Disjoining pressure and the energy of thin films

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The energy of a thin film

de Gennes, 2004 p 87f
Eqn Energy of thin film-I.png
Eqn Energy of thin film-II.png
  • Range of P(e):
    • Between surfaces: 1/e^2
    • For polymers: size of coils
    • Electrostatics 10"s nm in water, 100"s nm in oil

Disjoining pressure - liquid surfaces

A disjoining pressure (<math>\Pi</math>) describes the supplementary pressure that arises when the hydrostatic pressure of a thin layer differs from the pressure of a bulk phase. This pressure may have either positive or negative values.

Eqn Disjoining Definition.png

A decreasing P(e) gives rise to thicker films that are more stable. A positive disjoining pressure <math>\Pi</math> corresponds to spreading of the liquid. One physical example of a disjoining pressure helps make this clearer is the case of a surface sandwiched between two parallel plates. Suppose one had stacked two parallel plates separated by a distance h apart filled with a thin material. An equalibrium between the plates will be reached if the material between the plates satisfies the following properties: <math>\partial</math><math>\Pi</math>(h)/<math>\partial</math>h < 0 ie. the disjoining pressure increases as the seperation of the plates h goes to zero. In the reverse case, the interlayer will be unstable. Measurement of the disjoining pressure is only feasible if <math>\partial</math><math>\Pi</math>/<math>\partial</math>h < <math>\partial</math>N/<math>\partial</math>h. In words, one can only make a reasonable measurement of the disjoining pressure if small changes in the disjoining pressure result in large changes in an external pressure. One method for measuring disjoining pressures is featured in figure 4.1.

de Gennes, 2002, Fig. 4.1

In the Sheludko method, the film being studied is placed inside a porous ring filled with the same bulk material with density <math>\rho</math>. When hydrostatic pressure is applied to this ring <math>p_h</math> = <math>\rho </math> gH, the chemical potentials of the film and bulk liquid must be equal. This results in a correlation between the disjoining pressure and external pressure given below.

Eqn Disjoining Thermo.png

Curves of the disjoining pressure as a function of film thickness may be constructed by measuring the film's thickness for a range of disjoining pressures. Reference: Derjaguin (ch2). De Gennes (pg. 88-89)

Disjoining pressure - across films

The usual method to analyze the thermodynamics of interfaces is by excess properties (an idea developed by Gibbs precisely) (See a). However when two interfaces interpentrate the idea of an excess is imprecise (See b) and the thermodynamics better done with disjoining pressures. In figure a, Gibb's theory assumes that middle layer between the two shaded bounded regions has the intensive properties of the bulk phase. Once the boundary layers overlap (figure b), a surface no longer follows the laws of hydrostatics.

Derjaguin, 1987, Fig. 3. 2

Similar considerations arise when the interacial energy is calculated by Bakker's equation:

Eqn Bakkers.png

Derjaguin, 1987

Profile of a free film

The force of gravity acting on an elementary segment of the film between the levels, H and dH is balanced by a difference dσ between film tensions at the two levels due to a different thickness at the two levels. The tension is higher in the thinner regions.

Eqn Disjoing Pressure differential.png
Profile of a free film, drqwn from a liquid containg a surfactant. Derjaguin, 1987, Fig. 2.6

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