Stochastic Orientational Relaxation of a Plastic Crystal † Jun-ichi Koga* and Takashi Odagaki Department of Physics, Kyushu UniVersity, Fukuoka 812-8581, Japan ReceiVed: September 13, 1999; In Final Form: February 1, 2000 Dielectric properties of cyanoadamantane in the plastic phase are studied using a stochastic model for orientational relaxation. The jump rate distribution is assumed to be a power law function, and the master equation is solved within the coherent medium approximation. The orientational relaxation function is shown to be fitted well by a stretched exponential function. The exponent of the power law distribution function is related to temperature by comparing the Cole-Cole plot of the complex dielectric constant with experiments. The glass transition temperature is shown to correspond to the Vogel-Fulcher temperature where the relaxation time diverges. 1. Introduction Orientationally disordered crystals or plastic crystals consist of nonspherical (but close-to-spherical) molecules, in which the center of mass of each molecule forms a translational order and the orientation of each molecule changes randomly. Because of this characteristic structure, plastic crystals show plasticity as their name implies. When plastic crystals are cooled, the rotaional motion freezes and an orientational order appears at a certain temperature in most cases. However, under careful cooling, the rotational motion freezes keeping random orienta- tion of molecules. This state of plastic crystals, first recognized by Suga and Seki, 1,2 is now known as the orientaional glass. In the orientational glass forming process, plastic crystals show a dynamic and thermodynamic singularity at a certain temperature called the glass transition temperature. For example, cyanoadamantane (CNADM) exhibits the transition at T g ) 170 K. 3 The nature of the transition has been attracting wide interests because of its relation to the structural glass transition. 4 Although the singularities observed for the orientaional glass transition are similar to those known for the structural glass transition, it is important to find out what is the correspondence between these two kinds of glass transitions. In this paper, we study the orientaional glass transition of CNADM using a stochastic model for the tumbling motion of molecules. We exploit the master equation with a random jump rate to describe the stochastic motion and calculate the frequency dependent dielectric constant. By comparing the Cole-Cole plot with experiments, we deduce the temperature dependence of a model parameter and suggest that the stochastic motion freezes at the glass transition point. In section 2, we introduce the stochastic model for the tumbling motion and present the basic formalism to obtain the dielectric constant. We explain the coherent medium ap- proximation which is utilized to obtain the ensemble average of the dielectric constant. Numerical results are presented in section 5 and comparison with experiments is given in section 6. We discuss the results in section 7. 2. Stochastic Model It is known that the crystal structure of the plastic phase of CNADM is an fcc and the orientation of each molecule (direction of the dipole moment) is restricted to six cubic axes, and molecules perform tumbling motion among these six axes. 5-7 Because of the steric hindrance between neighboring molecules, the tumbling motion of a molecule must be highly correlated to the orientation of other molecules. Despite this complex mechanism of tumbling motion, however, one can still focus on the motion of a particular molecule and consider the one-body problem of the stochastic dynamics among the six orientations. We denote directions (0,0,1), (1,0,0), (0,1,0), (-1,0,0), (0,-1,0), and (0,0,-1) in a cubic lattice by 1, 2, 3, 4, 5, and 6, respectively, and the probability that the dipole moment is in the ith direction at time t by p i (t) when it was in direction 1 at time t ) 0. We employ a master equation which determines the time evolution of the probability vector p(t) ) (p 1 (t), p 2 (t), ..., p 6 (t)) Here, W is a 6 × 6 matrix whose component w ij (i * j) denotes the transition rate from the ith to the jth direction, and w ii ) -∑ j*i w ij which gurantees the conservation of probability. We assume that the direct transitions from one direction to the opposite direction are not allowed, and thus the jump rate matrix takes the form In the present one-body approach, the jump rates {w ij } are assumed to be random quantities whose distribution is deter- mined by the many-body effects. † Part of the special issue “Harvey Scher Festschrift”. * Present address: NTT Network Service Systems Laboratories. d dt p(t) ) p(t)W (1) W ) [ w 11 w 12 w 13 w 14 w 15 0 w 21 w 22 w 23 0 w 25 w 26 w 31 w 32 w 33 w 34 0 w 36 w 41 0 w 43 w 44 w 45 w 46 w 51 w 52 0 w 54 w 55 w 56 0 w 62 w 63 w 64 w 65 w 66 ] (2) 3808 J. Phys. Chem. B 2000, 104, 3808-3811 10.1021/jp993247v CCC: $19.00 © 2000 American Chemical Society Published on Web 03/30/2000