Hard hexagon model

Also the tricritical 3-state Potts exponents (for a phase diagram, see fig. 28c) can be obtained from conformal invariance (Cardy, 1987). But in this case the standard Potts critical exponents are related to an exactly solved hard core model, namely the hard hexagon model (Baxter, 1980), and not the tricritical ones. The latter have the values crt = 5/6, = 1/18, Yt = 19/18, <5t = 20, ut = 7/12, rjt = 4/21,

tricritical exponents coincide (den Nijs, 1979). 

The adsorption isotherm for the hard hexagon model has been derived by Joyce [144]. His results can be fitted to Fade approximants [129]. For the high density, ordered phase... 

Figure Bl.21.1. Atomic hard-ball models of low-Miller-index bulk-temiinated surfaces of simple metals with face-centred close-packed (fee), hexagonal close-packed (licp) and body-centred cubic (bcc) lattices (a) fee (lll)-(l X 1) (b)fcc(lO -(l X l) (c)fcc(110)-(l X 1) (d)hcp(0001)-(l x 1) (e) hcp(l0-10)-(l X 1), usually written as hcp(l010)-(l x 1) (f) bcc(l 10)-(1 x ]) (g) bcc(100)-(l x 1) and (li) bcc(l 11)-(1 x 1). The atomic spheres are drawn with radii that are smaller than touching-sphere radii, in order to give better depth views. The arrows are unit cell vectors. These figures were produced by the software program BALSAC [35]-...

Figure Bl.21.1. Atomic hard-ball models of low-Miller-index bulk-terminated surfaces of simple metals with face-centred close-packed (fee), hexagonal close-packed (hep) and body-centred cubic (bcc) lattices (a) fee... 

Nevertheless, Nagai s approach raised objections [54-57]. Later, however, Nagai presented proof [58] that his expression, our Eq. (67), is superior to the Rd expression derived from the ART approaeh. Using the hard hexagon adsorption model, Nagai showed that the ART approach underestimates the role of entropy changes as a factor affecting the kinetics of adsorption/desorption processes. 

In our model it was assumed that the bisulfate ion formed a x- /3template. This template leaves a honeycomb lattice of free sites for the adsorption of copper. The clear implication is that the first peak has 2/3 of a monolayer of Cu. The second peak corresponds to the replacement of the bisulfate by copper in the adlayer. We showed also that the broad foot of the first peak is due to a second order hard hexagon like transition, which is seen experimentally by Itaya[136] and Kolb [143]. We believe that the interpretation that the first peak corresponds to only 1/3 of a monolayer, based on the STM and LEED observations, is consistent with our model if it is the bisulfate ion that is seen. 

In our model [58, 129] the broad foot of the first spike in the Cu-Au voltammo-gram is due a second order surface phase transition, similar to the hard hexagon phase transition [59]. This transition was actually observed by Itaya [136] and by Kolb [143]. 

The atomic bonding in this group of materials is metallic and thns nondirectional in natnre. Conseqnently, there are minimal restrictions as to the nnmber and position of nearest-neighbor atoms this leads to relatively large nnmbers of nearest neighbors and dense atomic packings for most metallic crystal structnres. Also, for metals, when we nse the hard-sphere model for the crystal structure, each sphere represents an ion core. Table 3.1 presents the atomic radii for a number of metals. Three relatively simple crystal structures are found for most of the common metals face-centered cubic, body-centered cnbic, and hexagonal close-packed. 

Lattice gas models are simple to construct, but the gross approximations that they involve mean that their predictions must be treated with care. There are no long-range interactions in the model, which is unrealistic for real molecules the short-range interactions are effectively hard-sphere, and the assumption that collisions lead to a 90° deflection in the direction of movement of both particles is very drastic. At the level of the individual molecule then, such a simulation can probably tell us nothing. However, at the macroscopic level such models have value, especially if a triangular or hexagonal lattice is used so that three-body collisions are allowed. 

The extrusion curve for these materials hardly shows any release of mercury and reintrusion experiments exhibit a negligible intrusion, indicating the original hexagonal porous structure is damaged and as a consequence these materials are inappropriate as model materials in MIP. 

The three categories differ in how they represent the compressibility and expansion of the polymer systems under scrutiny. Volumetric changes are restricted to a change in cell volume in cell models. Lattice vacancies are allowed in lattice-fluid theory, and the cell volume is assumed constant. Cell expansion and lattice vacancies are allowed by hole models. The models also differ on the lattice type, such as a face-centered cubic, orthorhombic, hexagonal, and also in their selection of interpolymer/interoligomer potential such as Lennard-Jones potential, hard-sphere, or square-well. 

Two forms of the cell model (CM) are then developed harmonic oscillator approximation and square-well approximation. Both forms assnme hexagonal closed packing (HCP) lattice structure for the cell geometry. The model developed by Paul and Di Benedetto [13] assumes that the chain segments interact with a cylindrical symmetric square-well potential. The FOV model discnssed in the earlier section uses a hard-sphere type repulsive potential along with a simple cubic (SC) lattice structure. The square-well cell model by Prigogine was modified by Dee and Walsh [14]. They introduced a numerical factor to decouple the potential from the choice of lattice strncture. A universal constant for several polymers was added and the modified cell model (MCM) was a three-parameter model. The Prigogine cell EOS model can be written as follows. 

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