Anisotropic systems

Thus the patterns of hyperfine splittings observed in EPR spectra provide direct information about the numbers and types of nuclei with spin coupled to the electrons this information is exactly analogous to that obtained from coupling patterns in NMR spectra. The magnitudes of the hyperfine couplings can indicate the extent to which the unpaired electrons are delocalized, while g values could also show whether unpaired electrons are based on transition-metal atoms or on adjacent ligands. Another example is described in the online supplementary material for Chapter 5 (hyperfine splitting), and many other examples of the application of EPR to chemical problems are described in a series of annual reports [5]. 

So far, we have only considered EPR spectra obtained for isotropic systems, such as solutions. In practice, very many spectra are recorded for anisotropic systems, which can include unstable species prepared by irradiation of 

Litovitz, in Non-Crystalline Solids, edited by V. D. Frechette, Wiley, New York, 1960, p. 252. 

Philippoff, in Physical Acoustics, edited by W. P. Mason, Volume IIB, Academic Press, New York, 1965, p. 1. 

in Molecular Motions in Liquids, edited by J. Lascombe, Reidel Publishing Company, Dordrecht, 1974, p. 29 J. Rheol.. 22,317 (1978). 

Sternstein, in Treatise on Materials Science and Technology, Volume 10, Part B, Academic Press, New York, 1977, p. 541. 

Heijboer, Proc. Intern. Conf. on Physics of Non-Crystalline Solids, edited by J. A. Prins, North Holland Publishing, Amsterdam, 1965, p. 231. 

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In the case of anisotropic systems with a cylindrical synnnetry (such as rods or fibres), the scattered intensity can be expressed as (derivation to be made later in section Bl.g.dl ... 

The least recognized fonns of the Porod approximation are for the anisotropic system. If we consider the cylindrical scattering expression of equation (B 1.9.61). there are two principal axes (z and r directions) to be discussed... 

Bates M A and Luckhurst G R 1996 Computer simulation studies of anisotropic systems. 26. Monte Carlo investigations of a Gay-Berne discotic at constant pressure J. Chem. Phys. 104 6696-709... 

FIG. 21 Effective diffusion coefficients from Refs. 337 and 193 showing comparison of volume average results (Ryan) with models of Maxwell, Weisberg, Wakao, and Smith for isotropic systems (a), and volume averaging calculations (solid lines) and comparison with data for anisotropic systems (b). (Reproduced with kind permission of Kluwer Academic Publishers from Ref. 193, Fig. 3 and 12, Copyright Kluwer Academic Publishers.)... 

Ochoa-Tapia, JA Stroeve, P Whitaker, S, Diffusive Transport in Two-Phase Media Spatially Periodic Modles and Maxwell s Theory for Isotropic and Anisotropic Systems, Chemical Engineering Science 49, 709, 1994. 

The I term is of particular relevance since, in anisotropic media such as liposomes and artiflcial membranes in chromatographic processes, ionic charges are located on the polar head of phospholipids (see Section 12.1.2) and thus able to form ionic bonds with ionized solutes, which are therefore forced to remain in the nonaqueous phase in certain preferred orientations. Conversely, in isotropic systems, the charges fluctuate in the organic phase and, in general, there are no preferred orientations for the solute. Given this difference in the I term (but also the variation in polar contributions, less evident but nevertheless present), it becomes clear that log P in anisotropic systems could be very different from the value obtained in isotropic systems. 

However, it is clear that anisotropic systems become relevant in the presence of ionic species (see Section 12.1.4). 

A few examples of the moduli of systems with simple symmetry will be discussed. Figure 1A illustrates one type of anisotropic system, known as uniaxial orthotropic. The lines in the figure could represent oriented segments of polymer chains, or they could be fibers in a composite material. This uniaxially oriented system has five independent elastic moduli if the lines (or fibers) ara randomly spaced when viewed from the end. Uniaxial systems have six moduli if the ends of the fibers arc packed in a pattern such as cubic or hexagonal packing. The five engineering moduli are il-... 

A second type of anisotropic system is the biaxially oriented or planar random anisotropic system. This type of material is illustrated schematically in Figure 2A. Four of the five independent elastic moduli are illustrated in Figure 2B in addition there are two Poisson s ratios. Typical biaxially oriented materials are films that have been stretched in two directions by either blowing or tentering operations, rolled materials, and fiber-filled composites in which the fibers are randomly oriented in a plane. The mechanical properties of anisotropic materials arc discussed in detail in following chapters on composite materials and in sections on molecularly oriented polymers. 

For the general case of an anisotropic system the enhancement factor... 

The spin Hamiltonian for an anisotropic system is then given by... 

Because we are studying an anisotropical system where energy transport is possible along the channels and from one channel to another one, it is useful to consider the ratio / Ch/site between the number of parallel channels and the number of Uj-sites in a channel ... 

Self-diffusion and tracer diffusion are described by Equation 3-10 in one dimension, and Equation 3-8 in three dimensions. For interdiffusion, because D may vary along a diffusion profile, the applicable diffusion equation is Equation 3-9 in one dimension, or Equation 3-7 in three dimensions. The descriptions of multispecies diffusion, multicomponent diffusion, and diffusion in anisotropic systems are briefly outlined below and are discussed in more detail later. 

A calculation of the dipolar contribution to the magnetic anisotropy suggested that the diamagnetic substitution method was at least of qualitative value but probably cannot be trusted to provide quantitative values in highly anisotropic systems. 

This equation gives an estimation of the direction of the transition moment. Again, it should be noted that the CD of an anisotropic system can be falsified by interference with the effects of linear birefringence and linear dichroism. One should have in mind that the linear birefringent or linear dichroism contributions are three orders of magnitude larger than that of the intrinsic CD. 

We remind the reader that induced spectra, especially the low density spectra that are not affected by ternary contributions, are feeble. Spec-troscopists use, therefore, high pressures and wide open slits as much as possible for good signal-to-noise ratios. Van der Waals molecules are highly anisotropic systems and their prominent rotational lines show substantial pressure broadening. Moreover, their band spectra typically... 

For an anisotropic system the differentiation should be applied in the individual directions so that the components of the magnetization vector are... 

Although this result was derived from the special case of an isotropic sphere, the above relationship between the total field and the polarization also applies to anisotropic systems. Furthermore, the volume of the sphere can be made arbitrarily small to envelop a single dipole. 

In the case of an anisotropic system, it is convenient to consider particular cases. Further on, expressions for characteristics of optical birefringence in two typical cases will be shown. 

The anisotropy introduces two new features (i) equations (6.305) and (6.306) cannot in general be transformed into each other, as the drift term V D may not be a gradient field. Equation (6.306) can describe systems where the directions of the principal axes depend on the spatial position, (ii) Detailed balance implies that the diffusion flow J vanishes everywhere in the stationary state. However, this is not automatically satisfied for anisotropic systems and one needs to exercise extra care in the modeling of such systems. Inhomogeneity does not affect the detailed balance, (iii) The diffusive part of the diffusion flow must be represented by J = VD73, while the drift is represented by (PV D). 

The choice of a particular technique is limited by factors such as availability, feasibility, and the nature of the information sought. Pharmaceutical scientists are more focused on the usefulness of a particular ME system for a drug delivery application and the influence of the microstructure on that, rather than on the fundamental understanding of aspects such as microstructure and phase behavior. Polarized light microscopy is a readily available technique that could be used at the early formulation development stage to differentiate between isotropic and anisotropic systems. Transmission electron microscopy (TEM) is another available technique that has been shown to provide microstructural as well as size-related information on droplet and bicontinuous ME systems. 

In this brief discussion, only radicals that have just one unpaired electron are mentioned. Obviously there are many systems that have several unpaired electrons but, especially for anisotropic systems, the ESR spectra for such species can be extremely complicated. More information on aspects of ESR spectroscopy of short-lived intermediates is given in two excellent reviews on the subject. ... 

The value of g-factor is a measure of the coupling between the spin of an unpaired electron and an external magnetic held. It is not only dependent on the spin species but also on its environment. A single numerical value of g is applicable only to systems that behave isotropically. With anisotropic systems, a modihed term that accommodates the variability of g with orientations relative to the external held is introduced as g-tensor. Three values, gx, gy, and gz, which represent principal gxx, gyy> and gzz values of the g-matrix, are important EPR parameters. 

An important EPR parameter, hyperfine coupling constant. A, is required to describe the hyperfine interaction. The A values can be obtained directly from EPR spectra when equally spaced spectral lines are exhibited because of the hyperfine interactions (Fig. 3). In many cases, a matrix, known as A-tensor, is required to describe the A values for a more complex anisotropic system. Because of the hyperfine interaction, EPR is very sensitive to the local surroundings. From a chemical standpoint, this property provides a wealth of information such as the identity and number of atoms that makes up a molecule or complex and their distances from the unpaired electron. This information can often be translated, in favorable cases, into the molecular structure of the sample. 

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