Statistical lattice model

KovalyovEV, Resnyanskii ED, Elokhin VI, Bal zhinimaev BS, Myshlyavtsev AV. 2003. Novel statistical lattice model for the supported nanoparticle. Features of the reaction performance influenced by the dynamically changed shape and surface morphology of the supported active particle. Phys Chem Chem Phys 5 784-790. 

In order to understand the thermodynamic issues associated with the nanocomposite formation, Vaia et al. have applied the mean-field statistical lattice model and found that conclusions based on the mean field theory agreed nicely with the experimental results [12,13]. The entropy loss associated with confinement of a polymer melt is not prohibited to nanocomposite formation because an entropy gain associated with the layer separation balances the entropy loss of polymer intercalation, resulting in a net entropy change near to zero. Thus, from the theoretical model, the outcome of nanocomposite formation via polymer melt intercalation depends on energetic factors, which may be determined from the surface energies of the polymer and OMLF. 

The theoretical and experimental investigations of rupture and permeability of amphiphile bilayers are valuable also for the understanding of some microstructural effects in interfacial layers and phases of small volumes. The interpenetration of macroscopically measured quantities, e.g. r and W, by means of molecular statistical models seems to be most interesting and useful. As first attempts in this respect, a molecular statistical lattice model of such bilayers has been proposed [427] and a lattice model of such bilayers has been studied by means of Monte Carlo simulation by Chowdhury and Stauffer [429]. The results obtained have been compared with some experimental data presented in this Section. Clearly, the combination of macro and micro considerations is a promising way to obtain a deeper insight into the properties of matter and, especially, of interfacial layers and phases of small volumes. 

Atomic Scale Imaging of Oscillation and Chemical Waves at Catalytic Surface Reactions Experimental and Statistical Lattice Models... 

There is, of course, a mass of rather direct evidence on orientation at the liquid-vapor interface, much of which is at least implicit in this chapter and in Chapter IV. The methods of statistical mechanics are applicable to the calculation of surface orientation of assymmetric molecules, usually by introducing an angular dependence to the inter-molecular potential function (see Refs. 67, 68, 77 as examples). Widom has applied a mean-held approximation to a lattice model to predict the tendency of AB molecules to adsorb and orient perpendicular to the interface between phases of AA and BB [78]. In the case of water, a molecular dynamics calculation concluded that the surface dipole density corresponded to a tendency for surface-OH groups to point toward the vapor phase [79]. 

If stationary phase interactions are negligible the lattice statistical thermodynamic model and the solvophobic model predict similar results. The strength of the lattice statistical thermodynamic model is that it can explain the shape selectivity observed for certain stationary phases and can accommodate silanophllic interactions. 

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. 

Extensive channeling measurements on 2H implanted into silicon have been published by Bech Nielsen (1988). These measurements also use the 3He-induced nuclear reaction in conjunction with extensive modeling using the statistical equilibrium model already described. The 2H implants were done at 30 K, and lattice location of the 2H was done as a function of annealing. 

Gibbs and DiMarzio [47] (GD) first developed a systematic statistical mechanical theory of glass formation in polymer fluids, based on experimental observations and on lattice model calculations by Meyer, Flory, Huggins, and... 

Further understanding of the kinetic of template polymerization needs consideration of the process entropy. Applying a well known lattice model, it is easy to see that entropy changes, AS, in free polymerization and the template polymerization, differs considerably. According to the principles of statistical thermodynamics, the entropy of mixing is given by the equation ... 

In the present article, we focus on the scaled particle theory as the theoretical basis for interpreting the static solution properties of liquid-crystalline polymers. It is a statistical mechanical theory originally proposed to formulate the equation of state of hard sphere fluids [11], and has been applied to obtain approximate analytical expressions for the thermodynamic quantities of solutions of hard (sphero)cylinders [12-16] or wormlike hard spherocylinders [17, 18]. Its superiority to the Onsager theory lies in that it takes higher virial terms into account, and it is distinctive from the Flory theory in that it uses no artificial lattice model. We survey this theory for wormlike hard spherocylinders in Sect. 2, and compare its predictions with typical data of various static solution properties of liquid-crystalline polymers in Sects. 3-5. As is well known, the wormlike chain (or wormlike cylinder) is a simple yet adequate model for describing dilute solution properties of stiff or semiflexible polymers. 

Recent developments in the theory of polymer solutions have been reviewed by Berry and Casassa (32), and by Casassa (71). Casassa, who has contributed very largely to these developments, has adopted a statistical mechanical approach using molecular distribution functions, as first outlined by Zimm (72), rather than using a lattice model like that used by Flory, Huggins, and many later workers. 

Verdier,P.H., Stockmayer, W.H. Monte Carlo calculations on the dynamics of polymers in dilute solution. J. Chem. Phys. 36, 227-235 (1962). See also Verdier,P.H. Monte Carlo studies of lattice-model polymer chains. 1. Correlation functions in the statistical-bead model. J. Chem. Phys. 45,2118-2121 (1966). 

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