Lattice fluid grand potential

The confined lattice fluid is coupled to an (infinitely large) external reservoir of heat and matter. Hence, from an equilibrium perspective, the grand potential is the relevant thermodynamic potential (see Elq. (1.58)) where N is equivalent to the number of occupied lattice sites. The exact diflerential of is given in Elq. (1.59), and the link to the microscopic level is provided by Eq. (2.81) where we need to replace the continuous variables Sq by their discrete analogs ria a = x, y, or z). Hence, the discrete analog of Eq. (2.120) may be cast as... 

According to our modular approach, the lattice fluid confined by hard repulsive substrates may be viewed as a bulk system, in which serves to introduce surfaces. We can then express the grand-potential density as... 

The possible morphologies of the lattice fluid can thus be determined by sandwiching a hard-substrate module of volume ri riy (rt — 2) sites between two hard-substrate modules of Ux% (Wz — 1) sites. Using the modular principle in this case, we may express the grand-potential density as... 

Table 4.1 Possible lattice fluid morphologies M" and associated grand potential fl (p) for T = 0.

The simultaneous calculation of the amount adsorbed and the distribution of adsorbate within the pores of a reconstructed solid can be impl ented afier the i lication of a Draisity Functional Theory (DFT) mean field model, which is particularly suited for on-lattice simulations on di tised structures [31]. In a lattice model the spatial distribution of adsorbate can be described at each site by the local density fimction. The equilibrium density profile for a given matrix realization and chemical potential ft (xin then be determined by minimising the grand potential 0 p with respect to the fluid density on each lattice site, / (x), leading to ... 

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