4.4 Physisorption	119

Figure 4.6. AES amplitude of Xe/graphite as a function of log(p), showing a ®rst order gas±solid phase transitions at p and T values indicated (after Suzanne et al. 1973, 1974, 1975, reproduced with permission).

for full 3D equilibrium: this point recurs when considering models of (epitaxial) crystal growth in chapter 5 and section 7.3. Chemisorption examples are discussed in section 4.5.

4.4Physisorption: interatomic forces and lattice dynamical models

4.4.1Thermodynamic information from single surface techniques

Once one applies `single surface' techniques to adsorbed layers with sub-ML sensitivity, several types of phase and phase transition can be observed on many materials; the following examples are highly selective towards rare gases on graphite. Figure 4.6 shows the AES amplitude for Xe/graphite as a function of log(p). These curves are adsorption isotherms, taken as the pressure is varied through the gas±solid transition. The ®rst order character of the transition is seen very clearly. At the same point that the AES amplitude jumps, spots appear in LEED (or other diVraction technique) characteristic

1204 Surface processes in adsorption

of an ordered ML solid. Understanding the thermodynamics of this 2D gas±solid transition enables one to measure both the cohesive energy of the 2D adsorbed solid, and the pre-exponential factor, which can be related to the entropy of adsorption. This results in an estimate of the change in vibration frequencies between the adsorbed 2D phase and the bulk 3D phase. In this case, the entropy is negative, corresponding to the eVective vibration frequencies being higher in the adsorbed state than the bulk phase (Suzanne et al. 1974, 1975).

We can see how this arises by reference to the vapor pressure equation introduced in section 1.3.1, coupled with the discussion of monolayer vibrations in section 4.2. The 2D gas±solid phase transition line on an Arrhenius plot has a slope measured experimentally as 2780 6 50 K/atom, and this corresponds approximately to the sublimation

energy L2 of the 2D solid phase. We note that this is considerably higher than the (T50) sublimation energy L051937 K/atom of 3D bulk Xe given in chapter 1, table 1.1, which is the basic reason why the adsorbed layer is stable.

The intercept of this 2D transition line on the log(p) axis at T2150 is actually higher than that of the 3D bulk sublimation line intercept, and the diVerence in ln(p) 52DS/k, where DS is the entropy diVerence between 2D and 3D solids. This has also been measured as DS5± 2 6 1 cal/mole/K, or in more useful units, DS/k521.0 6 0.5. In the high temperature limit of the Einstein model (see equations 1.14 and 4.6), we

can see that DS/k53ln(n/ne), where ne is the geometric average of three vibration frequencies in the 2D adsorbed solid. Thus, taking n50.73 THz from chapter 1, table 1.1,

we can estimate ne as ,1.0 THz, with of course a substantial error bar; the error limits indicate 0.86,ne,1.20 THz. The vertical vibration frequency has been measured by interpreting the hydrogen atom scattering Debye±Waller factor for Xe/graphite, as 0.90 THz (Ellis et al. 1985). Consequently, the thermodynamic DS/k estimate implies that

the lateral vibrations in the completed solid ML are also higher than bulk values. Although this is consistent with the compression of ML solid Xe at low temperatures, such a result is not inevitable. For example, a non-compressed sub-ML solid may well have a larger DS than the 3D counterpart at low temperatures, due to low-lying vibrational and translational modes.

The thermodynamics of these models are given in several places (e.g. Cerny 1983 or Price & Venables 1976); many have referred to earlier work by Lahrer (1970). This last reference is a relatively rare example of a Ph.D. thesis which was widely circulated, but which never appeared in the same form in the open literature. Because they are not generally available, some of the more useful thermodynamic relationships are reproduced here in Appendix E.2.

4.4.2The crystallography of monolayer solids

The crystallography of the 2D solid phase of Xe/graphite was observed by diVraction techniques (LEED and THEED). The THEED work had high enough precision to detect that this solid was incommensurate (I), having a lattice parameter some 6±7% larger than graphite under the conditions of ®gure 4.6. At lower T and p, these experiments showed that the layer was compressed into a commensurate (C) phase, i.e. an

4.4 Physisorption	121

Figure 4.7. I±C and monolayer±bilayer phase transitions for Xe/graphite as measured by THEED, where the mis®t is the diVerence in lattice parameter of the adsorbed layer relative to the commensurate Î3 structure. As the temperature is lowered at constant pressure

(p57.23 1028 Torr) within the ML regime, the mis®t decreases to zero at T,58 K. About 2 K lower, the bilayer condenses with a mis®t ,2% (after Kariotis et al. 1987, Hamichi et al. 1989; reproduced with permission).

I±C phase transition was observed, as shown in ®gure 4.7 (Schabes-Retchkiman & Venables, 1981, Kariotis et al. 1987, Hamichi et al. 1989, 1991). The opposite situation happens for Kr/graphite: Kr ®rst condenses into the C-phase, and then compresses into the I-phase, where the Kr lattice parameter is a bit smaller than the corresponding graphite spacing. This C±I transition was observed by both THEED and LEED.

The I-phase has a modulated lattice parameter; this gains energy from having more of the adsorbate in the potential wells of the substrate, but costs energy in the alternate compression and rarefaction of the adsorbate. For example, if we consider the substrate to provide a template on which the adsorbed monolayer sits, then the interaction potential varies periodically, and can be expressed as a Fourier series:

V(r)5V01og Vgexp(ig´r),

(4.14)

where the sum is over as many 2D reciprocal lattice vectors g as are needed to describe the corrugation of the potential adequately. Typically, only one term in Vg, consistent

1224 Surface processes in adsorption

with the symmetry of the underlying lattice, is retained; this does not mean that higher order components are not present in the potential, just that there is not enough detail in the model to ®nd out more by comparison with experiment.

If the geometry of all these phases is not clear, a pictorial description of the I-phase, and its representation in terms of domain walls, solitons or mis®t dislocations, is shown here in ®gure 1.17 (Venables & Schabes-Retchkiman 1978). There are in fact two types of I-phase: the aligned (IA) phase and the rotated (IR) phase, with another possibility of a phase transition. The IR phase was ®rst discovered for Ar/graphite using LEED (Shaw et al. 1978), and is even more pronounced in the case of Ne/graphite shown in ®gure 4.8 (Calisti et al. 1982).

In a rotated phase, the diVraction spots are split, corresponding to two domains rotated in opposite directions. Why does this happen? The mis®t is accommodated by compression and rarefaction; but typically shear waves cost less energy than compression waves, so it pays to include a bit of shear if the mis®t is large enough. This eVect was ®rst described quantitatively as a static distortion, or mass density, wave1 by Novaco & McTague (1977), and has been further developed by several other workers including Shiba (1979, 1980), as described by Bruch et al. (1997, chapters 3 and 5). The energies/atom gained by rotation for the various rare gases on graphite are indicated in ®gure 4.9. It is remarkable how small these energies can be, and still be suYcient to stabilize the rotated phase; this is because of the large numbers of atoms in each domain, and because the domain walls cannot act independently of their neighbors, unless they are far apart. In that limit, we enter new regimes, such as a domain wall ¯uid; but let's not get too complicated at this stage.

These observations mean we can get C±IA±IR transitions in sequence, which have been observed for both Kr and Xe/graphite. In the case of Xe/graphite, a large body of THEED data has been obtained at relatively low pressures, close to the C±IA and IA±IR transitions; one data set is shown in ®gure 4.7. These C, IA and IR phases have also been observed for Xe/Pt(111) using helium atom scattering (Kern et al. 1986). We can also get 1D incommensurate, or `striped' phases, where the mis®t is zero in one direction, and non-zero in the other. Then the symmetry is reduced, for example from hexagonal to rectangular as observed for Xe/Pt(111) at low mis®t. The reasons why such striped phases occur (or not) depend on details of the domain wall interactions, as discussed by Kern & Comsa (1988) and by Bruch et al. (1997, chapter 5).

Near to the C±IA transition, the lattice dynamics can be split into two components, involving low-lying vibrational modes of the domain walls, and faster vibrations of the atoms within their local cells. This is seen both in computer simulations (Koch et al. 1984, Schöbinger & Abraham 1985) and in various analytical models (Kariotis et al. 1987, 1988, Shrimpton & Joós 1989). For Xe/graphite, it was possible to use the position of the transition in the (T, p) plane, shown in ®gure 4.7, to determine the depth of the potential well DV52616 3 K/atom; similar analyses have been attempted for other adsorption systems. Here, DV is the diVerence in energy between the atoms in the

1Note that the acronym SDW is sometimes used for static distortion wave, but that SDW more usually means spin density wave in relation to magnetic materials.

		4.4 Physisorption		123
	(a)
				(b)

(c)

Figure 4.8. Rotated monolayer phases: the LEED pattern of (a) adsorbed Ne/graphite; (b) schematic diagram of the diVraction spots from the two domains; (c) measurements of lattice parameter and rotation (after Calisti et al. 1982, reproduced with permission).