Models partial averaging

From the NMR data of the polymers and low-molecular models, it was inferred that the central C—H carbons in the aliphatic chain in polymer A undergo motions which do not involve the OCH2 carbons to a great extent. At ambiet temperatures, the chemical shift anisotropy of the 0(CH2)4 carbons of polymer A are partially averaged by molecular motion and move between lattice positions at a rate which is fast compared to the methylene chemical shift interaction. 

It is convenient to consider a model of an anisotropic recombination region the reflecting recombination sphere (white sphere) with black reaction spots on its surface [77, 78], The measure of the reaction anisotropy here is the geometrical steric factor Q which is a ratio of a black spot square to a total surface square. Such a model could be actual for reactions of complex biologically active molecules and tunnelling recombination when the donor electron has an asymmetric (e.g., p-like) wavefunction. Note the non-trivial result that at small Q, due to the partial averaging of the reaction anisotropy by rotational motion arising due to numerous repeated contacts of reactants before the reaction, the reaction rate is K() oc J 1/2 rather than the intuitive estimate Kq oc Q. 

First, the diffusional radical reactions in solutions whose rates are proportional to I) sometimes have rate constants that are much smaller than their contact estimate for the isotropic black sphere, ki> = 4naD. It was proved that this is the result of chemical anisotropy of the reactants. Partially averaged by translational and rotational diffusion of reactants, this anisotropy manifests itself via the encounter efficiency w < 1, which enters the rate constant kn = w4nak) [249]. Even the model of white spheres with black spots is more appropriate for such reactions than the conventional Smoluchowski model. 

To deduce a model for the molecular motion exhibited by the peroxy radical several features of the data in Table V must be noted. First it is known (28) that the direction of the largest g value for a rigid peroxy radical lies parallel to the 0-0 bond direction. From the esr data at 77 K (supposedly the rigid form of the radical), this would be the direction of go. Secondly, the direction of the half-filled 2piT-orbital on oxygen at 77 K is given by the direction of gi. Thirdly, if rotation of the peroxy radical takes place about some axis at 300 K, then partial averaging results in an axially symmetric tensor. This is... 

The partially averaged potential (Eq. 29) can be used in the molecular theory of the nematic-isotropic transition (also being supplemented by the P4 term [19]). However, several other properties of nematics cannot be described in this way. For example, the full anisotropy of the Frank elastic constants can be accounted for only taking into account the explicit dependence of the interaction potential on the intermolecular vector [20]. In this case appropriate model potentials can be obtained using some more general expansion of the full potential y(a, Ty, a ). This potential can be expanded in terms of the spherical invariants... 

Figure 4 Time step dependence of the average total energy for two models of a partially hydrated dodecamer DNA duplex. Thinner traces show results for virtually harmonic conditions when temperature was lowered to 1 K. The DNA molecule has fixed bond lengths, rigid bases, and fixed valence angles except for the mtra- and extracyclic bond angles m sugars, (a) No modifications of inertia (b) inertia modified as explained m the text. (From Ref. 54.)...

For most applications, the engineer must instead resort to turbulence models along with time-averaged Navier-Stokes equations. Unformnately, most available turbulence models obscure physical phenomena that are present, such as eddies and high-vorticity regions. In some cases, this deficiency may partially offset the inherent attractiveness of CFD noted earlier. 

Patir, N. and Cheng, H. S., "Average Flow Model for Determining Effects of Three-Dimensional Roughness on Partial Hydrodynamic Lubrication, ASME J. Lubr. Technol., Vol. lOO.No. 1,1978,pp. 12-17. 

Patir, N., Effects of Surface Roughness on Partial Film Lubrication Using an Average Flow Model Based on Numerical Simulation. Ph.D. Thesis, Northwestern University, 1978. 

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