Surfaces slab model

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 ... 

The FeSa (100) surfaces are modeled using the supercell approximation. Surfaces are cleaved fi om a GGA optimized crystal structure of pyrite. A vacuum spacing of 1.5 nm is inserted in the z-direction to form a slab and mimic a 2D surface. This has been shown to be sufficient to eliminate the interactions between the mirror images in the z-direction due to the periodic boundary conditions. 

Surface models used a [1x1x1] crystal unit cell as a surface slab in the supercell. We have also investigated the effect of slab thickness on the calculation result. It shows that three Fe— S layer model can get almost the same result as four Fe— S layers model. 

In this surface-renewal model of gas exchange, the gas flux across the air-sea interfece is determined by the frequency at which the slab is replaced or renewed. Various parameterizations have been developed for this model. One example relates the net diffusive flux to the frequency of slab renewal (0) as follows... 

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. 

Section 4.8 The reconstructed Si(001) surface was modeled using a c(4 x 4) asymmetric slab with four layers. The bottom two layers were fixed at bulk positions. Reciprocal space was sampled with llxllxllfc points and the vacuum spacing was 17 A. 

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] ... 

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 ... 

While quantum-chemical calculations related to gas-phase reactions or bulk properties have become now a matter of routine, calculations of local properties and, in particular, surface reactions are still a matter of art. There is no simple and consistent way of adequately constructing a model of a surface impurity or reaction site. We will briefly consider here three main approaches (1) molecular models, (2) cluster models, and (3) periodic slab models. 

As to periodic slab models, their application is rather straightforward, but poses many problems in the cases when the activation energy should be found for a surface reaction at a bulky surface group. On the other hand, this is probably the best approach to determine the structural properties of an active site and the interaction energies between an active site and the reacting species. 

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