Advantages and disadvantages of lattice models

The discussion of the Joule-Thonison effect in the previous section clearly showed that it is advantageous in theoretical treatments of confined fluids to tackle a given physical problem by a combination of different methods. This was illustrated in Section 5.7 whore wo employed a virial expansion of the equation of state, a van der Waals type of equation of state, and MC simulations in the specialized mixed isostress isostrain ensemble to investigate various aspects of the impact of confinement on the Joule-Thomson effect. The mean-field approach was particularly useful because it could predict certain trends on the basis of analytic equations. However, the mean-field treatment developed in Sections 4.2.2 and 5.7.5 is hampered by the assump- 

The assumption of homogeneity can be abandoned if the continuous me m-field treatment is replaced by a discrete treatment where the positions of fluid molecules are restricted to nodes on a lattice. The discussion in Section 5.4.2 and 5.6.5 showed that the mean-field lattice density functional theory developed in Section 4.3 w as crucial in unraveling the c.om-plex phase behavior of fluids confined by chemically decorated substrate surfaces. A similar deep understanding of the phase behavior would not have been possible on the basis of simulation results alone. Nevertheless, the relation between these MC data and the lattice density functional results remained only qualitative on accoimt of the continuous models employed in the computer simulations. Thus, we aim at a more quantitative comparison between MC simulations and mean-field lattice density fimctioiial theory in the closing. section of this diaptcr. 

This being the primary goal of the subsequent discussion we would also like to emphasize two other, perhaps more practical, aspects. On account of the rigidity of the underl3dng lattice it seems inconceivable to develop mixed isostress isostrain ensembles suitable for lattice MC simulations. On the other hand, lattice simulations are computationally much less demanding because molecules can occupy only discrete positions in. space. Hence, the number of configurations possible on a lattice is greatly reduced compared with simulations of continuous model systems. 

Another aspect of lattice models concerns the determination of phase behavior. As far as continuous models were concerned we emphasized already that an investigation of phase transitions in such models usually requires a mechanical representation of the relevant thermod3marnic potential in terms of one or more elements of the micrascopic stre.ss teusor. The existence of sucli a mechanical representation was linked inevitably to symmetry considerations in Section 1.6, where it was also pointed out that such a mechanical expression may not exist at all. In this case a determination of the thermodynamic potential requires thermodynamic integration along some suitable path in thermodynamic state space, which may turn out to be computationally demanding. 

For lattice models mechanical expressions for thermodynamic potentials are out of the question regardless of whether the lattice fluid possesses a sufficiently high. symmetry in the sense of our discussion in Section 1.6. The reason is again the incompressibility of the lattice so that one alw ays has to resort to thennodynamic integration techniques in MC simulations of lattice fluids. 

A clear advantage of a lattice model, on the other hand, lies in the fact 

---