Local inhomogeneity parameter

On heterogeneous surfaces, the probability of metastable states increases due to local inhomogeneities. Therefore, the adsorption parameters calculated by MC should be always checked against the results obtained by other methods. 

The Seitz radius rs is the radius of a sphere which on average contains one electron, so rs —> 0 in the high-density limit valence electrons typically have 1 bohr < r, < 6 bohr. The relative spin polarization ( vanishes for a spin-unpolarized system, and equals + 1 or -1 where all the electron spins are up or down, respectively. The inhomogeneity parameter s measures how fast the density varies on the scale of the local Fermi wavelength valence electrons typically have 0 < s < 3. Both rs and s diverge in the low-density tail of an atom or molecule. 

Universal quantities T and p related to the whole system are not always appropriate for description of a nonequilibrium system due to its probable spatial inhomogeneity. In general, the nonequilibrium system needs to be characterized using local extensive parameters—for example, the Gibbs potential g(T(r, t), p(r, t)) per unit mass of the system matter. In this case. 

An identical mathematical description of the kinetics of curing of reactants different in chemical nature and that obtained on the basis of fundamentally different experimental methods allows us to assume that this apparent selfacceleration course of some rheokinetic parameters is common to the processes of formation of materials with a crosslinked structure. It should be emphasized once more that the self-acceleration" effect must not be identified with the self-catalysis of the reaction of interaction between epoxy monomers and diamines which is studied in detail on model compounds [116, 117]. For each particular curing process the self-acceleration effect is influenced by the mechanism of network formatic, namely, chemical self catalysis [118], the appearance of local inhomogeneities [120], the manifestation of gel eff t [78], parallel course of catalytic and noncatalytic reactions [68]. It is probably true that the phenomena listed above may in one form or another show up in specific processes and make their contribution into self-acceleration of a curing reaction. 

The free energy density/(r) of the inhomogeneous hnite system with a local order parameter /(r) was expressed by adding to Eq. (24), an even expansion of the gradient term, so that... 

The former reflects the slightly imperfect alignment of the domain directors, whereas the latter is a consequence of a smeared phase transition in the LCEs. Figure 9a,b serves as an example of how the H-NMR spectra are altered by the inhomogeneity of the domain-director aligmnent and of the local order parameter in side-chain LCEs. Both spectra were recorded on LCEs doped with deuterated probes and oriented with their principal aligning direction parallel to the magnetic field. 

Apart from the nonuniform director alignment, the inhomogeneity of the local order parameter is also encountered in LCEs. This is best observed in a H-NMR spectrum (Fig. 9b) recorded in the vicinity of the phase transition. The spread 5v of the spectral intensity between 0 and 20 kHz in this spectrum corresponds to a spread of the local order parameter 85 in the range between approximately 0 and 0.45. Two pronounced peaks can be noted in each half-spectrum, corresponding to the coexisting paranematic (lower S, inner peak at Vpn) and nematic components (higher S, outer peak at v ). 

It appears that the understanding of the local order parameter s inhomogeneity is closely connected with the understanding of the phase transition in LCEs. However, in comparison with the inhomogeneity of the domain-director alignment, the characterization of the heterogeneity of the local order parameter appears much less accessible. This is due to the lack of any general model that predicts the distribution of the local order parameter > 5(5) in LCEs. 

The general expression for the interdiHusion coefficients within the a-fi region has not been found yet. So far, different approaches [26] of introducing the transport parameters in a locally inhomogeneous system are used. But in many cases (if the diffusivity of a and phases does not differ that much), the model of phase parallel connection is considered a reasonable approximation... 

The whole system s entropy is defined here as simply the sum of entropies of all sites. If the characteristic inhomogeneity of the system is of the order of several nanometers, other methods should be applied. This refers, for example, to spinodal decomposition of supersaturated solid solutions, to nucleation of new phases, and to nano-crystalline alloy behavior. The simplest modification of local quasi-equilibrium approximation for strongly inhomogeneous systems implies adding the squared order parameter gradient into the set of local thermodynamic parameters (van der Waals [24, 25], Ginzburg-Landau [26, 27], Hillert [28, 29], Cahn-Hilliard [30, 31], Khachaturyan [32]). 

A vast majority of fuel cell aging studies consider cell degradation as uniform over the cell surface process (Borup et al., 2007). However, nonuniformities dramatically accelerate the rate of local aging. In a fuel cell, any local inhomogeneity of transport or kinetic parameters induces inhomogeneity in the distribution of the membrane potential over the cell surface. This, in turn, inevitably leads to nonuniform overpotentials of parasitic electrochemical reactions running in the cell. Domains where the parasitic overpotential increases die much faster. Thus, modeling of nonuniformities in cells is important in the context of cell durability. 

In the case of inhomogeneous fluid near the surface, the local order parameter Ap Az, t) at the distance Az from the surface may be defined similarly to the order parameter of bulk fluid ... 

The concentrations of reactants are of little significance in the theoretical treatment of the kinetics of solid phase reactions, since this parameter does not usually vary in a manner which is readily related to changes in the quantity of undecomposed reactant remaining. The inhomogeneity inherent in solid state rate processes makes it necessary to consider always both numbers and local spatial distributions of the participants in a chemical change, rather than the total numbers present in the volume of reactant studied. This is in sharp contrast with methods used to analyse rate data for homogeneous reactions in the liquid or gas phases. 

It should be also noted that the dynamic NMR line width Avayn = is always small (Fig. 11a) as compared to the static glassy line width induced by the inhomogeneous nature of the spectrum. Therefore the nanocluster dynamics can be locally seen only by T2 measurements, and not by ID line shape data which reflect the static glassy nature of the relaxor state characterized by the Edwards-Anderson order parameter. 

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