Effect of Local Anisotropy

Derived from linear approximation of the equations (3.37), the equilibrium correlation function (4.29), defines two conformation relaxation times r+ and r for every mode. The largest relaxation times have appeared to be unrealistically large for strongly entangled systems, which is connected with absence of effect of local anisotropy of mobility. To improve the situation, one can use the complete set of equations (3.37) with local anisotropy of mobility. It is convenient, first, to obtain asymptotic (for the systems of long macromolecules) estimates of relaxation times, using the reptation-tube model. 

It is not difficult to reproduce an expression for the correlation function Ma(t) and estimate times of relaxation due to the conventional reptation-tube model (see Section 3.5). Indeed, an equation for correlation function follows equation (3.48) and has the form 

These are exactly the known results (Doi and Edwards 1986, p. 196). The time behaviour of the equilibrium correlation function is described by a formula which is identical to formula for a chain in viscous liquid (equation (4.34)), while the Rouse relaxation times are replaced by the reptation relaxation times. In fact, the chain in the Doi-Edwards theory is considered as a flexible rod, so that the distribution of relaxation times naturally can differ from that given by equation (4.36) the relaxation times can be close to the only disentanglement relaxation time r[ep. 

We have introduced here, instead of index 2, an index x, value of which can be less than 2 according to the results of simulation (see the next subsection, x 0.5). 

The rates of relaxation r7(t) in the moment t, or, in other words, the current relaxation times of the macromolecular coil can be directly calculated as 

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The results discussed in the previous section are valid in linear approximation for any concrete representations of the memory functions / (s) and calculate relaxation times for macromolecular coil, one has to specify the memory functions and include the effect of local anisotropy. 

Nuclear magnetic resonance (NMR) in principle gives information about the environment of a nucleus that possesses a magnetic dipole. For solid samples, which of course include all LDH derivatives, special techniques must be employed to average out the effects of local anisotropies, and even so it is not possible to obtain the superb resolution typical of solution NMR spectra (301). Despite these limitations, the technique has been extensively applied to LDH derivatives, with H (287,329,330), (287), N (331), Cl (332), Se (333), Al... 

This method has been devised as an effective numerical teclmique of computational fluid dynamics. The basic variables are the time-dependent probability distributions f x, f) of a velocity class a on a lattice site x. This probability distribution is then updated in discrete time steps using a detenninistic local rule. A carefiil choice of the lattice and the set of velocity vectors minimizes the effects of lattice anisotropy. This scheme has recently been applied to study the fomiation of lamellar phases in amphiphilic systems [92, 93]. 

D11/Dj, from 1 to 10. Symbols correspond to synthetic experimental data generated assuming overall tumbling with rc = 5 ns and various degrees of anisotropy as indicated. Model-free parameters typical of restricted local backbone dynamics in protein core, S2=0.87, T oc =20 ps, were used to describe the effect of local motions. The H resonance frequency was set to 600 MHz. The solid lines correspond to the right-hand-side expression in Eq. (10). 

There were different generalisations of the reptation-tube model, aimed to soften the borders of the tube and to take into account the underlying stochastic dynamics. It seems that the correct expansion of the Doi-Edwards model, including the underlying stochastic motion and specific movement of the chain along its contour - the reptation mobility as a particular mode of motion, is presented by equations (3.37), (3.39) and (3.41). In any case, the introduction of local anisotropy of mobility of a particle of chain, as described by these equations, allows one to get the same effects on the relaxation times and mobility of macromolecule, which are determined by the Doi-Edwards model. 

For low naturally abundant nuclei, such as the nucleus, resonance lines reflecting the chemical shift anisotropy (CSA) can be observed for solid organic materials by eliminating the effects of local field with use of the so-called dipolar decoupling method. However, the CSA resonance lines are usually broad and, therefore, the respective CSA lines are superposed with each other for polymers composed of different C species. To separately measure these CSA lines, the following different methods were proposed ... 

In Section II, we discuss the effects of local arrangements of the electron cloud and other nuclei about a resonating nucleus in a molecule that are responsible for the effective magnetic field seen by this nucleus, and thus for its NMR absorption spectrum. Each interaction is discussed in turn, as if it were the only interaction present, the justification being that many timesthe NMR spectrum of a given nucleus appears to be the result of one or two interactions. A case in point is the NMR spectrum of a nucleus in a liquid, discussed in Section III. In this case, all information relating to anisotropy of the local environment effectively disappears, and the resulting spectrum is simply due to the isotropic portions of the chemical shift... 

The present review shows how the microhardness technique can be used to elucidate the dependence of a variety of local deformational processes upon polymer texture and morphology. Microhardness is a rather elusive quantity, that is really a combination of other mechanical properties. It is most suitably defined in terms of the pyramid indentation test. Hardness is primarily taken as a measure of the irreversible deformation mechanisms which characterize a polymeric material, though it also involves elastic and time dependent effects which depend on microstructural details. In isotropic lamellar polymers a hardness depression from ideal values, due to the finite crystal thickness, occurs. The interlamellar non-crystalline layer introduces an additional weak component which contributes further to a lowering of the hardness value. Annealing effects and chemical etching are shown to produce, on the contrary, a significant hardening of the material. The prevalent mechanisms for plastic deformation are proposed. Anisotropy behaviour for several oriented materials is critically discussed. 

The overall tumbling of a protein molecule in solution is the dominant source of NH-bond reorientations with respect to the laboratory frame, and hence is the major contribution to 15N relaxation. Adequate treatment of this motion and its separation from the local motion is therefore critical for accurate analysis of protein dynamics in solution [46]. This task is not trivial because (i) the overall and internal dynamics could be coupled (e. g. in the presence of significant segmental motion), and (ii) the anisotropy of the overall rotational diffusion, reflecting the shape of the molecule, which in general case deviates from a perfect sphere, significantly complicates the analysis. Here we assume that the overall and local motions are independent of each other, and thus we will focus on the effect of the rotational overall anisotropy. 

We conclude this subsection with several remarks on the interpretation of the anisotropy of Hc2. The largest in-plane anisotropy reported by Metlushko et al. (1997) coincides with the direction of the nesting vector (0.55,0,0). Another manifestation of strong local anisotropy effects is provided by deviations from the 9 (angular) dependence due to anisotropic effective masses (Fermi velocities)... 

These hybridisation variations are caused by anisotropy within the chemical bonds. This is due to the non-homogeneous electronic distribution around bonded atoms to which can be added the effects of small magnetic fields induced by the movement of electrons (Fig. 9.12). Thus, protons on ethylene are deshielded because they are located in an electron-poor plane. Inversely, protons on acetylene that are located in the C-C bond axis are shielded because they are in an electron-rich environment. Signals related to aromatic protons are strongly shifted towards lower fields because, as well as the anisotropic effect, a local field produced by the movement of the aromatic electrons or the ring current is superimposed on the principal field (Fig. 9.12). 

Figure 9.12—Effects of anisotropy and local inducedfields. The presence of it bonds is shown as zones in which there is a shielding (+)ora deshielding (—) effect. Ethylenic or aromatic protons are located outside a double cone of protection.

Abstract The discussion of relaxation and diffusion of macromolecules in very concentrated solutions and melts of polymers showed that the basic equations of macromolecular dynamics reflect the linear behaviour of a macromolecule among the other macromolecules, so that one can proceed further. Considering the non-linear effects of viscoelasticity, one have to take into account the local anisotropy of mobility of every particle of the chains, introduced in the basic dynamic equations of a macromolecule in Chapter 3, and induced anisotropy of the surrounding, which will be introduced in this chapter. In the spirit of mesoscopic theory we assume that the anisotropy is connected with the averaged orientation of segments of macromolecules, so that the equation of dynamics of the macromolecule retains its form. Eventually, the non-linear relaxation equations for two sets of internal variables are formulated. The first set of variables describes the form of the macromolecular coil - the conformational variables, the second one describes the internal stresses connected mainly with the orientation of segments. 

While introducing of the global anisotropy, the equation for the macro-molecular dynamics remains linear in co-ordinates and velocities, the introduction of the local anisotropy makes it non-linear in co-ordinates. Both global and local anisotropy are needed to describe the non-linear effects of the relaxation phenomena in the mesoscopic approximation. 

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