Harmonic mean model

Olaj and Zifferer [229] have performed Monte Carlo simulations in which they placed 300 polymer chains in a lattice, considering excluded volume effects. From the obtained configurations they evaluated the shielding factor which describes how severely the presence of a polymer coil retards the termination of its own radical compared to small (unshielded) radicals. For chains of unequal size Olaj and Zifferer showed that the geometric mean model provided a reasonable mathematical description of their results, although the harmonic mean model. 

All materials have both a polar (yi) and nonpolar (yl) or dispersion contribution to the total surface energy. There are two main models for determining the interfacial energy that consider both contributions to surface energetics for each material—the harmonic mean model and the geometric mean model (9). 

Both the harmonic mean model and the geometric mean model will be used to determine the surface energy of the solid surface aloug with the polar... 

While many data are suggestive of chain length dependence, the data are not usually suitable for or have not been tested with respect to model discrimination. Values of ,u have been determined for a variety of small monomeric radicals to be ca I09 M s 1.4 Taking kt0 as Jk,lj and a as 1.0 in the geometric expression yields values of ,iJ as shown in Figure 5.4a.49 Use of the Smoluchowski mean or the harmonic mean approximation prediets a shallower dependence of k 1 on the chain length (Figure 5.4b). All expressions yield the same dependence for j=i. 

L.133 Using two sets of backbone RDC data, collected in bacteriophage Pfl and bicelle media, they obtained order tensor parameters using a set of crystallographic coordinates for the structural model. This allowed the refinement of C -C bond orientations, which then provided the basis for their quantitative interpretation of C -H RDCs for 38 out of a possible 49 residues in the context of three different models. The three models were (A) a static xi rotameric state (B) gaussian fluctuations about a mean xi torsion and (C) the population of multiple rotameric states. They found that nearly 75% of xi torsions examined could be adequately accounted for by a static model. By contrast, the data for 11 residues were much better fit when jumps between rotamers were permitted (model C). The authors note that relatively small harmonic fluctuations (model B) about the mean rotameric state produces only small effects on measured RDCs. This is supported by their observation that, except for one case, the static model reproduced the data as well as the gaussian fluctuation model. 

Similar reasoning concerns also the R-band, if we shall apply for calculations the harmonic oscillator model (we mean now a rough qualitative description). 

Harmonic oscillator model of aromaticity (HOMA) — This is a geometry-based index of aromaticity that takes into account two effects. These are the increase in bond-length alternation (GEO term) and the increase in mean bond length in the system (EN term) such that HOMA= 1-EN-GEO <2004PCP249>. For examples see Sections 2.2.42.3, 2.3.42.3, and 244.2.3. 

Ca, is the fluid reactant concentration in the pore, Rp the pore radius. D,p in this model may be a harmonic mean of the bulk and Knudsen diflusion coefficient with real geometries it would be a true effective difTusivity including the tortuosity factor and an internal void fraction. D p is an effective diffiisivity for the mass transfer inside the solid and is a correction factor accounting for the restricted availability of reactant surface in the region where the partially reacted zones interfere. For Jt(y) < LJ2 (shown in Fig. 4.5-2) or j>2 < J f e factor ( = 1 for L/ > R y) > L/2 or >i < y < yj the factor = 1 — (40/x) where tgB = (2/L) Ji (y) - (L/2) for y < yi the factor C 0, where R(y) is the radial position of the reaction front. It is clear from Eq. 4.S-1 that no radial concentration gradient of A is considered within the pore. 

We consider first the outer-sphere electron-exchange reactions using a harmonic oscillator model for the solvent /40/ i.e., by assuming that the solvent molecules make small vibrations (restricted rotations)with the same effective frequency V The two ions are treated as two hard spheres /40a/ with different chargesjbeing stationary at a fixed separation (r = const) during the solvent fluctuation, which is necessary for the electron transfer. This means that the relative motion of the two ions is so slow that the vibrations of solvent medium change adiabatically in the course of the reaction. This adiabatic approximation implies that the ions are much heavier than the solvent molecules. 

Almost two decades previous to the Doering papers a reasonable model for substituent rate effects was proposed that was based on a geometric model for the MOE-J energy surface for the 3,3-shift. Thus, a hyperbolic paraboloid surface equation could be differentiated to obtain coordinates and the activation free energy for the saddle point (the transition state) cast in terms of the relative free energies for formation of the diyl and the two allyl radicals, the same independent variables of Eqs. (7.1) and (7.2). Equation 7.3, which relates the independent variables by the harmonic mean is based on the simplest hyperbolic paraboloid surface, that is, one with linear edge potentials. Slightly more realistic models were also explored. 

The thermal conductivity, k, and the dynamic viscosity of the fluid, p., can be modeled using the arithmetic and harmonic means of each component in accordance with the expressions [5] ... 

As the first point, the dynamics of the phenyl group in the poly-formal can be considered. Motional descriptions from the two segmental models can be compared as they have been before for the polycarbonates ( 5). In the three bond jump model the primary parameter is the harmonic mean correlation time, and in the... 

This is a mathematical law (law of logic), i.e. a proposition that can be rigorously proved without further assumptions. Interestingly, however, it is found that the effective shear and bulk moduli of multiphase materials always lie between the arithmetic and the harmonic mean. This is a physical law (law of nature), i.e. a finding which can be (and has been in micromechanics) rigorously proved for model materials with well defined microstructures. Of course, since its proof is based on model assumptions, its applicability to real materials is and remains, strictly speaking, a question of experience. 

Fig. 7.1.4. We see that the mean potential depends strongly on the value of the volume per particle. For small values of vjv, (o (r) is well fitted by a parabola. This occurs at low temperatures (generally in the solid state). We may then apply a harmonic oscillator model. The frequency of oscillations depends on the value of vjv. We shall come back to this model in 4.

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