Anisotropic mediums

Quite specific effects in the flow of dispersions of long fibers are connected with particles orientation in the flow. Indeed, the state of fibers during the flow changes greatly as compared the initial state, so that the material in a steady-state flow is an anisotropic medium. Therefore the viscosity of such a suspension may become independent of a fiber s length [30], The most strong effects caused by a deformation of anisotropic particles should be expected in transient flows, in particular if the particles themselves are flexible and deformed in the flow. 

Our first exploration of property space was focused on acetylcholine. This molecule was chosen for its interesting structure, major biological role, and the abundant data available on its conformational properties [15]. The behavior of acetylcholine was analyzed by MD simulations in vacuum, in isotropic media (water and chloroform) [16] and in an anisotropic medium, i.e. a membrane model [17]. Hydrated n-octanol (Imol water/4mol octanol) was also used to represent a medium structurally intermediate between a membrane and the isotropic solvents [17]. 

Conversely, in a membrane model, acetylcholine showed mean log P values very similar to those exhibited in water. This was due to the compound remaining in the vicinity of the polar phospholipid heads, but the disappearance of extended forms decreased the average log P value somewhat. This suggests that an anisotropic environment can heavily modify the conformational profile of a solute, thus selecting the conformational clusters more suitable for optimal interactions. In other words, isotropic media select the conformers, whereas anisotropic media select the conformational clusters. The difference in conformational behavior in isotropic versus anisotropic environments can be explained considering that the physicochemical effects induced by an isotropic medium are homogeneously uniform around the solute so that all conformers are equally influenced by them. In contrast, the physicochemical effects induced by an anisotropic medium are not homogeneously distributed and only some conformational clusters can adapt to them. 

In an anisotropic medium the physical properties depend on the direction of observation. The equation of propagation now reads ... 

The choice of method depends on the system to be investigated. The methods of intermolecular quenching and intermolecular excimer formation are not recommended for probing fluidity of microheterogeneous media because of possible perturbation of the translational diffusion process. The methods of intramolecular excimer formation and molecular rotors are convenient and rapid, but the time-resolved fluorescence polarization technique provides much more detailed information, including the order of an anisotropic medium. 

If the diffusion medium is isotropic in terms of diffusion, meaning that diffusion coefficient does not depend on direction in the medium, it is called diffusion in an isotropic medium. Otherwise, it is referred to as diffusion in an anisotropic medium. Isotropic diffusion medium includes gas, liquid (such as aqueous solution and silicate melts), glass, and crystalline phases with isometric symmetry (such as spinel and garnet). Anisotropic diffusion medium includes crystalline phases with lower than isometric symmetry. That is, most minerals are diffu-sionally anisotropic. An isotropic medium in terms of diffusion may not be an isotropic medium in terms of other properties. For example, cubic crystals are not isotropic in terms of elastic properties. The diffusion equations that have been presented so far (Equations 3-7 to 3-10) are all for isotropic diffusion medium. 

For binary diffusion in an isotropic medium, one diffusion coefficient describes the diffusion. For binary diffusion in an anisotropic medium, the diffusion coefficient is replaced by a diffusion tensor, denoted as D. The diffusion tensor is a second-rank symmetric tensor representable by a 3 x 3 matrix ... 

In the general case of three-dimensional multicomponent diffusion in an anisotropic medium (such as Ca-Fe-Mg diffusion in pyroxene), the mathematical description of diffusion is really complicated it requires a diffusion matrix in which every element is a second-rank tensor, and every element in the tensor may depend on composition. Such a diffusion equation has not been solved. Because rigorous and complete treatment of diffusion is often too complicated, and because instrumental analytical errors are often too large to distinguish exact solutions from approximate solutions, one would get nowhere by considering all these real complexities. Hence, simplification based on the question at hand is necessary to make the treatment of diffusion manageable and useful. 

In an isotropic medium, D is a scalar, which may be constant or dependent on time, space coordinates, and/or concentration. In anisotropic media (such as crystals other than cubic symmetry, i.e., most minerals), however, diffusivity also depends on the diffusion direction. The diffusivity in an anisotropic medium is a second-rank symmetric tensor D that can be represented by a 3 x 3 matrix (Equation 3-25a). The tensor is called the diffusivity tensor. Diffusivity along any given direction can be calculated from the diffusivity tensor (Equation 3-25b). Each element in the tensor may be constant, or dependent on time, space coordinates and/or concentration. 

The diffusion equation in an anisotropic medium is complicated. Based on the definition of the diffusivity tensor, the diffusive flux along a given direction (except along a principal axis) depends not only on the concentration gradient along this direction, but also along other directions. The flux equation is written as F = —D VC (similar to Fick s law F= -DVC but the scalar D is replaced by the tensor D), i.e.. 

For three-dimensional diffusion in an anisotropic medium, theoretically it is possible to transform the diffusion equation to a form similar to that in an isotropic system. However, in practice, the transformed equation is rarely used, and diffusion is often simplified to be along the fastest diffusion direction. 

More general ellipsoidal particles in an anisotropic medium, where there is no restriction on the principal axes of either the real or imaginary parts of the permittivity tensors, have been treated by Jones (1945). 

To give physical meaning to the principal dielectric functions, we consider propagation of plane waves E0exp(/k x — ioot) in an anisotropic medium that is, we ask What kind of plane waves can propagate in such a medium without change of polarization If we follow the same reasoning as in Section 2.6, we obtain from the Maxwell equations... 

For the sake of completeness we will state the equations for a three-dimensional anisotropic medium. In this case Darcy s law can be generalized to [5] ... 

A particle diffusing in a three-dimensional, inhomogeneous anisotropic medium with external force with potential U(r) obeys a generalization of (3.2),... 

ANISOTROPIC MEDIUM. An anisotropic medium has different optical or other physical properties in different directions. Wood and calcite crystals are anisotropic, while fully annealed glass and, in general, fluids at rest are isotropic. 

When plane polarized light passes through an anisotropic medium, the refractive indices of the two beams which emerge, which are right-hand and left-hand polarized, respectively, are not the same. This causes a phase difference between the two component beams and the resultant beam is rotated in its plane of polarization as it emerges from die medium. 

As shown in Fig. 4.19, in anisotropic medium, a surface acoustic wave represents a combined longitudinal (4.19a) and shear (4.19b) motion of the lattice in the y-(0)-z plane this is the saggital plane. In anisotropic media, in certain multilayer structures and at some interfaces, the surface wave velocity exceeds the velocity of the shear wave and the energy continuously leaks from the surface to the bulk of the material. In such cases, we talk about pseudo- or leaky waves. Various energy-loss... 

What happens to the diffracted beam in an anisotropic medium (e.g. as in an AOTF device) As given above, the birefringence induced by the acoustic wave causes the diffracted beam polarisation to change. So if, for example, the input beam is P-polarised, then... 

Thus the spectral function L(z) of an isotropic medium is represented as a linear combination of two spectral functions determined for an anisotropic medium pertinent to longitudinal ( ) ) and transverse (K ) orientations of the symmetry axis with respect to the a.c. field vector E. It is shown in GT, Section V, that these spectral functions are proportional to the main components of the dielectric-susceptibility tensor. 

Attention must now be paid to the exponential factor, exp( 2nir (n iij)/A), in Equation 6.5, where (n it) is known as the complex refractive index of a substance. It can be seen that the effect of this factor upon the electromagnetic wave increases with the distance Irl that the light travels in that medium. In the general case of an anisotropic medium, n and are referred to as a specific set of axes, usually chosen to coincide with the optical axes of the medium. For example, the axes of maximum and minimum transmittance are selected for anisotropic absorption. The extinction f for an anisotropic medium is related to the extinction coefficient through Equation 6.9. 

The extinction I is proportional to the concentration C of light absorbers in the medium and the directional nature of absorption in an anisotropic medium is denoted by the vector g, known as the molar extinction. Because Equation 6.8 was derived for a wave propagating in the vacuum, the contribution of the extinction to the light intensity is lost in Equation 6.7. For an optical wave propagating in an isotropic medium, Equation 6.7 can be rewritten in the form of Equation 6.10. 

Within the dielectric continuum model, the electrostatic interactions between a probe and the surrounding molecules are described in terms of the interaction between the charges contained in the molecular cavity, and the electrostatic potential these changes experience, as a result of the polarization of the environment (the so-called reaction field). A simple expression is obtained for the case of an electric dipole, /a0, homogeneously distributed within a spherical cavity of radius a embedded in an anisotropic medium [10-12], by generalizing the Onsager model [13]. For the dipole parallel (perpendicular) to the director, the reaction field is parallel (perpendicular) to the dipole, and can be calculated as [10] ... 

The magnetic tensors for a molecule embedded in an anisotropic medium can be calculated on the basis of an effective Hamiltonian which, in addition to the Hamiltonian of the isolated molecule, contains a contribution describing the response function of the reaction potential [32]. The general aspects are presented in the contribution by Sadlej and Pecul. 

The system of entangled macromolecules becomes anisotropic when velocity gradients are applied, and one can assume that each Brownian particle of the chain moves in the anisotropic medium. The expressions for the discussed quantities (7.5) for case, when one can neglect the hydrodynamic interaction... 

The value of the refractive index n of light in the anisotropic medium depends on the direction of propagation s and on the direction of the polarisation of the light. For the given relative permittivity tensor ji, the refractive index can be determined from the relation (Born and Wolf 1970 Landau et al. 1987)... 

Correlated charge fluctuations between anisotropic bodies acting across an anisotropic medium create torques as well as attractions or repulsions. The formulae derived here for semi-infinite media can also be specialized to express the torque and force between anisotropic small particles or between long rodlike molecules. (For example, Table C.4 and Subsection L2.3.G.)... 

AOTFs (101, 102) work on a different principal. This type of filter functions by the interaction of light with a traveling acoustic wave in an anisotropic medium. Both crystals and polymers have been used for the anisotropic medium. An acoustic transducer is bonded to one end of the material and an... 

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