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J . Phys. Chem. 1989, 93, 7026-7031
and use eq 17 to find, asymptotically at long times, that
The t 3 , result corresponds to walks with the mean time ( t ) spent in traversing a path being infinite. This case is known as Richardson's law of turbulent diffusion. The second case has ( t ) finite, but ( t Z )infinite, and the last case with the Brownian motion result occurs when ( t 2 ) is finite. This last case shows that Kolmogorov scaling ( z = 1/3) in itself is not sufficient to product turbulent diffusion. The condition @ I1 / 3 is also necessary. V. Conclusions We have generalized the Montroll-Weiss continuous-time random walk to include multistage processes. Our analysis of the random walk equations focused on the asymptotic behavior of an
unbiased walker. We investigated how, when probability distributions without characteristic scales are introduced, one can avoid the Gaussian limit. We studied cases where the walk was mostly waiting at a trapping site and where the walk was mostly transporting. The more detailed description provided by the multistage random walk will be important for first passage time calculations which are needed for a theory of reactions. The CTRW would have the walker wait at a site until it is time for an instantaneous transition to the next site, while in practice the walk could undergo a reaction on its way to the new site. The study of reactions in multiwalker systems and systems with biases fall easily into the multistage method which we have presented and will be the subject of further study. Acknowledgment. The authors thank Dr. Eric Kunhardt for suggesting the multistage random walk problem in regard to the motion of excess electrons in liquids.
Longitudinal Dielectric Relaxation Daniel Kivelson* Department of Chemistry and Biochemistry. University of California, Los Angeles, Los Angeles, California 90024
and Harold Friedman Department of Chemistry, State University of New York, Stony Brook, New York 11794 (Received: April 28, 1989; In Final Form: June IO, 1989)
Because of some apparent misunderstanding in the literature of the significanceof longitudinal dielectric relaxation, we discuss this property. We emphasize that because the ratio of the longitudinal ( T L ) to the transverse ( T ~correlation ) time is e(m)/e(O), where ~ ( 0 is ) a thermodynamic property of the system and c(m) an electronic property of the molecules, it follows that, for any dielectric material, both correlation times must contain the exact same dynamical information. Nevertheless, for nonexponential decay very rapid molecular motions will be easier to detect in longitudinal than in transverse relaxation; we show why this is so, and why oscillatory motions such as librations (torsional oscillations connected with Poley absorption) and dipolarons (inertial oscillations) are detectable in longitudinal relaxation. We relate the collective, low k, or continuum dielectric relaxation times to molecular reorientation times, but many interesting molecular phenomena depend heavily on high k properties.
1. Introduction
To chemists, dielectric relaxation in liquids is of interest to the extent that it yields information concerning the dynamical behavior of individual molecules. The principal use of dielectric phenomena has been directed toward studies of molecular rotation, but there has also been much interest of late in understanding the effect of the fluctuating dielectric environment upon the course of chemical reactions. The treatment of dielectric relaxation and its relationship to molecular phenomena is a subtle subject, one that has been fraught with uncertainty, error, and controversy. In this article we comment on one aspect of the problem, the interrelationship of the transverse or Debye (TD) and the longitudinal ( T ~ dielectric ) relaxation times, both of which are macroscopic or collective quantities, and the molecular quantities to which they are connected. That at least these two relaxation times are relevant to the theory of dielectric relaxation in isotropic, nonionic liquids has been indicated by numerous recent studies.'-9a The transverse (1) (2) (3) (4)
Berne, B. J. J. Chem. Phys. 1975, 62, 1154. Hubbard, J. B.; Onsager, L. J. Chem. Phys. 1977, 67, 4850. Hubbard, J. B. J. Chem. Phys. 1978,68, 1649. Fulton, R. L. J. Chem. Phys. 1975,63, 77; Mol. Phys. 1975,29,405.
0022-3654/89/2093-7026$01.50/0
time TD can be associated with the relaxation of the displacement vector D after a jump in the electric field E, while the longitudinal time T L is associated with the relaxation of E after a jump in D.Z6*'O Alternatively, rD can be associated with [ ~ ( u-)e(-)], where ~ ( o ) is the dielectric permittivity, while T~ can be associated with the functional [c(m)-' To formulate the problem on the molecular level, we use the fact that D and E are vector fields which are the transverse and longitudinal parts, respectively, of the polarization P. Here longitudinal and transverse means parallel or perpendicular to the wave vector k, respectively. We are interested in the source of the anisotropy which makes TD differ from T L in isotropic media, and how, in molecular fluids, these two times are connected to the molecular motions. e(u)-1].499ab
(5) Frohlich, H. Theory of Dielectrics, 2nd ed.; Oxford University Press: London, 1958. (6) Friedman, H. L. J. Chem. SOC.,Faraday Trans. 2 1983, 79, 1465. (7) Pollack, E. L.; Alder, B. J. Phys. Reu. Lett. 1981, 46, 950. (8) Impey, R. W.; Madden, P. A,; MacDonald, I. R.Mol. Phys. 1982,46, 513. (9) Madden, P.; Kivelson, D. Adu. Chem. Phys. 1984, LVI, 467-566; (a) p 485-490; (b) p 522; (c) pp 490-501; (d) pp 473, 478; (e) pp 520-524; (f) 523; (9) pp 524-538; (h) pp 479-485; 6 ) pp 490-493. (10) Friedman, H. L.; Newton, M. D. Faraday Discuss. Chem. SOC.1983, 74, 73.
0 1989 American Chemical Society
The Journal of Physical Chemistry, Vol. 93, No. 19. 1989 1021
Longitudinal Dielectric Relaxation In media with high dielectric constants c(O), one finds T~ 0, and (IMx(k,O)lz)< N ( p ; p i ) . Furthermore we expect ( p i p i ) = ( p i p : ) = (p1-pI)/3, so that
We conclude that (M,(k,O) M,(-k,O)) is small because of "cancellation effects" arising from antiparallel dipoles, and this suggests that the net M,(k) is small. This is not true of Mx(k). The fact that M,(k) is small because of cancellation effects indicates that (M,(k,t)M,(-k,O)) should be very sensitive to small fluctuations in dipolar orientations; i.e., very small fluctuations can upset the balanced orientational distributions which lead to cancellation. Said differently, rotations of, say, only 2 O or 3 O , which take place in a time very short compared to T,, do little to reorient an individual molecule but can change the sign of the nearly cancelled collective dipole, M,(k,t), thereby rotating it by 1 8 0 O ; the rotational time for individual molecules is T~ and that , we see that T~
