Branching density theory

The moduli, measured at crosslinking temperature T, which are given in the last two columns of Table IV, are abou two to three fold greater than those computed from phantom theory. Except for the samples with the lowest branching densities, the observed values agree satisfactorily with those for an affine network. 

The remaining question is, how the deviations from phantom network theory at high branching densities can be explained. 

Figure 7. Ratio of experimentally observed and theoretically calculated modulus, using phantom network theory with f2 and v2, versus branching density z-...

Huang, Y.-L. and Brown, N. (1991) Dependence of slow crack growth in polyethylene on butyl branch density - morphology and theory. J. Polym. Sci. Polym. Phys. Ed., 29,129. 

The available methods in molecular electronic structure theory are illustrated in Figure 1 with a family tree of quantum chemistry labeled with the acronyms of some of the most often used methods. The variety is a bit daunting to newcomers, who might be cautioned by a comment by Levine If you learn enough abbreviations you can convince some people that you know quantum chemistry. Flowever, as for most areas of science, electronic structure theory looks much worse from the outside than from the inside. The tree has three main branches density functional theory (DFT), quantum Monte Carlo (QMC), and Rayleigh-Ritz variational theory (RRV). Each of these leads to additional branches. In addition there are a number of interbranch connections indicated by dotted lines. We give a brief description below of the DFT and RRV branches and their relation to QMC, which is described in sections follow-... 

Huang, Y.L., Brown, N. Dependence of slow crack growth in polyethylene on butyl branch density morphology and theory. J. Polym. Sd. Part B Polym. Phys. 29,129-137 (1991) Humbert, S., Lame, O., Chenal, J.-M., Rochas, C., Vigier, G. Small stiain behavior of polyethylene in situ SAXS measurements. J. Polym. Sd. Part B Polym. Phys. 48, 1535-1542... 

Storer model used in this theory enables us to describe classically the spectral collapse of the Q-branch for any strength of collisions. The theory generates the canonical relation between the width of the Raman spectrum and the rate of rotational relaxation measured by NMR or acoustic methods. At medium pressures the impact theory overlaps with the non-model perturbation theory which extends the relation to the region where the binary approximation is invalid. The employment of this relation has become a routine procedure which puts in order numerous experimental data from different methods. At low densities it permits us to estimate, roughly, the strength of collisions. 

The quantum theory of spectral collapse presented in Chapter 4 aims at even lower gas densities where the Stark or Zeeman multiplets of atomic spectra as well as the rotational structure of all the branches of absorption or Raman spectra are well resolved. The evolution of basic ideas of line broadening and interference (spectral exchange) is reviewed. Adiabatic and non-adiabatic spectral broadening are described in the frame of binary non-Markovian theory and compared with the impact approximation. The conditions for spectral collapse and subsequent narrowing of the spectra are analysed for the simplest examples, which model typical situations in atomic and molecular spectroscopy. Special attention is paid to collapse of the isotropic Raman spectrum. Quantum theory, based on first principles, attempts to predict the. /-dependence of the widths of the rotational component as well as the envelope of the unresolved and then collapsed spectrum (Fig. 0.4). 

It should be noted that there is a considerable difference between rotational structure narrowing caused by pressure and that caused by motional averaging of an adiabatically broadened spectrum [158, 159]. In the limiting case of fast motion, both of them are described by perturbation theory, thus, both widths in Eq. (3.16) and Eq (3.17) are expressed as a product of the frequency dispersion and the correlation time. However, the dispersion of the rotational structure (3.7) defined by intramolecular interaction is independent of the medium density, while the dispersion of the vibrational frequency shift (5 12) in (3.21) is linear in gas density. In principle, correlation times of the frequency modulation are also different. In the first case, it is the free rotation time te that is reduced as the medium density increases, and in the second case, it is the time of collision tc p/ v) that remains unchanged. As the density increases, the rotational contribution to the width decreases due to the reduction of t , while the vibrational contribution increases due to the dispersion growth. In nitrogen, they are of comparable magnitude after the initial (static) spectrum has become ten times narrower. At 77 K the rotational relaxation contribution is no less than 20% of the observed Q-branch width. If the rest of the contribution is entirely determined by... 

As can be seen from the above, the shape of the resolved rotational structure is well described when the parameters of the fitting law were chosen from the best fit to experiment. The values of estimated from the rotational width of the collapsed Q-branch qZE. Therefore the models giving the same high-density limits. One may hope to discriminate between them only in the intermediate range of densities where the spectrum is unresolved but has not yet collapsed. The spectral shape in this range may be calculated only numerically from Eq. (4.86) with impact operator Tj, linear in n. Of course, it implies that binary theory is still valid and that vibrational dephasing is not yet... 

One possibility for this was demonstrated in Chapter 3. If impact theory is still valid in a moderately dense fluid where non-model stochastic perturbation theory has been already found applicable, then evidently the continuation of the theory to liquid densities is justified. This simplest opportunity of unified description of nitrogen isotropic Q-branch from rarefied gas to liquid is validated due to the small enough frequency scale of rotation-vibration interaction. The frequency scales corresponding to IR and anisotropic Raman spectra are much larger. So the common applicability region for perturbation and impact theories hardly exists. The analysis of numerous experimental data proves that in simple (non-associated) systems there are three different scenarios of linear rotator spectral transformation. The IR spectrum in rarefied gas is a P-R doublet with either resolved or unresolved rotational structure. In the process of condensation the following may happen. 

Let us consider the quasi-classical formulation of impact theory. A rotational spectrum of ifth order at every value of co is a sum of spectral densities at a given frequency of all J-components of all branches... 

Only the structures of di- and trisulfane have been determined experimentally. For a number of other sulfanes structural information is available from theoretical calculations using either density functional theory or ab initio molecular orbital theory. In all cases the unbranched chain has been confirmed as the most stable structure but these chains can exist as different ro-tamers and, in some cases, as enantiomers. However, by theoretical methods information about the structures and stabilities of additional isomeric sul-fane molecules with branched sulfur chains and cluster-like structures was obtained which were identified as local minima on the potential energy hypersurface (see later). 

By ab initio MO and density functional theoretical (DPT) calculations it has been shown that the branched isomers of the sulfanes are local minima on the particular potential energy hypersurface. In the case of disulfane the thiosulfoxide isomer H2S=S of Cg symmetry is by 138 kj mol less stable than the chain-like molecule of C2 symmetry at the QCISD(T)/6-31+G // MP2/6-31G level of theory at 0 K [49]. At the MP2/6-311G //MP2/6-3110 level the energy difference is 143 kJ mol" and the activation energy for the isomerization is 210 kJ mol at 0 K [50]. Somewhat smaller values (117/195 kJ mor ) have been calculated with the more elaborate CCSD(T)/ ANO-L method [50]. The high barrier of ca. 80 kJ mol" for the isomerization of the pyramidal H2S=S back to the screw-like disulfane structure means that the thiosulfoxide, once it has been formed, will not decompose in an unimolecular reaction at low temperature, e.g., in a matrix-isolation experiment. The transition state structure is characterized by a hydrogen atom bridging the two sulfur atoms. 

The description of fuzzy, local density fragments is facilitated by the use of local coordinate systems, however, some compatibility conditions of such local coordinate systems must be fulfilled, reflecting the mutual relations of the fragments within the complete molecule. Manifold theory, topological manifolds, and in particular, differentiable manifolds [153-158], are the branches of mathematics dealing with the general properties of compatible local coordinate systems. 

Freed et al. [42,43], among others [44,45] have performed RG perturbation calculations of conformational properties of star chains. The results are mainly valid for low functionality stars. A general conclusion of these calculations is that the EV dependence of the mean size can be expressed as the contribution of two terms. One of them contains much of the chain length dependence but does not depend on the polymer architecture. The other term changes with different architectures but varies weakly with EV. Kosmas et al. [5] have also performed similar perturbation calculations for combs with branching points of different functionalities (that they denoted as brushes). Ohno and Binder [46] also employed RG calculations to evaluate the form of the bead density and center-to-end distance distribution of stars in the bulk and adsorbed in a surface. These calculations are consistent with their scaling theory [27]. 

A limiting case is constituted by many branches grafted to an inflexible line. The scahng theory is pertinent to this case. Then the density of segments is... 

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