Physical Adsorption Project Section 2.1

2.1 Free Energy

The free energy of an adsorbed gas is a function of the temperature T of the substrate and the pressure p of the (3-dimensional) gas, which in equilibrium is at the same temperature. The basic thermodynamics is spelt out in section C2, following a previous introduction in section 1.3, of the Surface Physics Course. The Free Energy has to be defined with respect to a reference state, and, in the case of rare gases on graphite, it is reasonable that this reference state should be the commensurate (C) state, where the rare gas atoms occupy every third graphite hexagon. This state is chosen because, for unconstrained solid Xe and Kr monolayers at least, the ‘natural’ lattice parameters are close to the (square root-3) x (square root-3) Rotated 30 degrees (the so-called root-three) C-structure.

This structure has become so well known that it is not drawn in the more recent papers. However, an illustration is included here for reference. The present discussion follows Venables and Schabes-Retchkiman (1977), where the interest centered on the free energies of the C and incommensurate (I) phases, and the phase transitions between them.

We worked out the Free Energy (F/atom) for an adsorbed density n, where n = 1 corresponds to the C-phase. If n is not equal to 1, then atoms have to exchange between the adsorbed layer and the gas phase. The Free Energy of the system can be expressed in terms of the free energy per atom, Fa of the adsorbed layer as

Since the phase with the lowest free energy is the most stable, we can see from equation (1) that either the C-phase (n = 1), I-phase (n > 1 in this case) or the gas phase (n = 0) can be most stable at a given T and p; thus equation (1) also allows one to compute the position of phase transitions lines (p as a function of T), for both C-I solid, and gas-solid (either C or I) transitions; once one gets into this, one starts to appreciate the power of statistical mechanics. Without wishing to confuse anyone, those interested in the structure of statistical mechanics can note that equation (1) corresponds to the Helmholtz free energy of the adsorbed phase, plus the Gibbs free energy of the gas phase. This arises because the thermodynamic variables are the area of the adsorbate, the pressure of the gas, and T. Another way to arrive at the same conclusion is to focus only on the adsorbate, but to consider it as an open system, in which particle number is not conserved. Then the problem is posed within the Grand Canonical ensemble, in which the potential, (capital) Omega = F-G. These various approaches are described in every Stat Mech book: adsorption is covered particuarly thoroughly by Hill, Introduction to Statistical Thermodynamics, chaps 7-9.

The input for a calculation of the free energy are the various potential energies between the gas and the substrate and between the adsorbed gas atoms. There are many possible types of interaction, but in the case of rare gas adsorption pair interactions are the most important. A detailed listing and discussion of these interactions is given by Vidali et al. (1991). In addition one needs a model of the vibrations of the atoms. In the calculations done here, we use an Einstein model for vibrations perpendicular to the substrate, plus a cell model for vibrations parallel to the substrate plane. There are many types of vibrational models, appropriate in different circumstances. Here the cell model is appropriate to the heavier adsorbed gases which behave classically; it is very successful at modelling the thermal expansion of the lattice, which is very large for adsorbed (rare) gases, especially in comparison to 3D solids. For Ne/ graphite, the most successful calculations have used the quantum cell model (Bruch et al., 1984). This is not included in the present calculations, but we may develop this later in collaboration with Prof Bruch.

The results are shown as before in the form of a table and a plot below.

This data table only indicates the low T end of the data shown on the plot, but shows that the free energy has a value ~ -550 K/atom for all values of the lattice parameter studied; this is due to the dominance of the Ne-graphite potential energy. In the following sections, however, it is the gradient and curvature of these plots, with respect to T and lattice parameter, which are important. One can see that the T-dependence at fixed a is roughly parabolic; by taking three a-values only, we can ensure that the a-dependence at fixed T is analytically parabolic.

To see how this works out, continue to the following section 2.2, or alternatively return to section 1.