Orientational Order and Rotational Relaxation in the Plastic Crystal

344

J. Phys. Chem. B 2008, 112, 344-357

Orientational Order and Rotational Relaxation in the Plastic Crystal Phase of Tetrahedral Molecules† Rossend Rey* Departament de Fı´sica i Enginyeria Nuclear, UniVersitat Polite` cnica de Catalunya, Campus Nord B4-B5, Barcelona 08034, Spain ReceiVed: July 11, 2007; In Final Form: September 12, 2007

A methodology recently introduced to describe orientational order in liquid carbon tetrachloride is extended to the plastic crystal phase of XY4 molecules. The notion that liquid and plastic crystal phases are germane regarding orientational order is confirmed for short intermolecular distances but is seen to fail beyond, as long range orientational correlations are found for the simulated solid phase. It is argued that, if real, such a phenomenon may not to be accessible with direct (diffraction) methods due to the high molecular symmetry. This behavior is linked to the existence of preferential orientation with respect to the fcc crystalline network defined by the centers of mass. It is found that the dominant class accounts, at most, for one-third of all configurations, with a feeble dependence on temperature. Finally, the issue of rotational relaxation is also addressed, with an excellent agreement with experimental measures. It is shown that relaxation is nonhomogeneous in the picosecond range, with a slight dispersion of decay times depending on the initial orientational class. The results reported mainly correspond to neopentane over a wide temperature range, although results for carbon tetrachloride are included, as well.

I. Introduction phase1

The plastic crystal (also referred to as rotator phase or ODIC, orientational disorder in crystals) is an intermediate phase between the totally ordered crystal and the liquid phase, characteristic (among other compounds2,3) of globular molecules such as the XY4 cases studied here. It is complementary to the more popular liquid crystal phase, and it is usually described as being characterized by crystalline positional order (for the centers of mass) and liquidlike orientational order.4 The latter aspect is mainly inferred from the fact that rotational relaxation is known to take place within the time scale typical of the liquid phase. Therefore, in addition to its intrinsic interest, it constitutes a convenient scenario in which to probe rotational relaxation, being in principle free from some complicating features of the liquid, such as solvation shell exchange, although there certainly exists an entanglement with translational distortion.5 The recent renewed experimental6 and theoretical7 interest in molecular level details of rotational dynamics is one of the main motivations to revisit it. An additional source of interest comes from the rotational freezing that can take place with sufficiently fast cooling, which avoids complete orientational ordering.8 The orientational relaxation resembles in this limit the relaxation of conventional glass formers so that the plastic crystal constitutes in many aspects a simpler model of the glass transition. Whether these similarities can be fully exploited largely depends on a satisfactory understanding of orientational order in both the plastic and liquid phases. Unfortunately, the highsymmetry characteristic of tetrahedral molecules has hindered the efforts in this direction, mainly focused until now on the liquid phase. Recently,9 a clear-cut classification of mutual orientations that has facilitated a quantitative understanding of orientational order in the liquid phase of carbon tetrachloride †

Part of the “James T. (Casey) Hynes Festschrift”. * E-mail: [email protected].

has been proposed, although the methodology is not limited to this phase or compound. This approach produces, for orientational correlations, functions similar to the radial distribution function, which is so useful for the characterization of the distance dependence of positional order. It seems adequate to use the same tools for a quantitative reassessment of the basic assumption just described; namely, that there are no significant differences with the liquid-phase concerning orientational order, which is what will constitute the main topic of this work. This program has been complemented with the study of two intimately related issues. First, the existence of a preferred molecular orientation with respect to the crystal structure has been revisited. New quantitative methods have been developed to measure the known dominance of a D2d molecular orientation. An additional benefit of this approach is that it facilitates the study of rotational relaxation in a finer level of detail. In particular, it has been possible to check whether it depends on the initial orientation with respect to the crystal field. The confidence in the results to be reported is based on the very good degree of agreement with the experimental data for the plastic crystal phase of neopentane. There have been several reasons to focus on C(CH3)4. Probably, this has been the compound, from all apolar tetrahedral molecules, for which the experimental effort on its plastic crystal phase has been most important. Its dynamics has been addressed by NMR,10-15 neutron scattering,16-21 and Raman scattering.22-24 In addition, there have also been a number of theoretical (simulation) studies.25-30 It is also important to note that, upon cooling the liquid, its fcc plastic crystal phase spans a remarkable temperature range (from 256.5 down to 140.5 K), which allows one to study the temperature dependence in detail. Finally, an accurate model for the simulation of the methylchloromethane series (C(CH3)4-nCln) that provides thermodynamic, structural and dynamical32 properties in excellent agreement with experimental data for the liquid phase was developed some time ago.31

10.1021/jp0754177 CCC: $40.75 © 2008 American Chemical Society Published on Web 11/01/2007

Plastic Crystal Phase of Tetrahedral Molecules

J. Phys. Chem. B, Vol. 112, No. 2, 2008 345

TABLE 1: Temperature and Density/lattice Constants for Each Simulation Point neopentane temperature, K lattice constant (Å) density (g/cm3) a

150 8.56a

173 8.66a

225 8.76a

CCl 4 253 8.80a

270 (liq) 0.6159c

235 8.34b

298 (liq) 1.584c

Reference 19. b Reference 37. c Reference 38.

Sample results for carbon tetrachloride have been included, as well, although (in contrast with neopentane) its fcc plastic crystal phase is metastable with respect to a rhombohedral structure and exists within a reduced margin of some 20 K below the melting point. It has been studied by X-ray,33 NMR,34 neutron scattering,35,36 and simulation.33,35-37 Comparison with neopentane will highlight which features are common to the plastic phase and which are peculiar to each compound, providing some insight, for instance, into their different stability ranges. The paper is organized as follows: Computational details are described in Section II; Section III contains the results for orientational order and its comparison with the liquid state; Section IV addresses preferential orientation with respect to the crystal structure; Section V is devoted to the dynamics of rotational relaxation; and finally, Section VI contains some concluding remarks. Two mathematical appendices are also included. II. Models and Computational Details Molecular dynamics (MD) simulations have been performed for several state points (see Table 1 for a summary of temperatures and density/lattice constants). The models used are basically those developed in ref 31, with a modification for carbon tetrachloride based on the more complex model developed in ref 39 (see details below). These are rigid, nonpolarizable, five-site models that accurately reproduce a variety of liquid-state properties. It has been demonstrated that, as far as the liquid phase is concerned, neglecting polarizability,9,32 internal motion,9 or both has no significant effects for the methylchloromethane series. Geometries, charges, and LennardJones parameters can be found in ref 31. Geometrical constraints have been handled with the “shake” algorithm,40 with a relative accuracy of 10-5. For electrostatic forces, the Ewald summation method has been used, except for the largest system sizes, with R ) 5/L and nmax2 ) 16. NVT conditions have been maintained during equilibration,41 whereas temperature control was turned off during the production runs, with a time step in both cases of 2 fs. For the intermolecular interaction, a spherical truncation scheme between molecular centers has been used. The potential effect of periodic boundaries and treatment of long range forces has constituted a main concern for the plastic phase. In an effort to minimize them and to spot any possible artifacts, two rather different box sizes have been used for each state point, with different treatment of long range forces. The smaller simulation box contains 4000 molecules, includes Ewald summation, and the cutoff is set at 14 Å (a simulation with a cutoff extended to 25 Å has also been performed at 150 K to check for cutoff effects). A second set of simulations has been run for each state point with a box containing 16 384 molecules. In this case, no Ewald sum has been included, and the cutoff has been maintained at 14 Å. In all cases, molecules have been initially located on the ideal fcc sites and assigned random orientations and velocities. This initial configuration has been propagated during a time of 200 ps for the smaller box (50 ps for the larger one) with tight temperature control (velocity rescaling). The equilibration run has been followed

by production runs of a length between 200 and 300 ps for the smaller box (40 ps for the larger one) without any sort of temperature control. No temperature drifts have been observed in any case. Although the total simulated time is substantial (≈0.5 ns for the smaller system), the fcc structure has been found to be perfectly stable in the case of neopentane (see below for carbon tetrachloride). No differences whatsoever have been observed, for any of the static and dynamical quantities monitored, between both sets of simulations. The basic conclusion is that no signs of artifacts introduced by simulation parameters are discernible. Although the presence of overall biases cannot be discarded (most likely stemming from the possible coupling of periodic boundary conditions and the periodic arrangement of the plastic crystal), the tests described lend some confidence to the size independence of the results, reinforced by the excellent agreement with experimental results for several properties. Regarding the simulations of the liquid phase, box sizes with 4000 molecules have been used, with equilibration runs of 30 ps and production runs of 200 ps. A comprehensive test of parameters and models for liquid carbon tetrachloride9 showed that these choices were more than sufficient to guarantee the negligible effect of simulation parameters. The study of the plastic phase of carbon tetrachloride presents some rather subtle aspects, which probably are a reflection of its metastability. The plastic crystal phase (Ia) exists only from the liquid-solid (fcc) transition at 244.6 K down to 225 K,42 where it transforms into a rhombohedral structure (Ib). When heated, Ib directly transforms into the liquid at 250 K without passing back through the Ia phase, which highlights the metastable nature of the latter. The model developed in ref 31 was initially tried for the plastic phase. This model, which does not include polarizability and has an octupolar moment lower than experiment, provides excellent agreement for thermodynamic, structural, and dynamical quantities31,32 in the liquid phase. The unimportance of the rough treatment of electrostatic forces is directly related to the fact that the electrostatic energy represents only 1% of the total energy.31 Remarkably, it is unable to form a stable plastic crystal phase at the known density and temperature. All attempts resulted in a totally disordered (liquid) simulation box. In ref 39, this model was enhanced with the exact experimental polarizability and correct octupolar moment. Although these improvements have barely noticeable effects on most structural and dynamical properties of methylchloromethanes,32 they were indispensable to accurately reproduce the collision-induced absorption spectrum of liquid carbon tetrachloride.39 It turns out that such subtle effects are critical, as well, to attain stability of the plastic crystal phase. The initial fcc structure was maintained with no signs of melting for the longest runs (for a box containing 256 molecules). Unfortunately, the polarizable model is computationally demanding, and it is unfeasible to use it with the large simulation boxes previously described. To bypass this difficulty, the mean charges were computed during the run with 256 molecules. The fluctuations are rather small,32,39 with the chlorine charges centered at ∼0.3e (to be compared with 0.17e for the initial model), which yields almost exactly the experimental octupole

346 J. Phys. Chem. B, Vol. 112, No. 2, 2008

Rey, R.

moment.37,39 Simulations of the plastic crystal were tried again, with the only change of the aforementioned increase in the octupole moment up to its experimental value. It is pleasing that this reasonable improvement resulted in a stable plastic phase. A series of different equilibration runs (with 4000 molecules) were performed to ensure that the stability did not depend on a fortunate choice of initial conditions: (a) the same equilibration procedure was applied but with all molecules starting with the same orientation; (b) a fcc crystal was equilibrated at a lower temperature (150 K) and then heated up to the plastic crystal temperature of 235 K; (c) during the equilibration, all molecular centers were attached to the fcc sites by springs to dampen center-of-mass displacements. These equilibration runs were followed by production runs in the NVE ensemble without any sort of external perturbations. In all cases, the results were statistically indistinguishable for the properties studied. Finally, a configuration with the density corresponding to the crystal phase was equilibrated at a high temperature (350 K) so that it attained liquidlike disorder. Subsequently, it was cooled down to 235 K during a run that lasted 0.6 ns, with a temperature coupling constant41 of 50 ps. No signs of an fcc crystal were found in this case, and the box remained disordered for the full length of the run, that is, the crystal structure was not trivially attained from any starting configuration, not even in this case in which there is a very small energy difference (≈6 kJ/mol) between the liquid and plastic crystal phases. III. Mutual Orientation A. Lattice Distortion and Radial Distribution Function. Figure 1 displays two different views of a configuration obtained during the simulation of the plastic crystal phase of neopentane at 150 K (most of the results in this section and beyond will focus on this temperature because it is the one with more pronounced differences with the liquid). Simple inspection of Figure 1a shows that there is a remarkable degree of orientational disorder, which will be the subject of analysis in the following sections. Translational distortion is also evident in Figure 1b, in which only the centers of mass are displayed. Their arrangement differs from a perfect crystal structure, more so if we consider that this is the lowest temperature studied, and therefore, thermal effects are at their minimum. This distortion was already noticed in the pioneering simulations of Mountain and Brown25 and can be given a numerical description. The mean square displacement has been computed for all systems studied and displays for the plastic crystal the typical plateau characteristic of a solid (not shown). For neopentane, this plateau yields the following values of x: 1.36 Å (at 150 K), 1.43 Å (173 K), 1.54 Å (225 K), 1.67 Å (253 K); and for carbon tetrachloride, 1.43 Å (at 235 K). Therefore, substantial excursions from the ideal positions occur, on the order of 1.5 Å, to be compared with the minimum separation of some 4.5 Å for the dimer. It has been checked that these excursions are temporary and correspond to oscillations around the ideal position, with no permanent distortion of the fcc lattice. Additional insight can be obtained from the comparison between liquid and plastic crystal phases of the radial distribution function for the centers of mass, displayed in Figure 2 for neopentane. As it was the case for liquid carbon tetrachloride,9 translational order in the liquid disappears at a distance of ≈20 Å. In contrast, the solid phase shows the typical behavior of the fcc structure, with more marked oscillations that survive up to the largest distances, although they are gradually attenuated. This latter effect underlines the difficulty of this angle-

Figure 1. (a) Sample configuration of simulated plastic crystal of neopentane at 150 K for a box containing 256 molecules; (b) same, but displaying only the centers of mass.

averaged function to convey (in real space) the infinitely periodic arrangement of the plastic crystal, as will be discussed in more depth. B. Short Range Order. We now turn to the main topic of this work; namely, the study of orientational order using the tools already applied to liquid carbon tetrachloride.9 The high symmetry inherent to tetrahedral molecules has hindered a detailed understanding of orientational order. Until recently, only qualitative descriptions gathered with indirect methods were available. The solution proposed concerning this unsatisfactory situation consists of a classification of mutual orientations based on a well-defined mathematical procedure. Figure 3 displays the simple geometrical construct on which it is based and the classes that result from it. Given a pair of tetrahedral molecules, we consider two parallel planes (represented by vertical lines in Figure 3), each one of them including one of the molecular centers (the carbon atoms in the case of the carbon tetrachloride molecule) and perpendicular to the line joining them. The space


Figure 2. Radial distribution function for the centers of mass. Solid line, plastic crystal neopentane (150 K); dashed line, liquid neopentane (270 K).

between both planes will contain a number of chlorine atoms, which are used as the basis for the classification: For the pair on the upper left of Figure 3, there is only one chlorine from each molecule (the minimal number possible), so this class is denoted corner-to-corner. Six classes result from this procedure, that is, all the combinations of simple geometrical objects: corner (one chlorine in between both planes), edge (two chlorines), face (three). As detailed in ref 9, there are a number of advantages to this approach: easy implementation, a substantial flexibility regarding the range of existence for each class, the assimilation in a natural way of many of the classes used until now, and remarkably, the case of random orientations can be solved analytically. This aspect is important because it allows one to clearly distinguish random from nonrandom distributions for such highly symmetrical compounds. The following percentages result for each class in the case of random orientations (see Appendix in ref 9): corner-to-corner and face-to-face account for 3% each; corner-to-edge and edge-to-face are characterized by 23%; corner-to-face by 6%; and finally, the most populated class is edge-to-edge, with 42%. Figure 4 displays the percentage obtained for each class, as a function of intermolecular distance, for plastic crystal neopentane at 150 K. They are compared with those of the liquid phase at 270 K. The range of distances roughly corresponds to the 12 closest molecules, the exact number of nearest neighbors for the fcc lattice, and approximately the number of molecules within the first solvation shell for the liquid phase.31 Maybe the most important aspect to notice is the strong degree of orientational order present. Suffice it to note that, if there were no orientional order, horizontal lines would be obtained at the percentages just mentioned for random orientations. Instead, there are wild oscillations of the different classes, with, for instance, the face-to-face class starting at 100% and going down to almost 0% within a rather narrow range. The second aspect that merits being highlighted is the strong similarity between the results for the liquid and those for the plastic crystal. It constitutes an unambiguous quantitative confirmation of the shared characteristics concerning orientational order between the liquid and plastic crystal phases. As previously stated, this close relationship is a basic piece of the current understanding, complemented with the notion that this order is of short range, which was slightly corrected for the liquid phase.9 It was shown that the range of orientational order is highly similar to that of positional order and that remnants of it are still present within

J. Phys. Chem. B, Vol. 112, No. 2, 2008 347 the fourth solvation shell. More dramatic changes will be described in the next section for the plastic crystal. Before addressing the behavior at larger distances, short-range order will be analyzed in more detail. A feature that stands out is the overwhelming weight of face-to-face configurations for very short distances, to the point that all the molecules that make it into the very immediacy of the central molecule belong to this class. This is followed by a steep increase in edge-to-face orientations, followed by the edge-to-edge class. It should be considered, though, that these are distance-dependent percentages, and since the corresponding volume grows quadratically, the integrated weight (over the full volume corresponding to the range considered) provides an alternative perspective. Indeed, if one looks at the closest molecule, only a mere ≈15% of the pairs belong to the face-to-face class, but ≈50% of the dimers belong to the edge-to-face class, followed by an ≈30% of edge-to-edge configurations. Put another way, the (rather broad) range in which the face-to-face configuration dominates is seldom visited. If one should pick a single configuration for the closest molecule, the edge-to-face class would be the most representative for both the liquid and plastic crystal phases. The percentages just mentioned depend only slightly on temperature and compound. Table 2 contains the detailed results, which reinforce the notion that differences between the plastic crystal and the liquid are rather small (at contact separations). The analysis can be further extended to the full first shell. In terms of the distance-dependent percentages, the more relevant aspect might be the succession of maxima (i.e., different configurations dominate for different distances). The initial maximum (with a record 100%) corresponds to the face-to-face configuration, which is followed by the maximum of the edgeto-face configuration (≈60%); the edge-to-edge class (≈50%); and finally, the corner-to-edge class (≈40%). The latter displays the highest differences between plastic crystal and liquid (with the maximum attaining almost 50% for the solid and roughly 30% for the liquid). It is worth noting in passing that the maxima for the plastic crystal are in all cases higher than those for the liquid; that is, there is a slightly stronger orientational order within the plastic phase. In terms of the two planes that are the basis of the present approach, the succession of maxima can be summarized by stating that the dominant configuration has a decreasing number of methyl groups (chlorines) between both planes for increasing distance. In addition, the dominant role is increasingly disputed by other configurations as percentages corresponding to the maxima tend to diminish. Finally, and still within the first shell composed of 12 molecules, ∼5 molecules belong to the edge-to-edge class (i.e., 41% of the cases), ∼3 correspond to the edge-to-face configuration (24%), at least 2 to corner-to-edge (22%), and at most 1 to corner-to-face (7%). Face-to-face (3.4%) and corner-to-corner (2.5%) configurations share the remaining molecule, so that according to the present definitions, these classes play an almost negligible role. These numbers show even slighter variations, depending on temperature and compound, as compared to those for the closest molecule. C. Long Range Order. Figure 5 displays the distancedependent percentages obtained for the largest simulation box. Substantial fluctuations below 10 Å correspond to the strong orientational order within the first solvation shell just discussed. This is followed by oscillations of smaller amplitude for the plastic phase that reach up to ≈30 Å. In contrast, those for the liquid phase are indistinguishable from the horizontal lines, characteristic of random orientations, well below this distance (as was the case for liquid carbon tetrachloride9). Finally,


Rey, R.

Figure 3. The six classes used for the classification of molecular dimers. Vertical lines denote the planes that delimit the region between both molecules.

Figure 4. Percentatges (as a function of intermolecular distance) obtained for each of the orientational classes defined in Figure 3. The results displayed correspond to plastic crystal neopentane at 150 K (solid lines) and to liquid neopentane at 270 K (dashed lines). The green curves correspond to corner-to-face configurations (“Apollo’’), and the black curves, to corner-to-corner configurations.

TABLE 2: Percentage for Each Orientational Class for the Closest Molecule neopentane temperature, K face-to-face edge-to-face edge-to-edge

150 17 55 27

173 15 53 29

225 13 51 32

253 13 50 32

CCl4 270 (liq) 16 50 31

235 6 50 38

298 (liq) 17 51 29

orientational order is still present for the plastic crystal up to the largest distances (≈65 Å). Notice particularly the oscillations of the edge-to-edge curve. Although these oscillations are highly damped as compared to those of the radial distribution function (see Figure 2), the remarkable finding is that the range of orientational order is substantially larger than that for the liquid. Thus, the notion that orientational order in the plastic phase is very similar to that of the liquid is not supported (at long distances) by the present results. Actually, inspection of angle-averaged functions may not be the most pertinent way to study long range order (positional or orientational) within the plastic phase. As previously noted, the

Figure 5. Same is in Figure 4, but for the full range of distances studied. For clarity, the results for the liquid are not included. Horizontal dashed lines denote the percentages for random orientation.

increasingly damped fluctuations of the radial distribution function for this phase are due to angle averaging; they should not be interpreted as a decay of positional order. This can be seen more clearly by looking at the density along a given direction of the crystal. To be more precise, Figure 6 contains a sketch of the crystal, with a central molecule and those that surround it in the fcc lattice (represented by filled circles). The axes representing the most relevant crystallographic directions ([100], [110], [111]) are also displayed. The one-dimensional number density (n(x)) has been computed along two different directions, starting from the central molecule. Figure 7 displays the results for the [110] direction (corresponding to the axes that go through first neighbors) and for the [100] direction (second neighbors). The infinitely periodic arrangement is now perfectly seen, with slightly higher peaks in the close vicinity of the central molecule that level off for distances larger than ≈20 Å without any signs of the damping present in g(r). The distance-dependent percentages, computed as a function of the radial distance in Figures 4 and 5, can now be computed along the crystallographic directions. The results are displayed in Figure 8a,b. Notice that (for clarity) only a portion of the



Figure 6. Sketch of the fcc crystal. The central molecule is surrounded by 12 nearest neighbors, represented by filled circles (only nine of them are visible in this view). Also displayed are the most relevant crystal directions. Circles filled with darker gray represent molecules (almost) aligned with the four bonds of the central molecule, which is oriented according to D2d symmetry.

Figure 7. Number density along the [110] axes (s) and along the [100] set (- -).

largest distances has been displayed. Long range orientational correlations are now clearly visible, with no noticeable damping (in contrast with Figure 5). The oscillations are not as marked as those of the short-range section (Figure 4), but nevertheless, they are pretty well defined, particularly along the directions that go along second shell molecules (Figure 8a). This difference might be due to a higher degree of orientational averaging for those displayed in Figure 8b, because they result from averaging along 12 (semi)axis. In short, long range orientational correlations are found for the plastic phase that clearly differ from liquidlike behavior, a result that to the best of our knowledge is new and unexpected. As previously stated, no differences have been spotted between the distributions obtained with different simulation boxes or cutoff lengths. Unfortunately, these checks do not guarantee by themselves that long range orientational correlations might not result from a numerical artifact. The simulation boxes used still fall many orders of magnitude away from a macroscopic sample. Before looking into the reasons that might explain this behavior, it is interesting to ask whether it might be disproved right away by experiments that directly probe structure (X-ray and neutron diffraction). Although the measured scattering functions contain a mixture of contributions from all partial distribution functions, it is possible in some cases to extract them by isotopic substitution. Carbon tetrachloride was the first liquid for which these functions became available,43,44

Figure 8. Percentages for each of the orientational classes, computed along (a) [100] and (b) [110] directions. Color assignments as in Figure 4. Only the farthest portion of intermolecular distances is displayed. Noisy zones correspond to low probability regions (see Figure 7).

although there may be some inconsistencies in the results because a high degree of accuracy is required.37 The question we ask is whether these functions (in case they can be accurately extracted) may contain a signature of orientational long range order. To this end, the comparison with random orientations seems, again, a straightforward way to look at this question. Random orientations have been generated for a pair of molecules, with intermolecular distances sampled according to the radial distribution function between centers of mass31 (taken from the simulation results). In this way, one generates the partial distribution functions that would be obtained if there were no orientational order while keeping the same radial distribution function for the centers of mass. The results are displayed in Figure 9 for the carbon-methyl pair, corresponding to plastic crystal neopentane at 150 K. Noticeable differences are found for short and medium range (i.e., up to 20 Å), reflecting the fact that strong nonrandom order exists within this range, as discussed in the previous section. In contrast, the differences for larger distances are barely discernible and, therefore, would be not visible in scattering experiments. This is most probably due to the high symmetry of tetrahedral molecules and can be made even more evident. If the orientation of one of the molecules is fixed, for instance, in the edge configuration (and the second is still assigned random orientations), no corner-to-


Figure 9. Carbon-methyl radial distribution function corresponding to (black) neopentane plastic crystal at 150 K; (green) random orientations, with intermolecular distances sampled according to the center of mass radial distribution function for the plastic crystal; (red) same, but excluding corner-to-corner, corner-to-face, and face-to-face configurations.

corner, face-to-face, and corner-to-face configurations will be able to contribute to the partial distribution functions. As displayed in Figure 9, no signature of the bias introduced is visible for the carbon-methyl radial distribution function, confirming that long range orientational order may be difficult to infer from partial distribution functions. IV. Preferential Orientation within the Crystal Lattice The complex long range orientational correlations displayed in Figure 8a,b might be explained by the interplay between short range order and preferential orientation induced by the crystal field, that is, molecules might be prone to some particular orientations induced by the periodic structure (and, thus, result in long range order), which would be modulated by the strong steric interactions with nearest neighbors discussed in Section 3B. Actually, the ordering effect of the periodic structure was demonstrated theoretically in a landmark paper for deuterated solid methane.45 It was shown that a model with no center of mass distortion (an ideal lattice), with only octupolar interactions between molecules, could produce a series of orientational phase transitions, including orientationally disordered crystals, partial orientational ordering (one molecule in four is randomly oriented), and totally ordered phases (perfect long range orientational order). Given the substantial distortions of the lattice (Figure 1b), it is doubful that this simplified model captures all the relevant aspects for neopentane and carbon tetrachloride, but it certainly demonstrates that the periodic structure can induce a variable degree of long range orientational order, with only feeble intermolecular interactions, for similar (XY4) systems. A. Preferential Orientation. In principle, preferential orientation was already noticed by McDonald et al.37 in their simulations of plastic crystal carbon tetrachloride, as carbonchloride bonds tended to be directed along the [110] direction. Closer inspection shows, though, that this result does not demonstrate by itself the existence of preferential order. Totally random orientations would also produce a substantial majority of bonds pointing in the [110] direction, as it is analytically demonstrated in Appendix A. For randomly oriented molecules, a substantial 44.4% of the bonds will point in the [110] direction,

Rey, R.

Figure 10. Percentages of carbon-methyl bonds found along each crystal direction. The vertical dashed line marks the frontier between plastic crystal and liquid phases. Horizontal lines in the liquid phase denote analytic results for random orientations. Curves along the plastic crystal phase are just a guide to the eye. Symbols on the left vertical axis correspond to plastic crystal carbon tetrachloride at 235 K (note the broken horizontal axes).

28.1% will be found along the [111] direction, and 27.5% along [100]. However, any deviation from the previous figures would constitute an unambiguous sign of such preferential orientation. Figure 10 displays the percentages obtained for each direction as a function of temperature. Almost all the results correspond to neopentane. Those for plastic crystal carbon tetrachloride (for a temperature of 235 K) have been displayed on the left vertical axis. We note first that, as expected, the liquid-phase results perfectly match the analytic percentages deduced in Appendix A for random orientations (represented by horizontal segments). At the frontier between liquid and plastic crystal phases, there is a well-defined jump for each crystallographic direction. Although it is not a dramatic change, it demonstrates unambiguously the existence of preferential ordering within the plastic crystal. There is an increase for the [110] direction, followed by a smaller one for the [100] direction, both compensated by a more substantial decrease for the [111] direction. The broad range of temperatures over which the plastic crystal phase of neopentane exists allows the study of the temperature dependence of these changes. In general terms, there are only small variations of the figures just given. As the temperature decreases, the dominance of the [110] direction is increased up to ≈50 %, while the [111] class loses some weight, down to ≈20 %. No significant temperature dependence for the [100] direction is detected. Finally, we turn to the results for carbon tetrachloride. Again, the percentages obtained for the liquid are indistinguishable from those for random orientations. Within the plastic phase, the results for both the [110] class (the one that accounts for more than half of the cases) and the for [111] class (the one that is depleted) can be pictured as a continuation of the trends found for neopentane for lowering temperature. That is, the distortion induced by the lattice is maximum for carbon tetrachloride, surpassing that for neopentane at the lowest temperature (itself also rather close to the transition to a perfectly ordered crystal). A similar behavior will also be found for the results to be discussed below, and in a way, it provides some insight into the metastability of plastic crystal carbon tetrachloride. One might conjecture that its orientational distortion is close to the


Figure 11. (a) Contour line plot of the stereographic projection of a uniform distribution over the sphere; (b) same for the distribution obtained for plastic crystal neopentane at 150 K.

maximum possible for a plastic crystal made of XY4 molecules so that any further temperature decrease can only result in a phase transition to an ordered crystal. B. Stereographic Projections. A more detailed understanding can be gained from the stereographic projections19 of the unit vectors along the bonds. Appendix B contains a description of this approach and might be a required reading prior to the following discussion. Basically, the probability distribution defined over the sphere by the directions of the carbon-methyl bonds is projected onto the x-y plane. The advantage is that this projection can be easily displayed (similarly to a cartographic representation of the Earth), although this comes at the price of an unavoidable distortion of the true distribution. Appendix B contains the deduction of the analytic density obtained for a uniform distribution over the sphere (random molecular orientations). A contour line plot is displayed in Figure 11a. It has a peak on the vertex (001) and diminishes (in a nonuniform way) up to the border at r ) 1 (which goes through 100 and 110). Figure 11b displays what is obtained for plastic crystal neopentane at 150 K. The distortion introduced by the lattice is clearly visible compared to Figure 11a and is roughly indicative of an increase in the [110] direction and a decrease in the [111] direction. To bypass the effects of distortion, the present analysis will focus on the change (in terms of percentages) over the distribution for random orientations (Figure 11a). The results are displayed in Figure 12 for the highest and lowest temperatures studied. The broken line corresponds to the points for which there is no change and, in agreement with the results of


Figure 12. Changes (in terms of percentages) over the distribution for random orientations. The dashed line denotes the points for which there has been no variation. Dark gray corresponds to negative changes, and light gray, to positive ones. The spacing is of 10%, and the broken continuous lines denote the frontiers between directions. All the results correspond to plastic crystal neopentane for the highest (253 K) and lowest (150 K) temperatures studied.

the previous section, it completely encloses the region corresponding to the [111] direction, with a net decrease for all the points belonging to the [111] tile. The increases previously discussed for the other two directions can now be discussed in greater detail. Most of the directions (within the [100] and [110] areas) suffer a net increase, but for those in the outskirts, there is a depletion so that the net values displayed in Figure 10 result from a balance between (mainly) positive and negative contributions. Much the same picture is obtained for all temperatures. The main difference might be that the maximum depletion of the [111] direction occurs at 150 K and reaches 50%, whereas for the temperature closest to the liquid state (253 K), this figure is 30%. The pattern for carbon tetrachloride is similar but contains some distinct features (Figure 13). There is a higher increase in the [110] direction (≈40% compared with a maximum of ≈30 % in neopentane; see Figure 12) and a lower decrease in the [111] direction (≈40% compared with ≈50 % in neopentane). The [100] direction is subject to both negative and positive changes, which are almost equilibrated. In addition, the dashed line corresponding to a null change now delimits two zones instead of one. C. Orientational Classes with Respect to the fcc Lattice. From a qualitative analysis of stereographic projections, it was concluded in refs 19 and 27 that the main orientation, with respect to the crystal lattice, corresponded to a configuration called D2d. It is sketched in Figure 6 for the central molecule


Rey, R.

Figure 13. Same as in Figure 12 for plastic crystal carbon tetrachloride at 235 K.

and can be described as having all its carbon-methyl bonds roughly pointing in the [110] direction. This has been emphasized by using a different color (dark gray) for the molecules to which the bonds point to (only three out of four are visible). Given the cubic symmetry, there are six equivalent orientations that correspond to the same D2d arrangement. It was noted,19 though, that the results could not be interpreted as all molecules having this orientation within the crystal, but no attempt was made to determine the relative proportions, an issue that will now be inspected. Seeking a quantitative approach, molecules have been grouped in different classes, defined by their orientation with respect to the crystal. For each molecule, the two bonds having a maximum alignment with a crystal direction have been selected and are the basis of the classification. The procedure starts determining (for each molecule) the crystal direction to which each bond points. The two bonds with a better degree of alignment are the ones selected. If, for instance, both point in the [110] direction (what would correspond to the D2d configuration, which is thought to be dominant), the molecule is assigned to a class denoted (110-110). Since there are three crystal directions, molecules can be grouped into six classes. The results are displayed as a function of temperature in Figure 14. As usual, the percentages obtained for the liquid-state correspond to random orientation. The dominant class in this case is (110-111) closely followed by the (110-110) class, both with nearly 25%. Classes (100-111) and (100-110) are less populated (with ≈20 % each), and finally, classes (111111) and (100-100) have a rather small weight (≈5 % each). The jumps that take place at the frontier between liquid and plastic crystal phases are again a signature of preferential ordering induced within the crystal network and can be understood focusing on the two classes that together account for more than 50% of the molecules. Classes (110-111) and (110-110) undergo a change (of almost an identical 4%) in opposite directions, swapping their roles. Now the (110-110) class is dominant, confirming the notion that the D2d configuration is the one representative of the plastic crystal, at the expense of the (110-111) class. As the temperature is lowered, this effect is increased: the (110-111) class becomes the fourth most populated at 150 K (from being the most populated in the liquid phase), and (110-110) increases its weight up to a maximum of 34%. To summarize, a maximum of one-third of all configurations belong to the dominant (D2d) class within the plastic phase. The changes that take place can be rationalized in terms of a slight rotation of one of the bonds for some of the molecules belonging to the (110-111) class. This rotation would

Figure 14. Percentages of molecules found with two bonds preferentially aligned in directions (xxx-yyy). Symbols on the left vertical axis correspond to plastic crystal carbon tetrachloride at 235 K. In color, the configurations for which the changes with respect to the liquid phase are more important. For the curves in black, squares correspond to (100-111), and circles, to (100-100).

shift the bond from [111] to the [110] direction. Of course, this is just a rough picture; the rest of the configurations also change their weights at the transition (and as a function of temperature), although to a lesser extent. The exception is the (111-111) class, which is considerably depleted at the transition, a process that is increased with lowering temperature, consistent with the general depletion of bonds pointing into the [111] direction (see Figure 10). However, this class has a very low population at any temperature. Finally, carbon tetrachloride displays again signs of belonging to an extreme scenario. If we concentrate on the most populated class (110-110), the corresponding percentage within the plastic crystal phase can be understood again as a continuation of the curve for neopentane, reaching a maximum value of ≈35 %. V. Rotational Relaxation The experimental work on the rotational relaxation of neopentane has been tentatively explained through free rotations,2,17 diffusion models,22 and instantaneous jumps between potential wells.18 The preferential order with respect to the crystal lattice, which has just been discussed, implies that rotational motion takes place in the presence of free energy wells,29 favoring in principle the latter mechanism. These wells, though, are rather shallow29 (on the order of kBT), and therefore, it is not evident that they might have a noticeable influence on rotational relaxation. Here, an attempt will be made to clarify this issue. A. Comparison with Experimental Data. A preliminary step will be the comparison of the simulated rotational dynamics with experimental data. A good level of agreement has been reported for liquid carbon tetrachloride,32,44 but there seems to be no comparison for the plastic crystal phase of neopentane or carbon tetrachloride. This might be related to the fact that previous models used for the plastic crystal were described, in the case of neopentane, as either “unrealistically soft’’29 or, to the contrary, giving rise to slow dynamics,25 with, for instance, two-thirds of the molecules never undergoing reorientation for as long as 20 ps (while it is known that rotational relaxation takes place within the 1 ps time scale18). Similar problems were


Figure 15. Rotational autocorrelation functions for plastic crystal neopentane at 150 K. Thin line, cJ(t); thick line, c1(t); dashed line, c2(t).

reported for the simulation of the rotator phase of carbon tetrachloride.37 The link with experimental measures is provided by the autocorrelation functions of the spherical harmonics of first and second order,47 which in terms of the unit vector (u b) along one of the carbon-methyl bonds are expressed as c1(t) ) b(t)‚u b(0))2 - 1]>. In the and c2(t) )

Orientational Order and Rotational Relaxation in the Plastic Crystal

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