J. Phys. Chem. 1996, 100, 6935-6941
6935
Molecular Dynamics Study of the Freezing of Clusters of Chalcogen Hexafluorides Kurtis E. Kinney, Shimin Xu, and Lawrence S. Bartell* Department of Chemistry, UniVersity of Michigan, Ann Arbor, Michigan 48109 ReceiVed: December 14, 1995; In Final Form: February 5, 1996X
Because knowledge about homogeneous nucleation in supercooled molecular liquids is largely indirect, a molecular dynamics investigation of the freezing of liquid clusters was initiated to furnish a plausible account of the molecular behavior involved. Results of the first stage of research are reported. Clusters with free boundaries were chosen instead of bulk systems in order to avoid the interference introduced by periodic boundary conditions. Systems of 150-molecule clusters of SF6, SeF6, and TeF6 were examined. Analyses of Voronoi polyhedra in the warm systems prepared confirmed that the clusters were genuinely liquid, containing no crystalline seeds capable of initiating freezing. As the clusters cooled, random structural fluctuations created short-lived embryonic nuclei. At deeper supercooling, a nucleus of critical size ultimately appeared in each cluster and freezing began. When cooled at a rate of 2 × 1010 K/s or more slowly, all clusters froze to bcc single crystals and these transformed to monoclinic single crystals upon further cooling. Voronoi polyhedra gave much more delicate and definitive analyses of the presence of solid nuclei than did other common indices such as the Lindemann δ. The polyhedra, however, were quite blind to the solid-state transition to monoclinic. It was found that the threshold value of the Lindemann index for freezing decreased systematically with increasing size of the molecules. The reported failure of similar systems to freeze in prior molecular dynamics simulations may have been due to the faster cooling rates adopted.
Introduction A recent perspective in the journal Science1 lamented how little is known about the molecular behavior underlying one of the commonest and simplest transformations of matter, namely that of freezing. Although the molecular mechanisms of many complex chemical reactions have been elucidated by detailed studies of the kinetics of the reactions,2 a comparable enlightenment has not been achieved for freezing. This is partly because of severe technical difficulties encountered in experimental studies of the kinetics of freezing. The ground-breaking research by Turnbull and colleagues3-5 on the dynamics of freezing began in an industrial research laboratory and was largely devoted to metallic systems. It was found that freezing is ordinarily initiated heterogeneously, i.e., by the catalytic effects of foreign particles. What was sought was the transformation initiated by homogeneous nucleation, a process in which random fluctuations in the structure of a pure supercooled liquid phase generate a nucleus of the solid phase large enough to form a template for continuous, spontaneous growth of the solid. Turnbull and his co-workers showed that, by subdividing the liquid sample into sufficiently small droplets, the fraction of droplets containing a heterophase contaminant could be reduced to an acceptable level. Although this was a major advance, it still left many difficult experimental problems to be overcome. A recent technique that avoids some of these difficulties while introducing others has supplanted the very small droplets by large molecular clusters formed in supersonic jets.6-8 Although the nucleation is genuinely homogeneous and the probing of the cluster structures by electron diffraction is successful in monitoring the rate of formation of critical nuclei, the technique is nevertheless blind to the structural fluctuations that create the critical nuclei. A promising alternative path in nucleation research, and one which does enable the detailed behavior of the molecules to be monitored, is the computer simulation of cooling clusters by X
Abstract published in AdVance ACS Abstracts, April 1, 1996.
0022-3654/96/20100-6935$12.00/0
molecular dynamics (MD) techniques.9-11 If a realistic intermolecular interaction function can be constructed, it is possible to follow the molecular trajectories that lead to nucleation. Indeed, many MD investigations of phaselike changes in extremely small clusters of atoms have already been made,12 and several have been carried out on clusters of polyatomic molecules.9-11,13-21 Quite a few have also been performed on the freezing of bulk atomic systems.22 As shown in a classic paper by Swope and Andersen,22 however, artifacts arising from the imposition of periodic boundary conditions have distorted conclusions derived from most simulations of the freezing of bulk systems. Enormous systems are required if these artifacts are to be avoided. Swope and Andersen also confirmed that the behavior of the system during the freezing of atomic liquids in MD simulations closely conformed to the behavior postulated by the classical theory of homogeneous nucleation. Although a number of reports of melting of molecular systems have appeared,14-16,19,21 virtually no reports of successful MD simulations of the freezing of such systems by unequivocally homogeneous nucleation have been published to date. To bypass the interference associated with periodic boundary conditions when moderately small systems are studied, and to avoid the penalty of working with truly large systems, it has been found advantageous to carry out simulations upon clusters whose boundaries are free. Although clusters are subject to substantial size effects in their thermodynamic properties, and despite the fact that only a modest fraction of the volume of a cluster can be considered to be effective in nucleation, nevertheless it has been shown that clusters can serve as convenient and realistic models of the bulk.8 Even though atomic systems are poorly modeled by clusters unless the clusters are very large,23-26 experience has shown that interiors of clusters of polyatomic molecules mimic the bulk surprisingly well in structure, configurational energy, and rotational diffusion, for clusters as small as 100 molecules. Therefore, our aim in the present investigation was to find whether liquid molecular clusters could spontaneously freeze to truly crystalline aggregates in MD runs. © 1996 American Chemical Society
6936 J. Phys. Chem., Vol. 100, No. 17, 1996
Kinney et al.
TABLE 1: Potential Parameters for the Seven Site, 6-12-1 Atom-Atom Interaction Functions molecule SF6 SeF6 TeF6
atom pair
σ, Å
, κJ/mol
S-S S-F F-F Se-Se Se-F F-F Te-Te Te-F F-F
3.405 3.165 2.943 3.780 3.322 2.920 4.294 3.553 2.940
1.260 0.2752 0.2184 1.380 0.5422 0.2130 1.376 0.531 0.2049
F chargea
r, Åb
0.175
1.561
0.185
1.67
0.25c 0.0d
1.815
a Partial charges on fluorines, in excess electrons. Charge on central atoms to preserve electrostatic neutrality. b Effective intramolecular bond length. c Partial charge consistent with those of the other molecules. d Partial charge adopted in the runs illustrated in the text.
Several properties of the chalcogen hexafluorides suggested that their clusters might make attractive systems to study. Their intermolecular interactions are known fairly well,27 and they display rapid solid-state transitions that can be followed, both in heating and in cooling runs, on the time scale of molecular dynamics simulations. In an earlier MD study,14 attempts to freeze clusters of SF6 were reported to be unsuccessful despite much trial. Subsequently, in this laboratory in an MD investigation of six TeF6 clusters of various sizes,17,19 five of the clusters froze upon cooling although the most likely one, the largest, did not. These findings were challenged on the basis that seeds of the crystals surviving the process which generated the liquid clusters might still have been present at the outset of the cooling runs of the molten clusters. Such a criticism could not be ruled out because the solid clusters had not been heated to temperatures far in excess of the melting points, and no definitive analyses had been performed to verify the absence of solid nuclei. In the present MD study, liquid clusters of SF6, SeF6, and TeF6 were successfully frozen. The methods used to prepare the liquid clusters and to confirm that they contained no seeds capable of initiating the crystallization are described in the following sections. Computational Details Molecular Dynamics Simulations. Simulations were carried out on 150-molecule clusters of sulfur, selenium, and tellurium hexafluoride. Additional analyses were performed on the outputs of two prior runs on 150-molecule tellurium hexafluoride clusters in which freezing had taken place. Molecules in all simulations were taken to be rigid octahedra with lengths of 1.561, 1.67, and 1.815 Å, for the S-F, Se-F, and Te-F bonds. Potential functions were of the form of seven-site, pairwiseadditive 6-12-1 atom-atom intermolecular interactions (Table 1) with no cutoffs imposed. Partial charges had been calculated by the proprietary program BioGraph/Polygraph but were close to those derived as outlined elsewhere27 from the observed crystal structure of TeF6 together with considerations of the electronegativities of S, Se, Te, and F. Runs for SF6 and SeF6 incorporated the BioGraph/Polygraph partial charges but, for trivial reasons, runs for TeF6 did not. In the earlier runs on TeF6,19 partial charges had also been taken to be zero. Each cluster was constructed to be as spherical as possible with monoclinic packing (C2/m) for SF6 and trigonal (P3hm1) for the others. It had seemed simpler to generate liquid clusters by melting the solid rather than by some other technique, and the initial solid structure was presumed to be immaterial to the melt. Molecular dynamic runs were performed with a modified version of the program MDMPOL,28 using time steps of 10 fs in all calculations. The crystalline clusters were heated from a
low temperature to 200 K (for TeF6) or to 180 K (for the others) in 10 degree increments per 5000 time steps. Of these time steps, 1000 were spent in a heat bath to equilibrate to the new temperature and 4000, at constant energy while thermodynamic averages were calculated. All clusters transformed to bodycentered cubic (bcc) before melting, then melted ∼20-50 degrees before the heating was terminated. Appreciable evaporation was experienced during the later stages of heating and also during the initial stages of cooling. Cooling was carried out at rate an order of magnitude lower than the heating to increase the probability of forming crystalline nuclei. Although the temperature steps remained 10 degrees in magnitude, the number of time steps at a given temperature was increased to 50 000 for SF6 and TeF6 (including 2000 in the heat bath) and 100 000 for SeF6 (with 1000 in the heat bath) until well after the freezing was completed. For SeF6 the cooling rate was maintained down to 10 K but for the other two clusters the cooling rate was increased 10-fold below 70 K (SF6 during the bcc to monoclinic transition) or 60 K (TeF6, well below the bcc to monoclinic transition). Because the solid state nucleation rate is much higher than that for freezing the abrupt change of cooling rate for SF6 was of little consequence. Coordinates and velocities were saved every 400 to 500 time steps. Analyses of Phase Changes and of Completeness of Melting. The standard techniques described in prior papers on phase changes were implemented, including Pawley projections,14 MACSPIN images of the clusters, and plots of configurational energies, Lindemann δ indices, pair correlation and angular distribution functions. The first two techniques afford the most intuitively appealing assessments of transformations and the last two were of marginal utility. None were sufficiently sensitive to structural characteristics of very small regions to recognize with adequate discrimination the presence in a melt of potential nuclei for freezing. Prior research had demonstrated that analyses based on Voronoi polyhedra,29-31 by contrast, are well suited for such purposes in atomic systems. Tests indicated that the polyhedra based on molecular centers of mass would also work well for our hexafluoride systems, even though they make no use of the orientational order that is an important property of molecular crystals. On the other hand, an earlier study19 had found Voronoi polyhedra to be ineffective in diagnosing the transition from the bcc to the monoclinic phase. In the present investigation, however, a Voronoi analysis was able to detect the presence of bcc-like aggregates in molten clusters even when the aggregates were much smaller than critical nuclei. Although fluctuating fcc, hcp, and icosahedral aggregates might also have appeared in our systems as they did in atomic systems that ultimately froze to fcc and hcp regions22 we did not test for their presence because they are of no importance in the freezing process of the hexafluorides. Because prior diagnoses based on Voronoi polyhedra had been applied to bulk systems which tend to be more homogeneous than clusters, we relaxed the criterion for bcc recognition to cover the possibility that regions clearly identifiable as crystalline in small clusters might conform to less strict rules than those for the bulk. Inasmuch as our purpose was to be able to rule out, if possible, the presence of crystalline seeds in our melts, it seemed better to err on the side of including false nuclei rather than to miss bona fide nuclei. To accomplish these ends we proceeded empirically. An indication of the structural environment of a given molecule is the shape of the Voronoi polyhedron encompassing it. The polyhedron is characterized by the number of its faces and the number of edges of its various faces. For a perfect, infinite, bcc crystal the polyhedra have 14 faces, six with four edges and eight with six edges.
Freezing of Chalcogen Hexafluorides
J. Phys. Chem., Vol. 100, No. 17, 1996 6937
TABLE 2: Voronoi Polyhedra Found in 150-Molecule bcc Reference Cluster of SF6 mode of prep I
II
III
no. of faces
n3
n4
index n5
n6
n7
percent
12 13 13 13 14 12 12 13 14 12 13 13 14 14
0 0 0 0 0 0 0 0 0 0 0 0 0 0
5 3 5 6 6 4 5 5 6 4 3 5 4 6
2 6 2 3 0 4 2 2 0 4 6 2 4 0
5 4 6 4 8 4 5 6 8 4 4 6 6 8
0 0 1 0 0 0 0 0 0 0 0 0 0 0
