Boundary field

The effects due to the finite size of crystallites (in both lateral directions) and the resulting effects due to boundary fields have been studied by Patrykiejew [57], with help of Monte Carlo simulation. A solid surface has been modeled as a collection of finite, two-dimensional, homogeneous regions and each region has been assumed to be a square lattice of the size Lx L (measured in lattice constants). Patches of different size contribute to the total surface with different weights described by a certain size distribution function C L). Following the basic assumption of the patchwise model of surface heterogeneity [6], the patches have been assumed to be independent one of another. 

The behavior of an adsorbate on a single patch of size L has been represented by the familiar two-dimensional lattice gas model Hamiltonian with the added term resulting from the presence of a boundary field ... 

In the above, U] ] is the nearest neighbor interaction energy, V is the adsorption energy and Fb is the boundary field acting on the particles located at the patch boundary... 

In the systems characterized by more complex surface structure, consisting of patches of different size and different magnitude of the boundary field, the properties of adsorption isotherms have been found to depend on the... 

Finally, we must consider the contribution of the electrostatic work required to transfer one electron into free space. After overcoming the short range chemical forces, the electron must be moved a certain distance against the electric field in the surface. Under the assumption that the lines of force of the electric field are located between the ion defects in the boundary layer and the surface charges represented by the chemisorbed gas atom, we obtain the expression afi for this electrostatic work term. is the boundary field strength represented in Equation (11), and a is the distance between the surface of the oxide and the centers of charge of the chemisorbed atoms in the a-phase. 

Assuming that the boundary field strength will be changed only by the concentration of the chemisorbed particles, and that the structural and electrostatic terms will be practically uninfluenced, we get for the difference between the work functions for a free surface, and for one occupied by chemisorbed particles, i/ coupied, the expression... 

Fig. 18. Snapshot pictures of a nearest-neighbor Ising ferromagnet on the square lattice with bulk field H = 0 and boundary fields H] — Hi. — 0 (this model is isomorphic to the c(2x2) ordering at coverage 0 = 0.5) at the temperatures T — 0.95 Tc (a), T = Tc (b) and T = 1.05 Ft (c), for a L x M system with L — 24, M — 288 and two free boundaries of length M, while periodic boundary conditions arc used along the strip. Up spins (adatoms on sublattice 1) are shown in black, down spins (adatoms on sublattice 2) are shown in white. Domain formation at T 5 Tc can be clearly recognized. From Albano ei a . (1989b).

Fig. 58. Snapshot pictures of a lattice gas model of adsorption on an L x M terrace (fig. 3) choosing L - 24, M = 288 and the two-dimensional version of eq. (1) with H = 0, 7S = J, and antiparallcl boundary fields H = -37, 7/l — T37, at the time step r = 24000 MCS/site of a Monte Carlo simulation, and three temperatures T = 0.68 fc (a), T = 0.78(7) ) (b), and T — 0.887) (c). Sites taken by adsorbed atoms are shown in black, empty sites are left white. From Albano et al. (1989a).

Figure 15 (a) Phase diagram of a binary polymer blend N= 32) as obtained from Monte Carlo simulations of the bond fluctuation model. The upper curve shows the binodais in the infinite system the middle one corresponds to a thin film of thickness D=2.8/ e and symmetric boundary fields [wall = 0.16, both of which prefer species A (capillary condensation). The lower curve corresponds to a thin film with antisymmetric surfaces (interface localization/delocalization). The arrow marks the location of the wetting transition. Full circles mark critical points open circles/dashed line denotes the triple point, (b) Coexistence curves in the (T, A/y)-plane. Circles mark critical points, and the diamond indicates the location of the wetting transition temperature. It is indistinguishable from the temperature of the triple point. Adapted from Muller, M. Binder, K. Phys. Rev. 2001, 63, 021602. ... 

By adding a boundary field to the Ising Hamiltonian, that is a term of... 

---