Isotropic systems

The above radius of gyration is for an isotropic system. If the system is anisotropic, the mean square radius of gyration is equal to... 

It has been discovered recently that the spectrum of solutions for growth in a channel is much richer than had previously been supposed. Parity-broken solutions were found [110] and studied numerically in detail [94,111]. A similar solution exists also in an unrestricted space which was called doublon for obvious reasons [94]. It consists of two fingers with a liquid channel along the axis of symmetry between them. It has a parabolic envelope with radius pt and in the center a liquid channel of thickness h. The Peclet number, P = vp /2D, depends on A according to the Ivantsov relation (82). The analytical solution of the selection problem for doublons [112] shows that this solution exists for isotropic systems (e = 0) even at arbitrary small undercooling A and obeys the following selection conditions ... 

We have already commented on the equivalence between peripheral PCA and the problem of directed percolation (see footnote on page 343). It is easy to show that the pi and P2 of the isotropic system are given in terms of site- and bond- directed percolation probabilities - ps and pb, respectively - by the expressions pi = PaPb and P2 = PsPfe(2 — p ) see [domany84] and [kinzel85]. 

Usually, dilute polymer solutions are isotropic systems, i.e. macromolecular chains can exist in these solutions independently of each other with a random distribution of orientations of the long axes of coils. The solutions of flexible-chain polymers remain isotropic when the solution concentration increases whereas in concentrated solutions of macromolecules of limited flexibility the chains can no longer be oriented arbitrarily and some direction of preferential orientations of macromolecular axes appears, i.e. the mutual orientations of the axes of neighboring molecules are correlated. This means that... 

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.)... 

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. 

It is our opinion that, among isotropic systems, alongside the standard octanol-water, the alkane-water system (partihoning between water and different alkanes is relahvely independent of the alkane used [14]) is the only system that can be successfuUy used in ADMET predichon, because of its completely different nature from octanol-water. The situahon is much more confused for arhsohopic systems (see Ref. [7] for a brief review) since no standard system has been defined to date. 

The response function K can depend only on r — r for a uniform isotropic system. On Fourier transformation,... 

The usual methods for characterization of anisotropic patterns involve the calculations of the scattering patterns and their further analysis [167,168] in the Fourier space. Thus, for an isotropic system, the scattering intensity, S(k) is symmetric, its maxima of the same amplitude are arranged at the circle k = k ... 

Assuming an isotropic system, the following Maxwell relation can be derived from eq. (2.19), since dA is an exact differential ... 

In the remaining part of the chapter we will use the term y for interfaces that involve solids. It should then implicitly be understood that we are here considering bulk solids that are treated as isotropic systems and that the surface energy thus defined is the average value of the surface energies for different crystal surfaces. Furthermore, we will consistently use superscripts to denote the phases adjacent to the interface in the rest of Section 6.1 and in Section 6.2. 

Figure 5.2. Simulated fluorescence anisotropy decays for (a) an isotropic system, (b) a system such as a lipid bilayer with short and long rotational correlation times, and (c) a system in which one of the rotational correlation times is infinite, and there is therefore a residual anisotropy or r .

The general case of mass transfer includes both diffusion and convection. Hence, there are both diffusive flux and convective flux for a component. Therefore, the total flux is the sum of the two fluxes. For a given component in a binary and isotropic system, the total flux is... 

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. 

The recognition of a ligand at the surface of a natural or artificial membrane by a specific receptor is governed by kinetic and thermodynamic parameters. In a non-isotropic system, perturbations occur to the classical kinetics and thermodynamics of the reaction ... 

Cahn, J. W. Phase separation by spinodial decomposition in isotropic systems. J. Chem. Phys. 42, 93 (1965). 

The case of a pure dilational transformation strain in an inhomogeneous elastically isotropic system has been treated by Barnett et al. [10]. For this case, the elastic strain energy does depend on the shape of the inclusion. Results are shown in Fig. 19.9, which shows the ratio of A(inhomo) for the inhomogeneous problem to A<7 (homo) for the homogeneous case, vs. c/a. It is seen that when the inclusion is stiffer than the matrix, AgE (inhomo) is a minimum... 

The intermediate scattering function for an isotropic system is given by... 

The observed hyperfine shifts could come from contact coupling or pseudocontact interactions between the electrons and the protons. Contact shifts arise when a finite amount of unpaired electron density is transferred to the observed protons. The contact shifts of the proton resonances for isotropic systems are given by Bloembergen s (9) expression... 

Evans, R. M., S. J. Farr, N. A. Armstrong, and S. M. Chatham. 1991. Formulatiomariblo evaluation of pressurized inhalation aerosols containing isotropic systems of lecithin and Wfetam. Res. 8 629-635. 

General Form of the Raman Tensor for Transversely Isotropic Systems... 

This expression is identical in form to equation (5.20). In the case of Raman scattering, however, it is necessary to compute the average Raman tensor, (. For a transversely isotropic system, the segment is free to spin about the r. axis, and the vector ni is averaged over the unit circle normal to r . In addition to (n() = 0 and equation (5.22), we require the result,... 

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