Cube model

Approximating the real potential by a square well and infinitely hard repulsive wall, as shown in figure A3.9.2 we obtain the hard cube model. For a well depth of W, conservation of energy and momentum lead [H, 12] to the very usefiil Baule fomuila for the translational energy loss, 5 , to the substrate... 

This interpretation is largely based on the results of cube models for the surface motion. It may also be that... 

Two environment and melting ice cube models—Ng and Rippin (1965) and Suzuki (1970)... 

Scheme 3 Reaction of perrhenate on aluminosilsesquioxane cube (model for silica-alumina) with SnMe4 is predicted to form grafted MeReOs.

Transmetalation of the perrhenate/aluminosilsesquioxane cube model with SnMe4 is considerably more exothermic than for the perrhenate/silsesquioxane cube model. A similar grafted trimethyltin fragment is formed, as is MeReOs however, the latter is not liberated. It remains bound to the aluminosilsesquioxane cube via the Lewis acid-base interaction with the A1 center. The optimized structure also contains a Lewis acid-base interaction between Re and an adjacent framework oxygen... 

Figure 3.6. Schematic of the cube model for energy transfer ( ) of an atom/molecule of mass m incident with energy Et to the lattice represented by a cube of mass Ms. The atom/molecule adsorption well depth is W. The double arrow labeled Ts emphasizes that the cube also has initial thermal motion in the scattering.

For the cube model, when Eq>E , trapping into the adsorption well occurs. Therefore, there is a critical energy Ec such that the trapping coefficient a = 1 for E direct inelastic scattering occurs for En>Ec. For the stationary cube,... 

Figure 3.12. Inelastic scattering of Ar from Pt(lll) at the various input energies listed in the figure and for an initial angle of incidence 0, = 45° and Ts = 800 K. Results are plotted as EfIE vs. the final scattered angle . Points are the experimental results and the lines marked adjacently in the label are results of molecular dynamics simulations on an empirical PES. The long dot-dashed curve is the prediction of a cube model of energy transfer, while the dashed curve is the prediction from hard sphere scattering. From Ref. [135].

This description is elaborated below with an idealized model shown in Figure 17. Imagine a molecule tightly enclosed within a cube (model 10). Under such conditions, its translational mobility is restricted in all three dimensions. The extent of restrictions experienced by the molecule will decrease as the walls of the enclosure are removed one at a time, eventually reaching a situation where there is no restriction to motion in any direction (i.e., the gas phase model 1). However, other cases can be conceived for a reaction cavity which do not enforce spatial restrictions upon the shape changes suffered by a guest molecule as it proceeds to products. These correspond to various situations in isotropic solutions with low viscosities. We term all models in Figure 17 except the first as reaction cavities even... 

A check of lack of fit in control points No. 8 and 9 showed that the incomplete cube model is adequate. Graphic interpretation of this model as contour lines is shown Fig. 3.11. 

The given designs are used for fitting a three-component simplex-centroid (or an incomplete cube model) with main effects of process factors ... 

Figure 4.1. Cube model for RGB system. The RGB cube model illustrates the definition of colors by the three primary components along the three axes R, G, and B. Each color point is represented by a triple (r, g, b). The three primary colors are red (1, 0, 0), green (0, 1, 0), and blue (0, 0, 1). Other binary-status (0/1 for r, g, b) colors are cyan (0, 1,1), magenta (1, 0, 1), yellow (1,1, 0), white (1, 1, 1), and black at origin (0, 0, 0). Different colors are expressed by a combination of r, g, and b values varied between 0 and 1. For example, gray colors correspond to the main diagonal between black and white.

There is a very simple model for estimating the trapping probability in atomic adsorption due to a phonon-excitation mechanism. In the hard-cube model (HCM) [6, 7], the impact of the atom on the surface is treated as a binary elastic collision between a gas phase atom (mass m) and a substrate atom (mass Mc) which is moving freely with a velocity distribution Pc(uc). This model is schematically illustrated in Fig. 1. If the depth of the adsorption well is denoted by Ead, the adsorbate will impinge... 

Figure 1 Schematic illustration of the hard cube model. An atom or molecule with mass m is impinging in an attractive potential with well depth Fad on a surface modeled by a cube of effective mass Mc. The surface cube is moving with a velocity uc given by a Maxwellian distribution.

Assuming a weighted Maxwellian velocity distribution for uc, the trapping probability in the hard-cube model can be analytically expressed as [7]... 

Figure 8 Trapping probability of 02/Pt(l 11) as a function of the kinetic energy for normal incidence. Results of molecular beam experiments for surface temperatures of 90 and 200 K (Luntz et al. [81]) and 77 K (Nolan et al. [87]) are compared to simulations in the hard-cube model (HCM).

Similar modelling has been performed for both of these systems, based on the cube model. Following Hand and Harris, the molecular motion was coupled to the surface oscillator via a rigid shift of the Z-coordinate in the PES, i.e. V(Z, r,. .., y) = V(Z — y, r...), where y is the oscillator coordinate. For the H2/Pd system [80], six molecular degrees-of-freedom were included in a classical treatment, while four molecular degrees-of-freedom were included in a quantum solution for the H2/Cu system [81, 82]. In the classical calculations, the surface temperature dependence was introduced by sampling the surface vibration from a Boltzmann distribution. In quantum calculations, this is not possible, and many calculations were required, each in a different initial surface oscillator state. The results... 

Figure 4 Results from classical trajectory calculations for in-plane scattering of Ar from Ag(l 11) with an incidence angle of 40° measured with respect to the surface normal. In the panels a and c results for the relative final energy Ef/Ei are shown, where E is the initial energy. Lines indicate the energy transfer computed with the cube model (parallel momentum conservation) and a binary collision model. In panels b and d angular distributions are shown. Calculations for 0.1, 1,10 and lOOeV are shown. The panels a and b are calculated for a zero temperature, static lattice panels c and d for Ts = 600 K. From Lahaye et al. [43].

Figure 5 Angular width measured for various systems at incidence angles of 38-45° and systems as indicated in the figure. The dotted, dashed and dash-dotted lines are results from calculations with the hard cube model. The dotted line represents calculations with a mass ratio of 32/195 and Ts = 400 K (02/Pt), the dashed line with a mass ratio of 32/150 and Ts = 600 K (C>2-Ag), and the dash-dotted line represents calculations with a mass ratio of 40/195 and Ts = 500 K (Ar-Pt). Details about the sources of the various datasets can be found in the paper by Wiskerke and Kleyn, from which the figure is taken [54].

Since supercritical fluids were chosen for their ability to penetrate small cracks and crevices, additional tests were performed to evaluate this characteristic. A test cube modeled after a similar fixture fiibricated by Ferranti Aerospace, was developed and manufactured to md in this study. The cube had a number of blind holes, tapped holes, dtannels and crevices to simulate actual hardware. Beryllium, 300 Series stainless steel and aluminum cubes were constructed to simulate the conunon metals found in the instrument. In addition, the sides of the cube were removable to facilitate deposition of the contaminants into these blind holes and crevices and later analysis of cleaning effectiveness. The base of the cube was equipped vrith a scanning electron microscope (SEM) mount so that the cube could be examined directly in the SEM. Figure 3 is a photograph of a test cube. Extensive evaluations with these test cubes indicated that supercritical fluids were indeed effective at removing contaminants from cracks and crewces. 

Fig. 23. The distribution of the (scalar) velocity of atoms at different times in a molecular dynamics simulation of the impact of a 125 atom Ar cluster at a surface where the surface is simulated by the hard cube model at the temperature of 30 K. The Impact velocity is 20 km s or 1 A per 5 fs where the range parameter of the Ar-Ar potential is 3.41 A. The mean free path is very roughly of the same magnitude. Thermalization is essentially complete by 80 fs or, after roughly four collisions.

---