Slab model

It is normal to set the cell vector lengths based on the bulk optimized structure at the level of theory to be used in the slab calculations. This choice gives lattice parameters with zero strain in the bulk which is a likely constraint on the surface repeat unit. 

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. 

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Figure 2.1 Dependence of the effectiveness factor on the Thiele modulus for a first-order irreversible reaction. Steady-state diffusion and reaction, slab model, and isothermal conditions are assumed.

Suppose we would like to carry out calculations on a surface of an fee metal such as copper. How might we construct a slab model such as that depicted in Fig. 4.1 It is convenient to design a supercell using vectors coincident with the Cartesian x, y, and z axes with the z axis of the supercell coincident with the surface normal. Recall that for fee metals, the lattice constant is equal to the length of the side of the cube of the conventional cell. The supercell vectors might then be... 

Figure 4.3 View of a five layer slab model of a surface as used in a fully periodic calculation. In making this image, the supercell is similar to the one in Fig. 4.1 and is repeated 20 times in the x and y directions and 2 times in the z direction.

In the example above, we placed atoms in our slab model in order to create a five-layer slab. The positions of the atoms were the ideal, bulk positions for the fee material. In a bulk fee metal, the distance between any two adjacent layers must be identical. But there is no reason that layers of the material near a surface must retain the same spacings. On the contrary, since the coordination of atoms in the surface is reduced compared with those in the bulk, it is natural to expect that the spacings between layers near the surface might be somewhat different from those in the bulk. This phenomenon is called surface relaxation, and a reasonable goal of our initial calculations with a surface is to characterize this relaxation. 

Figure 4.11 Schematic illustration of relaxation of surface atoms in a slab model. The top three layers of atoms were allowed to relax while the bottom two layers were held at the ideal, bulk positions.

Section 4.5 Surface relaxations were examined using asymmetric slab models of five, six, seven, or eight layers with the atoms in the two bottom layers fixed at bulk positions and all remaining atoms allowed to relax. For Cu(100), the supercell had c(2 x 2) surface symmetry, containing 2 atoms per layer. For Cu(l 11), (y/3 X /3)R30 surface unit cell with 3 atoms per layer was used. All slab models included a minimum of 23 A of vacuum along the direction of the surface normal. A 6x6x1 /c-point mesh was used for all calculations. 

Calculate the activation energy for diffusion of a Pt adatom on Pt(100) via direct hopping between fourfold sites on the surface and, separately, via concerted substitution with a Pt atom in the top surface layer. Before beginning any calculations, consider how large the surface slab model needs to be in order to describe these two processes. Which process would you expect to dominate Pt adatom diffusion at room temperature ... 

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 ... 

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