Angular velocity correlation function

Fig. 1.15. Translational and angular velocity correlation functions for nitrogen. MD simulation data from [70], T = 122 K, densities are indicated in the figure. Reduced units for time t = (e/cr2), for density p" = p

Anderson J. E., Ullman R. Angular velocity correlation functions and high-frequency dielectric relaxation, J. Chem. Phys. 55, 4406-14, (1971). 

Kluk E., Monkos K., Pasterny K., Zerda T. A means to obtain angular velocity correlation functions from angular position correlation functions of molecules in liquid. Part I. General discussion and its application to linear and spherical top molecules, Acta Physica Polonica A 56, 109-16 (1979). 

C. Excited Initial Distribution and Corresponding Angular Velocity Correlation Function... 

Figure 1 Structural (left column) and dynamical (right column) properties of the systems investigated. Upper left centre-of-mass radial pair distribution function gooo( ) lower left spherical harmonic expansion coefficient g2oo(r) upper right angular velocity correlation function lower right orientational correlation function. Dotted lines CO, 80 K, 1 bar thin lines CS2, 293 K, 1 bar thick lines CS2, 293 K, 10 kbar.

The twist viscosity is consequently inversely proportional to the TCFI of the director angular velocity correlation function. The physical interpretation of this relation is, that the twist viscosity is low when there are large fluctuations in the director orientation. This is usually the case when the order parameter is low. When the order parameter increases it becomes harder for the director to reorient, so that the twist viscosity increases. 

Equation (239) may also be used to calculate the angular velocity correlation function (AVCF) in the fractional dynamics. From Eq. (239) with n = 1 and q= 0, we have... 

Figure 25. Normalized angular velocity correlation function cm(t)/cm(0) for y = 1 and various values of a 0.5 (curve 1), 1.0 (curve 2), and 1.5 (curve 3).

Moreover, we have from Eq. (342) the equilibrium velocity correlation function cv(t) [cf. Eq. (264) for the angular velocity correlation function]... 

The complex rotational behavior of interacting molecules in the liquid state has been studied by a number of authors using MD methods. In particular we consider here the work of Lynden-Bell and co-workers [60-62] on the reorientational relaxation of tetrahedral molecules [60] and cylindrical top molecules [61]. In [60], both rotational and angular velocity correlation functions were computed for a system of 32 molecules of CX (i.e., tetrahedral objects resembling substituted methanes, like CBt4 or C(CH3)4) subjected to periodic boundary conditions and interacting via a simple Lennard-Jones potential, at different temperatures. They observe substantial departures of both Gj 2O) and Gj(() from predictions based on simple theoretical models, such as small-step diffusion or 7-diffusion [58]. Although we have not attempted to quantitatively reproduce their results with our mesoscopic models, we have found a close resemblance to our 2BK-SRLS calculations. Compare for instance our Fig. 13 with their Fig. 1 in [60]. 

It describes the evolution in time of rotational velocities. It also very closely approaches the angular velocities correlation function C j and the memory function Kj(t), or generalized friction coefficient of the diffusive rotational molecular motion, a property constituting our main concern in section IV.3 below. 

As for their respective definitions, the angular velocity correlation function is... 

For t = o this function is identical to the ordinary second moment. At larger times M2(t), a cross orientation-angular velocity correlation function, resembles the angular velocity correlation function C (t) for systems close to the diffusion limit 46-48. Contrary to the memory functions discussed above, M2(t) has the... 

Pig 14 Cross orientation-angular velocity correlation function for CH,J at various pressures 31 ... 

The same result is obtained for linear molecules. Equation (26) suggests that a useful way to compare correlation functions with different L values is to plot[ilnC (t)J/L(L + 1) versus t. More importantly it shows that the angular velocity correlation function determines the lowest order contribution to CL(t). Before we look... 

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