dp+Ad\sigma +\sum{\left( n_{i}-n_{i}^{\alpha }-n_{i}^{\beta } \right)}d\mu _{i}=0$ Defining the excess quantities: $S^{\sigma }=S-S^{\alpha }-S^{\beta }$ $S^{\sigma }=S-S^{\alpha }-S^{\beta }$ $S^{\sigma }=S-S^{\alpha }-S^{\beta }$ Substitution and subtraction gives: $Ad\sigma +S^{\sigma }dT-V^{\sigma }dp+\sum{n_{i}^{\sigma }d\mu _{i}}$

Finally:

 Gibbs adsorption isotherm: $-d\sigma =\sum{\frac{n_{i}^{\sigma }}{A}}d\mu _{i}=\sum{\Gamma _{i}}d\mu _{i}$ The surface excess: $\Gamma _{i}=\frac{n_{i}^{\sigma }}{A}\text{ mol m}^{\text{-2}}$ For a 2-component system: $-d\sigma =\Gamma _{2}d\mu _{2}\simeq kT\Gamma _{2}d\ln c_{2}$

 Air-water surface Air-oil surface Oil-water interface Strong adsorption, substantial lowering of surface tension. Little adsorption, little lowering of surface tension. Strong adsorption, substantial lowering of interfacial tension.

Ratio of the observed velocity of ascent of a bubble to the calculated Stokes’ velocity in solutions of various concentrations of

• (a) polydimethylsiloxane in trimethylolpropane–heptanoate;
• (b) polydimethylsiloxane in mineral oil;
• (c) N-phenyl–1–1napthylamine in trimethylolpropane–heptanoate.

Each figure shows the transition from the Hadamard to the Stokes regime.

Suzin and Ross, 1985

Suzin, Y.; Ross, S. Retardation of the ascent of gas bubbles by surface-active solutes in nonaqueous solutions, J. Colloid Interface Sci. 1985, 103, 578 – 585.

## Adsorption by a solid surface

The surfactant must be soluble in the liquid !

 Solid-water interface Solid-oil interface The adsorption is driven by both strong tail/solid interaction and entropy – the hydrophobic effect. The adsorption is driven by strong head group/solid interaction.