114 4 Surface processes in adsorption

Figure 4.2. Thermal expansion of a 2D Lennard-Jones triangular monolayer solid on a smooth substrate, computed for Xe interaction potential parameters (Bruch et al. 1997, after Phillips et al. 1981, replotted with permission). Results for the quasi-harmonic theory (QHT) are good at low temperature, with the (quantum-corrected) cell model agreeing closely with classical Monte Carlo calculations at high T. The triple point of 2D Xe (on graphite) is 99 K.

the corresponding quantity in 3D is a free volume, Vf. In a high-density adsorbate with large U(r) at moderate T, the integral is negligible except at positions r close to the equilibrium spacing. This approximate, classical theory, is very eVective in computations for solids at high temperature, since it includes thermal expansion ± the response to the spreading pressure exerted by the anharmonic vibrations ± which many, apparently more sophisticated models, ignore. For these reasons at least, it deserves to be better known and more widely used as a teaching aid.

One anharmonic model which aims to have to have the correct low temperature limit, and to be useful at higher T, is the quantum (or quantum-corrected) cell model (Holian 1980, Barker et al. 1981, Bruch et al. 1997, chapter 5). A comparison of lattice dynamical models for a 2D solid monolayer interacting via the approximate LennardJones potential, with parameters appropriate to xenon, is shown in ®gure 4.2.

4.3Phase diagrams and phase transitions

One of the intriguing aspects of both physiand chemisorption is the large number of phases that can exist at the surface, and the transitions that occur between these phases.

4.3 Phase diagrams and phase transitions	115

Figure 4.3. Sub-ML isotherms for Xe/graphite between 97 and 117 K. The isotherms, from left to right, are at 97.4, 100.1, 102.4, 105.4, 108.3, 112.6 and 117.0 K. Between 110.1 and 117 K, the layer undergoes two ®rst order phase transitions, showing 2D gas (G), liquid (L) and solid

(S) phases, whereas at 97.4 K only the G to L transition occurs; the 2D triple point is at 99 K (after Thomy et al. 1981, reproduced with permission).

There is a comparable richness of structure to that displayed in high pressure physics, where there is both a density r, and a corresponding structure, at a given p and T. The relation r5f(p,T) is called the Equation of State (EOS) in the (3D) physics of bulk matter, or the (p,V,T) relation.

4.3.1Adsorption in equilibrium with the gas phase

The corresponding equilibrium equation for (2D) adsorbed layers is u5f(p,T), and

since we have already used mv5m01kT ln (p), and mv5ma, we can think of u5f(m,T) equivalently. For u, read Na if we do not de®ne the ML unit in the standard way. So, as we compress a 2D gas or localized adlayer by increasing the (gas) pressure p, the

adatoms will come within range of their mutual attractive or repulsive forces, and phase transitions may result, ®rst within the ML, and subsequently from ML to multilayer.

If the substrate and adsorbate are well ordered, the condensation may proceed in well de®ned steps, as shown in ®gures 4.3 and 4.4 for physisorbed Xe and Kr/graphite respectively at ML and sub-ML coverages. As studied by several French groups especially (Thomy et al. 1981, Thomy & Duval 1994, Suzanne & Gay 1996), these volumetric studies, using high quality exfoliated graphite, established the existence of 2D solid, liquid and gaseous layers. The (p,T) positions of the phase transitions (including multilayer transitions) and ®xed points such as triple points and critical points in these layers were also accurately measured.

More recent experimental thermodynamic work has automated the measurement process, and has achieved very high accuracy for quantities which depend on the slope

116 4 Surface processes in adsorption

Figure 4.4. Sub-ML isotherms for Kr/graphite between 77 and 110 K. The isotherms are at 77.3, 79.8, 82.3, 84.8, 86.0, 88.0, 91.8, 96.6, 102.6 and 109.5 K. These show 2D gas, liquid (maybe) and two solid phases, with a presumed solid-solid phase transition at point A10, which is not ®rst order (after Thomy et al. 1981, reproduced with permission).

of the isotherm, such as the isothermal compressibility. Examples for Kr in the multilayer region are given by Gangwar & Suter (1990) and for Xe near ML melting by Gangwar et al. (1989) and Jin et al. (1989). An example showing the much improved precision of these data is given in ®gure 4.5 which can be directly compared with ®gure 4.3.

Before we examine these results, we can note the diVerent forms of graphs and phase diagrams that can be plotted. The problem arises because we have three variables, T, ln(p) or m, and u, but we typically want to output onto paper, so that one of these three is not plotted; the corresponding (third piece of) information may either not be known, or may be discarded, or it may be given as a parameter.

An isotherm is a graph of u against ln(p) or m, with T as the parameter, as used in the previous section. A phase diagram using log(p) and 1/T as axes is very convenient for (physisorption) experimentalists, because the pressure can be varied over many orders of magnitude, and this plot results in straight lines for phase transitions (e.g. gas±solid or monolayer±bilayer transitions) which show an Arrhenius behavior ± the slope of these lines give the corresponding energies. But typically the coverage information is lost. Theorists are fond of phase diagrams as a function of T and m: this gives them the chance to investigate the adsorbed phase, and ignore the 3D gas phase, which provides the value of m, typically Dm with respect to the bulk 3D phase, when the comparison with experiment is made later.

4.3 Phase diagrams and phase transitions	117

Figure 4.5. (a) Isotherm and (b) compressibility on a logarithmic scale in the monolayer and bilayer region of Xe on graphite at T5105.4 K. Note the large range and high measured accuracy of compressibility, and the reproducibility of two runs taken at T5105.456 and 105.435 6 0.001 K, with absolute values 6 0.1 K (from Gangwar et al. 1989, reproduced with permission).

An isobar is a graph/cut/contour at constant pressure, giving a plot of u(T), with ln(p) or m as the parameter. The meaning is the same as used on weather charts, but the context is a little diVerent (is the weather an equilibrium phenomenon?). In many single surface experiments, a more or less directed beam is aimed at the substrate to establish a steady state concentration which is almost a true equilibrium, but not quite. In particular, the temperature of the beam Tb is typically not the same as that of the adsorbate Ta; the question of whether or not to correct the pressure for this thermomolecular eVect, of order (Ta/Tb)1/2, recurs in the experimental literature.

An isostere is a contour on a p(T) plot at constant coverage. Typically log(p) varies as 1/T, and the energy associated with such an Arrhenius plot is called the isosteric heat of adsorption. This is the energy associated with the adsorbed phase at that coverage, and it comprises the adsorption energy and lateral binding energies, their derivatives