J. Phys. Chem. 1993, 97, 10645-10648
10645
Molecular Dynamics Studies of Solid-state Phase Transitions. 1. Inference of Low-Temperature Phase of tert-Butyl Chloride Jian Chen and Lawrence S. Bartell' Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109 Received: June 15, 1993'
A molecular dynamics simulation was carried out on a 188-molecule cluster of tert-butyl chloride to investigate whether it would transform upon cooling from the known tetragonal phase I11 to phase IV, a phase of unknown structure. The cluster was found to exist stably in the tetragonal phase a t an appropriately low temperature and to undergo a change to a new form when cooled. The transition was shown to be reversible and to occur a t a temperature nearly 100 deg lower than in the bulk, as would be expected for such a small crystal. The new form corresponded to a monoclinic structure, space group P21/m, no. 11. Powder diffraction patterns calculated from the molecular coordinates, after minor structural refinement, agreed convincingly with experimental electron and neutron patterns which had previously resisted indexing. This demonstrates the utility of molecular dynamics techniques in the study of solid-state structures in cases where transitions are facile.
Introduction Molecular dynamics (MD) techniques have been developed into a powerful means of investigating condensed matter.' When applied with realistic intermolecular interactions, MD simulations are able to provide a detailed view of the molecular behavior underlying various properties of specific substances. One of the more important phenomena examined, and one whose mechanism has resisted a fully satisfactory understanding to date, is the evolution of phase changes. Such processes are conveniently studied in clusters.* Clusters have become increasingly popular subjects for investigation, partly because of their novel status intermediate between molecular and bulk systems but partly because they are particularly tractable systems to treat by MD. An advantage of such systems is that clusters, when undergoing transformations, are not subject to the severeconstraints imposed by the periodicboundary conditions that are invoked in treatments of bulk phases. Most of the detailed studies of transitions in clusters to date have been concerned with very small atomic systems,3-5 and they have mainly been concerned with the transition between liquidlike and solidlike structures which we shall refer to, for simplicity, as phases. Even when van der Waals clusters of atoms are comparatively large (with up to 1000 atoms) the atomic packing arrangements in the solidlike phases differ from those found in the bulk.6 Therefore, while they contribute valuable insights into mechanisms of change, atomic clusters constitute somewhat marginal models for processes occurring in the bulk. Systems of polyatomic molecules are more ubiquitous than those of simple atoms and offer considerably richer diversity in their solid-state transformations. Moreover, it turns out that their clusters naturally adopt solid-state structures characteristic of the bulk even when they are considerably smaller than 1000 molecules. Several dozen molecules may suffice.' It is the purpose of the present research to exploit this property of small molecular clusters in an attempt to learn more about the structural and dynamic aspects of phase transitions in condensed matter. Particularly noteworthy is the fact that certain molecular systems have been found which are able to undergo transitions even in the solid state a t rates that are so extraordinarily fast that they can be monitored on the (extremely short) characteristic time scale of MD simulations. These systems include the hexafluorides of selenium,s sulfur: and tellurium.' Because measurements of Abstract published in Advance ACS Absrracfs, September 15, 1993.
nucleation rates for solid-state transition in pure, one-component systems had eluded experiment until recently,g the opportunity to follow the course of such transformations by MD in molecular detail was irresistible. Finding systems amenable to treatment by the MD technique had been helped enormously by experiments carried out in this laboratory on large molecular clusters produced by the condensation of vapors in supersonic flow. In these experiments the characteristic time scale of transitions is of the order of microseconds, a time span prohibitively long for MD studies. Nevertheless, it has proven to be feasible to apply the theory of homogeneous nucleation to extrapolate rates obtained experimentally for large clusters as a means of obtaining crude estimates of rates corresponding to more extreme undercoolings. Such extrapolations in the case of SeF6 were sufficiently successful to forecast the correct order of magnitude later observed in MD simulations.8 It should be noted that undercooling does not always lead to rates of transition fast enough to be seen in computer simulations. For example, experimental observations of the freezing of CCld clusters suggested that the maximum rate of nucleation should be of the order of lO*9nuclei/(m3 s).IO Deeper undercooling would be expected to lead to more sluggish freezing. Molecular dynamics computations cover too brief a time span to monitor nucleation of the predicted order of magnitude for CCl,, and the freezing of liquid clusters to the crystalline solid was not, in fact, seen in MD simu1ations.l' One system projected to be promising on the basis of cluster studies was that of tert-butyl chloride. Clusters of this substance were observed to transform from phase 111 to phase IV in supersonic flow, and extrapolations of the nucleation rate to lower temperatures8 suggested that the rate might well become large enough for nucleation to be seen in MD simulations. An additional element of interest in this system lay in the fact that the structure of phase IV was unknown when the study began. Neutron diffraction studies of tert-butyl chloride powder had been carried out,l2 but a definitiveassignment of reflections had eluded analysis. The purpose of the present paper is to report the modeling of the potential energy surface and the resultant simulation of the transition which automatically generated the structure of the unknown phase IV. An analysis of the kinetic results of a series of runs from which the nucleation rates J(T) for the I11 to IV transition could be derived will be presented in a subsequent report.'3
0022-365419312097-10645$04.00/0 0 1993 American Chemical Society
Chen and Bartell
10646 The Journal of Physical Chemistry, Vol. 97, No. 41, 1993
TABLE I: Lennard-Jones Parameters and Fractional Charges for the Five-Site Intermolecular Potential Function for tert-Butyl Chloride Molecules’ carbon chlorine methyl charge (le11 +0.25 4.25 0.0 Q
c 0
(A)
(J/mol)
3.800 209.00
3.472 1115.60
3.960 731.52
Combining rules adopted, algebraic mean for (I,geometric for c.
Physical Properties of tert-Butyl Chloride At 15 OC liquid tert-butyl chloride has a density14 of 0.8470 g/cm3 and a heat of vaporization of 29.53 kJ/mol. The solidstate behavior is complex, paralleling to some extent the behavior of the other ~hloromethylmethanes.~5At 248.2 K the liquid freezes to a disordered fcc structure (phase I), which transforms at 219.5 K to an unknown structure (phase 11) that is stable over only a 1.8-deg range.16 Upon further cooling a tetragonal structure (phase 111, space group P4/nmm, no. 129) is encountered. The symmetry implied by the diffraction intensities for phase 111 requires the methyl groups to be statistically disordered about the C-Cl bonds. At 182.9 K this phase undergoes a transition to phase IV, the uncharacterized phase of particular concern in the present paper. Although the structure of phase IV was unknown prior to the present investigation, it had been concluded from indirect evidence that the transformation from 111 to IV must hinge upon a change from disorder about the C-Cl bonds to order.8 Reported molecular properties for tert-butyl chloride are as follows: In thevapor phase the C3, molecule has mean internuclear distances of 1.528 and 1.828 A for C-C and C-Cl bonds, respectively, and the C-C-Cl angleis 1O7.4O.l7 In the tetragonal crystal the bond lengths had been assumed to be 1.54 and 1.80 A by Silver and Rudman in successful analyses of diffracted intensities.’* Dipole moments of the molecule in the gas phase and in solution in CCI, were determined to be 2.131, and 2.15 D.19 The construction of an intermolecular potential energy function, taking into account the above properties, is outlined in the next section. Procedure Model Potential Energy Function. The tert-butyl chloride molecule was assigned a rigid C3, structure with C-C and C-C1 bond lengths of 1.53 and 1.80 A, respectively, and a C-C-C1 angle of 108’. Intermolecular interactions (with the exception of the methyl groups) were modeled by pairwise-additive atomatom Lennard-Jones functions incorporating partial charges. Following Jorgensen, we treated methyl groups as pseudoatoms. Lennard-Jones parameters for the central carbon and chlorine were taken from a simulation of CC14,11and those for methyl groups, with minor modification, were taken from Jorgensen. Partial charges of f0.25 electron placed on the central carbon and chlorine atoms reproduced the molecular dipole moment. Although a more realistic modeling would have apportioned some of the charge to the methyl groups, the extra computational expense such a distribution would haveentailed was not considered worthwhile. As a partial test of potential parameters, a Monte Carlo calculation for 128 molecules of tert-butyl chloride was carried out, imposing periodic boundary conditions, to simulate properties of the bulk liquid at 15 O C and 1 atm. Results of runs totaling 3 million cycles at a rejection rate of 42% gave a molar volume that was 0.8%too low and an average configuration energy of -28.2 f 0.3 kJ/mol, a value that is about 3.8% higher in magnitude than the energy ofvaporization at 15 O C . These results were considered satisfactory for the purpose, and molecular dynamics simulations were begun. The potential parameters adopted are listed in Table I.
Starting Structure of Cluster. A cluster containing 188 molecules, with a shape as spherical as possible, was carved out of the phase 111 lattice for MD computations. Cell parameters a and c for the starting structure were assigned values of 7.08and 6.14 A, respectively. According to the space group, P4/nmm, the two molecules per cell are related by a center of symmetry. Central carbon atom 1 was given x, y , and z coordinates at 1.77, 1.77, and 1.20 A. Its chlorine partner was placed 1.80 A below while the methyl groups above were distributed randomly about the C-CI axis. Molecular Dynamics Simuladons. Molecular dynamics runs were carried out on an IBM RISC workstation using a modest modification of the CCPS program MDMPOL of Smith and Fincham.20 The primary purpose was to investigate whether the transition of clusters of tert-butyl chloride from phase 111to phase IV might be realistically modeled by MD techniques. To do this, it was first necessary to find whether a cluster governed by the model intermolecular interactions adopted can exist stably in the tetragonal phase. Although this phase exists in the bulk over the temperature range from 183 to 218 K, the thermal range of stability for a small cluster would be expected to lie significantly lower. For example, in MD studies of clusters of TeF6,Xu found that solid-state transition temperatures fell linearly with the reciprocal of the cluster radius, in the same way as melting points do.2’ Therefore, runs were initiated a t several different starting temperatures. Time steps were set to 5 fs for all runs. When the tetragonal cluster of tert-butyl chloride was equilibrated at 180 K it melted. In the first run at 140 K it changed to a structure which appeared to be the precursor of the fcc hightemperature phase, but the regular organization of the bulk crystal was not fully developed. In later runs beginning at 140 K this behavior was not repeated. At 100 K the tetragonal cluster was stable over the time spans examined. The cluster was placed in a heat bath at 100 K for 1000 time steps to equilibrate, then maintained at constant energy for 4000 more time steps, returned to the heat bath at 100 K for another 1000 timesteps, and allowed to relax at constant energy for an additional 9000 time steps. After this equilibration, a series of 10-deg cooling stages began, each stage consisting of 1000 steps in a heat bath at constant temperature, followed by 9000 steps a t constant energy. Before the cluster reached 50 Kit transformed to a new structure. Upon heating back to 120 K it became tetragonal again. Six such cycles confirmed, beyond any doubt, the reproducibility of the behavior within the limits expected for the stochastic process of nucleation initiating the phase change. Refinement of Crystal Lattice. Because of the small number of unit cells spanning a cluster of 188 molecules (of which approximately 62% of the molecules are in the surface and inclined toward disorder), results for lattice parameters derived in the M D simulations are somewhat noisy. Therefore, it seemed worthwhile to refine the crystal structure spontaneously adopted by the cluster when it cooled, by carrying out a crystal packing analysis based on potential energy minimization for a bulkcrystal composed of molecules bound by the same potential function. For this purpose Williams’ program PCK8322 was used after modification by KinneyZ3to eliminate problems encountered in the ordering of matrix elements in the routine “Givens” when many parameters were varied simultaneously. Results Clusters of 188 molecules of tert-butyl chloride existed at equilibrium in a tetragonal structure with complete disorder of the tert-butyl groups about the C-CI bonds and significant excursions of the C-Cl bond directions from the c axis. Surface molecules were rather disordered. Projections of images of molecules in the cluster viewed from the principal direction are shown in Figure 1. In a dozen cooling cycles tetragonal clusters were observed to undergo a transition to a new crystalline phase,
Low-Temperature Phase of tert-Butyl Chloride
.
The Journal of Physical Chemistry, Vol. 97,No.41, 1993 10647
TABLE II: Cell Parameters of the Monoclinic Phast? energy minimum
MD
5.99 7.21 6.18 90 88.7 90 O The MD values are estimated from MD simulations*at 50 K. The values derived from minimization of packing energy via PCK83 correspond to stationary molecules and, hence, denser packing.
TABLE 111: Fractional Coordinates of Sites in the Monoclinic Phase
Figure 1. Images of tert-butyl chloride molecules in the tetragonal phase of 188-molecule clusters at 120 K viewed down the principal lattice direction. Amplitudes of molecular motions are large, and the orientations of tert-butyl groups about the C-Cl bonds are random. Surface molecules are disordered.
a
*AyAvAY+
A7#4*PA*
*JiFn+!flrrk
site
X
C" ClO methyl" methyl* methylo Cb Clb methyl6 methylb methylb
0.131 -0.162 0.207 0.207 0.207 0.176 -0.124 0.252 0.257 0.257
Y
'14 '14 '14
0.425 0.075 '14
'14 '14
0.425 0.075
z 0.199 0.199 -0.037 0.3 17 0.317 0.187 0.198 -0.05 1 0.302 0.302
From M D simulations. From energy minimization.
arranged into a reasonably well-ordered lattice. The most important change in the solid-state structure is that of the tertbutyl groups from disorder to order. Lattice angles which all began at 90° in the tetragonal structure distorted mildly but not markedly from 90° in the course of the transition. Potential energy minimization was carried out to establish more precisely than the MD results could whether the intermolecular interactions lead naturally to an orthorhombic packing or whether the ordered tert-butyl groups distort the lattice to monoclinic. It was confirmed that the most efficient packing of molecules with the ordering found in the MD simulations induces a small but definite tilt making the new structure monoclinic, space group P21/m, no. 11. Lattice constants and site coordinates from the analysis are listed in Tables I1 and 111. Discussion
4
Figure 2. Images of tert-butyl chloride molecules in the monoclinic phase of 188-molecule clusters at .50 K viewed down the principal lattice directions: (a) a direction, (b) b direction, (c) c direction. tert-Butyl groups are ordered. The molecular bonds are shown in the figure.
sometimes at 90 and sometimes at 80 K. Enough information was acquired for reasonable estimates of nucleation rates to be made, the analysisof which will be presented e1~ewhere.l~Figure 2 illustrates the structure of the new phase. Although surface molecules are still somewhat disordered, interior molecules are
Two important results were obtained in this work. First, it proved to be possible to deduce the unknown structure of a phase by application of the molecular dynamics technique. More important than the identification of a specific phase, however, was the confirmation of an earlier study that phase changes in the solid state are able to occur, under some conditions, with such extraordinary speed that they can be followed by MD simulations.8 Here we conclude that the speed is not simply an artifact of a mathematical procedure. We believe that it corresponds to an actual physical process, provided that realistic interaction energies are incorporated into the simulation. Note that the rate of the transformation in the current runs had to be of the order of 1036 nucleation events/(m3 s) for the process to be observed on the time scale of the run. Such a rate is approximately 8 orders of magnitude faster than the rate measured in the electrondiffraction studies of the same phase change in molecular clusters cooling in supersonic flow,g which rate, in turn, was 4-8 orders of magnitude faster than the typical nucleation events measured in more conventional studies of n ~ c l e a t i o n . Before ~~ we return to the first point, namely, the inference of the structure of the lowtemperature phase, we shall briefly review the mechanism that allows the solid-state transition to occur so rapidly. Before the structure of the colder phase was known, Dibble8 analyzed the experimental data for tert-butyl chloride in terms of the Turnbull-Buckle theory of nucleation rates.25-2' He
Chen and Bartell
10648 The Journal of Physical Chemistry, Vol. 97, No. 41, 1993
concluded that the only mechanism consistent with the rate observed in the electron diffraction study of clusters in a supersonic jet was for the transformation to be nonreconstructive, that is, one in which molecular jumps from the old phase to the new are principally reorientational, not translational. Extrapolating neutron diffraction rates of reorientation about the C-Cl axis in the tetragonal phase,28 he could account for the rate observed in the electron diffraction experiments and predict that, a t even higher supercoolings, the rate should increase to a level observable on the MD time scale. Both his assumption that the phasechange was one from orientational disorder of tert-butyl groups to order and his projection of the increase in rate with supercooling turned out to be correct. It is fair to ask whether the internal organization of a mere speck of 188 molecules has much to say about the structure of bulk matter. As mentioned in the Introduction, atoms in small monatomic van der Waals clusters do not arrange themselves into the cubic closest packing found in large crystals for reasons that have been analyzed extensively.6 In the cases examined to data, however, crystalline clusters of polyatomic molecules have been found to adopt bulk structures even when they were only one-third the volume of the present c l ~ s t e r .Why ~ more efficient packings than bulk closest-packed structures can be found by spherical atoms in small clusters when quasispherical or aspherical molecules in even smaller clusters can only manage to follow the pattern of molecules in the bulk is an interesting question beyond the scope of the present paper. Of crucial importance is the fact that the transformation between tetragonal and monoclinic structures was reversible in cooling and heating cycles, leaving no doubt that the computed structures are true equilibrium structures for the model potential function adopted. Confirming that the new computed structure also corresponds to that of the real substance is the diffraction pattern it yields. Powder diffraction patterns calculated from the molecular coordinates (refined by packing analyses) satisfactorily reproduced peak positions and intensities of experimental powder patterns of phase IV. These had been obtained in electron diffraction studies of large molecular clusterss and neutron powder diffraction investigations of the bulk perdeuterio compound.l* Experimental diffraction patterns, to be published elsewhere, had not been indexed successfully prior to the MD simulations. In the new crystal structure, molecules are packed in layers in which each C-Cl bond directed upward into a layer is surrounded by four parallel C-Cl bonds directed downward. This packing satisfies electrostatic interactions and fills space. On each side of each layer, tert-butyl groups fit together, tongue in grove. A more efficient packing is attained by tilting the molecules away from the normals to the layers. Structural aspects of phases I11 and IV of (CH3)3CCl are similar to those found in crystalline Cl3CCH3, a molecule whose size, shape, and dipole moment are comparable (methyl groups and chloride atoms being approximately the same size). The latter substance also has a tetragonal phase with disorder about the molecular 3-fold axis, and its colder phase possesses ordered layers of tilted molecules packed together just as those found in monoclinic tert-butyl chloride. The colder phase of CCl3CH3 is orthorhombic, however. It achieves its 90° lattice angles by alternating tilt directions and CC13orientations every layer, in contrast to tert-butyl chloride which does not undergo analogous reversals. A packing analysis for tert-butyl chloride found a local minimum corresponding to the orthor-
hombic structure of the trichloride, but its energy was distinctly higher (less stabilizing) than that of the monoclinic form. Presumably, the difference between the colder phases of the two similar molecules is related to the fact that the C-Cl bond is longer than the C-C bond. This makes the tert-butyl compound prolate and the other, oblate. The present study corroborates the utility of clusters as model systems in explorations of structural and dynamic properties of condensed matter. Molecular dynamics simulations of clusters can provide detailed information about how molecules organize a t equilibrium and how they reorganize under different thermal conditions. The present results show that surprisingly small clusters can realistically model bulk systems. Despite severe limitations on the size of systems that can be treated and on the duration of affordable runs, it is clear that molecular dynamics simulations of clusters have added a fruitful new dimension to the study of phases and phase changes. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for the support of that part of the research covering the Monte Carlo simulations of the bulk liquid. The remainder of the research was supported by a grant from the National Science Foundation. We thank Drs. E. J. Valente and B. M. Powell for making available their unpublished neutron diffraction patterns of tert-butyl chloride and their computations of powder patterns based on the present MD structure. We also thank Mr. K. Kinney, Dr. S.Xu,and Dr. F. Dulles for their assistance in computations. References and Notes (1) Klein, M. L.; Lewis, L. J. Chem. Rev. 1990, 90. 459. (2) Bartell, L. S.;Dibble, T. S.;Hovick, J. W.; Xu,S. In The Physics and Chemistry of Finite Systems: From Clusters to Crystals; Jena, P., Rao, B. K., Khanna, S. N., Eds.; Kluwer Academic Publishers: Dordrecht, 1992; Vol. 1, p 71. (3) Honeycutt, J. D.; Anderson, H. C. J . Phys. Chem. 1987, 91,4950. (4) Etters, R. D.; Kaelberer, J. B. Phys. Rev. A 1975, 1 1 , 1068. (5) Beck, T. L.; Jellinek, J.; Berry, R. S.J . Chem. Phys. 1987,87, 545. (6) Raoult, B.; Farges, J.; De Feraudy, M. F.; Torchet, G. Philos. Mag. 1980, 860, 881. (7) Bartell, L. S.;Xu, S.J . Phys. Chem. 1991, 95, 8939. (8) Dibble, T. S.;Bartell, L. S . J . Phys. Chem. 1992, 96, 8604. (9) Fuchs, A. H.; Pawley, G. S. J . Phys. (Paris) 1988, 49,41. (10) Bartell, L. S.;Dibble, T. S.J. Phys. Chem. 1991, 95, 1159. (11) Bartell, L. S.;Chen, J. J . Phys. Chem. 1992, 96, 8801. (1 2) Powell, B. M.; Valente, E. J. Private Communication. (13) Bartell, L. S.;Chen, J. In preparation. (14) CRC Handbook of Chemistry and Physics, 63rd ed.; CRC Press: Boca Raton, FL, 1982. (15) Rudman, R.; Post, B. Mol. Cryst. 1968, 5, 95. (16) Ohtani, S.;Hasebe. T. Chem. Lett. 1986, 1283. (17) Hilderbrandt, H. L.; Wieser, J. D. J . Chem. Phys. 1971,55,4648. (18) Silver, L.; Rudman, R. J . Chem. Phys. 1972,57, 210. (19) Minkin, V. I. Dipole Moments in Organic Chemistry;Plenum Press: New York, 1970; p 200. (20) Smith, W.; Finchman. D. Program MDMPOL, CCP5 Program Library, SERC Daresbury Laboratory, Daresbury, UK. (21) Xu, S.Ph.D. Thesis, University of Michigan, 1993. (22) Quantum Chemistry Program Exchange, QCPE Program No. 481, Chemistry Department, Indiana University. (23) Kinney, K. Private communication. (24) Dibble, T. S.;Bartell, L. S.J. Chem. Phys. 1992, 95, 2317. (25) Turnbull, D.; Fisher, J. C . J. Chem. Phys. 1949, 17, 71. (26) Buckle, E. R. Proc. R. Soc. London 1961, A261, 189. (27) Turnbull, D. In Solid Stare Physics; Academic Press: New York, 1956; Vol. 3. (28) Goyal, P. S.;Nawrock, W.; Urban, S.; Domoslawski, J.; Natkaniec, I. Acta Phys. Pol. A 1974, 46, 399.
