Wavenumber coupling theory

Note that the above study is performed for a simple system. There exists a large body of literature on the study of diffusion in complex quasi-two-dimensional systems—for example, a collodial suspension. In these systems the diffusion can have a finite value even at long time. Schofield, Marcus, and Rice [17] have recently carried out a mode coupling theory analysis of a quasi-two-dimensional colloids. In this work, equations for the dynamics of the memory functions were derived and solved self-consistently. An important aspect of this work is a detailed calculation of wavenumber- and frequency-dependent viscosity. It was found that the functional form of the dynamics of the suspension is determined principally by the binary collisions, although the mode coupling part has significant effect on the longtime diffusion. 

Gottke et al. [5] offered a theoretical treatment of collective motions of mesogens in the isotropic phase at short to intermediate time scales within the framework of the Mode coupling theory (MCT). The wavenumber-dependent collective orientational time correlation function C/m(, t) is defined as... 

A little work seems to have been carried out on the wavenumber-dependent orientational correlation functions C/m(, t). These correlation functions can provide valuable insight into the details of microscopic dynamics of the system. A molecular level understanding of C/m(, t) would first require the development of a molecular hydrodynamic theory that would have coupling between C/m(, t) and the dynamic structure factor S(k, t) of the liquid. A slowdown in C/m(, t) may drive a slowdown in the dynamic structure factor. This would then give rise to a two-order parameter theory of the type develops by Sjogren in the context of the glass transition and applied to liquid crystals by Li et al. [91]. However, a detailed microscopic derivation of the hydrodynamic equations and their manifestations have not been addressed yet. 

Section 3 is a theoretical discussion that covers the free-ion Hamiltonian, crystal-field theory, and intensities. We discuss the effect of interelectronic Coulomb interactions and the spin-orbit interaction in determining the structure of the free-ion levels. Several additional interactions that are weaker than the above shift the free-ion levels by a few hundred wavenumbers and are necessary to give good agreement between calculated free-ion levels and experiment. These include spin-independent configuration interaction by the Coulomb interaction, spin-dependent configuration interaction by the spin-orbit interaction, and relativistic interactions that couple orbital and spin angular momenta of different electrons. 

To compare spectroscopic data relative to different push-pull polyenes and centrosymmetric polyenes it is useful to select the normal mode with the largest fl content, which corresponds to the v mode in apolar polyenes, and to apply ECC theory considering both frequency and intensity. Also for push-pull polyenes it is often possible to find another normal mode, with large fl character, in the wavenumber range 1000-1200 cm, which corresponds to 1 3 of polyenes (Tables 28.1 and 28.2). However, 1 3 in many cases is heavily affected by coupling with the vibrations of the end or side groups. For this reason, comparison between different molecules is made simpler by focusing only on I l. 

Figure 2.24 shows the temporal evolution of the compositional structure factor S<(, (a) and the orientational structure factor Ss (b) for the temperature quench into the lu region (T/Tn, = 0.6, (f)Q = 0.55) in Figure 2.22. The structure factor for concentration has a maximum at q, which corresponds to the peak wavenumber of coi(q). With time the corresponding mode grows exponentially and the peak position qm is invariant. Then the time evolution of the structure faaor S<, is lhe same as that of the Cahn-Hilliard theory for isotropic SD [102]. The amplitude of the peak at q = 0 decreases with time because s > 0 and another peak appears at q. In this quench, the concentration fluctuation initially induces the SD and the orientational ordering within the domain subsequently takes place due to the coupling between the two order parameters concentration-induced SD.

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