Flat-lattice model

The other illustration concerns oligomers, adsorbed flat in a monolayer. For an r-mer we derived [II.2.4.33] using a lattice model ... 

Figure 3. Adsorption isotherms predicted by Ouidelli s et at. three-dimensional model (—) for (A) a polar dimeric solute adsorbed vertical or flat at cr = 0, 0,05, 0.1 C m (from left to right), and (B) a polar monomeric solute at cr = 0.04, 0, -0.04, -0.08, -0.12 C m (from left to right). Broken lines represent the best Frumkin s isotherms. (Solid lines were reprinted from J. Electroanal. Chem., 329, R. Ouidelli, and O..Aloisi, A three Dimensional Lattice Model of TIP4P Water Molecules and of Polar Monomeric and Dimeric Solute Molecules Against a Charged Wall., p.39. Copyright 1992, with permission from Elsevier Science).

Up to now the model has been applied with monomeric, dimeric and trimeric solute molecules. Although the study of these cases is not complete, possibly due to computational difficulties, it seems that some of the adsorption features are satisfactorily predicted only in the case of non-polar monomeric and polar dimeric solute molecules, provided that the latter exhibit certain orientations on the electrode surface. " In the case of polar monomeric and dimeric molecules that may adsorb either vertically or flat, the model does not give satisfactory predictions. This is shown in Figure 3 where the solid lines represent adsorption isotherms predicted by the model and the dotted lines represent the best Frumkin s isotherms that describe them. In the case of the trimeric solutes, the model predicts the existence of a surface phase transition. However, the transition properties, due to the use of an inappropriate statistical mechanical treatment, contradict thermodynamic and experimental data. Thus, despite its novelty the three-dimensional lattice approach has not given the expected results yet. 

Whereas in approach 1 lattice models are used, we will work in the continuum, making extensive use of interface thermodynamics. The advantage of such an approach, as it turns out, is that detailed properties such as the size distribution of microemulsion droplets and the interfacial tension of a flat monolayer separating a microemulsion and an excess phase can be predicted. On the other hand, the lattice approaches as summarized in item 1 predict global phase behavior, which is not (yet) possible with the thermodynamic formalism reviewed in the following section. The reason is that currently a realistic model for the middle phase is lacking. A more detailed discussion regarding this issue is presented in Sec. VIII. 

The quasi-lattice model was developed by Roe (13) and Scheutjens and Fleer (14) (SF theory) The basic analysis considered all chain conformations as step-weighted random walks on a quasi-crystalline lattice that extends in parallel layers from the surface. This is illustrated in Figure 16.2 which shows a possible conformation of a polymer molecule at a flat surface. The partition function was written in terms of the number of chain configurations that were treated as connected sequences of segments. In each layer parallel to the surface, random mixing between the segments and solvent molecules was assumed, i.e. by using... 

In turn, porous space of many real and model porous materials can be considered as a lattice of expansions cavities (or sites), connected with narrower windows or necks (bonds). With such a definition of sites and bonds it is acceptable to have the whole volume of pores concentrated only in cavities of different sizes and forms. In this case, windows are considered as volumeless figures that correspond to the flat sections in places of the smallest narrowings between the neighbors (as well as bond in a lattice of particles) [3,61], This approach seems to be the most... 

Ideal Surfaces, A model of an ideal atomically smooth (100) surface of a face-centered cubic (fee) lattice is shown in Figure 3.13. If the surface differs only slightly in orientation from one that is atomically smooth, it will consist of flat portions called terraces and atomic steps or ledges. Such a surface is called vicinal. The steps on a vicinal surface can be completely straight (Fig. 3.13a) or they may have kinks (Fig. 3.13b). 

The lattice-gas model provides effective ways on atomic-molecular level for allowing the mechanical equilibrium of all solid atoms and adsorbed particles on any (flat or structural-rough) surface if one expresses... 

Quite recently the ferromagnetic ground state in a class of Hubbard systems on decorated lattices that contain flat or nearly flat lowest energy bands has been found [42,43]. Obviously such multiband systems have more degrees of freedom than the one-band model considered by Nagaoka, and we can expect the appearance of unusual effects even in the case of infinite electron repulsion. 

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