Missing neighbor

The same theory, i.e. Eqs. (86) and (87), allows us to understand why CO and similar molecules adsorb so much more strongly on under-coordinated sites, such as steps and defects on surfaces. Since the surface atoms on these sites are missing neighbors they have less overlap and their d band wUl be narrower. Consequently, the d band shifts upwards, leading to a stronger bonding. 

Fig. 3. The scheme of the directions toward the nearest missing neighbors, cut by the (001), (201), and (101) planes for an atom of the fee lattice.

Fig. 5, The scheme of the arrangement of atoms around the (111) plane of the fee lattice and around the (0001) plane of the hep lattice. The arrows represent those directions toward the nearest missing neighbors, which are sticking out perpendicularly to a given plane. No other directions are indicated. Only the atoms on edges are reproduced here.

The reactivity of a surface depends on the number of bonds that are unsaturated. An unsaturated bond is what is left from a former bond with a neighboring metal atom that had to be broken in order to create the surface. Thus we want to know the number of missing neighbors, denoted by Zs, of an atom in each surface plane. One can infer from Fig. A.l that an atom in the fee (111) or hep (001) surface has 6 neighbors in the surface and three below, but lacks the three neighbors above that were present in the bulk Zs= 3. An atom in the fee (100) surface, on the other hand, has four neighbors in the surface, four below, and lacks four above the surface Zs=4. Hence, an atom in the fee (100) surface has one more unsaturated bond than an atom in the fee (111) surface and is slightly more reactive. 

Table A.l Number of missing neighbors in the surface, Zs, and neighbors in the bulk, N. 

Atoms at solid surfaces have missing neighbors on one side. Driven by this asymmetry the topmost atoms often assume a structure different from the bulk. They might form dimers or more complex structures to saturate dangling bonds. In the case of a surface relaxation the lateral or in-plane spacing of the surface atoms remains unchanged but the distance between the topmost atomic layers is altered. In metals for example, we often find a reduced distance for the first layer (Table 8.1). The reason is the presence of a dipole layer at the metal surface that results from the distortion of the electron wavefunctions at the surface. 

Atoms at the surface miss neighbors on the vacuum side. Hence, in the direction perpendicular to the surface the atoms have more freedom to vibrate than bulk atoms ... 

Of course, some general aspects of our treatment could be easily extended to a general form of f b ireJ as in the semi-infinite case [226],but for explicit numerical work a specific form of fs(b ire) ((()) is needed. Equation (10) can be justified for Ising-type lattice models near the critical point [216,220], i.e. when ( ) is near ( >crit=l/2, as well as in the limits f]>—>0 or <()—>1 [11]. The linear term —pj( ) is expected due to the preferential attraction of component B to the walls, and to missing neighbors for the pairwise interactions near the walls while the quadratic term can be attributed to changes in the pairwise interactions near the walls [144,216,227]. We consider Eq. (10) only as a convenient model assumption to illustrate the general theoretical procedures - there is clear evidence that Eq. (10) is not accurate for real polymer mixtures [74,81,82,85]. 

When we now consider a thin film of thickness D, Eq. (41) must be supplemented by boundary conditions of the same type as in the polymer blend case, Eqs. (7) and (10), i.e. we add a (bare) surface free energy contribution to the free energy that accounts for preferential attraction of one kind of monomers to the walls, missing neighbors in the pairwise interactions, and possible changes in the pairwise interactions near the surface. As in the blend case, this surface contribution is taken locally at the walls only and expanded to second order in the local order parameter /(z). Per unit area of the wall, this free energy is written as... 

The quadratic in (j)s form is preserved with gj - originating now mainly due to the difference in surface-polymer contact energies and g - entirely specified by missing neighbor effect. Both coefficients appearing in the linear form of the surface energy derivative (-dfs/d( ))s=p1+g( )s maybe expressed as [177]... 

We now consider the interface between a vacuum and a system that undergoes a first-order (i.e., discontinuous) order-disorder transition in the bulk at a temperature Tc. Due to missing neighbors at a surface, we expect that the order parameter at temperatures T < Tc is slightly reduced in comparison with its bulk value (fig. 67). If this situation persists up to T > T, such that both the bulk order parameter < >(z oo) and the surface order parameter 4> = z = 0) vanish discontinuous the surface stays ordered up to Tc, a situation that is not of very general interest. However, it may happen (Lipowsky, 1982, 1983, 1984, 1987 Lipowsky and Speth, 1983) that the surface region disorders somewhat already at T < Tc, and this disordered layer grows as T T and leads to a continuous... 

Density profiles in the wetting phase (liquid near a strongly attractive surface) and in the drying phase (vapor near a weakly attractive surface) are not affected by the surface transitions. These profiles reflect the competition between the missing neighbor effect and the fluid-wall interaction and may be described in the framework of the theory of the surface critical behavior (see Section 3). In particular, a gradual density adsorption or a density depletion decays exponentially toward the bulk... 

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