3918
J. Phys. Chem. 1995, 99, 3918-3923
Molecular Motions in Plastically Crystalline Clusters of Tetragonal tert-Butyl Chloride. A Molecular Dynamics Study Jian Chen and Lawrence S. Bartell" Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109 Received: June 4, 1994; In Final Form: November 14, 1994@
The interior of a 188-molecule cluster of tert-butyl chloride was examined in detail by molecular dynamics computations covering the range of temperature over which the cluster's orientationally disordered phase IV is stable. The aim was to characterize the molecular behavior in a bulklike region in order to interpret the kinetics of the spontaneous transformation to the ordered monoclinic phase occurring at colder temperatures. Of particular concern were the rotational motions about molecular 3-fold axes dictating molecular jump rates in the transition. The activation energy for rotational diffusion, mean librational frequencies, coefficients of rotational diffusion, and relaxation times were derived from the simulations by analyzing results in terms of a model based on a single rotational force constant and coefficient of friction. Prior experimental evidence about these quantities obtained by NMR, cold neutron diffraction, and dielectric techniques had been somewhat contradictory. A reasonable agreement was obtained between computational results and a subset of the experimental results.
Introduction The structure of plastic crystals and the molecular motions responsible for their distinctive properties have attracted attention for several decades.' Many physical methods have been applied to help characterize these rather soft crystals, which are also known as orientationally disordered solids by virtue of the disarray in their molecular alignments. Perhaps the most intimate views of the behavior of molecules in such systems obtainable by current techniques are those provided by molecular dynamics (MD) simulations carried out with realistic intermolecular interaction functions. Pawley and co-workers, in particular, have published a number of illuminating conclusions based on MD studies of plastically crystalline substance^.^-^ The aim of the present paper is to examine what can be learned about molecular motions in the tetragonal phaselo." of terfbutyl chloride from an MD investigation. Our attention was drawn to this orientationally disordered phase I11 of the material when we found it possible to monitor the kinetics of the solidstate phase change between phase I11 and phase IV (ordered monoclinic) in clusters undergoing evaporative cooling in supersonic flow.'? From the extraordinarily fast nucleation rate we observed by electron diffraction, we projected that it might be possible to see the same transition at greater undercoolings on the picosecond time scale of MD simulations. This proved to be true, and the MD method not only yielded information about the nucleation kinetics but also revealed the hitherto unknown structure of phase IV." However, inconsistencies arose when it was attempted to derive the interfacial free energy of the boundary between the tetragonal and monoclinic phases by applying the capillary theory of nucleation. The trouble was associated with extrapolations to low temperatures of frequencies of rotational jumps of molecules derived for phase I11 from lowresolution inelastic neutron diffraction studies.14 Therefore, it appeared worthwhile to find how molecules move in the tetragonal phase in our MD runs to make the analysis of interfacial free energy self-consistent and, perhaps, to clarify some of the features of molecular motions left unresolved by a
Abstract published in Adi"tce ACS Ahrrrucrs, March 1, 1995.
0022-365419512099-3918$09.00/0
the experimental investigations. Our inferences about the molecular motions in the tetragonal phase are presented in the following.
Properties of Crystalline tert-Butyl Chloride tert-Butyl chloride freezes at 250 K into phase I, an orientationally disordered face centered cubic (fcc) structure. I I On cooling, it transforms at 219.5 K to a phase III5 of unknown structure that is stable only over a 2 deg temperature range before the tetragonal phase I11 is reached. Measurements of the dielectric c ~ n s t a n t ' ~ -revealed l~ that the freedom of molecules to rotate randomly in all directions in phase I disappears in phase 111. On the other hand, diffraction studies showed that it is only the directions of dipole moments which become ordered in phase III.Io Although the C-Cl bond directions attain a regular alignment, the orientations of the tertbutyl groups about their 3-fold axes remain disordered, consistent with the 4-fold site symmetry of molecular positions in space group P4/nmm implied by the diffraction intensities. When cooled below 183 K, the substance transforms into a monoclinic phaseI3 similar in structure to the tetragonal phase in gross aspect but with ordered orientations of the tert-butyl groups. Rotational motions of molecules in the cubic and tetragonal phases have been the focus of investigations by a variety of methods, including NMR,'5.20-27X-ray diffraction,I0.'I incoherent quasielastic neutron scattering, 14.28.29 infrared specand calorimetry.16.' 8.3 I .32 troscopy,'8.30 dielectric measurement,) These studies established the qualitative validity of the above description but posed certain contradictions and left unanswered questions about librational and diffusional motions of molecules in phase 111.
Procedure Molecular Dynamics Simulations. Before describing the thermal history imposed upon the cluster, a few comments should be made about the size of the system chosen for examination. The system was a cluster of 188 molecules whose starting configuration was an approximately spherical section carved out of an idealized tetragonal structure. Although such
0 1995 American Chemical Society
Crystalline Clusters of Tetragonal tert-Butyl Chloride
TABLE 1: Lennard-Jones Parameters and Fractional Charges for the Five-Site Intermolecular Potential Functions for tert-Butyl Chloride MoleculesP carbon chlorine methyl charge (14) +0.25 -0.25 0.0 fJ (A) 3.800 3.472 3.960 E (J/mol) 209.00 1115.60 731.52 Combining rules adopted, algebraic mean for 0, geometric for E . an aggregate is small in comparison with some atomic clusters studied by molecular dynamics, the number of interaction sites it contains is almost 1000. Taking all of the sites into account during the number of cooling and heating cycles required to develop the statistics needed to establish the rate of the tetragonal-to-monoclinic phase change (a stochastic process) is a computer-intensive operation. A determination of the kinetics of nucleation, to be reported in the next paper of this series, was the primary target of the investigation. Compensating for the smallness of sample is the fact that, as demonstrated in many dozens of simulations including a variety of hexafluorides and other polyatomic molecule^,^.'^^'^^^^^^^ van der Waals clusters of such molecules spontaneously adopt the structures of bulk systems when they are 1 or 2 orders of magnitude smaller than van der Waals systems of atoms that naturally adopt the structure of the bulk matter. Clusters of rare-gas atoms need upward of 1000 atoms to begin to pack into the bulk fcc structure (instead of into amorphous polytetrahedral aggregate^)^^ whereas even clusters as small as 53 quasispherical molecules (of TeF6) have organized into the bulk bcc structure with a lattice constant very close to that of large crystals.34 Moreover, the interior molecules in small crystalline clusters experience environments quite similar to those encountered in their bulk counterparts. One substantial difference, however, arises from the rather large depressions of transition temperatures that occur in phase changes when clusters are small. A consequence is that clusters are stable in a given phase only when they are much cooler than bulk crystals in that phase.36 Therefore, the dynamic parameters may differ from those in the bulk. As related in the following, we attempt to extrapolate our results to the temperature of the bulk measurements. Molecular dynamics computations were carried out with a modified version of the program MDMPOL37on an IBM RISC workstation using the potential function listed in Table 1, with time steps of 5 fs each. The idealized tetragonal cluster was first placed in a heat bath at 180 K, at which temperature it quickly melted. At 140 K the tetragonal cluster began to undergo a transition to a warmer phase after a few thousands of time steps. When started at 100 K, the 188-molecule cluster remained stably in the tetragonal structure over 15 000 time steps. Therefore, cooling stages were begun, taking 10 deg steps until 50 K was reached. At each temperature the cluster was equilibrated in a heat bath for loo0 steps and then run at constant energy for another 9000 steps. During this cooling process the cluster transformed from the tetragonal phase I11 to the monoclinic phase IV, a transition which was found to be reversible by repeating 10 deg cooling and heating stages over the transition range many times (although hysteresis-associated with the element of chance in the nucleation during cooling-did occur). This reversible phase change was evidence that the model system in the MD runs mimicked the experimental system faithfully enough once the size effect was taken into account, to provide insight into the dynamics of the molecular motions in the tetragonal phase. The procedure to obtain the information was to heat the cluster to 130 K, to equilibrate it in a heat bath for 1000 time steps, and then allow it to relax at constant energy for 9000 times steps over which dynamic information was
J. Phys. Chem., Vol. 99, No. 12, 1995 3919 accumulated. Cooling in 10 deg stages was continued until the transition to phase IV was complete. Results of five such cooling runs, in all, were averaged to obtain the values reported in the following. Each such run was begun with the final configuration of the previous run, after an additional equilibration of 10 000 time steps. Methods of Analysis. In the tetragonal phase subjected to analysis, only the interior molecules (those 41% of the total which lay within 14 8, of the cluster center) were selected for studies of libration and diffusion. It was supposed that exclusion of the less rigidly bound surface molecules would yield results more nearly comparable with those of the bulk phase 111. Displacements of two orientational coordinates were monitored, one (e)corresponding to the deviation of the C-C1 bond from the c axis of the crystal. and the other (4) to the rotation of a molecule about its C-C1 bond. The first, B(t-!o), was taken to be the angle between the C-C1 bond directions at t and at to, where to is any independent time origin in the run. Of particular significance in our investigation was the other displacement, r#@-to). For each time origin to considered, the displacement &(t-to) at a later time from the orientation at to was determined for each of the core molecules, and the meansquare amplitude (&*) averaged over this sample was calculated. To reduce noise, (@*)was averaged over as many time origins as possible. Both the oscillatory amplitudes ($/i2)1ib of molecules librating in the local cages of their neighbors and the steady diffusion (42)dif were extracted from a plot of the time evolution of ([4(t-t0)l2). After an elapsed time of approximately a picosecond the distribution over (&2)llb was filled out, and the coefficient of diffusion D was derived from the slope of the subsequent linear portion of the mean-square amplitude, via 2 0 = d([#)(t-to)]*)/dt
In determining the coefficient D by least squares, the points were weighted by ascribing to them standard deviations directly and inversely proportional to the proportional to ([~#~(t-to)]~) square root of the number of independent time origins (which number decreased as t increased). Mean-square librational amplitudes (&)lib (relative to initial positions, not to equilibrium positions) were taken to be the intercepts of the line fitting the steadily rising portion of the curves ([&(t-to)]*). To facilitate comparison with the neutron diffraction frequency of libration, an effective librational force constant k@ was computed from (I#J2)lib. For nondiffusing, damped librators it can be shown that at thermal equilibrium lib= mkBT/k6
where m = 2 for an ensemble of oscillators with random phases. If the equilibrium position of the local potential well is itself considered to diffuse, and to be governed by the same coefficient of friction as that damping the librator, the simplest model differential equation describing a damped, diffusing librator implies that m = 3/2 (see Appendix). Even though the model grossly oversimplifies the physics involved, we adopted the latter value to compute the angular frequency of libration o from
w = (kJZ)'l2
(3)
where I is the moment of inertia about the principal molecular axis. A more incisive analysis of the damping and spectrum of frequencies could have been derived from the autocorrelation function of the angular velocities. Unfortunately, such an analysis could not be carried out before it was necessary to terminate the project.
Chen and Bartell
3920 J. Phys. Chem., Vol. 99, No. 12, 1995
TABLE 2: Comparison of Results of MD Simulation with Those from Experiment (Estimated Errors in Parentheses) method T, K E,, kJ/mol w , rad/ps D33.O rad2/ps ~ 3 3 , 'ps ~ t ~ , 'ps .~ tlh, ps trot, ps Dil,d rad2/ps MD' NMR' ND NDA Langevin stochastic dielectric/
130 20.59 20.5 200 205
6.2(8)h -6 6.3(13)
4.6(2)' [3.6(6)1
0.017(5) LO.12(7)1
0.018(5) [0.08(4)1
0.38(8)
0.26
0.7(2)
negligible [negligible] negligible 0.035
2.9
0.5(2)
0.2(1) negligible
Pertains to rotation about C-C1 bond axis. Value of Ut. From eq 7. Pertains to rotation about axes perpendicular to the C-C1 bond. Present investigation. f From eq 3, I = 1.86 x kg m2, 9 Extrapolated from low-temperature results. Calculated from slope [d In D(StokesEinstein)/d(l/T)] at 205 K. Reference 21.' Reference 14. Reference 28, interpreted in terms of model of Langevin rotational diffusion or twostep stochastic model of free rotatioddiffusion. / References 16- 19.
Because the thermal range over which the tetragonal phase is stable is much colder in small clusters than in the bulk, it was natural to try to extrapolate the coefficient of diffusion D(T) and effective force constant k,#,(nto the temperature of the most definitive neutron study. For the force constant an extrapolation of the MD values at the four warmest temperatures was made, using the monotonic function
to take into account the anharmonic effect of lattice expansion, optimizing parameters a, b, and c by least squares. For the coefficient of diffusion two different extrapolations were carried out. The first simply applied the Arrhenius law
D = A exp(-E,lRT)
(5)
as is commonly done. The second associated D with the generalized Stokes-Einstein equation
D = kBTl(
(6)
25
-
Q
where, as is typical, the friction coefficient 6, not D, was considered to be governed by the Arrhenius law. There is little to chose between the two laws insofar as the accuracy with which they represent the MD data is concerned, but the two laws do give modest differences in the extrapolated diffusion coefficient and activation energy. In results reported below we use the latter method.
Results Mean-square displacements for 4 and 8 behave entirely differently. As shown in Figure la, the C-Cl bonds oscillate about the crystallographic axis, attaining their equilibrium but somewhat disordered distributions. By contrast, the distribution of the azimuthal angle 4 plotted in Figure l b steadily broadens. As is to be expected for molecules disordered about their principal axes, the mean-square displacement ([4(t-to)12) continues to rise as rotational diffusion proceeds, as is illustrated more clearly in Figure 2 . In this figure are indicated the leastsquares lines representing the diffusion slopes. The deterioration of the fit at large times is a result of the large noise intrinsically associated with random walks and the very few independent time origins available when t - to approaches the magnitude of the time of the runs, themselves. Values of the rotational diffusion coefficients and effective rotational force constants derived from the data of Figure 2 are plotted in Figures 3 and 4 for cooling runs carried out at the simulation temperatures 130, 120, 110, 100, 90, and 80 K. It should be pointed out that, in runs at 80 K and in some of the runs at 90 K, nucleation t o the ordered phase had begun, a phenomenon which always
0
90
180
270
360
cp (degree) Figure 1. Rotational distribution functions for rert-butyl chloride in tetragonal phase at 130 K. (a) Distribution about axis perpendicular to mean C-C1 bond direction at times of 1 ps (circle), 12.5 ps (filled circle), 25 ps (square), 37.5 ps (diamond), and 50 ps (triangle). The time-independent aspect is characteristic of libration without diffusion. (b) Distribution about the C-C1 axes at times of 1, 25, and 50 ps. The distribution broadens as the molecules undergo rotational diffusion.
occurred in the cluster's interior. Therefore, because D tends to be too small and k, too large in comparison with values corresponding to pure phase I11 at these temperatures, we did not include the illustrated points at 80 and 90 I< when deriving D(T) and k 4 ( n for the tetragonal phase. From extrapolations of k4 and D to the 205 K temperature of the neutron diffraction study of ref 28 were calculated estimates of the librational frequency w and the coefficient of friction E , from which the relaxation time 2 3 3 of rotational diffusion about the C-C1 axis was computed. Values of w,D, and t 3 3 are compared with their experimental counterparts in Table 2 .
J. Phys. Chem., Vol. 99, No. 12, 1995 3921
Crystalline Clusters of Tetragonal tert-Butyl Chloride 40
35
30 A
N
€+ V
0.6
25
0.4
20
0.2
0.0
5
0
70
80
90
100
110
T
120
130
140
Figure 4. Decrease of mean force constant k, (kJ rad-2 mol-’) for libration as the lattice expands with increasing temperature. The curve represents a least-squares fit via eq 4 that was used to extrapolate results to the temperature of experimental measurements. For the lowest two temperatures k+ is too high because the transition to the ordered phase has begun.
10 15 20 25 30 35 t(PS)
(a)
Figure 2. Time evolution of mean-square displacements (@2) of molecules from their initial orientations, averaged over many independent time origins. Progressively steeper curves are for temperatures of 80, 90, 100, 110, and 130 K, respectively. The straight lines are from weighted least-squares fits of the diffusional portions of the curves. The intercepts yield the mean rotational force constants via eq 2.
-2.0
e
h
0
-
i
-2.5
M
I
-3.0 -3.5
-4.0
I
j,, ,
7
ia
1
L
,
,
8
, , ,
,
/
9
,
,
,
,
,
, , ,
10
,
,
,
11
,
, ,
, 12
,
,I
I
I*
a
I
)I
,o,
13
lOoO/T Figure 3. Temperature dependence of coefficients of rotational diffusion (D in rad*/ps). At the lowest two temperatures D is too low because nucleation to the ordered phase has occurred during some of the runs. Therefore, these points were disregarded in determining the activation energy of diffusion.
Discussion Although the rotation of methyl groups about their C-C axes in phase I11 was readily followed by NMR spectroscopy, only one of the many NMR investigations provided information about the rotation of special concem in the present study. Stejskal et a1.*‘ inferred that the activation energy for rotation about the C-C1 bond is -6 kJ/mol (or 1.5 kcdmol, stated to be “probably only an approximate value”) and that the tumbling of molecules about other axes is considerably more restricted. Several cold neutron diffraction studies observed aspects of the molecular motions about the C-C1 bonds, but none combined the energy resolution, temperature dependence, and rigorous analysis required to characterize the molecular behavior with fidelity. The low-resolution study of Goyal et a1.,I4while unable to fumish details about librational frequencies and diffusion, did provide a rough estimate of the activation energy for molecular jumps in rotational diffusion. As shown in Table 2, Goyal’s result is of the same magnitude as that of the NMR analysis and of the MD computations. The higher resolution experiments
Figure 5. Schematic representations of the molecular packing in the tetragonal phase: (a) looking down the c axis; (b) viewed approximately down the b axis. The dominant interactions associated with rotations are those typified by the contacts between the tert-butyl groups of molecule B’ with those of A and C. See text.
of Larsson et a1.28 for this phase were done at only one temperature and therefore afforded no determination of the activation energy. It is instructive to consider the magnitude implied by our potential function for the strongest intermolecular interactions contributing to the barrier to rotational motions. These interactions correspond to contacts between the tert-butyl groups of nearest neighbors (A--B’ or B’--C pairs in Figure 5). Forces between a single A--B pair oriented with the methyls of one (regarded as fixed) straddling a methyl of the other (regarded as the diffuser or librator) would give a contribution to the activation barrier to rotation and to the force constant k,p roughly 4 times larger than the average barrier and net effective force constant found in the simulation, if the axes of A and B were constrained to their average positions. It is geometrically impossible, of course, for all four A molecules about each B to
3922 J. Phys. Chem., Vol. 99, No. 12, 1995
Chen and Bartell
adopt perfectly straddling positions, and some of the neighboring molecules must introduce convex potential functions for B that oppose the concave functions restraining B's rotation. In the perpetually evolving, chaotically oriented aggregate of molecules, the fluctuating cage that a given molecule experiences permits fleeting librations to take place about a momentary equilibrium position before a new potential minimum spontaneously forms elsewhere. The net response of the molecules in our simulations to this ferment is expressed in terms of the parameters for libration and diffusion summarized in Table 2, where the results of the simulation are compared with those determined by experiment. Some comments about the meaning of the MD parameters listed should be made before comparing them with experimental values listed in the same table. The frequencies w listed were calculated via eqs 2-4 and not from observed periodic displacements of molecules. Indeed, as Figure 2 graphically verifies, no oscillatory behavior of the aggregate as a whole occurs. If all molecules librated with more or less the same frequency for several periods before damping, the curves of Figure 2 would display oscillations even if the molecules librated with random phases. If individual molecules oscillate for a few periods, the spectrum of their frequencies is required by the temporal evolution of ([@(t-to)l2) to be extremely broad. The time 233 listed in Table 2 represents the relaxation time for rotational Langevin diffusion about the C-C1 bond axis, in the notation of Larsson et a1.28 It was computed, following those authors, as Ut,where 6 was inferred from the coefficient of diffusion (eq 6). At low temperatures in the MD runs it is very short in comparison with the librational damping time seen in Figure 2. A more intuitively agreeable relaxation time of the diffusing, oscillating molecules is ZD, derived according to the model of a diffusing Langevin librator (see Appendix) where
zD = 2Z/[(
-
(E2 - 16k,J)"2]
(7)
At 130 K the result ZD = 0.7 ps is 35-fold larger than t 3 3 and corresponds plausibly with the damping evident in Figure 2 (which damping was not used in the evaluation of ZD). When the temperature rises to 205 K, the temperature of the highresolution neutron study, however, the extrapolated value of (E2 - 1 6 k d becomes negative, implying a damped oscillatory behavior of the quantity ([&t-to)]*), and a only a single relaxation is given by the model, that corresponding to 2 3 3 . Perhaps the most detailed comparisons of MD results that can be made are with those derived by Larsson et a1.,28whose energy resolution was sufficient to permit an evaluation of several alternative models of molecular motion proposed for the tetragonal phase. For this phase the authors concluded that neither free rotations nor well-developed librations are present. Of course, the absence of free rotations had been firmly established in the much earlier investigations of dielectric relaxation, I 6 - I 9 NMR spectra,21and X-ray diffraction intensities.I0 Moreover, the known chaotic orientations made welldeveloped librations extremely unlikely. Larsson et al. concluded that two other models accounted for the rotational motions relatively well. These were a two-parameter Langevin rotational diffusion and a two-step libration-diffusion stochastic model. For the former model were derived the (not mutually independent) coefficients of friction E, rotational diffusion coefficients D,and relaxation times 2 3 3 and 511 for motions about the C-Cl axis and for motions perpendicular to it. For the latter stochastic model were determined frequencies and relaxation times for libration and rotation. The parameters so deduced are compared with those of the present simulations extrapolated to 205 K in Table 2.
Our simulations, to be sure, are based on a simplified interaction potential energy, and our extrapolations to 205 K are subject to uncertainty. On the other hand, the models invoked in analyses of experimental data are simplified, as well. All in all, it is gratifying that interpretations of the experimental data by Larsson et ~ 1 . , *and ~ by Stejskal et aL2' and by Goyal et a1.l4as well, are in substantial accord with those of the present simulations in the motions of greatest concern in the fmhcoming analysis of nucleation rate of phase 1V. Coefficients of friction and of diffusion, relaxation times, activation energies, and frequencies of libration agree as well as might be expected for the rotational motions about the molecular C-Cl bonds, whether the Langevin or the stochastic model is compared with the MD resulu, and the friction coefficients deduced from the MD rotational diffusion do indeed account in magnitude for the damping of the librations. If the coefficient of friction for libration is the same as that for rotational diffusion (our simplifying assumption), it is plain that the relaxation times are so short that librations are damped in a fraction of a librational period, and any rotational jumps are halted in a small fraction of a cycle. If this is true, the Larsson model of successive steps of distinct librations and of free rotations cannot be strictly adhered to, and both experiment and simulation agree on this. The MD results clearly show that concerted rotational jumps are far smaller in amplitude than the 120" jumps proposed by Goyal et al,. and the times between successive reorientations are far briefer than the 12 ps suggested. On the other hand, the present results and those of Larsson et a1 indicate that in 12 ps the rotational diffusion is indeed of the magnitude of 120". The above features of agreement notwithstanding, marked discrepancies occur in other aspects of the dynamics probed by the simulations and experiments. What the simulations do not support is the end-over-end rotational diffusion perpendicular to the molecular C-Cl bonds suggested by an analysis of the high-resolution neutron experiments.28 In this, our simulations corroborate the dielectric measurements,I6-l9 NMR investigation,21 and X-ray diffraction studylo in finding quite well localized C-C1 bond directions. Because the neutron data seem to have been interpreted in terms of an existing model in which molecular motions were assumed to be isotropic, the derived magnitude of the diffusion perpendicular to the molecular 3-fold axes may simply be an artifact of the model of analysis. Now that the characteristic times of molecular reorientation have been established for the interior molecules in our 188molecule clusters; it is possible to complete a self-consistent interpretation of the kinetics of the formation of critical nuclei of phase IV in undercooled clusters of tert-butyl chloride. Results will be presented in a forthcoming paper.
Acknowledgment. This research was supported by a grant from the National Science Foundation. Appendix Behavior of Diffusing Librator. In order to interpret the complex motions of tert-butyl chloride molecules in the chaotically evolving structure of the plastically crystalline tetragonal phase, we simplified the problem to one characterized by a single mean rotational force constant ,k$ and single coefficient of friction E. According to our model, a rigid molecule with moment of inertia I executes librations in the potential well formed by interactions with neighboring molecules. These neighbors also perturb the librator by exerting Langevin fluctuating torques L(t). It is assumed that &, the position of the minimum of the potential well governing the librations, is itself undergoing diffusion with a coefficient of
J. Phys. Chem., Vol. 99, No. 12, 1995 3923
Crystalline Clusters of Tetragonal ferf-Butyl Chloride rotational diffusion D = kBT/t. The differential equation for the rotation acceleraton, then, is
14 = -k&4 - f$e) - [($
+ L(t)
(All
where it is assumed that
Equation A1 can be rewritten and averaged over time to get
where the fluctuationg torque averages to zero. From energy conservation we write
1
-k 2@ ((4 - #e)’>
1
+ ~ 1 ( $=~ k> g
~
044)
and take (&(4 - Qe)) = 0 because 4 - 4, and 4, are unconrelated. We reckon 4 from its initial position 4(0) so that, by this convention, the mean-square displacement of 4 from its initial position is the function (42)of eq A3. By this convention the quantity (4e(0)2)is nonvanishing and presumably related to the initial mean potential energy (V(0)) = ‘/2k&,(O)’) = ‘/2kBT. Noting that (&&(O)) = (~$,(o)~), we rearrange eq A3 to obtain
whose solution is
(4’)
= Cle-‘lt
+ C2e-‘*‘ + (1/2k,)(4kB - [D) + 2Dt
(A6)
where
with d2 = tz- 16k& and the constants CI and C2 are determined by the initial conditions. If d2 > 0, as it is for the low-temperature solutions, 6 is real, and no oscillatory behavior of the solutions arises. When MD results are extrapolated to higher temperatures, it appears that d2 becomes negative, a circumstance that yields oscillating solutions. If, for sake of simplicity, we assume that the same coefficient of friction that govems damping of the librator also govems diffusion, and DC
= kBT, making the intercept of the extrapolated line of steady diffusion 3k~T/2k,.
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