Response to ‘‘Comment on ‘Long-time tails in angular momentum correlations’ ’’ [J. Chem. Phys. 104, 7363 (1996)

Abstract

~Received 29 December 1995; accepted 22 January 1996!@S0021-9606~96!02216-4#In their comment on our paper, Cichocki and Felderhofpoint out that our conclusion about the shape independenceof the long-time tail in the angular momentum correlationfunction must be incorrect because it contradicts a ‘‘rigor-ous’’ result that they have obtained previously. Moreover,they stress that we have incorrectly stated that their theoreti-cal results are limited to the case that the orientation of theparticle is fixed.To start with the second point: It is obvious from Ref. 1,that Cichocki and Felderhof do not consider the case of largeamplitude reorientational motion. In fact, they say as muchin the preamble to the derivation in Ref. 1 where they statethat they consider the case of ‘‘small amplitude motion.’’ Inour paper, we referred to this case as the limit that the ori-entation is ‘‘fixed.’’ Admittedly, it would have been moreaccurate to speak about an orientation that is ‘‘essentiallyfixed,’’ rather than literally fixed. We thought this to be un-necessarily verbose, because truly fixed orientations areunrealistic—in this limit the angular velocity autocorrelationfunction ~AVACF! does not exist.Let us then compare the two cases of real interest,namely the Cichocki–Felderhof limit of small amplitude an-gular motion and the limit of large amplitude reorientationthat we considered in our paper.Cichocki and Felderhof point out that for small ampli-tude motion, the AVACF does depend on the shape of theparticle, and we completely agree. More interestingly, wewere happy to notice that the theoretical predictions ofCichocki and Felderhof for this limit are in excellent agree-ment with the numerical results that for nonspherical objectsunder conditions where the angular displacement is negli-gible. For small amplitude motion, the shape and frequencydependent friction coefficient is proportional to the memoryfunction of the AVACF and gives a shape dependent tail.Thus, so far as the calculation of the friction coefficient isconcerned, the particle is fixed ~‘‘essentially fixed’’ in ourpresent terminology!. The calculation described in Ref. 1neglects the effect of the particle’s orientational motion onthe frequency dependent rotational friction coefficient. Oursimulations show quite convincingly, that for a particle thatis ‘‘essentially fixed,’’ a shape dependent decay of theAVACF is indeed observed. It is reassuring that our model,which uses a relatively crude lattice representation of a non-spherical object, is in essentially quantitative agreement withthe relevant theoretical predictions. This gives us great en-couragement that we can reliably extend our simulations tothe case of concentrated suspensions of non-spherical ob-jects.Next, let us consider the case discussed in Ref. 2, viz.the effect of reorientation on the dynamics of the particle.Reorientation is important because, on a sufficiently longtimescale, i.e., in the truly asymptotic regime, reorientation isalways important. In Ref. 2 we stressed that the theory con-tained in Ref. 1 assumes that reorientation is negligible andis therefore not appropriate ~let alone ‘‘rigorous’’! in thislimit. However, the microscopic theory contained in Ref. 4should apply in the limit of large-amplitude reorientation. Itpredicts that reorientation changes the decay of the AVACFin such a way that it becomes identical to the result for aspherical object with the same moment of inertia. A carefulmicroscopic analysis produces kinetic equations that, in theBrownian limit, reduce to the Navier–Stokes equations ~plusappropriate boundary conditions!. Not all the mode-couplingor kinetic theory treatments, which have been proposed, givethis limit correctly.