Statistical Orientation Models

For noninteracting dipole moments and for a single crystal there exists a straight relationship between the microscopic and macroscopic hyperpolarizabilities. This is obtained by the transformation of the corresponding tensor components from the molecular to the laboratory reference frame  

At higher temperatures (above or slightly below the room temperature), when molecules may rotate freely (free gas model) one uses the Gibbs-Boltzmann distribution function for G(0) given by  

It is straightforward to express the order parameter P2 in terms of Langevin function l2(x) [167]  

From the knowledge of the order parameter from the measurement of the variation of the optical absorption spectrum due to poling, one can estimate the x parameter (Eq. (43)) intervening in the above developments. For poled polymers one can use also another alternative description of linear and nonlinear optical susceptibilities by expanding the orientation distribution function in the series of Legendre polynomials, where the expansion coefficients are order parameters  

The following recurrent relations obey the modified spherical Bessel functions  

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Tier-3 The tier with complex specific approaches. In the third tier, a mechanistic or semimechanistic model is adopted in addition to an empirical, statistical-modeling part. For example, the use of SSDs is a statistically oriented approach, but the use of this model can be refined by using mechanism-oriented insights (i.e., by considering separate species groups that are inherently more sensitive than others for the individual substance or the components of a mixture). In this way, the approach becomes more complex to handle, but likely also yields more accurate results than the moderately simple generic approach. 

On the other hand, solids are characterized by a very ordered structure in which each ion or molecule is surrounded by a fixed number of neighbors whose nature and orientation are determined by the interparticle forces in the crystal. These may be chiefly ion-ion interactions, as in an ionic crystal, or intermolecular forces, as in a molecular crystal. Because of the high state of order in crystals it is a reasonably straightforward problem to calculate their thermodynamic properties on the basis of quite simple statistical mechanical models. 

Databases. The key to success of any statistics-oriented predictive modeling is the availability of a large set of quality data used as a training set. Usually, at least several tens of structures... 

Stein (67) has used the basic theory of Nishijima for developing equations for predicting the values of various ratios of the fluorescence intensities of differing polarization expected from a stretched ideal rubber. Stein s theoretical efforts made use of Roe and Krigbaum s (55) expression for the complete orientation distribution function derived from the Kuhn and Griin (58) statistical segment model. Stein s equations have yet to be verified by experiment. 

Many models have been proposed (117) to explain the electrical conductivity of mixtures composed of conductive and insulating materials. Percolation concentration is the most interesting of all of these models. Several parameters, such as filler distribution, filler shape, filler/matrix interactions, and processing technique, can infiuence the percolation concentration. Among these models, the statistical percolation model (118) uses finite regular arrays of points and bonds (between the points) to estimate percolation concentration. The thermodynamic model (119) emphasizes the importance of interfacial interactions at the boimdary between individual filler particles and the polymeric host in the network formation. The most promising ones are the structure-oriented models, which explain condnctivity on the basis of factors determined from the microlevel stmctin-e of the as-produced mixtures (120). 

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]. 

A microscopic description characterizes the structure of the pores. The objective of a pore-structure analysis is to provide a description that relates to the macroscopic or bulk flow properties. The major bulk properties that need to be correlated with pore description or characterization are the four basic parameters porosity, permeability, tortuosity and connectivity. In studying different samples of the same medium, it becomes apparent that the number of pore sizes, shapes, orientations and interconnections are enormous. Due to this complexity, pore-structure description is most often a statistical distribution of apparent pore sizes. This distribution is apparent because to convert measurements to pore sizes one must resort to models that provide average or model pore sizes. A common approach to defining a characteristic pore size distribution is to model the porous medium as a bundle of straight cylindrical or rectangular capillaries (refer to Figure 2). The diameters of the model capillaries are defined on the basis of a convenient distribution function. 

For highly exothermic SN2 reactions, which have a central barrier significantly lower in energy than that of the reactants, association of the reactants may be the rate controlling step in TST.1 The SN2 rate constant can then be modeled by a capture theory9 such as VTST,10 average dipole orientation (ADO) theory,11 the statistical adiabatic channel model (SACM),12 or the trajectory capture model.13... 

Additional software has been developed to merge data from various data collection steps and to model the data using suitable statistical distribution functions. We are working on software to perform corrections for absorption, specimen shape, and misalignment. Library routines for 2-diraensional data smoothing and integration are being adapted to the calculation of orientation functions and other moments of the probability distributions. 

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