J . Phys. Chem. 1987, 91, 4990-4994
4990
Quadrupolar Freezing In (KBr)
,-,(KCN), Mixed Crystals
Laurent J. Lewist and Michael L. Klein*t Division of Chemistry, National Research Council of Canada, Ottawa, Ontario, Canada K l A OR6 (Received: February 23, 1987)
We have used constant-pressure molecular dynamics to investigate the process of quadrupolar freezing in (KBr),,(KCN), mixed crystals. In agreement with experiment, a system with x = 0.25 was seen to remain cubic at all temperatures and form an orientational glass. The ground state is analyzed in terms of absolute and relative ordering of the CN- molecular anions. We find that neighboring quadrupoles exhibit a definite preference for ferroelastic ordering, though a broad distribution of relative orientationsis also present. These results are in sharp contrast to those in a system with x = 0.73,which is unequivocally found to freeze into an orientationally ordered (crystal) state at low temperature.
I. Introduction (KBr) ,_,(KCN), are site-disordered solid solutions in which Br- is randomly substituted for CN- on the anion sublattice. At high temperature, these systems possess overall cubic symmetry (space group Fm3m) with the CN- ions undergoing rotational diffusion. As the temperature is lowered, however, a remarkably intricate pattern of phases is revealed; often this is accompanied by softening of transverse acoustic phonons (shear modes). Because they are the simplest ionic molecular salts, a considerable effort-experimental, theoretical, and computational-has been devoted to rationalizing the observed experimental properties in terms of models for the coupling of the rotational degrees of freedom of the dumbell-shaped CN- molecular ions to other lattice excitations. On cooling from the high-temperature rotator phase, pure KCN undergoes a cubic orthorhombic ferroelastic transition at 168 K, followed at 83 K by an antiferroelectric ordering transition.' The first of these transitions is anomalous, as phonons of T,, symmetry exhibit dramatic ~ o f t e n i n g ; ~translation-rotation .~ coupling has been invoked to explain such b e h a ~ i o r . ~Upon ,~ dilution of the CN- with Br-, as noted earlier, a complex tangle of phases is observed. For x 2 0.85, orthorhombic and triclinic phases are found to c o e ~ i s t . ~For 0.6 Ix I0.85, on the other hand, the ground-state structure is m o n ~ c l i n i c ,though ~~~ a rhombohedral phase is also present at intermediate temperatures for certain concentration^.^,^ Finally, in the Br--rich regime, x I0.6, the crystal is found to remain cubic at all temperatures, even though a strong ferroelastic anomaly is seen in the dielectri~,~,~ and neutron responses9,",12 This anomaly has been interpreted as evidence for the formation of a quadrupolar glass state."*13 There is now evidence that, in these systems, dipolar (head-to-tail) and quadrupolar (orientational) freezing temperatures differ.14Js In order to rationalize the effect of dilution on the formation of an orientational glass state, theoretical work has invoked the coupling of the rotational degrees of freedom of the CN- anions to the random strain fields generated by the chemical disorder of the anion sublattice.I6 We have reached similar conclusions on the basis of computer simulation studies which indicated that the ground-state structure of (KBr) I,(KCN), and related systems was determined principally by a competition between translation-rotation and strain-rotation couplings.17 Even though the existence of an orientational glass state for x I 0.6 is now well established, there are still questions regarding its nature. This is the issue we address here. We report on constant-pressure molecular dynamics (MD) simulations, i.e. in the (N,P,H) e n ~ e m b l e , ' ~for , ' ~a system in the glass-forming range of CN- ion concentrations, namely (KBr)o,75(KCN)o 25. We
-
'Present address: Department de Physique et Groupe des Couches Minces, Universitt de MontrCal, Case Postale 6128, Succursale A, Montreal, Qutbec, Canada H3C 357. *Presentaddress: Department of Chemistry, University of Pennsylvania, Philadelphia, PA 19104-6323.
TABLE I: Parameters for the Model Buckingham-Type Potentials, Vu&) = Au8 exp(-eU8r) - Ba8/r6,Used in the Present Simulations" ff-8 A,@, k J mol-' a,@,A-' B,@, A6 kJ mol-' K-K 151 000 2.967 1464
c-c
N-N Br-Br
174 300 192000 429 600
3.400 3.600 2.985
2569 1718 12410
"Parameters for the cross interactions are obtained by using the combining rules A,, = (AuUABB)'/*, B,, = (B,,BBB)'/*,and a,@ = (aa, + a@@).
present an analysis of the results in terms of orientational ordering of individual CN- ions with respect to the crystal axes and the relative ordering between neighboring pairs. To serve as a guide, a similar analysis is performed on a system which freezes into an orientational ordered (crystal) state, that is (KBr)o.27(KCN)o.73.17 Various conceptual models of the orientational glass phase have been proposed; among these are the favored orientation model,"-" where CN- ions freeze into random pockets, e.g. { 11 1)directions, and the ferroelastic domain mode1,20*21 where quadrupoles are aligned within domains but with disordered domain wall regions also present. Anticipating our MD results, we find a definite (1) Rowe, J. M.; Hinks, D. G.; Price, D. L.; Susman, S.; Rush, J. J. J . Chem. Phys. 1973, 58, 2039. Rowe, J. M.; Rush, J. J.; Prince, E. J . Chem. Phys. 1977, 66, 5147. (2) Haussiihl. S. Solid Stare Commun. 1973. 13. 147. (3j Rowe, J. M.;Rush, J. J.; Chesser, N.;Michel, K.H.; Naudts, J. Phys. Rev. Lett. 1978, 40, 455. (4) Michel, K. H.; Rowe, J. M. Phys. Rev. B 1985, 32, 5827. (5) Rowe. J. M.: Rush. J. J.: Susman, S. Phvs. Reu. B 1983, 28, 3506. Rowe, J. M.; Bouillot, J.; Rush, J. J.; Liity, F. Physica (Amsterdam) 1986, i36B, 498. (6) Knorr, K.; Loidl, A. Phys. Rev. B 1985, 31, 5387. (7) Knorr, K.; Loidl, A.;Kjems, J. K. Phys. Reo. Lett. 1985, 55, 2445. (8) Bhattacharya, S . ; Nagel, S. R.; Fleishman, L.; Susman, S . Phys. Reu. Lett. 1982, 48, 1267. Knorr, K.; Loidl, A. Z . Phys. B 1982, 46, 219. (9) Loidl, A.; Feile, R.; Knorr, K. Phys. Reu. Lett. 1982, 48, 1263. Feile, R.; Loidl, A,; Knorr, K. Phys. Rev. B 1982, 26, 6875. (IO) Garland, C. W.; Kwiecien, J. 2.;Damien, J. C. Phys. Rev. B 1982, 25, 5818. (1 1) Rowe, J. M.; Rush, J. J.; Hinks, D.G.; Susman, S . Phys. Rev. Lett. 1979, 43, 1158. (12) Loidl, A,; Feile, R.; Knorr, K.; Kjems, J. K. Phys. Reu. B 1984, 29, 6052. Loidl, A,; Feile, R.; Knorr, K.; Renker, B.; Daubert, J.; Durand, D.; Suck, J. B. Z.Phys. B 1980, 38, 253. (13) Michel, K. H.; Rowe, J. M. Phys. Reo. B 1980, 22, 1417. (14) Volkmann, U. G.; Bohmer, B.; Loidl, A.; Knorr, K.; Hochli, U. T.; Haussiihl, S . Phys. Reu. Left. 1986, 56, 1716. (15) Doverspike, M. A,; Wu, M.-C.; Conradi, M. S . Phys. Reo. Lett. 1986, 56, 2284. (16) Michel, K. H. Phys. Reu. Lett. 1986, 57, 2188. Michel, K. H. Phys. Rev. B 1987, 35, 1405. 1987, 35, 1414. (17) Lewis, L. J.; Klein, M. L. Phys. Reu. Lett. 1986, 57, 2698. (18) Parrinello, M.; Rahman, A. Phys. Rev. Lett. 1980, 45, 1196. (19) Nost, S.; Klein, M. L. J . Chem. Phys. 1983, 78, 6928. Phys. Reu. Letf. 1983, 50, 1207. Impey, R. W.: NosC, S.; Klein, M. L. Mol. Phys. 1983, 50, 243. (20) Knorr, K.; Civera-Garcia, E.; Loidl, A. Phys. Rev. E , in press. (21) Ihm, J. Phys. Reu. B 1985, 31, 1674.
0022-3654/87/209l-4990$01.50/0 0 1987 American Chemical Society
Orientational Freezing in (KBr),_,(KCN),
The Journal of Physical Chemistry, Vol. 91, No. 19, 1987 4991 I
preference for neighboring CN- ions to order ferroelastically (i.e. either parallel or antiparallel), though a broad distribution of relative orientations is also observed. Our findings therefore seem to support the favored orientation picture, though, because of the system size used in the M D experiment, it is not possible to rule out the model in which domain walls play an important role.20 11. The Simulations Constant-pressure MD s i m ~ l a t i o n s lwere ~ ~ ' ~carried out on two different neutral, periodically replicated systems of 5 12 ions (N+ = N- = 256). In one, the anion sublattice contained, randomly distributed, 70 Br- ions and 186 CN- molecular ions [(KBr)027(KCN)073], whereas the other was made up of 192 Br- and 64 CN-ions [(KBr)07s(KCN)02s].The ions were initially arranged in the observed crystal structure at the experimental volume and temperature. Thus, the high-temperature Fm3m phases were formed by placing the ions on the sites of two interpenetrating face-centered cubic lattices with sides of length 4ao (aobeing the appropriate lattice constant) and giving each CN- a random orientation. Encouraged by earlier successes in the study of phase transitions in (KBr) ,-,(KCN), mixture^,'^^^^^^^ and in the corresponding optimization of pair potential^:^ we again chose the ions to interact via two-body potentials of the Buckingham type, whose parameters are given in Table I. The internal degrees of freedom of the CNions were neglected, and the C-N bond length was fixed at 1.17 A. The anion charge distribution was represented by a three-site model, with charges -0.81eI at the N site, +0.81el in the C-N bond a distance 0.204 from the C atom, and -le( outside the C-N bond a distance 0.222A from the C atom. Each simulation was initiated with a run performed using conventional (constant volume, constant shape) MD, in which the pressure was calculated. The latter was then used to determine Pext,the external pressure applied to the M D cell in subsequent constant-pressure runs.I9 P,,, was corrected to allow for the truncated portion of the interionic potentials. For (KBr), 27(KCN), 73 and (KBr)o75(KCN)o25 respectively, we found P,,, to be 0.115 and 2.73kbar, i.e. modest pressures for these materials. The generalized Parrinell~-Rahman'*~'~ equations of motion were integrated by using a third-order Gear algorithmZSfor the translational and M D cell degrees of freedom, while the angular equations of motion were evaluated in quaternion form by using a fourth-order Gear algorithm.26 The electrostatic interactions were handled by the Ewald method. For each MD run (Le. at each point in temperature), the evolution of the system was followed for 3500 time steps at intervals At = 2 fs, the first 500 of which are reserved for equilibration. Starting from the high-temperature cubic rotator phase, the effect of isobarically cooling each sample was then examined; Table I1 summarizes this temperature evolution.
-
(22) Impey, R. W.; Sprik, M.; Klein, M. L. J . Chem. Phys. 1985, 83, 3638. (23) Bounds, D. G.; Klein, M. L.; McDonald, I. R.; Ozaki, Y . Mol. Phys. 1982, 47, 629. (24) Bounds, D. G.; Klein, M. L.; McDonald, I. R. Phys. Rev. Lett. 1981, 46, 1682. Klein, M. L.; McDonald, I. R. J . Chem. Phys. 1983, 79, 2333. Ferrario, M.; McDonald, I. R.; Klein, M. L. J . Chem. Phys. 1986, 84, 3975. (25) Gear, C . W. Numerical Initial Value Problems in Ordinary Differe n l i d Equations; Prentice-Hall: Princeton, NJ, 197 1 . (26) Evans, D. J.; Murad, S. Mol. Phys. 1977, 34, 327.
1
-I
1.8
1
i
i 11111
1.5 la6
0
100
300
200
400
Figure 1. Temperature dependence of the positions of the (1 1 l), (200), At) ~high . ~ ~temperature, . the and (220)Bragg peaks for ( K B T ) ~ , ~ ~ ( K C N system is cubic and all lines of the same family are degenerate. The splitting of the (1 11) and (220) lines at T = 120 K signals the presence of a cubic rhombohedral structural transition. (This transition is given the label (11) to be consistent with ref 17. The prefix "d" identifies mixtures with a chemically disordered anion sublattice.)
-
2.9 2.8 2.7 2.6 n
2.0 Y
w++ + + +
+
100
200
1.9
12001 +
1.8 1.7 1.6
111. Low-Temperature Structures
Figures 1 and 2 show the temperature dependence of the positions of the { 1 1 l}, {200),and (220}Bragg diffraction lines for the two systems studied. At high temperature, both systems are cubic and all lines of a given family are degenerate. However, as cooling of the (KBr), 27(KCN)073 sample proceeds, Figure 1, the lines are clearly seen to split at a temperature T = 120 K; the cubic symmetry is broken. In fact, as reported e l ~ e w h e r e , ' ~ the system undergoes a cubic rhombohedral transition at that point, in excellent accord with the experimental transformation at T = 112 K . ~ The situation in (KBr), 75(KCN)025is radically different, as shown in Figure 2. There, again in agreement with experiment,
2.6 n
1.5
0
T
300
400
(to
Figure 2. Same as Figure 1, but for (KBr)o,,5(KCN)o,25.Here, no splitting is observed: the system remains cubic at all temperatures.
the system is found to remain cubic at all temperatures. Note, however, that the lines exhibit substantial dispersion as the temperature is lowered. Even though this behavior is probably amplified by the finite size of the MD cell, it does signal the presence of a high degree of frustration in this system. It is a manifestation of the competition between translation-rotation coupling which tends to induce a structural transition, and strain-rotation coupling, which opposes it.I7 Our studies of other orientational glass formers reveal similar beha~ior,~' as do diffraction experiments on these
system^.^,^ Quadrupolar freezing will be fully discussed below, but we may already note here that dispersion effects reach a maximum at a (27) Lewis, L. J.; Klein, M. L., unpublished.
4992
Lewis and Klein
The Journal of Physical Chemistry, Vol. 91, No. 19, 1987
0.3
p
0.3
p
0.2
P
E g
a
b P
0.2
8
E
e
xb
0.1
0.1
E 0
0 0.0
0.0
-0.1
-0.1 0
100
200
300
400
0
Figure 3. Temperature dependence of the CN- orientational order parameters, as defined in the text, for the (KBr)02,(KCN)07, mixture.
temperature substantially above the freezing point. This is consistent with our observation that the structural transition, in systems exhibiting one, precedes the freezing transition. Thus, it appears that the net coupling between the CN- rotational degrees of freedom and phonons or strains is at a maximum before the orientational freezing temperature is reached. The point at which maximum coupling occurs coincides with the structural transition and the maximum anomaly in the acoustic behavior.
IV. Freezing Transitions and Orientational Ordering We now examine in detail the orientational freezing process and consider various order parameters for the molecular orient a t i o n ~ .In ~ particular, we examine quantities of the form [( Y(C))] and their fluctuations ([ ( Y(C))2])1/2, where (...) denotes a time average for a given CN- ion and [...I an average over all sites. Here, C = (x,y,z) is the unit vector along the C-N bond; the cubic crystal axes are used as a reference frame. Thus, dipolar (D)order was monitored through the three functions
yr’) = (3/4a)1/2x fi’)= (3/47r)”2y
6’)= (3/4?r)@Z whereas quadrupolar order was studied via two functions of E, symmetry
u2)= ( 1 5 / 4 ~ ) ’ / ~ -( 31)~ ~ 6,)= (15/4a)’i2(x2 - y 2 )
and three functions of T,, symmetry
= (15/4~)’/~xy ~
2
=) ( 1 5 / 4 ~ ) I / ~ y z
~
2
=) (15/4a)’/2zx
This characterization is an e ~ t e n s i o n of ’ ~ the approach used by Edwards and Anderson to treat the problem of magnetic impurities in spin glasses.28 Figure 3 shows the temperature dependence of these parameters for the system exhibiting a cubic rhombohedral transition,” ( K B I - ) ~ ~ ~ ( K C N )We ~ , , .consider this case first, because it is easier to interpret. From the behavior of the fluctuations it is clear that, at low temperature, the CN- ions have frozen-in orientations, with
-
(28) Edwards, S. F.; Anderson, P.W. J . Phys. F 1975, 5, 965.
100
200
300
400
T (K)
T (K)
Figure 4. Same as Figure 3, but for KBr)075(KCN)025.
quadrupoles ordering essentially along (1 111 directions. (Perfect (111) would yield ( [ ( D ) 2 ] ) ’ /=2 0.282, ([(E,)2])1/2= 0 and ([ (T2,)2])1/2= 0.364; the E, fluctuation, however is a very sensitive measure of how much the average orientation departs from, here, perfect (1 1I].) Unfortunately, these order parameters cannot distinguish between the cases where CN- ions freeze into totally random (uncorrelated) ( 111)pockets, or well-defined (correlated) (111) pockets. To resolve this, we examine the other quantities indicates that plotted in Figure 3. While the value of [(D)] head-to-tail disorder is present (the net dipole moment of the sample is zero), it can be inferred from the behavior of [ ( T2,)] that only the [ l l l ] and [TTT] directions are occupied. Thus, quadrupoles have ordered with C-N bond vectors essentailly parallel to the [ 1111 direction of the parent cubic phase (Le. along the rhombohedral axis), but with head-to-tail disorder (Le. antiferroelectrically on average; we note here that the time scale for observation of true dipole orderingI4 is far beyond the reach of present simulations). As can be. seen in Figure 4, the situation for (KBr)o,s(KCN)02s, which we have seen does not exhibit a structural transition, is somewhat different. Here, C N axes again align along { 11 1] directions, as indicated by the values of ( [ ( D ) 2 ] ) ’ and / 2 ([(T2,)2])1/2, but only in an average sense, the large value of ([ ( E,)2])’/2signally substantial departures from this particular direction. In fact, as can be deduced from the values of the other parameters plotted in Figure 4, the distribution of orientations appears to be fairly wide (again the sample has zero net dipole moment). These results already suggest that (KBr), 75(KCN)o,,5may well have frozen into an orientational glass state. We defer further discussion of this point to section V. where the relatiue orientations of the CN- ions with respect to their neighbors is examined. Information about the orientational probability distribution flu) can also be obtained by examining its expansion in terms of cubic harmonics: 4a
F(U)
= 1 + C4K4
+ c6K6 + ...
The coefficient C, = (K4) = (21/15)1/2(5(x4+ y 4 + z4) - 3 ) is of particular interest, since it can be determined from neutron diffraction.’ The temperature dependence of C4 is listed in Table 11 for our two systems. At high temperature, it is found to approach zero from the negative side, characteristic of a state of rotational diffusion with, perhaps, a slight preference for { I 11) directions. This preference in ordering becomes unambiguous at low temperature, as C4 becomes strongly negative (of order -1) in both systems, though more strongly so in the case x = 0.73. I t should be noted here that perfect (1 1 1) order would yield C4 = 1.53.
Orientational Freezing in (KBr),,(KCN),
The Journal of Physical Chemistry, Vol. 91, No. 19, 1987 4993
TABLE 11: Temperature Evolution of the Simulation, Performed in the (N,P,H) Ensemble, for (KBr),,(KCN), x = 0.73 x = 0.25 T, K
(Econr), kJ mol-'
( V ) ,AS
c4
T, K
(EFonf), kJ mol-'
360 267 181 159 148 140 133 127 119 113 102 95 85 55
-681.39 -684.1 3 -686.77 -687.46 -687.79 -688.07 -688.28 -688.55 -688.77 -689.09 -689.51 -689.89 -690.32 -69 1.44
282.54 279.30 275.96 275.10 274.77 274.45 274.16 273.87 273.62 273.38 272.97 272.75 272.48 271.58
-0.03 -0.07 -0.14 -0.17 -0.16 -0.20 -0.21 -0.23 -0.25 -0.31 -0.39 -0.52 -0.66 -0.97
300 204 151 122 100 81 70 59
-676.80 -679.69 -680.77 -681.53 -682.1 1 -682.63 -682.91 -682.62 -683.47 -683.61 -683.75 -683.90 -684.03 -684.18
50 45 41 35 30 25
c4
( V ) ,A' 28 1.05 277.99 276.45 275.53 274.86 274.24 273.94 273.56 273.30 273.15 272.99 272.84 272.68 272.51
-0.05 -0.12 -0.23 -0.22 -0.33 -0.40 -0.38 -0.50 -0.54 -0.64 -0.68 -0.72 -0.79 -0.81
'For each run, i.e. at each point in temperature, (EFonf) is the average configurational potential energy of the system, ( V ) is the average volume of the unit cell, and C, is an orientational order parameter discussed in section IV.
3 t
7 J 1
t 1
0.3
n"
E
O
0
E
loo
200 f
300
0
0
loo 200 To9
O
300
400
Figure 5. Temperature dependence of the single-ion reorientation times 77, and 7;' (see text), and of the orientational diffusion constant, Dn, for
0
0
0
loo
200
300
To9
Figure 6. Same as Figure 5 , but for (KBr)0.27(KCN)0,73.
the (KBr)075(KCN)o25 mixture.
We now consider the dynamical aspects of the freezing transition. In Figures 5 and 6 we plot, for the two systems, the angular diffusion constant Dn, as well as the single-ion reorientation times and ~ 5 relevant 3 ~ to N M R experiments and defined as
loo 200 300 400 T 0
0
(Io
d-(KBr),n(KCN).,3 1
T = 360 K
I
I
7...................... 1 0.M
where
.t .................
I'
In
*. 9% 102 0..
These are the only two time correlation functions of interest for a diatomic molecular ion in a cubic crystal field. In both systems examined, the E, and TZgrelaxation times have similar behavior: each becomes very long as the freezing transition is approached. The fact that both relaxation times seem to diverge at low temperature is indicative of orientational glass formation on the time scale of the MD experiments. Upon comparing the values of Do for the two samples, now, we observe that the freezing transition is more sharply defined N ) ~ ,freezing ~~, in (KBr)o,,7(KCN)o73 than in ( K B T ) ~ , ~ ~ ( K Cwhere seems to be approached more gradually; the transition appears to be continuous. This behavior is characteristic of glasses. N M R experiments on a sample with x = 0.50 seem to confirm the continuous nature of this t r a n ~ i t i o n . ' ~ V. Nature of the Orientational Glass State In order to assess the existence of an orientational glass state and to understand its nature, we now examine orientational correlations between neighboring CN- molecular ions. Thus we plot, in Figures 7 and 8, the temperature dependence of P(cos e),
*.******'*
................
i
**.*-*]
**(xo.2)
'
(x0.2)
*.
-1
-0.5
0
0.5
1
cos 8 Figure 7. Evolution, as a function of temperature, of P(cos e), the relative probability of finding a pair of neighboring molecules such that 0 is the angle between them, for the (KBr)027(KCN)o,7, system. Note that the lowest two panels have been scaled down by a factor of 5.
which gives the relative probability of finding a pair of neighboring CN- ions such that 8 is the angle between them. At high temperature, both samples exhibit rotational diffusion. The molecular anions interact with one another only weakly (Le.
4994
The Journal of Physical Chemistry, Vol. 91, No. 19, 1987
I
T = 300 K
I
I
........................
Lewis and Klein
1
b/
e
Q
........................ .................... .. 0 . .
25
-0.5
0.5
0
COS
g
5
8
0.02
t 'a i
0.00 -1.0
0
-0.5
0.0
0.5
1.0
cos 8
Figure 9. Time dependence of P(cos 6') at low temperature (cf. Figure 8) for the system exhibiting an orientational glass state, namely (KBr)07J(KCN)0 25.
1 -..................................... ........ -1
.I
16 ps
1
e
Figure 8. Same as Figure 7 , but for (KBr)075(KCN)025.
they behave as independent rotors) and there is no preferred orientation: the distribution is flat. However, as cooling proceeds, both systems develop a tendency for neighboring molecules to align with their axes parallel (cos 0 = + I ) or antiparallel (cos 0 = -1) (with equal probability so that the net dipole moment of the sample vanishes). As expected on the basis of the results of section IV, this behavior is unequivocally marked in the (KBr), 27(KCN)07 3 sample, Figure 7 . Here, as temperature is lowered, those peaks at cos 0 = f 1 become increasingly sharp, at the expense of other possible relative orientations. In fact, at the lowest temperature examined, the quadrupoles have completely ordered in a ferroelastic pattern. This is in distinct contrast with the situation in (KBr),,,(KCN), 2 5 , Figure 8. In this system, even though a number of pairs (roughly 30%) show a preference for parallel or antiparallel alignment, there still exists at low temperature a broad, more or less featureless distribution of relative orientations. Further cooling is not likely to affect this situation significantly, as orientational freezing has already taken place at this point, at least on the MD time scale. It is not clear which orientations give rise to the broad background, though we have been able to identify contributions from particular (1 111 directions correlated with ( I IO}, (IOO),and other { 1 11] directions. Interestingly, it appears that ideal quadrupolar (perpendicular) ordering (cos 0 = 0) is not a particularly favorable configuration, as can be seen in the lowest panel of Figure 8. We tested the time dependence of our results for the frozen-in state of (KBr)075(KCN)025 by extending the last run from 6 to 16 ps and recomputing P(cos 0). This is shown in Figure 9. We find essentially no difference between t h e two runs, t h e small
discrepancies being most likely due to statistical error. Thus, even
on the shorter of the two time scales (which are relevant to neutron scattering experiments), the system indeed seems to be frozen, i.e. relaxation processs have ceased. Our results therefore suggest a picture of the orientational glass state of (KBr),-,(KCN), mixtures (and presumably of other related systems as well) as consisting of CN- molecules loosely aligning along (11 I ) directions, while exhibiting a definite preference for ferroelastic ordering. This is by no means exclusive, though, as a fair proportion of molecules do not follow this pattern and choose to sit in apparently random orientations. VI. Conclusion We have used constant-pressure molecular dynamics to study the low-temperature structure, and in particular the characteristics of orientational ordering, of two (KBr) ,,(KCN), mixed crystals. For the sample with x = 0.73, which undergoes a cubic rhombohedral transition,'? we found that the CN- ions freeze into an orientationally ordered (crystal) state with ferroelastic order along [ 11 13 but with dipole (head-to-tail) disorder. In contrast, the sample with x = 0.25 was observed to remain cubic at all temperatures, in good accord with experiment. Here, the CNions were again found to align along (111) directions, but only on the average. Closer examination of relative orientational distribution revealed a preference for ferroelastic ordering, though a broad distribution of relative orientations was also observed. Interestingly, ideal quadrupolar ordering (perpendicular orientations) does not appear to be a favorable configuration. Even though our findings generally support the favored orientation conception of (KBr),-,(KCN), glass formers4*" (Le. dipoles freezing into random (111) pockets), we cannot, on the basis of the present study, dismiss other hypotheses such as those invoking domain walls.20
-+
Acknowledgment. We thank K. Knorr, A. Loidl, K. H. Michel, J. M. Rowe, and D. Walton fpr informative discussions and R . W. Impey for invaluable assistance in numerous computational aspects of this work. L.J.L. gratefully acknowledges support from the NSERC at the time during which this work was performed. Registry No. (KBr),_,(KCN), x = 0.25, 107957-86-8; (KBr)i-x(KCN), x = 0.73, 106389-68-8.
