Three-dimensional lattice model

A second way of dealing with the relationship between aj and the experimental concentration requires the use of a statistical model. We assume that the system consists of Nj molecules of type 1 and N2 molecules of type 2. In addition, it is assumed that the molecules, while distinguishable, are identical to one another in size and interaction energy. That is, we can replace a molecule of type 1 in the mixture by one of type 2 and both AV and AH are zero for the process. Now we consider the placement of these molecules in the Nj + N2 = N sites of a three-dimensional lattice. The total number of arrangements of the N molecules is given by N , but since interchanging any of the I s or 2 s makes no difference, we divide by the number of ways of doing the latter—Ni and N2 , respectively—to obtain the total number of different ways the system can come about. This is called the thermodynamic probabilty 2 of the system, and we saw in Sec. 3.3 that 2 is the basis for the statistical calculation of entropy. For this specific model... 

A. Ciach, J. S. Hoye, G. Stell. Microscopic model for microemulsion. II. Behavior at low temperatures and critical point. J Chem Phys 90 1222-1228, 1989. A. Ciach. Phase diagram and structure of the bicontinuous phase in a three dimensional lattice model for oil-water-surfactant mixtures. J Chem Phys 95 1399-1408, 1992. 

These problems are also readily modelled in terms of lattice gas systems, which now need to be semi-infinite three-dimensional lattices rather than two-dimensional ones, with an appropriate boundary condition at the free surface to model the effects due to the substrate. A model which has been studied... 

The Flory-Huggins theory begins with a model for the polymer solution that visualizes the solution as a three-dimensional lattice of TV sites of equal volume. Each lattice site is able to accommodate either one solvent molecule or one polymer segment since both of these are assumed to be of equal volume. The polymer chains are assumed to be monodisperse and to consist of n segments each. Thus, if the solution contains TV, solvent molecules and TV2 solute (polymer) molecules, the total number of lattice sites is given by... 

We turn now to the results of computer simulation. Early work in this field was based on the lattice model in which each molecule is assumed to be located at a point of a three-dimensional lattice and the variables are the orientational coordinates. Until relatively recently the computers available could not tackle a more realistic model. However, the lattice model does, to some extent, assume the answer to the problem which one is examining and is unable to make predictions about the interesting questions concerning the formation and stability of particular mesophases. 

Ueda Y, Taketomi H, Go N (1978) Studies on protein folding, unfolding and fluctuations by computer simulation. A three-dimensional lattice model of lysozyme. Biopolymers 17 1531-1548... 

With lattice models it is assumed that the molecules in the fluid phase repose on the lattice points of three dimensional lattice, while entering into an exchange effect with the adjacent molecules. 

Figure 34 shows the temperature dependencies of the static fractal dimensions of the maximal cluster. Note that at percolation temperature the value of the static fractal dimension Ds is extremely close to the classical value 2.53 for a three-dimensional lattice in the static site percolation model [152]. Moreover, the temperature dependence of the stretch parameter v (see Fig. 34) confirms the validity of our previous result [see (62)] Ds = 3v obtained for the regular fractal model of the percolation cluster [47]. 

In lattice models, the location of each element on the lattice can be stored as a vector of coordinates [(X, F,), (X2, Y2), (X3, Y3),..., (Xn, F )], where (X Y,) are the coordinates of element i on a two-dimensional lattice (a three-dimensional lattice will require three coordinates for each element). Since lattices enforce a fixed geometry on the conformations they contain, conformations can be encoded more efficiently by direction vectors leading from one atom (or element) to the next. For example in a two-dimensional square lattice, where every point has four neighbors, a conformation can be encoded simply by a set of numbers (Lu L2, L3,..., L ), where L, g 1, 2,3,4 represents movement to the next point by going up, down, left, or right. Most applications of GAs to protein structure prediction utilize one of these representations. 

The percolation probability (q) for the lattice models is defined as the probability that a given site (or bond) belongs to an infinite open cluster (47). It is fundamental to percolation theory that there exists a critical value qc of q such that 9(q) = 0 3t q < qc, and (q) > 0 if > qc. The value qc is called the critical probability or the percolation threshold. Mathematical methods of calculating this threshold are so far restricted to two dimensions, consistent with the experience in the field of phase transitions that three-dimensional problems in general cannot be solved exactly (12,13). Almost all quantitative information available on the percolation properties of specific lattices has come from Monte Carlo calculations on finite specimens (8,11,12). In particular. Table I summarizes exactly and approximately known percolation thresholds for the most important two- and three-dimensional lattices. For the bond problem, the data presented in Table I support the following well-known empirical invariant (8)... 

Comparing Eqs. (8) and (3), one can conclude that the Bethe model is appropriate for describing the three-dimensional lattices only at z = 3 and 4. If z 3= 5, the percolation probability for the Bethe lattice differs considerably from that for regular lattices. 

To evaluate Q for a binary solution, let us consider a particular model in which we imagine the molecules in the solution to occupy lattice points in a three-dimensional lattice. Suppose there are N sites and that each molecule occupies one site. Q is then the number of ways of arranging the molecules of 1 and the N molecules of 2 on the N =Ni +N2 sites. This number is equal to the number of distinct combinations... 

Soil structure, antecedent soil moisture and input flow rate control rapid flow along preferential pathways in well-structured soils. The amount of preferential flow may be significant for high input rates, mainly in the intermediate to high ranges of moisture. We use a three-dimensional lattice-gas model to simulate infiltration in a cracked porous medium as a function of rainfall intensity. We compute flow velocities and water contents during infiltration. The dispersion mechanisms of the rapid front in the crack are analyzed as a function of rainfall intensity. The numerical lattice-gas solutions for flow are compared with the analytical solution of the kinematic wave approach. The process is better described by the kinematic wave approach for high input flow intensities, but fails to adequately predict the front attenuation showed by the lattice-gas solution. 

In this chapter, we use the results of numerical infiltration experiments in dual porosity media performed with a three-dimensional lattice-gas model to characterize preferential flow as response to rainfall intensity. From the temporal and spatial evolution of the water content during infiltration and drainage, we evaluate the adequacy of a kinematic wave approximation to describe the flow. We also discuss the conceptual basis of the asymptotic kinematic approach to Richards equation in comparison with the macropore kinematic equation. 

Three-dimensional regular lattices do not have enough symmetry to ensure macroscopic isotropy. To model three-dimensional fluids, a suitable three-dimensional projection of a four-dimensional model, the face-centered-hypercube (FCHC), is used (see, for instance, Sommers Rem, 1992). Each node in the lattice is connected via links to 24 nearest neighbors. In this case, up to 24 moving particles may occupy the cells. 

The versatility of lattice models to describe encounter-controlled reactions in systems of more complicated geometries can be illustrated in two different applications. In this subsection layered diffusion spaces as a model for studying reaction efficiency in clay materials are considered and in the following subsection finite, three-dimensional lattices of different symmetries as a model for processes in zeolites are studied. Now that the separate influences of system size N, dimensionality d (integral and fractal), and valency v have been established, and the relative importance of d = 3 versus surface diffusion (and reduction of dimensionality ) has been quantified, the insights drawn from these studies will be used to unravel effects found in these more structured systems. 

Although the above diagram is a simple model, this is not how it occurs in Nature. We have presented the above concept because it is easier to understand than the actual conditions which occur. It should be clear that the overall solid state reaction is dependent upon the rate of diffusion of the two (2) species, and the two rates may, or may not, be the same. The reason that A and/or B do not react in the middle, i.e.- the phase AB, is that AB has a certain ordered structure which probably differs from either A or B. But there is a more important reason which is not easily illustrated in any two-dimensional diagram, particularly since each compound has a three-dimensional lattice structure. 

Even though we have made considerable effort to explore the reciprocal lattice because it allows us to determine the actual location of the bonding electrons in any given three-dimensional lattice, it should be clear that the electron-band model is easier to understand and easier to use than the Reciprocal lattice band model. To date in this discussion, we have covered ... 

Lattice models have provided many insights into the behaviour of polymers despite the obvious approximations involved. The simplicity of a lattice model means that many states can be generated and examined very rapidly. Both two-dimensional and three-dimensional lattices are used. The simplest models use cubic or tetrahedral lattices in which successive monomers occupy adjacent lattice points (Figure 8.7) The energy models are usually very simple, in part to reflect the simplicity of the representation but also to permit the rapid calculation of the energy. 

In our studies, we consider several types of aggregated structures such as bispheres, linear chains, plane arrays on a plane rectangular lattice, compact and porous body-centered clusters embedded on the cubic lattice (bcc clusters, the porosity was simulated by random elimination of monomers), and random fractal aggregates (RF clusters). To generate RE clusters, a three-dimensional lattice model with Brownian or linear trajectories of both single particles and intermediate clusters was employed for computer simulations of aggregation process. At the initial time moment, = 50,000 particles are generated at... 

A three-dimensional lattice model (section 3.1.1) was used to simulate aggregation kinetics, in which single particles and intermediate clusters move on Brownian or linear trajectories. Initially, = 50,000 particles are placed randomly in a cubic lattice of size L = 215x 215x 215. A combined cluster is formed whenever a particle or a cluster moves to a lattice point adjacent to another particle or intermediate cluster. This model produces DLCA clusters [77] with fractal dimension around 1.8 (Brownian trajectories) and 2 (linear trajectories). A sequence of two integers is used to describe the... 

The two-dimensional lattice model disregards completely any direct correlation between the adsorbed water molecules and the adjacent water molecules of the solution phase. In order to overcome this weakness Guidelli and Aloisi have developed a three-dimensional lattice model, the main features of which are the described next. 

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. 

On this issue Guidelli et al. expressed the view that phase transitions take plaee when the shape of the solute molecules hinders H-bond formation between water (solvent) molecules. In this case the water molecules are squeezed out of the adsorbed layer, leaving behind a compact film of solute molecules. This view seems to be verified by the three-dimensional lattice model, which in the presence of non-polar trimeric solute molecules does predict the occurrence of a phase transition. However, due to an inappropriate statistical mechanical approach based on the use of the grand ensemble H instead of the generalized ensemble A, it is not possible to know whether this model predicts correctly or not the properties of the phase transitions. ... 

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