The Slab Model

The slab model consists of a film formed by a few atomic layers parallel to the (hkl) crystalline plane of interest. The film, of finite thickness, is limited by two surface planes, possibly related by symmetry. For sufficiently thick slabs, this kind of model can provide a faithful description of the ideal surface. The adequacy of the model must be checked by considering convergence of geometry, energy, and electronic properties with an increasing number of atomic layers included in the slab. 

In actual calculations, two different schemes can be envisaged to deal with a slab model  

When we use a plane wave basis set, which requires a 3-D Fourier representation of many intermediate quantities, such as the charge density, only model (b) can be adopted. On the contrary, when a local basis set is adopted, no problems occur in the implementation of both schemes. 

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Since the logarithm of the retention factor is proportional to the free energy of adsorption, we hnd for the slab model that... 

A complete description of the protein-stationary phase interaction involves many complications and a general model is extremely difficult to formulate. The slab model described above is very simple, yet it gives interesting physical insights and may be a useful starting point for more elaborate theories. Here, we shall only briefly discuss some of the challenges a more complete model meets. For more complete discussions see Refs. [1,24]. 

The Poisson-Boltzmann equation. The slab model is based on a solution of the linearized Poisson-Boltzmann equation that is valid only for low electrostatic surface potentials. As... 

The geometry. It is clear that the geometry of the system is much simplified in the slab model. Another possibility is to model the protein as a sphere and the stationary phase as a planar surface. For such systems, numerical solutions of the Poisson-Boltzmann equations are required [33]. However, by using the Equation 15.67 in combination with a Derjaguin approximation, it is possible to find an approximate expression for the interaction energy at the point where it has a minimum. The following expression is obtained [31] ... 

Only a very limited number of manufactured catalysts could be approximately described by the slab model but there appear to be many which conform to the shape of a cylinder or sphere. Utilising the same principles as for the slab, it may... 

The effectiveness factor for the slab model may also be calculated for reactions other than first order. It turns out that when the Thiele modulus is large the... 

If the reactions were not influenced by in-pore diffusion effects, the intrinsic kinetic selectivity would be kjk2(= S). When mass transfer is important, the rate of reaction of both A and X must be calculated with this in mind. From equation 3.9, the rate of reaction for the slab model is ... 

A classical example of this type of competitive reaction is the conversion of ethanol by a copper catalyst at about 300°C. The principal product is acetaldehyde but ethylene is also evolved in smaller quantities. If, however, an alumina catalyst is used, ethylene is the preferred product. If, in the above reaction scheme, B is the desired product then the selectivity may be found by comparing the respective rates of formation of B and C. Adopting the slab model for simplicity and remembering that, in the steady state, the rates of formation of B and C must be equal to the flux of B and C at the exterior surface of the particle, assuming that the effective diffusivities of B and C are equal ... 

The dispersion modeling of the resultant ammonia vapor hazard zones to the ERPG-2 value for ammonia of 200 ppm was done using the SLAB model. The isopleths for the D/5 and F/2 meteorological conditions found in these examples are shown in Figure 7.17 and 7.18. 

The whole point of the slab model is to reproduce an isolated surface so that overlap of electron density between the slab and its periodic images must be vanishingly small. This means that k-point sampling used to describe interactions between unit cells in the direction perpendicular to the surface can be achieved with a single k-point. In the direchons parallel to the surface, however, k-point sampling comparable to the reference bulk calculahon must be maintained. 

It is important to test the validity of the slab model to reproduce the surface properties of isolated surfaces. The two main parameters are clearly the vacuum gap introduced and the thickness of the slab employed. Both of these are system dependent but Figure 8.12 shows the effect of the vacuum gap on the calculated surface energy for the unrelaxed surface of a-AbOs taken from localized basis set GGA-DFT calculations using the DSOLID code. At small gap distances the surface energy is underestimated, since in the limit of a zero vacuum gap Equation 8.26 would give = 0. The surface energy in this case is clearly converged above a... 

The slab geometry also suffers from other finite-size effects. If the extent of the unit cell parallel to the interface is too small, artificial strain effects axe introduced, because the metal and ceramic axe forced to be coherent by the periodic boundary conditions. Of course, this may be eliminated by enlarging the unit cell, which unfortunately leads to very computerintensive calculations, as is the case with the cluster models. However for the slab model, the oscillations in the electronic density of states are not as dramatic when varying the number of atoms as in the case with clusters. This is because the slab is infinite parallel to the interface. This implies the spectrum is continuous, and the metal slab does not have an artificial band gap, unlike the metal cluster. 

Figures 15, 16, 17 and 18 show examples of the paths of air masses estimated in the manner just described. The arrival times in El Monte are used to identify each trajectory in any subsequent references in this report. The date of Sept. 29, 1969, is chosen because of the variety of data that we have available for that day. The earlier morning meandering patterns give way to the dominant onshore flows for all trajectories 0900 to 1000 hours Pacific Standard Time (PST). The meteorological formulation that we have adopted takes the time and location information from these trajectories to establish the initial conditions and the boundary conditions. Initial conditions are specified as vertical profiles of concentration and boundary conditions as time histories of surface-based pollutant emissions. These trajectories are not used in the tests of the slab model they are used to test the moving air parcel model.

Atmospheric dispersion concentrations are obtained by solving the SLAB model with the tracking method. However, the existing SLAB model assumes steady state in case of the continuous releases. This assumption makes a simple calculation procedure possible, but brings inaccuracy. We modified the SLAB model to make it simulate the unsteady releases as well (see Fig. 3 for details). We seek for source information from... 

Figure 11. Distribution coefficient K for the slab model. Rigid rod (solid line) and once-broken rod (dotted line). Reproduced with permission from Ref. 36, Figure 2, Copyright 1972, John Wiley Sons, Inc.

The experimental values of the effective diffusivities are clearly lower than the values deduced from the theoretical models, even taking into consideration the internal convective flow. Of course, the experimental values depend on the pseudohomogeneous model chosen to represent the alumina particle, but even if the spherical model were used, the values obtained (1,8 times those obtained with the slab model by identification of the variance) would be less than the theoretical values. Thus, the theoretical models based on the porous structure of the particles cannot be used for... 

Figure 1. Geometrical arrangement of the slab model. The periodic supercell is replicated into three directions one side of the contains the metal slab, the other side the vacuum or water phase. The top view of the surface shows a typical arrangement for a V3xV3 unit cell, in this case representing an alloy AjB.

If the catalyst is deposited as a thin layer on the inside or outside of a tube, the slab model can be used if the thickness of the catalyst layer is much less than the tube radius. The slab model is also used to analyze the performance of eggshell catalysts, which have a layer of active catalyst near the outer surface of the pellet, and of catalyst monoliths, which have a thin layer of catalyst on the inside of square, triangular, or hexagonal passages. Flowever, when the catalyst layer is very thin, pore diffusion effects are... 

By imposing 2-D periodic boundary conditions. The slab model is really two-dimensional, with a 2-D unit cell (Figure 33a). 

Figure 33 Three-layer slab models of the MgO (100) surface, (a) With 2-D periodic boundary conditions, (b) 3-D supercell approximation of the slab model as adopted in plane wave calculations.

Within the slab model approach, the surface formation energy is computed as... 

Apart from the simulation of ideal surfaces, increasing interest in real 2-D crystals now exists, which are quasi-periodic structures in two dimensions but only a few atomic layers thick, and which may present new and useful properties precisely because of their limited thickness. This branch of nanoscience is then an ideal ground for application of the slab model. 

An interesting example of the application of the slab model approach to the study of interfaces is the modeling of ultrathin oxide films on metallic substrate, which has been the subject of recently published papers by Pisani et It deals with a model of the epitaxially grown MgO (100) thin... 

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