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

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.