Lattice model coordination number

The first model of porous space as a 2D lattice of interconnected pores with a variation of randomness and branchness was offered by Fatt [220], He used a network of resistors as an analog PS. Further, similar approaches were applied in a number of publications (see, e.g., Refs. [221-223]). Later Ksenjheck [224] used a 3D variant of such a model (simple cubic lattice with coordination number 6, formed from crossed cylindrical capillaries of different radii) for modeling MP with randomized psd. The plausible results were obtained in these works, but the quantitative consent with the experiment has not been achieved. 

We use a square lattice with coordination number = 4. In principle the model can be described by equations... 

We have studied the phase and micellization behavior of a series of model surfactant systems using Monte Carlo simulations on cubic lattices of coordination number z = 26. The phase behavior and thermodynamic properties were studied through the use of histogram reweighting methods, and the nanostructure formation was studied through examination ofthe behavior ofthe osmotic pressure as a function of composition and through analysis of configurations. 

The Plory-Huggins theory is a generalization of the BRAGG-WILLIAMS" approximation in the lattice model of binary solutions. The polymer is considered to consist of X segments equal in size to a solvent molecule. Hence x is the ratio of molar volumes of the polymer and solvent. N2 polymer molecules and Nj solvent molecules are placed randomly on a lattice of coordination number z. The volume fractions of solvent and polymer are then... 

The COORDINATION NUMBER of an atom is its number of nearest-neighboring atoms. For a given crystal lattice, the coordination number is established by inspection of the model. In the SIMPLE CUBE, each atom touches six adjacent atoms. Note that in Fig. 9.3a atom X is closest to the six atoms numbered 1-6. 

One particular concept employed in the original works must be mentioned, since it is still important. In the theoretical developments, Flory used a lattice model , constructed as shown schematically in Fig. 3.13. The A- and B-units of the two polymer species both have the same volume and occupy the cells of a regular lattice with coordination number >2 . It is assumed that the interaction energies are purely enthalpic and effective between nearest neighbors only. Excess contributions kTx which add to the interaction energies... 

Effective transport coefficients Lattice representations of space provide a convenient means for representing porous materials. As shown in the previous subsection, some important material properties (critical porosity, accessible porosity, cluster size) can be predicted given a suitable lattice model for the structure. In order to determine the rate of solute transport in the structure, h(9l) must be evaluated to find the effective diffusion coefficient. For diffusion on a Bethe lattice, analytical expressions for the effective diffusion coefficient are available [43,44]. For a Bethe lattice with coordination number the effective diffusion coefficient is found from ... 

Figure 12 Representation of the Gibbs-DiMarzio model. A lattice of coordination number Z is used to calculate the number of complexions for a system of molecules each of length X. Energy a is associated with holes and energy Ae is associated with flexing of a bond from its low energy shape (after ref. 16, with permission)...

There is nothing unique about the placement of this isolated segment to distinguish it from the placement of a small molecule on a lattice filled to the same extent. The polymeric nature of the solute shows up in the placement of the second segment This must be positioned in a site adjacent to the first, since the units are covalently bonded together. No such limitation exists for independent small molecules. To handle this development we assume that each site on the lattice has z neighboring sites and we call z the coordination number of the lattice. It might appear that the need for this parameter introduces into the model a quantity which would be difficult to evaluate in any eventual test of the model. It turns out, however, that the z s cancel out of the final result for, so we need not worry about this eventuality. 

In spite of the absence of periodicity, glasses exhibit, among other things, a specific volume, interatomic distances, coordination number, and local elastic modulus comparable to those of crystals. Therefore it has been considered natural to consider amorphous lattices as nearly periodic with the disorder treated as a perturbation, oftentimes in the form of defects, so such a study is not futile. This is indeed a sensible approach, as even the crystals themselves are rarely perfect, and many of their useful mechanical and other properties are determined by the existence and mobility of some sort of defects as well as by interaction between those defects. Nevertheless, a number of low-temperamre phenomena in glasses have persistently evaded a microscopic model-free description along those lines. A more radical revision of the concept of an elementary excitation on top of a unique ground state is necessary [3-5]. 

The structure of a simple mixture is dominated by the repulsive forces between the molecules [15]. Any model of a liquid mixture and, a fortiori of a polymer solution, should therefore take proper account of the configurational entropy of the mixture [16-18]. In the standard lattice model of a polymer solution, it is assumed that polymers live on a regular lattice of n sites with coordination number q. If there are n2 polymer chains, each occupying r consecutive sites, then the remaining m single sites are occupied by the solvent. The total volume of the incompressible solution is n = m + m2. In the case r = 1, the combinatorial contribution of two kinds of molecules to the partition function is... 

Let us first derive the regular solution model for the system AC-BC considered above. The coordination numbers for the nearest and next nearest neighbours are both assumed to be equal to z for simplicity. The number of sites in the anion and cation sub-lattice is N, and there are jzN nearest and next nearest neighbour interactions. The former are cation-anion interactions, the latter cation-cation and anion-anion interactions. A random distribution of cations and anions on each of... 

The model of clusters or ensembles of sites and bonds (secondary supramolecular structure), whose size and structure are determined on the scale of a process under consideration. At this level, the local values of coordination numbers of the lattices of pores and particles, that is, number of bonds per one site, morphology of clusters, etc. are important. Examples of the problems at this level are capillary condensation or, in a general case, distribution of the condensed phase, entered into the porous space with limited filling of the pore volume, intermediate stages of sintering, drying, etc. 

Model of a granule of a porous solid as a lattice (labyrinth) of pores and particles, which takes into account the average values of coordination number of bonds and distribution of sites and bonds over the characteristic sizes. 

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