An equation of state for pure confined fluids

As we see from Ex. (2.112)j the key quantity is the configurational energy U (r ), which we henceforth separate into the potential energy of an unperturbed (reference) system, Uo (r ), and its perturbation represented by Ui (r ). We may then rewrite Eq. (2.112) as 

Henceforth, we consider a Lennard-Jones(12,6) (LJ) fluid between two plane parallel solid substrates. The ba.sic. setup of our model is sdiemati-rally depicted in Fig. 4.3. We assume the fluid-substrate interaction to be a pair-wise additive sum of L.I potentials. As the reforenee system we take a hard-sphere fluid (diameter at) between hard-sphere substrates (diameter (7a). Moreover, we smear the hard spheres of the surface layer of each 

is the areal density of the solid substrate. The fluid-fluid perturbation is just the attractive term of the LJ pair potential. Likewise, the (original) fluid-substrate perturbation is the L.T attraction -4efg (afs/r) . To be consistent with the smooth-wall approximation to the reference potential, we average the fluid-substrate attractions over the (x, y) positions of the substrate atoms in the planes in which they lie. We suppose each substrate to comprise an infinite half-space of atomic planes, separated successively by distance d. Approximating the sum over these planes by the Euler-MacLaurin formula [37] yields the expression in Eq. (4.15). 

Moreover, neglecting fluid-substrate correlations, we take the fluid to be homogeneous throughout the pore volume that is, we approximate the local density by 

It is well established [38] that a fluid confined to a slit-pore is stratified (i.e., the fluid molecules order themselves in strata parallel with the substrates see Section 5.3.4 for a more detailed discussion). Stratification diminishes rapidly with increasing distance from the substrates, because of the decay of the fluid-substrate interaction beyond a few molecular diameters at, the fluid is essentially homogeneous, as computer simulations have repeatedly 

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