The Disjoining Pressure Derivation

The disjoining pressure theory by Churaev et al. [6] begins with the definition of the disjoining pressure, n. There is a quantity that is a function of the coverage, F, or adsorbed film thickness , t, defined by the equation (for the theory t and F can be used interchangeably) 

This physical quantity is interpreted to be the pressure needed to separate two parallel plates from each other when there is an intervening hquid phase. In this case, the hquid phase is interpreted to be an adsorbed phase. Thus, the chemical potential of this intervening phase may be specified by the pressure of the gas phase. IT can be related to the difference in the chemical potential, A u, between the pure hquid phase at the saturation pressure, 1, and the chemical potential of the adsorbate, or 

The maximum incremental energy released by the adsorption process should be at this minimum point. In other words, for two plates held 

The conclusion is that there is a minimum in , when there is exactly one monolayer of adsorbate between two plates. This, however is on average exactly 1/2 a monolayer for one plate. Using the symbol Fj for a mono-layer surface excess, then = 1/2 Fj within the first approximation assumed with the hard sphere approximation. Using this together with Eq. (79), (80) and (82) one arrives at 

Although in this form it looks different from the x theory equation, it is identical if V ,no = in the x theory. 

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