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In order to locate the single particle thin-thick adsorbed film transition line, one approach is to calculate density profiles at a series of state points along a particular path in the phase diagram. If one uses as a starting approximation in the numerical (iterative) procedure for calculating the density profiles those profiles corresponding to the previous point on the path, then one finds at some state point there is a jump from a thin to a thick adsorbed film. However, this jump does not occur at the equilibrium transition, rather it occurs at the "spinodal" for this single particle thin-thick adsorbed film transition. In order to locate correctly the equilibrium transition line, one must calculate the grand potential corresponding to the profiles with a thick and those with a thin adsorbed film. The proper location for the transition is where these two grand potential curves cross. Near the thin-thick adsorbed film critical point there is little hysteresis and therefore very little difference between the location of the line of equilibrium transitions and the spinodal lines. However, for the present system well away from the critical point, there can be very substantial hysteresis. In Ref. 9 a spinodal line was calculated rather than the true transition line, plotted here in Fig. 1. Similar problems of metastability were elucidated in recent DFT studies of layering transitions for a model colloid-polymer mixture adsorbed at a hard wall (Ref. 31).
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