The Lattice Model

Analytical Expressions for Lattice Modeis. Concerning the aforementioned paracrystalline lattice, an analytical equation has first been deduced by Hermans [128], His equation is valid for infinite extension. Ruland [84] has generalized the result for several cases of finite structural entities. He shows that a master equation 

Hermans equation for the infinitely extended lattice is obtained. For a material built from finite structural entities containing an average of N) particles Ruland obtains 

In analogy to the treatment of the stacking model Jq (5) = 0 is valid, if the structural entities are embedded in matrix material. Compact material, again, may require a correction because of the merging of particles from abutting structural entities 

Independently, Burger [231] develops analytical equations for lattice models without substitutional disorder. His results are special cases of the models presented by Ruland. 

As has already been mentioned in the discussion of the stacking model, such equations are particularly useful for the analysis of nanostructured material with weak disorder in order both to assess the perfection of the material and to discriminate among lattice and stacking models (cf. Sect. 8.8.3). 

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C2.5.3.4 EXPLORING THE PROTEIN FOLDING MECHANISM USING THE LATTICE MODEL... 

The lattice model that served as the basis for calculating ASj in the last section continues to characterize the Flory-Huggins theory in the development of an expression for AHj . Specifically, we are concerned with the change in enthalpy which occurs when one species is replaced by another in adjacent lattice sites. The situation can be represented in the notation of a chemical reaction ... 

In this section and the last, we have examined the lattice model of the Flory-Huggins theory for general expressions relating AHj and ASj to the composition of the mixture. The separate components can therefore be put together to give an expression for AGj as a function of temperature and composition ... 

It is not particularly difficult to find macroscopic measures of interactions between small molecules of the same type, that is, quantities which are proportional to Wii and W22 in Eq. (8.40). Among the possibilities, we consider the change in internal energy AU for the vaporization process for component i. This can be related to Wjj in terms of the lattice model by the expression... 

Simplified models for proteins are being used to predict their stmcture and the folding process. One is the lattice model where proteins are represented as self-avoiding flexible chains on lattices, and the lattice sites are occupied by the different residues (29). When only hydrophobic interactions are considered and the residues are either hydrophobic or hydrophilic, simulations have shown that, as in proteins, the stmctures with optimum energy are compact and few in number. An additional component, hydrogen bonding, has to be invoked to obtain stmctures similar to the secondary stmctures observed in nature (30). 

Finally, we assume that the fields 4>, p, and u vary slowly on the length scale of the lattice constant (the size of the molecules) and introduce continuous approximation for the thermodynamical-potential density. In the lattice model the only interactions between the amphiphiles are the steric repulsions provided by the lattice structure. The lattice structure does not allow for changes of the orientation of surfactant for distances smaller than the lattice constant. To assure similar property within the mesoscopic description, we add to the grand-thermodynamical potential a term propor-tional to (V u) - -(V x u) [15], so that the correlation length for the orientational order is equal to the size of the molecules. 

Using the lattice model, the approximate value of W in the Boltzmann equation can be estimated. Two separate approaches to this appeared in 1942, one by P. J. Rory, the other by M. L. Huggins, and though they differed in detail, the approaches are usually combined and known as the Rory-Huggins theory. This gives the result for entropy of mixing of follows ... 

There are different mechanisms for explaining exfoliation of organically modihed clays in the polymer matrices. One such mechanism is the Lattice model proposed by Vaia and Giannelis [200]. The mechanism is explained as follows. 

An Important Tool for Understanding Folding the Lattice Model... 

The entropy of mixing disoriented polymer and solvent may be obtained, according to the original assumptions pertaining to the lattice model, by subtracting Eq. (9) from (8). The result reduces to ... 

In contrast to the lattice models discussed below, off-lattice models allow the chemical species under consideration to occupy in principle any position in space, so that important information concerning the relaxation and space distribution of the constituents of the system can be obtained. We discuss next some applications of these models to electrochemical problems. 

We now discuss the translation of the MC time-step into physical time units. It is desirable to map the mobility of the lattice model (due to jumps of the effective monomers) onto the average jump rate of the torsional degrees of freedom, since these motions dominate the relaxation of the overall configuration of the chain. This means that we must allow for a temperature-depen-dent time unit tmc(T) which one attempted MCS per monomer corresponds to, via the formula ... 

The stacking model (Fig. 8.42c) does not carry this inconsistency [128,229], It cannot be discriminated from the lattice models if the polydispersity is strong. For small polydispersity even the lattice models make physical sense, because then the mutual penetration is negligible. Computation and fitting of stacking and lattice models are described in Sects. 8.7.3.4 and 8.133. 

Properties and Application. The two independent statistical distributions of the two-phase stacking model are the distributions of amorphous and crystalline thicknesses, h (x) and ii2 x). Both distributions are homologous. The stacking model is commutative and consistent. If the structural entity (i.e., the stack as a whole) is found to show medium or even long-ranging order, the lattice model and its variants should be tested, in addition. As a result the structure and its evolution mechanism may more clearly be discriminated. 

Figure 8.48. Best fits of stacking model and lattice model to the data from Fig. 8.48. The lattice model fits much better. Data sets are shifted for clarity...

Figure 9.5 Two-dimensional illustration of the lattice model for polymer solutions. Black sites are occupied by the polymer chain, white sites by solvent monomers.

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