Small-step diffusion

Consider for a moment a rod-shaped particle of unit length. The orientation of the rod, u, can be specified by a unit vector u directed along its axis with spherical polar coordinates, D - id, random walk along the surface of the unit sphere. Debye [16] in 1929 developed a model for the reorientation process based on the assumption that collisions are so fiiequent that a particle can rotate throu only a very small angle before having another reorienting collision (i.e., small step diffusion). Debye began with the diffusion equation... 

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]. 

Debye (1929) developed a model for the reorientation processes based on the assumption that collisions are so frequent in a liquid that a molecule can only rotate through a very small angle before suffering a reorienting collision (small-step diffusion). We give here a heuristic treatment of the Debye model. 

In the limit of small-step diffusion where Xj J-diffusion reduces to Hubbard s relation (2.12). Thus in this limit the extended J-diffusion model agrees with the classical theory. This is not true for the M-diffusion model. In the other limit when Xj becomes large, as in a dilute gas, where many rotations occur between... 

However for the other cases in Table 2 it is possible that the high frequency process may be of a different character, since the local motions relax such a substantial part of . Warchol and Vaughan (73) proposed a model of small-step diffusion in a cone to account for limited motions in such supercooled systems. Their model has been extended by Wang and Pecora (74). The essential result of this model (73, 74) is that for motions of a dipole vector limited within a cone of cone-angle 0, where 0Q < 40 , we have... 

Note that the steady-state a.c. Kerr-effect is not simply related in the general case to the transient Kerr-effect in the time-domain (86-88), but exact relations can be given if the model for reorientation is specified. For example Benoit gave the necessary relations for the small-step diffusion model for axially-symmetric molecules (85). In this case Kjjj(t) = exp[-m(nH-l)Dj t] where Dr is the rotational diffusion coefficient (see also refs. 86,89). Thus for the case where the permanent dipole moment contribution to K greatly exceeds the "induced dipole moment contribution, the effective relaxation time for step-on response,... 

It was also shown (81) that Bk - 0.90 (near single relaxation time) at the highest temperatures so these results suggest that TnBP ion-pairs undergo small-step diffusion at higher temperatures. The fact that lower temperatures, and tend to... 

The correlation time tj is the angular momentum correlation time, the time the molecule changes its angular momentum, usually the time between collisions. The terms C and Cj are components of the spin-rotation interaction tensor and I is the moment of inertia of the molecule. For small step diffusion, and r are connected by the Hubbard relation ... 

Reorientation dynamics in liquids is described by either diffusion constants, D., or reorientational correlation times, since these two parameters are closely correlated. D. is the diffusion rate about a given molecular axis while is the time period required for the angular correlation function to decay to 1/e of its initial value [34,35]. For symmetric-top molecules, such as two diffusion constants, and D, are usually required to characterize the overall motion. and represent rotational diffusion about and of the top axis, respectively. The overall motion is now characterized by an effective reorientational correlation time, that, in the limit of small-step diffusion, is given by [36]... 

For CSA relaxation, 0 is the orientation of the CST tensor relative to the molecular symmetry axis. In principle it is possible to determine DZ and DX for a symmetric-top molecule provided xCeff and 0 values are known for different nuclei in the molecule. We employed the solution and solid-state experimental correlation times, along with the Gaussian generated 0 values, in (5) to simultaneously solve an equation for each carbon and obtained the best-fit values for DZ and DX at each temperature. While (5) assumes a small-step diffusion process, one can employ this equation to characterize reorientational motion outside this limit provided the calculated values are viewed as rough or base-line estimates of the diffusion process. 

Thus small-step diffusion gives a single relaxation time process with relaxation time x= 2Dr), ... 

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