THE GRAND CANONICAL ENSEMBLE MONTE-CARLO SIMULATION OF KRYPTON ADSORBED ON GRAPHITE
This section details a computer simulation study of krypton adsorbed on the cleavage face of graphite at 90.12K in the region of the commensurate (CM)-incommensurate (IC) monolayer transition. This transition has been observed by many experimentalists in different fields but eluded Spurling and Lane  in their computer simulations. Attempts are made to overcome two of the possible short-comings in Lane and Spurling's work, the small system size (a maximum of 49 atoms in the monolayer) and the exclusion of terms higher than second order in the krypton/graphite interaction potential. In this work the system size is increased to accommodate a 722 atom monolayer and later a correction to the krypton/krypton interaction potential derived from the accumulation of all the krypton/krypton/carbon three body interaction terms is introduced. The assumption that the "effectiveness" of the Lennard-Jones atom/atom potential - LJ(12-6) - precludes the need for any explicit inclusion of purely krypton three body terms in the calculations is retained though it is by no means obvious that this approximation, although it works very well in the homogeneous fluid, will be valid in the highly heterogeneous environment of an adsorption system.
The cleavage (or 0001) face of graphite is one of the most popular surfaces on which to study adsorption, principally because it is the easiest to obtain with relatively large atomically flat and energetically homogeneous surfaces. This is an essential pre-requisite of accurate adsorption studies as a more inhomogeneous surface would produce unpredictable effects due to adsorption at edges, peaks and in crevices. This adsorption may mask many real but small scale transitions in the nature of the adsorbate. The 0001 face of graphite is the one parallel to the plates of hexagonally arranged carbon atoms that make up the solid. This would appear to the observer above it as a tessellation of hexagons with the carbon atoms forming the vertices. This image is worth remembering as it will reappear quite often in this section. When the potential energy for an atom above this surface is calculated, by summing all the individual LJ(12-6) interactions between this atom and the carbon atoms in the solid (Steele ), the minimum energy positions (adsorption sites) are found to be above the centres of the hexagons and the maxima above the carbon atoms (vertices) with saddle points midway between the atoms. A CM monolayer is one where the adsorbed atoms lie on or about these sites whereas the IC monolayer is the one where the adsorbate pays little attention to the underlying potential and its density is determined purely by the adsorbate/adsorbate potential. In the cases of argon and krypton the CM phase is the less dense, though in the case of krypton the difference is only a few percent. Xenon however is the other way around with the CM phase being the more dense. The existence of both phases has been claimed for all three gases (for argon, Millot  and for xenon, Venables et al ) but is only really well documented for krypton.
The possibility of two different solid-like phases in the first monolayer of krypton adsorbed on graphite was first proposed by Thinly and Duval . In their paper they present isotherms for krypton, xenon and methane at a variety of temperatures in the region of the formation of the first layer. The accuracy of their measurements and the strict control of the temperature throughout the experiments (to within 0.1K) enabled transitions involving only a few percent change in density to be detected. Thanks to this care they did indeed detect a sharp increase in the quantity of krypton adsorbed from about 0.95 to 1.00 statistical monolayers in the isotherms. At the time other possible explanations for this small step were offered based on adsorption at other types of sites on the graphite sample such as edges and peaks. Stevens et al's suggestion (1973] that the adsorption was taking place on residual Iron atoms left over from the preparation of exfoliated graphite (the heating of an inclusion compound with Ferric Chloride) prompted Duval and Thorny  to produce a letter where they convincingly disproved this claim, presenting further isotherms measured on a variety of differently prepared graphite samples all showing very similar transitions at much the same pressure. Since then the transition has been confirmed by many others in the field - Putnam and Fort , Larher  and Larher and Terlain  - and its existence is now in no doubt.
Isotherms only give the number of atoms adsorbed under certain conditions of temperature and pressure and therefore are only able to detect those transitions involving a change in density. They do give the densities of the two phases involved but this is not sufficient to determine their actual structures. In this case the densities gave a very good indication that the step observed by Thorny and Duval was in fact a c-ic transition but it was left to diffraction studies, involving a variety of radiations, to furnish the proof. By observing the way radiation is diffracted by a collection of particles it is possible to deduce how they are arranged in space. X-rays have been used for years to study the arrangement of atoms in crystals but a monolayer of atoms presents much more difficult problem due to the small number of particles involved and successful techniques are a much more recent developement. Low and transmission high energy electron diffraction (LEED and THEED) have been the most popular techniques applied to the krypton/graphite system (Schabes-Retchkiman & Venables , Venables & Schabes-Retehkiman , Fain et al , Chinn & Fain  and Kramer & Suzanne ) and along with the X-ray work of Horn et al  confirm that the transition is indeed between the CM and the IC monolayers. The work of Fain et al  also shows that (at 52K) as the density or the krypton layer is increased from the CM towards the IC the layer begins to rotate with respect to the graphite structure. This rotation starts at a misfit
(where d and d0) are the spacings of adsorbate and sites respectively) of about -2% and reaches an angle of about 0.5 degrees at a misfit of about -5%, proceeding in an apparently 2nd. order transition.
Another powerful tool in delimiting the phases involved is the study of heat-capacity anomalies. Heat capacities rise to abnormally high values in the vicinity of a transition, enabling its position to be pinpointed. As these heat capacity anomalies aren't masked by other effects taking place in the sample simultaneously, Butler et al  were able to produce a phase diagram for the first adsorbed layer up to a coverage of about 2 statistical monolayers, which is reproduced in figure 2.1. A statistical monolayer is the amount of adsorbate required to form a perfect close packed (CP) monolayer. The θ values quoted later are expressed as fractions of this quantity.