Orientational time correlation function

Figure 4. Water dipole (a) and the water H-H vector (b) orientational time correlation functions. In both panels, the dotted line is for the first layer, the solid thick line is for bulk water, the dashed line is for the second layer, and the thin solid line is for the third layer from the Pt( 100) surface (T = 300 K).

The mode coupling theory of molecular liquids could be a rich area of research because there are a large number of experimental results that are still unexplained. For example, there is still no fully self-consistent theory of orientational relaxation in dense dipolar liquids. Preliminary work in this area indicated that the long-time dynamics of the orientational time correlation functions can show highly non-exponential dynamics as a result of strong in-termolecular correlations [189, 190]. The formulation of this problem, however, poses formidable difficulties. First, we need to derive an expression for the wavevector-dependent orientational correlation functions C >m(k, t), which are defined as... 

The Second ingredient is the expression of the rotational friction in terms of the orientational time correlation functions. We have earlier derived an expression for this which was based on Kirkwood s formula [190]. The full expression should be derived by following an approach similar to that of Sjogren and Sjolander [9]. In addition, the coupling to rotational currents (the vortices) have not been touched upon. 

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... 

Despite extensive investigation of phase behavior of liquid crystals in computer simulation studies [97-99], the literature on computational studies of their dynamics is somewhat limited. The focal point of the latter studies has often been the single-particle and collective orientational correlation functions. The Zth rank single-particle orientational time correlation function (OTCF) is defined by... 

Figure 9. Orientational relaxation in the model liquid crystalline system GB(3, 5, 2, 1) (N = 576) at several densities across the I-N transition along the isotherm at T = 1. (a) Time dependence of the single-particle second-rank orientational time correlation function in a log-log plot. From left to right, the density increases from p = 0.285 to p = 0.315 in steps of 0.005. The continuous line is a fit to the power law regime, (b) Time dependence of the OKE signal, measured by the negative of the time derivative of the collective second-rank orientational time correlation function C t) in a log-log plot at two densities. The dashed line corresponds to p = 0.31 and the continuous line to p = 0.315. (Reproduced from Ref. 112.)...

Figure 12. Orientational dynamics of the discotic system GBDII (N = 500) at several temperatures across the isotropic-nematic transition along the isobar at pressure P — 25. (a) Time evolution of the single-particle second-rank orientational time correlation function in a log-log plot. Temperature decreases from left to right, (b) Time dependence the OKE signal at short-to-intermediate times in a log-log plot. Temperature decreases from top to bottom on the left side of the plot T = 2.991,2.693,2.646, and 2.594. The dashed lines are the simulation data and the continuous lines are the linear fits to the data, showing the power law decay regimes at temperatures. (Reproduced from Ref. 115.)...

The average (over the water molecules in the layer) orientational time correlation function also shows markedly non-exponential decay with a 3-7-times slower time constant than that of the bulk for lysozyme [2]. Simulation studies show that the slow molecules belong to those water molecules that had longer residence times near the lysozyme. Thus, residence time correlates with the rotational correlation time. 

The basic problem associated with elucidating a relaxation mechanism from experimental data is most clearly seen in terms of the time-dependent conditional orientational distribution function p(0,t/l2Q,0). The quantity p(0,t/0Q,0)d0d0Q is the probability of obtaining the orientation of the body (molecule, chain segment) in the element of solid angle dO around Q at time t given its orientation was in dOo around JIq at t = 0. We may expand the distribution function in terms of the elements Dj jj(J2) of Wigner rotation matrices and the orientational time-correlation functions... 

The experimentalist does not ordinarily measure a conformational state time correlation function. Usually a technique is sensitive to some first or second order tensor, an orientational quantity such as a dipole moment along or perpendicular to the chain, the direction of a C-H bond, a polarizability, or a transition moment of a chromophore. It is even more difficult to analyze these objects orientational time correlation functions. Furthermore they may exhibit both rapid relaxation due to conformational transitions and slow relaxation due to coupling to long wavelength modes. We have suggested fitting the short time part to the simplest function which accounts for single and pair transitions, ... 

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