Isotropic models

The range of systems that have been studied by force field methods is extremely varied. Some force fields liave been developed to study just one atomic or molecular sp>ecies under a wider range of conditions. For example, the chlorine model of Rodger, Stone and TUdesley [Rodger et al 1988] can be used to study the solid, liquid and gaseous phases. This is an anisotropic site model, in which the interaction between a pair of sites on two molecules dep>ends not only upon the separation between the sites (as in an isotropic model such as the Lennard-Jones model) but also upon the orientation of the site-site vector with resp>ect to the bond vectors of the two molecules. The model includes an electrostatic component which contciins dipwle-dipole, dipole-quadrupole and quadrupole-quadrupole terms, and the van der Waals contribution is modelled using a Buckingham-like function. 

The probability that an improvement in the fit compared to the isotropic model could occur by chance is 2xl0-13. The rotational diffusion axis is oriented along the a-helix axis, and the angle between the two is about 20°. This plot also illustrates a clear separation between two sets of NH vectors, those belonging to the helix (hence oriented nearly parallel to the diffusion axis) and those in the /7-strands. Because of the architecture of the PH domain fold, the latter NH vectors are almost all orthogonal to the helix and hence to the diffusion axis. 

In the absence of accurate structural information, the analysis based on anisotropic diffusion as discussed above cannot be applied. The use of the isotropic overall model is still possible (see below) because it does not require any structural knowledge. However, the isotropic model has to be validated, i.e. the degree of the overall rotational anisotropy has to be determined prior to such an analysis. 

The isotropic model is justified when the estimated degree of the overall rotational anisotropy is small. A D /D l ratio of less than 1.1-1.2 could probably be considered as a reasonable value for the isotropic model, although an anisotropy as small as 1.17 can be reliably determined from 15N relaxation measurements, as demonstrated in Ref. [15]. 

In the isotropic model, the overall rotational diffusion is characterized by a single parameter, the overall correlation time zc. The following steps could be used to determine zc. 

The corresponding expression for coupling B of the isotropic model is not given as it was not used in the final analysis of the stress-optical data. 

Analysis of Stress—Optical Data. The slight, if indeed real, improvement of the isotropic model over the Takayanagi model would be of little consequence were it not for a more pronounced difference between the two models in their ability to describe the stress-optical data. When the parameters obtained from the dynamic data (Table IV) are substituted into Equations 8 and 9, Equation 8 produces results which are uniformly too low. Equation 9 also underestimates the magnitude of Ka but only by an average 7% (Figure 14). For most blends the discrepancy is less than 5%, and all calculated values show the characteristic elevation of the birefringence attributed to the multiphase structure. 

The values of the second-harmonic coefficient, d33, for samples 5-9 are listed in Table II. The values of d33 were obtained using the analysis of Jerphagnon and Kurtz (25), and were calculated under the assumption that the degree of alignment of the nonlinear optical chromophores can be described using the isotropic model. Hence, we assumed d33=3d31 (4). 

With this method, Thole fitted isotropic model polarizabilities for H, C, N and O to 16 experimental molecular polarizabilities, using a single screening length and only one polarizability for each atom type, regardless of the chemical environment. 

Obviously, such an isotropic model does not apply longer to Ba2CaCoFe2Fi4, where the presence of Co2+ ions in octahedral sites causes the susceptibility tensor to remain anisotropic even above TN, as it may be seen on the single crystal susceptibility measurements given in Fig. 24 [26],... 

Parameters and Ujj, like the isotropic temperature factor, are expressed in A, while are dimensionless. The accepted standard now is to report U for the isotropic model (Uiso) and Uij for the anisotropic one. ... 

The atomic environment within the crystal is usually far from isotropic, and the next simplest model of atomic motion (after the isotropic model just described) is one in which the atomic motion is represented by the axes of an ellipsoid this means that the displacements have to be described by six parameters (three to define the lengths of three mutually perpendicular axes describing the displacements in these directions, and three to define the orientation of these ellipsoidal axes relative to the crystal axes), rather than just one parameter, as in the isotropic case. Atomic displacement parameters, and their relationship to thermal vibrations and spatial disorder in crystals are covered in more detail in Chapter 13. 

Surface waves also provide a means for estimating upper-mantle anisotropy. There are three canonical ways of detecting anisotropy using surface waves. One comes from discrepancies in isotropic inversions of Love and Rayleigh phase velocities (e.g. Anderson 1961) and leads to transversely isotropic models or polar anisotropy. Another comes from azimuthal variations in phase velocities (e.g. Forsyth 1975) and leads to models of azimuthal or radial anisotropy. 

In a recent contribution [87] we studied the catalytic effect in polyelectrolyte-electrolyte mixtures by various theoretical techniques. For an isotropic model where the macroions, co-ions and counterions are pictured as charged hard spheres, we employed the HNC approximation, the modified PB and symmetric PB theories. The results for k/k° were compared with the computer simulations for the same quantity. Note that this quantity is much more sensitive to the details of the model and theory than thermodynamic properties like osmotic pressure studied before. The conclusion was that these theories are not well-suited to treat the problem they were capable of reproducing MC values only qualitatively and even this merely for low-charged macroions. 

Equation (1.380) is based on the assumption that all the normal stresses are equal thus representing an isotropic model for the Reynolds stresses. 

At 335.7K, the FAD curve shown in Figure 3 is particularly suitable for comparison with various molecular models of local chain dynamics. Indeed, at this temperature the decay is not too fast, in such a way that an accurate comparison with the models can be made over the full time window available, but nevertheless the FAD curve goes close to zero. It appears that the isotropic model (single exponential for the OACF) does not fit the data and that OACF expressions specifically proposed for polymer dynamics (for a review of them, see Ref. 3 and ) are required. The specific behavior of polymer chains is due to the chain connectivity requirement and the description of local dynamics requires consideration of the following features ... 

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