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Adsorption lowers surface energy

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

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Gibbs' adsorption isotherm

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


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

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Adsorption at interfaces

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.

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Adsorption on bubbles

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.

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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.

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