Anisotropy equation

It should be remembered that even for isotropic defects their elastic interaction is anisotropic, due to crystalline anisotropy, equation (3.1.4). A pair of the simplest Frenkel defects - vacancy and an interstitial atom - attract each other in the direction (100), but their interaction becomes repulsive, e.g., along (111) and (101) axes. 

The measurements are different in the frequency-domain. In this case we measure the phase shift between the parallel and perpendicular components of the emission, and a frequency-dependent anisotropy, which is analogous to the steady state anisotropy. These two types of data are used to determine the decay law for the anisotropy (equation 15). 

Orbital contributions in general determine departures of the g value from the free electron value, and introduce magnetic anisotropy. Equation (1) shows that the quantity XmT is independent of temperature. Sometimes instead of Xm a dimensionless quantity, the so-called effective magnetic moment /iefr=2.828 (xm7) is used, but in the recent literature the XmT value is usually reported. The spin-only XmT and //eff values for various numbers of unpaired electrons are given in Table 1. It is much more common for the magnetic susceptibility to use cgs rather than SI units and we will use the former throughout this chapter. 

Note that the effective aligning field Ep of the polymer network depends only on the density of the polymer fibers, but not on the dielectric anisotropy. Equation (11.70) shows that if the dielectric anisotropy of the liquid crystal is Ae, an electric field higher than Ep must be apphed in order to overcome the aligning effect of the polymer network such that the hquid crystal can... 

Polarized fluorescence data. As the emission anisotropy (equation 16) is not a directly measured quantity, polarized fluorescence data cannot be analysed by fitting calculated experimental values for r(t) with a simple convolution of the lamp profile and a model function (e.g. equation 17 for a rotating sphere). Instead, the polarized data are often analysed in two steps. First, the total fluorescence decay (I (t) -I- 2gl i(t) in the denominator of equation 16, and typically referred to as the sum data, S(t)) is fitted as described above. Once the appropriate expression F(t) for the intrinsic fluorescence decay has been obtained, the product F(t)r(t)Lij is then convolved with the lamp profile and fitted to the difference data D(t) = I (t) - glx(t) (the numerator of equation 16). Here, r(t) is the... 

A more natural way to account for the anisotropy is to treat tire parairreters in an interatomic potential, such as equation (A 1.5.64). as fiurctioirs of the relative orientation of the interacting molecules. Comer [131] was perhaps the first to use such an approach. Pack [132] pointed out that Legendre expansions of the well depth e and equilibrium location of the interaction potential converge more rapidly tirair Legendre expansions of the potential itself... 

As an illustration, we consider the case of SFIG from the (111) surface of a cubic material (3m. syimnetry). More general treatments of rotational anisotropy in centrosymmetric crystals may be found in the literature [62. 63 and M]- For the case at hand, we may detennine the anisotropy of the radiated SFl field from equation Bl.5.32 in conjunction with the fonn of -)from table Bl.5.1. We fmd, for example, for the p-in/p-out and s-... 

Figure Bl.5.7 displays results of a measurement of the rotational anisotropy for an oxidized Si(l 11) surface [65]. For the case shown in the top panel, the results confonn to the predictions of equation Bl.5.42 (with/...

C2.2.12 and Ae is the anisotropy in pennittivity in the nematic liquid crystal. Note that in equation (C2.2.16) the tlireshold voltage, that is the relevant quantity for display operation, is independent of cell thickness. 

Material parameters defined by Equations (1.11) and (1.12) arise from anisotropy (i.e. direction dependency) of the microstructure of long-chain polymers subjected to liigh shear deformations. Generalized Newtonian constitutive equations cannot predict any normal stress acting along the direction perpendicular to the shearing surface in a viscometric flow. Thus the primary and secondary normal stress coefficients are only used in conjunction with viscoelastic constitutive models. 

The referential formulation is translated into an equivalent current spatial description in terms of the Cauchy stress tensor and Almansi strain tensor, which have components relative to the current spatial configuration. The spatial constitutive equations take a form similar to the referential equations, but the moduli and elastic limit functions depend on the deformation, showing effects that have misleadingly been called strain-induced hardening and anisotropy. Since the components of spatial tensors change with relative rigid rotation between the coordinate frame and the material, it is relatively difficult to construct specific constitutive functions to represent particular materials. 

The deformation may be viewed as composed of a pure stretch followed by a rigid rotation. Stress and strain tensors may be defined whose components are referred to an intermediate stretched but unrotated spatial configuration. The referential formulation may be translated into an unrotated spatial description by using the equations relating the unrotated stress and strain tensors to their referential counterparts. Again, the unrotated spatial constitutive equations take a form similar to their referential and current spatial counterparts. The unrotated moduli and elastic limit functions depend on the stretch and exhibit so-called strain-induced hardening and anisotropy, but without the effects of rotation. 

It is remarkable that the spatial eonstitutive equations have the same form as the referential equations, exeept that spatial quantities appear instead of referential ones. In partieular, the moduli and elastie limit funetions (5.154), (5.I6O3), and (5.I683) have the same forms as their referential eounterparts (5.129), (5.132), (5.135) exeept for the dependenee of the spatial moduli and elastie limit funetions on F. If this dependenee on F is retained, then the spatial formulation is equivalent to the referential formulation and is fully eapable of ineluding anisotropie moduli and elastie limit funetions. Given one formulation, the other may be obtained from it. Clearly, when deformations... 

Wallace [15], [16] gives details on effects of nonlinear material behavior and compression-induced anisotropy in initially isotropic materials for weak shocks, and Johnson et ai. [17] give results for infinitesimal compression of initially anisotropic single crystals, but the forms of the equations are the same as for (7.10)-(7.11). From these results it is easy to see where the micromechanical effects of rate-dependent plastic flow are included in the analysis the micromechanics (through the mesoscale variables and n) is contained in the term y, as given by (7.1). 

The stored strain energy can also be determined for the general case of multiaxial stresses [1] and lattices of varying crystal structure and anisotropy. The latter could be important at interfaces where mode mixing can occur, or for fracture of rubber, where f/ is a function of the three stretch rations 1], A2 and A3, for example, via the Mooney-Rivlin equation, or suitable finite deformation strain energy functional. 

The quantitative assessment of the overall orientation of PET fibers is generally made on the basis of fiber optical anisotropy measurements, i.e., measurements of the optical birefringence of the fiber. The determination of the value of optical birefringence makes it possible to determine the value of Hermans function of orientation based on the equation ... 

In the more complex case of the saturated hydrocarbons, the anisotropic equations have not been solved. Rough estimates from approximate solutions indicated that the anisotropy of hydrogen and of saturated carbon atoms is small but an accurate check on this matter would be desirable. 

In Fig. 11, experimental results are shown for the PET films, together with two calculated relationships between B20/a2 and P20o°v< rall> based on the alternative hypotheses that the NMR anisotropy arises either from wholly trans oriented material, according to Equation (40b) or from wholly gauche oriented material. It is very satisfactory to note that the experimental results are very close to the first hypothesis. 

P 0 is the apparent limiting anisotropy and P is the anisotropy (17). The slope is read as the straight line portion of the curves in Figures la-f and applied in equation 10 to obtain the equivalent-sphere molar volume. The difference between the extrapolated intercept of the linear portion of the line on they ordinate and the extrapolated intercept of the curved line is attributed to the internal rotation of the fluorophores in the molecule (5). 

This simple relaxation theory becomes invalid, however, if motional anisotropy, or internal motions, or both, are involved. Then, the rotational correlation-time in Eq. 30 is an effective correlation-time, containing contributions from reorientation about the principal axes of the rotational-diffusion tensor. In order to separate these contributions, a physical model to describe the manner by which a molecule tumbles is required. Complete expressions for intramolecular, dipolar relaxation-rates for the three classes of spherical, axially symmetric, and asymmetric top molecules have been evaluated by Werbelow and Grant, in order to incorporate into the relaxation theory the appropriate rotational-diffusion model developed by Woess-ner. Methyl internal motion has been treated in a few instances, by using the equations of Woessner and coworkers to describe internal rotation superimposed on the overall, molecular tumbling. Nevertheless, if motional anisotropy is present, it is wiser not to attempt a quantitative determination of interproton distances from measured, proton relaxation-rates, although semiquantitative conclusions are probably justified by neglecting motional anisotropy, as will be seen in the following Section. 

The flow velocity, pressure and dynamic viscosity are denoted u, p and fj and the symbol (...) represents an average over the fluid phase. Kim et al. used an extended Darcy equation to model the flow distribution in a micro channel cooling device [118]. In general, the permeability K has to be regarded as a tensor quantity accounting for the anisotropy of the medium. Furthermore, the description can be generalized to include heat transfer effects in porous media. More details on transport processes in porous media will be presented in Section 2.9. 

The introduction of a cubic lattice grid leads to anisotropy in the equations at the Burnett and higher levels. 

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