Molecular dynamics study of the phase transitions in sulfur

Oct 1, 1993 - Potential Function for Tellurium Hexafluoride Molecules in the Solid. Kurtis E. Kinney and Lawrence S. Bartell. The Journal of Physical ...
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J. Phys. Chem. 1993,97, 10472-10477

10472

Molecular Dynamics Study of the Phase Transitions in Sulfur Hexafluoride Clusters of Various Size Franck M. BeniBre, Anne Boutin, Jean-Marc Simon, and Alain H. Fuchs' Laboratoire de Chimie-Physique des Matbriaux Amorphes, URA 1 1 04, Bat. 490, Uniuersitb de Paris-Sud, F 91 405 Orsay, France

Marie-Franqoise de Feraudy and Gerard Torchet Laboratoire de Physique des Solides, URA 2, Bat. 510, Universiti de Paris-Sud, F 91405 Orsay, France Received: April 27, 1993"

Molecular dynamics (MD) simulations have been performed on free clusters of sulfur hexafluoride (SFa) of various size: 229,137, and 5 5 molecules. Some preliminary simulations of 13-molecule clusters have also been performed. A systematic study of the melting and solid-solid transitions and the effect of cluster size on their temperature spreads was undertaken. The stable structures of the 229-, 137-, and 55-molecule clusters are bcc a t elevated temperatures and C2/m monoclinic a t low temperature, as in the bulk crystal. The melting transition is rounded and shifted to low temperatures when decreasing the cluster size. In the whole size range, from the thermodynamic limit down to 55 molecules, melting initiates a t the surface. No sign of another mechanism for melting, such as the "coexistence" between solid-like and liquid-like clusters, was found in our simulations. The structural monocliniGcubic transition is also initiated a t the surface. From 229 to 5 5 molecules the transition temperature decreases roughly linearly with N-1/3, Some experiments and simulation results seem to indicate that the transition temperature does not vary much in the size range of -500 up to the thermodynamic limit. In the size range between 5 5 and 13 molecules, a transition occurs between crystalline "bulk-like" and noncrystalline properties in which icosahedral arrangements can be observed. This is roughly 2 orders of magnitude below the range in which the same transition occurs in argon.

Introduction Polyatomic systems are known to exhibit a richer diversity of structures and dynamics than do the atomic systems.' An illustrative example of the difference in behavior between monoatomic and a very simple quasispherical molecule such as sulfur hexafluoride (SF6) is provided by the studiesof free clusters. Well-known electron diffraction experiments by the group of Farges2J have shown that small clusters of argon exhibit a noncrystalline,polyicosahedralthen multishell icosahedral, form and begin to adopt the bulk cubic structure when they reach a sizeofabout lOOOatomsor more. On theother hand,experiments and computer simulations have shown that clusters of SF6 exhibit a crystalline packing arrangement when they are much smaller (of the order of 100 molecules).610 Bartell and his group have shown that this was also true for other hexahalide molecules such as SeF6 and TeF6." A similar conclusion was drawn from experiments and computer simulations carried out on clusters of small quasispherical molecules such as (CH3)3CCl,l*CH3CC13,13 and CC14.14 Sulfur hexafluoride provides an interesting case of a small and highly symmetric molecule for which extensive experimental and theoretical data are available and, moreover, a realistic model potential exists. An equilibrium phase transition is known to occur between the stable low-temperature monoclinic and the high-temperature (plastic) body-centered cubic phase. This phase transition has been observed in bulk material by calorimetry,lSJ6 X-ray scattering," NMR,18 and neutron scattering.19 What emerges from these experiments is the picture of a first-order transition with little or no thermal hysteresis, occurring at -94.5 K. The possible existence of a low-temperature metastable hexagonal phase has been discussed in detail in a preceding paper20 and is somewhat beyond the scope of this work.

* Author to whom correspondence should be addressed.

e Abstract published in Advance ACS Abstracts,

September 1, 1993.

0022-3654f 93f 2097- 10472$04.00f 0

Experiments and molecular dynamics (MD) simulations performed on free clusters of several hundred of molecules have shown that a phase transition occurs between the same monoclinic and cubic phases as observed in bulk ~amples.~-"J A temperature spread of this transition has been shown to take place. Starting from the low-temperature structure the cluster progressively adopts, on heating, the high-temperature body-centered cubic (bcc)structure through molecular reorientations experienced first by surface and then by core molecules, whereas, on cooling, the growth of the monoclinic structure is initiated in the inner regions of the cluster.8 Intermediate experimental diffraction functions were in very good agreement with the calculated patterns and have thus been attributed to the evolution of single clusters rather than to the coexistenceof clusters with different structures as one could have suggested. The transition is spread over -30 K in a cluster containing 512 molecules. This phenomenon can be tentatively attributed to a finite size effect, but many questions concerning the size/temperature/structure behavior of SFa and molecular clusters in general remain unanswered.2' One of these questions can be put in the following way: for small cluster sizes is there a "transition" from a size domain in which essentially the macroscopic crystalline solid-state properties are observed to a size domain in which the stable arrangements are noncrystalline and specific size effects take place? In this work MD simulations have been performed on clusters of various decreasing size: 229, 137, and 55 molecules. Some preliminary simulations of a 13-moleculecluster have also been performed. A systematic study of the reversibility of the solid phase transition and the effect of cluster size on its temperature spread was undertaken. Finally, the melting transition of these clusters has been investigated in order to complete a previous MD work performed on larger clusters.22

Computational Method As in all the previous MD simulations on SF6, the potential used here is the one proposed by Pawley in 1981,23 in which a 0 1993 American Chemical Society

Phase Transitions in SF6Clusters single Lennard-Jones fluorine-fluorine atom-atom potential is used to compute the intermolecular interactions. Given the standard form of the pair potential,

The Journal of Physical Chemistry, Vol. 97, No. 40, 1993 10473 the surface effects. We define, somewhat arbitrarily, a “surface molecule” by the fact that it has less than 8 nearest neighbors. Using this criteria, -60% of the molecules of a 137-molecule cluster belong to its surface.

Results the value o f t = 0.14 kcal mol-’ has been used here, as in all the previous simulations. The value of u = 2.7 A was used in the simulations of refs 7, 8, 10, 20, and 23. This value had been slightly modified to u = 2.8591 A in 1987” as a result of a fit between a MD simulation and neutron diffraction measurements. This readjusted potential has been used in the simulations of refs 9 and 22. This change in u does not alter significantly the phase behavior of SF.5.25 The monoclinic structure obtained when using the readjusted potential is quantitatively closer to the experimental one, but the choice was made here to use the “old” potential in order to be able to compare the results obtained here with those of some previous works on SF6 clusters. A cut-off distance of 13 A was used which corresponds approximately to 4 intermolecular distances. Two clusters were made by taking a (bcc) plastic phase configuration at a temperature of 115 K and carving out spherical clusters containing 229 and 137 molecules, respectively. Both clusters have their outer shells completed. A third cluster was made at a lower temperature (60 K) by placing 55 molecules at the lattice sites of the bcc structure in such a way that the cluster was as spherical as possible. A fourth cluster was made at the same temperature by placing the 55 molecules at the sites of a regular two-shell icosahedron. The distance between the centers of mass of the nearest neighbor molecules in this latter configuration was initially fixed at a value of 5.5 %I which is slightly higher than the equilibrium intermolecular distance in the bcc structure. The 55-particle species corresponds to a “magic number” for the packing of soft spheres leading either to an icosahedral or to a cuboctahedral arrangement. The simulations of the third and fourth clusters were aimed at testing which was the most stable arrangement for a 55-molecule cluster of SF6. The first two clusters were cooled down from 120 to 50 K by steps of 5 K and then warmed up again in the same way. Both 55-moleculeclusterswere equilibrated a t -60K. Theone having initially an icosahedral arrangement transformed spontaneously and irreversibly into a crystalline cluster, while the initially crystalline cluster remained stable in the time scale of a MD run (a few thousand picoseconds). This latter cluster was slowly cooled down to 50 K and reheated by steps of 5 K up to 95 K. No boundary conditions were used. The positions and quaternions coordinates were time stepped using the Beeman algorithm which is accurate to the same level of approximation as the more commonly used Verlet algorithm. A velocity rescaling was applied in order to change the mean temperatureof theclusters. Except during theinitial stabilization stage (during which a very small time step was used), we have used a time step of 5 X 10-3 ps. The equilibration stage lasted between 100 and 300 ps, as indicated by the potential energy becoming stable, after which the simulations have been performed at constant energy for roughly 100 ps. The results shown and discussed below come out of the constant-energy simulations. The calculations have been performed on a massively parallel transputer-based computing surface. More details about the technique used for these simulations are given in ref 22. A ring array of 6 transputers was used for most of the calculations presented here. The rate of calculated interatomic interactions for such a configuration is roughly equal to 14 000 per second, the cycle time being 2.3 and 1.7 s for the 229- and 137-molecule cluster, respectively. In what follows we have distinguished the behavior of the core molecules from the whole cluster in order to take into account

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A. Structural Analysis. A structural analysis of the stable crystalline 229-, 137-, and 55-molecule clusters was undertaken on both sides of the structural transition, using the method described in detail in ref 9. The 229- and 137-molecule clusters were built in the bcc structure and remained stable in the same arrangement a t elevated temperature, in the time scale of a MD run. The bcc structure is characterized by identical mean molecular orientations. At 110 K, the cell parameter, a,, is identical in the center for both the 137- and 229-molecule clusters. The value of the cell parameter (averaged over 18 configurations) is 5.71 0.05 A. This value is slightly lower than the experimental value of 5.78 A at 100 K.17 This deviation from experiment is essentially due to a potential effect. In a previous simulation using the readjusted potential we found a value of 5.86 f 0.05 A26 at the same temperature for a 229-molecule cluster and 5.84 f 0.05 A for a sample with periodic boundary conditions. It has been shown elsewhere that both potentials lead to a thermal expansivity which is close to the experimental fir1dings.2~As already indicatedabove, the most stable arrangement for the 55-molecule cluster is crystalline. A direct comparison between the lattice parameters of the high-temperature form of the 55-moleculecluster and those of the 137- and 229-molecule clusters cannot be made at the same temperature since the former is already liquid a t 100 K. However, the structural analysis shows that it is also bcc with a, = 5.7 f 0.1 A at 75 K. When cooled down to low temperatures, the three clusters undergo a spontaneous structural transition. The reverse transition takes place when the systems are heated up again. After this heat treatment, identical values of the cubic cell parameter are found a t identical temperatures, within statistical uncertainties. This provides some further evidence that the most stablestructure for a SF6 cluster model at high temperature (using Pawley’s potential) is bcc, as long as the cluster contains more than 55 molecules. The low-temperature structures have been identified as being C2/m monoclinic ( Z = 3) for the three clusters studied here. The molecules possess two different molecular orientations in proportion 1:3 and 2:3. This low-temperature arrangement was also found in our earlier simulations on 5 12molecules and more details on this structure can be found in ref 9. The lattice parameters for the three cluster sizes at 50 K are given in Table I. The structural parameters are identical, within statistical fluctuations, for the 229- and 137-molecule clusters. These parameters were calculated for the core of the clusters. For the smallest 55-moleculecluster,the lattice parameters aregiven for the whole cluster. Thisis because thediameter ofthe “center”ofthiscluster, using the definition of surface and core molecules given above, is too small to perform a proper analysis of the monoclinic structure (-6 A as compared to 13 A for one of the cell parameters). This presumably accounts for the fact that the monoclinic cell of the 55-molecule cluster is slightly expanded with respect to the other two clusters a t the same temperature. The potential used here leads to a more compact monoclinic structure than the polycrystalline experimental (the cell volume is 8% lower). On the other hand, the readjusted potential leads to a more expanded structure (the cell volume is -3% higher than in experiments). In spite of these discrepancies, the main features of the cluster behavior are well reproduced by MD simulation, as evidenced by the excellent agreement between the calculated diffraction functions, whatever the potential used, and the experimental diffraction patterns.

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10474 The Journal of Physical Chemistry, Vol. 97, No. 40, I993 TABLE I: Monoclinic Unit Cell Parameters, Molecular Coordinates, and Unit Cell Volume for the 229-, 137-, and 55Molecule Clusters. For the 229- and 137-Molecule Clusters the Parameters Are Given for the Core of the Cluster and the Values Were Averaged over 18 Configurations. For the 55-Molecule Cluster the Parameters Are Given for the Whole Cluster and the Values Were Averaged over 60 Confieurations 229-molecule 137-molecule 55-molecule cluster cluster cluster cell parameters/A a 13.34 f 0.06 13.36 f 0.05 13.56 f 0.08 b 7.98 f 0.02 7.97 f 0.02 8.14 f 0.05 c 4.62 f 0.07 4.62 f 0.02 4.67 f 0.02 anglesldeg U 90.0 f 0.3 90.0 f 0.4 89.9 f 0.4 98 f 2 B 95.5 f 0.2 95.6 f 0.9 7 90.0 f 0.6 90.2 f 0.4 90.3 f 0.8 molecular coordinates ~ 2 1 0 0.333 f 0.001 0.335 f 0.003 0.332 f 0.003 Yzlb 0.000 f 0.002 0.001 f 0.006 0.002 f 0.008 z2Ic 0.325 f 0.005 0.325 f 0.009 0.4 f 0.2 xz‘/a 0.666 f 0.001 0.667 f 0.002 0.66 f 0.01 Yilb 0.000 f 0.003 0.001 f 0.004 0.001 f 0.006 z2’Ic 0.662 f 0.004 0.613 f 0.008 0.69 f 0.01 unit cell volumelbr’ V 490 f 2 490 f 4 511 f 4 B. Electron Diffraction Functions. In Figure 1 are shown two calculated diffraction functions for the 229-molecule cluster at 100and 50 K. Both functions are very similar to the experimental diffraction patterns recorded at high and low temperatures for a mean cluster size estimated to a few hundred molecules (Figure 2). This good agreement between simulation and experiment confirms what had been found earlier in the case of a 5 12-molecule cluster, using the readjusted ~ t e n t i a l .More ~ confidence is to be placed in the results presented here since the calculated diffraction functions have been obtained by averaging over a set of well-stabilized configurations. Sufficient care was taken in order not to obtain the oscillation of the data at low temperature observed in the previous work9 which was presumably due to a lack of equilibration. C. Temperature Spread of the SolidSolid Phase Transition and Cluster Size Effect. We have sketched in Figure 3 the evolution of the mean potential energy as a function of temperature for the 229-molecule cluster model. Structural information, as deduced from the calculated diffraction patterns, is represented in the following way: full symbols refer to a cubic type pattern, open symbols to the monoclinic structure, and half-full symbols to the intermediate (monoclinic cubic) structures. The squares correspond to the cluster configurations obtained on cooling the initial bcc model and the circles to those obtained on reheating the cluster model. Theinitiallycubicclustercan becooleddown to -92Kwithout observing noticeable changes in the diffraction pattern. At this stage a spontaneous transformation occurs which leads to an intermediate pattern similar to those obtained in our previous work on the 5 12-molecule model.9 This irreversible transformation can be interpreted as due to a spontaneous nucleation of the monoclinic structure in the core of the cluster.* However, the cluster model is not fully monoclinic a t this stage. A further lowering of the temperature leads to a progressive transformation of the patterns from an intermediate to a purely monoclinic structure. On reheating the cluster the same equilibrium thermodynamic path is followed and no “superheating” of the monoclinic structure is observed a t low temperature, presumably because the transformation initiates at the surface and thus needs no real “seed”. So, apart from the slight retardation observed on cooling the cubic cluster model, the phase transition, which is spread over a range of -20 K, seems to be reversible. Very similar diagrams were obtained in the case of the 137- and 55-

+

B6nibre et al.

0

5

s I A-‘

15

10

Figure 1. Calculated diffraction functions for the 229-molecule cluster at two temperatures. (A) 100 K, (B) 50 K.

\

5

B

10 s I ,&-I

Figure 2. Experimental diffraction patterns for a mean cluster size of a few hundred SFs molecules. (A) “warm” clusters, (B) “cold” clusters. Details on the experimental procedure are given in ref 9.

molecule cluster models, except that the phase transition takes place a t lower temperatures when decreasing the cluster size. Diffraction functions have been calculated a t every temperature. In order to account correctly for experimental patterns it has been necessary to perform an average over at least 6 configurations of the 229-molecule cluster, for T < 90 K and 15 configurations for T > 90 K. The calculations have been made separately for the whole cluster and for its core. The ratio of two line heights, namely, the (310) and (431) lines, measured on the diffraction patterns has proved to be a good index of the progressive structural change, especially in the intermediate range.s.9 This ratio is shown in Figure 4 for the 229-molecule cluster. In the whole cluster, thelower limit (the “beginning”) of the transition is roughly equal to 85 K but cannot be precisely determined because of the very smooth increase of the ratio of line heights when heating the cluster model. This results from the fact that the monoclinicto-cubic transition is a progressive disordering process initiating at the surface of the cluster, the core molecules being affected at a slightly higher temperature -93 K. The upper limit of the transition corresponds to the point at which the cluster is entirely cubic, and this point is much easier to detect. As shown in Figure

Phase Transitions in SF6 Clusters

The Journal of Physical Chemistry, Vol. 97, No. 40, 1993 10475

0 0 Cubic+Monoclinic 0 Monoclinic

M \ bE

-19

t

1 50

0

0.05

0.1

0.15

0.2

0.25

0.3

N -113 Figure 5. Clusters melting temperatures as a function of N-lI3. Nis the

numberofmoleculesinthecluster (965,531,229,137,55). Thestraight

line is the least squares fit.

N = 229 Q

whole cluster

0

Core molecules

,

0.0'' 40

"

I

60

'

"

"

80

"

" " "

100

120

" e140'

'

" 160

T/K

Figurel Ratio of two line heights measured on the calculateddiffraction function of the 229-molecule cluster as a function of mean temperature. Structures are reproduced as in Figure 3. The lines are guides to the eye (full line, whole cluster; dotted line, core of the cluster).

4, the upper limit for the 229-molecule model is equal to 105 K, the uncertainty in the absolute temperature being f 2 K. This value is identical to the value provided by the potential energy curve in Figure 3. The same computation made on the 137- and 55-molecule models showed the same qualitative behavior of the line-heights ratio, with an upper limit of the transition range of 95 and 66 K, respectively, in good agreement with the values provided by the potential energy curves. The lower limit of the transition is approximately 75 K for the 137-moleculecluster. In the case of the 55-molecule cluster, the low resolution of the diffraction functions, which is not surprising when dealing with such small systems, makes it more difficult to determine the beginning of the monoclinic-to-cubicchange. A rough estimation gives 55 f 10 K. The study of three clusters containing 229, 137, and 55 molecules has shown that the progressive transition between the monoclinic and the cubic phases is reversible. The spread of the transition is equal to roughly 20 K for the 137- and 229-molecule clusters. Surprisingly enough, this temperature spread seems to be lower for the 55-polecule cluster. It should be stressed, however, that only the upper limit of the temperature range of the transition is clearly detectable. Our simulations unambiguously show that the upper limit of the structural transition increases with an increasing cluster size (66 K for the 55-molecule cluster, 95 K for the 137-molecule cluster, and 105 K for the 229-molecule cluster). This is in agreement with the recent experimentssuggesting that small cubic clusters are produced at lower temperatures than larger ones.6

D. The Melting Transition. When heated above the solidsolid transition temperature the three clusters studied here undergo a melting transition. The melting process is similar to what had been previously observed in the 965- and 229-molecule clusters and described in detail in ref 22. An examination of the caloric curves, the orientational order parameter, and the results of the computation of the surface and core translational diffusion coefficients shows that melting initiates a t the cluster surface and then propagates into the core. This can also be seen in the evolution of the ratio of line heights shown in Figure 4 for the 229-molecule cluster. Between 110 and 125 K the ratio of the (310) to (431) line heights for the core of the cluster increases slowly with temperatureas expectedfor a crystallinesamplesince the Debye-Waller factor produces a larger damping for the (43 1) line. On the other hand, the value of the same ratio, calculated for the whole cluster, decreases continuously with temperature. In the range between 110 and 138 K the diffraction function of the whole cluster becomes structureless becauseof the occurrence of a liquid-like film at its surface. Even in the smallest 55-molecule cluster it is possible to distinguish between the orientational and translational disorder of the "surface" molecules, which, immediately above the structural transition, is higher (on the average) than in the "core" molecules. A marked change is detected in the slope of the caloric curves. At this point the cluster is entirely liquid. This point is taken as the melting point of the cluster. In Figure 5 we have reported the melting points of the 55-, 137-, and 229-molecule clusters, together with the previously observed value for the 229- and 965molecule cluster.22 Since these latter points were obtained with the use of the readjusted potential, we have performed some additional simulations of a 531-molecule cluster, using both potentials, inorder to test theeffect of the potentialon themelting point. As can be seen in Figure 5, the use of the readjusted potential leads to slightly higher melting points (less than 5 K higher). This difference lies in the range of uncertainties due to the statistical fluctuations. The lowering of the melting point is linear with N-l/3. This has also been observed in experiments2""J and simulations31 of metal clusters and can be understood in terms of the thermodynamic Gibbs-Thompson equation. An extrapolation of the fitted straight line to the thermodynamic limit leads to a value of 235 f 5 K for the equilibrium between the bulk bcc solid and liquid phases. This is in fair agreement with the known triple point temperature of 223 K.15 The simulation of melting of clusters of various sizes provides an interesting way of estimating the bulk equilibrium temperature of the model system since the direct computation of free energies is known to be a difficult task in plastic cry~tals.~2

BCnibre et al.

10476 The Journal of Physical Chemistry, Vol. 97, No. 40, 1993

Discussion A. The Phase Diagram of SF6 Clusters. The simulations performed in this work show, in agreement with experiment, that very small clusters of SFs exhibit bulk-like properties. In this section we shall try to put together the available simulation and experimental data. The aim is to determine what is still needed before a complete size-temperature phase diagram can be provided. Generally speaking, a lowering of a first order phase transition temperature with a decreasing cluster size is expected from theoretical considerations based on both finite-size and surface effects.33 This is what we have found for the melting transition (Figure 5). In the case of the solid-solid transition we have also observed a lowering of its upper limit in the size range 229-55 molecules. In a previous work on a 5 12-molecule cluster9we had found an upper limit of 120 K, but this latter result was obtained with the readjusted potential. In order to test the effect of the change in potential we have performed an extra simulation of a 53 1-molecule cluster (53 1 corresponds to a complete outer shell), using the so-called “old” potential.27 We have found an upper limit of 115 K. There is of course an uncertainty in all the transition temperatures given here, due to the statistical fluctuations in a finite-size system. This amounts to -5 K for a cluster containing several hundred molecules. So it seems that there is no significant difference between the transition temperatures obtained using either the “old” or the readjusted potential. From 55 to 53 1molecules, thestructural transition temperature increases regularly and roughly linearly with N-1/3.An extrapolation of the linear fitted straight line to the thermodynamic limit leads to a prediction of 165 K for the bulk equilibrium temperature between the two solid phases. This temperature is quite a bit higher than the known calorimetric value of 94.5 K.I5J6 This is a puzzling result since the same kind of extrapolation for the melting transition gave an acceptable equilibrium value for the model system. Since the bulk transition temperature for the model SF6 was not known, we have undertaken a simulation of a sample of 432 molecules with periodic boundary conditions. The details of the simulation and the full account of the results obtained are given e l s e ~ h e r e The . ~ ~ main result of this simulation, with respect to the matter discussed here, is that the estimation of the equilibrium transition temperature of the two solid phases is 110 f 15 K, a value which is in fairly good agreement with the calorimetric one. This seems to indicate that the structural transition temperature in clusters does not scale linearly with N-‘/3 in the entire size range between 55 molecules and the thermodynamic limit. The reason for this is not clear. However, it is consistent with experiments which seem to indicate that free clusters of SF6 having their sizes between roughly 500 and 3000 molecules have a similar phase transition temperature.6 Obviously more simulation work is needed in the size range of several thousand molecules in order to confirm these findings. Another point of interest is the size effect on the temperature spread of the transition. For finite systems, a first-order transition is expected to be shifted to lower temperatures, as discussed above, and is also expected to be spread over a certain range of t e m p e r a t ~ r e . ~As~ N tends to zero, the transition may then completely vanish. In the case of the melting transition, the temperature spread of the transition is difficult to estimate. An extensive MD investigation of the thermal disordering of the three low-index faces of SF6 has shown that, below the triple point temperature, a quasiliquid layer (QLL)wets the solid surfaces. This QLL has some liquid-like properties (for instance translational mobility) but retains some features of the solid (site-to-site molecular jumps and some long-range order). Only very near the triple point (at a reduced temperature between 0.90 and 0.98) does the surface behave, to all intents and purposes, as if it was a true liquid. The estimation of the onset of melting is somewhat arbitrary, since

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it will depend on the criteria used for its definition. A rather good criterion would be the one based on the first Fourier component of the density Pk for one of the smallest reciprocal wave vectors. This quantity must be equal to zero for a perfect bulk liquid. Unfortunately such a calculation is not possible for a shell of a free cluster which exhibits many different crystallographic orientations. This is why we have to rely on indirect evidence of liquid-like behavior such as surface thermal diffusivity. When using such a criterion, we agree with Bartell and Chen14 on the fact that melting takes place over a very broad range of temperature. These authors have pointed out, in the case of CC14, that the temperature spread of the transition is larger than predicted by the simple model of Reiss et al.35 In the case of the SF6 clusters studied here, immediately above the structural transition, the apparent activation enthalpy for surface translational diffusion is similar to the value observed in the bulk liquid. This means that the temperature spread of the melting transition is roughly equal to the range of existence of the bcc solid phase. This range of existence decreases with the decreasing cluster size, and thus the specific size effect on the temperature spread of melting cannot be estimated. However, it should be pointed out that this transition is still very well defined for the 55-molecule cluster (a marked change of slope of the caloric curve is clearly seen a t -89 K). The solid-liquid transition has not vanished at low cluster sizes. In the case of the structural transition, we have seen above that a close examination of the changes in the diffraction functions and in the ratio of line heights makes it possible to estimate the temperature spread of the transition. A temperature spread of -30 K was estimated for the 531-molecule cluster mode1.27 We have no information yet on the behavior of larger clusters. Surprisingly enough, the structural transition in smaller clusters seems to be spread over a smaller range of temperature (- 20 K for the 229- and 137-molecule clusters and 10 f 10 K for the 55-molecule cluster). One must be cautious not to draw definitive conclusions based on such tenuous evidence. It must be stressed, however, that, as in the case of melting, the structural transition has not vanished at low cluster sizes. M D simulations are in progress in order to complete this work on the temperature-size phase diagram of SF6. Whether or not two regimes exist in this phase diagram, a constant transition temperature above a size of 500 molecules and a decreasing transition temperature below this size, is a question which remains to be elucidated. B. Transition from Bulk-Like Crystalline Properties to NoncrystallineBehavior. As already mentioned above, the icosahedral arrangement is not stable for the 55-molecule cluster model of sF6. The most stable arrangement is a crystal in which the static and dynamic disorder is enhanced with regard to the larger clusters. This is not surprising since 80%of the molecules belong to the surface. However, when performing an average over a sufficiently large number of configurations (-60) in order to find the mean lattice parameters, there is no doubt that the highand low-temperature structures are similar to those observed in the 137-, 229-, and 5 12-molecule clusters. Moreover, an analysis of the local structures performed with the method of the Voronoi polyhedra36showed no sign of icosahedral arrangement (1 2-face polyhedra). This confirms that for this cluster size, SF6 still exhibits essentially the macroscopic solid-state properties with an enhanced disorder because of the very important surface-tovolume ratio. Conversely the 13-molecule cluster model appears to lie in the noncrystalline size domain. A cluster was made with an initial arrangement of the center of mass of the molecules corresponding to a perfect icosahedron. It remained stable for a few thousand picoseconds in a close-to-icosahedral arrangement, as evidenced by the existence of three peaks in the histogram of the center of mass distances. The ratios of these three average distances

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Phase Transitions in SF6Clusters correspond to what is expected for an icosahedron. A second cluster was made by placing the 13 molecules at the lattice sites of a bcc structure with a cell parameter which was guessed by extrapolatingthe high-temperature data. During the stabilization step this cluster transformed irreversibly into a closeto-icosahedral arrangement. These are only preliminary results. Work is also in progress in order to determine whether or not a 19-molecule cluster also shows noncrystalline properties. In this size range, electrondiffraction experimentsare no longer available. However, an indirect test of these results is provided by the experiments of Echt et al.3' on small ionized clusters [ ( ~ F ~ ) SFs+]. N - I These authors have found by mass spectrometry a pronounced peak at N = 13 but not at N = 55. This seems consistent with our MD simulations.

Conclusion

M D simulations of sF6, using a reliable model potential, have shown that the transition between noncrystalline and crystalline "bulk-like" properties occurs in the size range between 13 and 55 molecules. This is roughly 2 orders of magnitude below the range in which the same transition occurs in argon. That a cluster containing as few as 55 quasispherical molecules behaves like a small crystal is something which was not expected in the beginning of this work. The melting transition is rounded and shifted to low temperatures when decreasing the cluster size. In the whole size range, from 55 molecules to the thermodynamic limit, melting initiates at the surface. No sign of another mechanism for melting, such as the "coexistence" between solid-like and liquid-like clusters,3* was found in our simulation. Work is in progress in order to find whether or not such a mechanism exists in the 13-molecule cluster. The structural monocliniocubic transition is also initiated at the surface. Some questions remain open with regard to the effects of size on the rounding and shifting of the transition temperature. Some experiments and simulation results seem to indicate that the transition temperature does not vary much in the size range of -500 up to the thermodynamic limit, while, between 500 and 55 molecules, the transition temperature decreases roughly linearly with N-113. In the near future we hope to be able to provide a full temperature-size phase transition for complete shell SF6 clusters. References and Notes (1) Bartell, L. S.;Dibble, T. S.;Hovick, J. W.; Xu, S.NATO Advanced

Study Institute SeriesC374; Physics and Chemistry of FiniteSystems: From Clusters to Crystals; Kluwer Academic Publishers: Dordrecht, 1992; p 71.

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