J . Phys. Chem. 1993, 97, 13550-13556
13550
Molecular Dynamics Studies of Melting and Solid-state Transitions of TeFs Clusters Shimin Xu and Lawrence S. Bartell' Department of Chemistry, University of Michigan. Ann Arbor, Michigan 481 09 Received: July 28, 1993; In Final Form: October 20, 1993'
Clusters of TeF6 composed of 100, 128, 150, 250, and 350 molecules were examined by molecular dynamics simulations to analyze structural, energetic, and kinetic aspects of phase changes. Crystalline phases studied included body-centered cubic and four different, particularly compact structures. As had been observed experimentally for large clusters, the bcc phase transformed at a n extremely rapid rate, upon fast cooling, to a monoclinic phase unknown in the bulk, for reasons that are now well understood. Melting and solid-state transitions were reproducible when clusters were heated but not when they were cooled, owing to the element of chance in the nucleation step involved. Transitions were spread out over an appreciable range of temperatures and began at the surface when clusters were heated and in the interior when they were cooled. Conventions for defining characteristic transition temperatures were introduced. These transition temperatures decreased approximately linearly with the reciprocal of cluster radius, for the solid-state phase change as well as for melting. All of the above behavior of clusters is in good qualitative but poor quantitative accord with conventional capillary theory based on molecularly sharp interfaces. Limited comparisons are made between simulations including and excluding effects of partial electrostatic charges on the atoms.
Introduction Clusters offer fresh insights into the properties of condensed matter, For this reason they are the subject of increased attention by workers in many areas of physics and chemistry. One particularly promising area is that concerned with the nature of nucleation and the dynamics of phase transitions.' Clusters have yielded new information in this area both in experimental studies and in computer simulations. A technique has been developed in this laboratory for measuting nucleation rates in phase changes in unsupported molecular clusters monitored by electron d i f f r a ~ t i o n . ~By - ~ observing the time evolution of transitions in highly undercooled condensed matter in a supersonic jet, nucleation rates could be determined on a microsecond time scale both in freezing and in solid-state transformations. Observed rates were many orders of magnitude faster than those found by more conventional methods in less highly undercooled systems of larger particles. Faster yet by many more orders of magnitude were nucleation rates determined in clusters in the course of molecular dynamics (MD) simulation^.^-' Simulations had been undertaken because the electron diffraction investigations, while informative about structures and rates of change, yielded little information about very small clusters or about the critical nuclei believed to be the precursors of change in phase transitions. Size/temperature properties of small clusters were thought to hold the key to the diversity of cluster structures that could be generated under different conditions by condensation of vapors in supersonic expansions. The possibility of obtaining direct information about critical nuclei was an even greater incentive for carrying out realistic simulations of cluster properties. In MD simulations, clusters are more amenable than bulk systems for several reasons. One of the most important is that, unless the bulk system is extremely large, periodic boundary conditions place severe constraints upon the nature of the transition allowed if crystalline phases are considered. We have noticed no such constraints in clusters where we have seen the same transitions in simulations that happen in the laboratory in bulk systems (e.g., cubic' and tetragonal' to monoclinic, orthorhombic to cubic*). Of course, the high proportion of molecules in clusters that are in the surface phase introduces certain problems of interpretation, but this Abstract published in Aduance ACS
Abstracts.
December 1, 1993.
penalty is tolerable. What is derived is direct information about the molecular basis of properties of liquids and solids. Although a considerable number of computer simulations of clusters have been described in the literature, most of them have been performed on systems of atomic clusters, particularly clusters of Lennard-Jones sphere^.^-^^ Studies of molecular clusters offer elements of interest not found in clusters of atoms. Even small, quasi-spherical polyatomic molecules such as SF6 exhibit a qualitatively different behavior in clusters than do Lennard-Jones sphere^.^^^^^"^^ For some reason, molecules can spontaneously form stable crystalline structures characteristic of the bulk when they are 1 or 2 orders of magnitude smaller than the smallest clusters of argon that are able tocrystallize. Moreover, molecular clusters exhibit a much richer diversity of solid phases as well as other features deserving consideration that are not found in atomic clusters, such as rotational diffusion and rotational melting. Therefore, it seemed worthwhile to initiate computer simulations of small molecular clusters in an effort to shed light on the molecular activity that is presently too difficult to deduce from experiments. An attractive system to begin with is tellurium hexafluoride. Molecules of TeF6 are simple and symmetrical, and their intermolecular interactions are fairly well known.6*8329*33 Puzzling features had been observed in experiments on clusters of TeF66,33that MD simulations promised to be able to clarify. Moreover, as discussed in the previous paper in this series,34over a dozen different stable or metastable solid phases of TeF6 have been e n ~ o u n t e r e d .It~ ~seemed worthwhile to acquire a deeper perspective on the packing of a simple octahedral molecule, a member of a series comprising what are arguably the most symmetrical polyatomic molecules that exist. The aim of the present paper is to study the effect of cluster size and to a more limited extent, the effect of assumed intermolecular interaction law upon the systematics of melting and solid-state transitions in clusters of tellurium hexafluoride. The relative stabilities of several crystalline packing arrangements will also be assessed. Subsequent papers will address more directly the role of molecular rotation in transitions and the phenomenon of nucleation in freezing and in solid-state transitions. Also included will be the determination of nucleation rates and the identification of critical nuclei.
0022-3654/93/2091- 13550%04.00/0 0 1993 American Chemical Society
Solid-state Transitions of TeF6 Clusters
The Journal of Physical Chemistry, Vol. 97,No. 51, 1993 13551
TABLE 1: Lennard-Jones Parameters for Seven-Site Intermolecular Potential Function for TeF6 Mokcdes F-F
Fe-Te
2.940 0.2049
4.294
atom pair Q,
e,
A
kJ/mol
1.376
Fe-F 3.553 0.531
Computational Details Simulations were performed on 100-, 128-, 150-, 250-, and 350-molecule clusters of tellurium hexafluoride. The molecules were taken to be rigid octahedra with effective bond lengths of 1.8 15 A. Pairwise-additive atom-atom intermolecular interactions based on Lennard-Jones potential functions were employed for the clusters. The parameters of the potential functions are listed in Table 1. In some of the runs to be presented partial charges on atoms were included in the model potential function in an attempt to make it more realistic and to find whether the associated ionic character has a significant effect on the phase behavior of the system. Fluorines were assigned a charge of -0.1 Seand telluriums, +0.90e, based on a fitting of the accurately determined structure of the orthorhombic phase of the bulk.8 As borne out elsewhere,6,8even the model neglecting partial ionic character gives a fairly good account of the structures and the transformations of condensed TeF6. Clusters were generated to be as spherical as possible, starting from a fragment of a perfect crystalline lattice of the desired phase. A substantial fraction of the runs began with a bcc configuration with idealized molecular orientations, i.e., with all T e F bonds of the molecule aligned to be parallel to the bcc cell axes. The bcc cell parameter was initially taken to be 6.3 Molecular dynamics runs were based on a modified version of the program MDMPOL of the CCP5 Program Library, Daresbury Laboratory. The simulation processes were typically as follows. When starting from the bcc configuration, the bath temperature was set at 140, 150, or 160 K (for smaller to larger clusters, at which temperatures the bcc phase is stable). For all other phases the initial temperature was 50 K. An initial bath temperature was maintained by rescaling each of the first 1000 time steps, step sizes always being 10 fs. After this period of equilibration, temperature rescaling was switched off and constant energy MD trajectories were followed for 4000 time steps. Thermodynamic averages such as potential energy and temperature were accumulated, and coordinates, quaternion parameters, etc., of all the molecules were saved every 50 time steps (for the loo-, 150-, and 250-molecule clusters) or 20 time steps (for the 128- and 350molecule clusters) during the constant-energy period. The bath temperature was changed by 10 deg (dropped for bcc, raised for the others) after each period of 5000 time steps, and the process was repeated with the current configuration and rescaled velocities and angular momenta. Runs were continued over a temperature span of 100 deg or more. In runs beginning with bcc, the cooling process was reversed when low temperatures were reached, after the bcc phase had transformed to a single crystal of monoclinic, a transition which never failed to occur. Cluster structures were monitored for possible phase changes after each period of 5000 time steps. Several diagnostic functions were applied to characterize the phase transitions, including caloric curves, the Lindemann index 6 of vibrational amplitudes, and self-diffusion. For 6, we followed the convention of ref 29 in including only contacting pairs of molecules and, for coefficient of diffusion, the definition of Berry et Mean-square displacements of centers of molecular mass are given by
where the ensemble average is over as many independent time origins as possible to improve the statistics. The translational coefficient of diffusion is related to the mean-square molecular
c
d
Figure 1. Images of 10 configurationsof a 150-moleculecluster of TcF6 at various stagesofcoolingand heating. Projectionsof centersofindividual molecules are represented by sequencesof dots plotted at 500-fsintervals, at (a) 150,(b) 50, (c) 150, and (d) 200 K.
displacements by
It was calculated over limited time intervals to avoid spurious effects due to the finite cluster sizes.23 In simulations of the 100-, 150-, and 250-molecule clusters, the initial angular velocities of clusters were nonzero bcause of random but noncanceling initial velocities and independently rescaled linear and angular velocities during equilibration. Therefore, it was necessary to perform backrotations to calculate translational diffusion coefficients correctly at various temperatures. The 350-molecule cluster was prepared not to rotate.36 The liquid phase and five solid phases of clusters of TeF6 were examined by M D simulations in the present work. The solid phases included bcc, orthorhombic, rhombohedral, and two monoclinic phases characterized in some measure in the previous paper of this series34and whose internal organization is specified in the following section. RC!SultS
We will first consider the transitions that are reversible on the time scale of M D simulations and characterize the transition temperatures for the phases involved. We will then examine the relative stabilities of several other solid phases and the effect of electrostatic interactions. When the clusters constructed from 100,150,250, and 350 molecules with neutral atoms were cooled from the bcc phase, they all transformed to the monoclinic phase in the vicinity of 100-80 K. When the monoclinic clusters so formed were warmed, they transformed back to the bcc phase. Upon further heating, the clusters melted. In none of the runs were any of the moleculesobserved toevaporate. Figure 1 presents images of the 150-molecule cluster showing projections of the centers of mass of individual over 10 configurations. The phase changes are readily recognized in the configurational energies of the clusters plotted as a function of temperature in Figure 2. The irregular plot for the 100-moleculecluster is a consequence of the fact that the cluster was molten at the beginning of the cooling stage because of an inappropriate starting configuration. Nev-
13552 The Journal of Physical Chemistry, Vol. 97, No, 51, 1993 -15
I
c
Xu and Bartell
I
i
0
0
55
110
165
56
220
166
220
a
T (K) Figure 2. Configurationalenergy per molecule in (TeF& clusters as a function of temperature in cooling (solid curves) and heating (dashed curves) stages: filled circles, N = 100;triangles, N = 150; solid squares, N = 250; diamonds, N = 350.
ertheless, upon cooling, it transformed to the bcc phase at around 110 K before undergoing a further change to monoclinic ("mono 1 ") . Changes in the slopes of the cooling and heating curves in Figure 2 are associated with the phase transitions herein investigated. Because the phase transitions of small clusters are not sharp, no single, unique temperature corresponds to a given transition. Nevertheless,it is convenient to define a characteristic temperature for each transition. We suggest several alternative definitions in the following and investigate the degree to which they are mutually consistent. At this preliminary stage of research on clusters, each of the definitions has a significant degree of arbitrariness. The first definition selects the midpoint of a jump in the confaurational energy that occurs during the transition while the cluster is being heated. The cooling curve is unsatisfactory for the definition because the stochastic nature of the nucleation initiating a phase change makes the temperature unpredictableand unreproducible. Heating curves are much more reproducible because the precursor for the higher-temperature phase appears already to exist in the loosely held, more disordered surface moleculesof the cluster. Other measures of phase change, to some extent more sensitive than the configuration energy, are considered next. For melting and freezing, in particular, the coefficient of diffusion is a natural choice because the ability to flow is the most obviouscharacteristic of a liquid. The principal drawbackof this measure is the intrinsic noisiness characteristic of diffusion (and random-walk processes,in general). This noisiness is conspicuous in Figure 3, which illustrates the cluster translational diffusion coefficients D, averaged over all the molecules in each of the clusters. At each temperature, the translational diffusion coefficient was calculated from the molecular trajectories for the time period between 2 and 15 ps, during which the translational mean-square displacement curves were almost linear after the correction for cluster rotation. Because the coefficient D, for individual molecules is considerably greater for the surface molecules than for those in the interior, it is inappropriate to infer the melting temperature of clusters from a simple arithmetic mean of D, values taken over all the molecules in a cluster. On the other hand, the coefficients0,for individual molecules provide a satisfactory gauge for the melting temperature, as will be shown. It has been demonstrated in many studies that the Lindemann index 6 is a successfulgaugeof melting.13J4~20~21~29 For consistency, we invoke a cutoff distancez9of 7.5 A in the computation of 6. This distance, which corresponds approximately to the first minimum of the pair-correlation functions of Te atoms in the bcc and liquid phases, was used for all of the clusters at all temperatures. Two virtues of the index 6 are that it is by its very nature much less noisy than D, (although it offers a less fundamental characterization of the phase) and that it does not vary as steeply with position in a cluster as does D,. Because of
110
T (K)
Om4
0.3
r-----I
h
&
7 ;Y
tab
0.2
0.1 0
0
55
110
166
220
T (K) b Figure 3. Translational diffusion coefficients averaged over all the molecules in (TeF6)~clustersas a function of temperature in (a) cooling and (b) heating stages: filled circles, N = 100; triangles, N = 150; solid squares, N = 250; diamonds, N = 350. Solid lines, left scale; dashed, right.
the latter property, the average of S over an entire cluster is very nearly as suitable for defining a transition temperature as is an analysis of S values corresponding to individual molecules. The temperature dependence of the average of 6 over all molecules in each of the clusters is shown in Figure 4. It will be noted that the shapes of these curves are quite similar to those of the caloric curves in Figure 2 and lead to similar transition temperatures. We tentatively adopt 6 < 0.09 as the criterion for s~lidlikeclusters~~ and 6 < 0.05 as the criterion for a monoclinic phase. Because each molecule in a cluster has a different environment, particularly when the cluster is as small as those we were simulating, it is useful to examine the behavior of molecules in different regions of the cluster. To do so, we divided clusters into several shells, defined to correspond to the regions of the peaks of pair-correlation functions of Te atoms in the monoclinic phase. Four, five, six, and seven shells were defined for the loo-, 150-, 250-,and 350-moleculeclusters, respectively. Dividing lines were drawn at the minima between peaks in the pair-correlation functions at 7.3, 9.2, 11.5, 14.0, 16.5, and 18.5 A. The shells (from innermost to outermost) contain 15,12,32,49,61,77, and 104 molecules for the 350-molecule cluster at 100 K. Figure 5 depicts the translational diffusion coefficients and Lindemann indices for the different shells, as a function of temperature, for the 150-molecule cluster as it is being heated. In all four clusters, the outermost shells have the largest Lindemann indices and diffusion coefficients, verifying that they are the first to transform from monoclinic to bcc,and the first to melt, as clusters are heated. The curves of the Lindemann indices 6( T ) and translational diffusioncoefficientsD,( T ) for the different shells of cooling clusters have qualitatively the same features as those of clusters being heated except for the more erratic behavior associated with the element of chance of when nucleation occurs. To formulate a definition of melting temperature in terms of the values of 6 and D, for individual molecules, we estimate the fraction of a cluster that is liquidlikeas a function of temperature.
Solid-state Transitions of TeF6 Clusters
The Journal of Physical Chemistry, Vol. 97, No. 51, 1993 13553 1
0.09
0.76
0.07 b e
&
0.5
0.05
0.26
0.03 0.01
0
66
110
165
0
220
55
0
T (K)
110
166
220
166
220
T (K)
a
a
0.11,
i
1
0.09
0.76
0.07 0.5
Lo
0.05
0.25
0.03
0
0.01 0
66
110
166
56
0
220
b Figure 4. Lindemann indices 6 averaged over all molecules in (TeF& clusters as a function of temperature in (a) cooling, and (b) heating stages: filled circles, N = 100, triangles, N = 150; solid squares, N = 250; diamonds, N = 350.
/-I
1
0.24
110
T (K)
T (K)
b Figure 6. Fraction of molecules in liquid state in (TeF6)N clusters as a functionof temperature. The fraction was based on translationaldiffusion coefficients (a) or Lindemann indices 6 (b) of individual molecules: filled circles, N = 100; triangles, N = 150; solid squares, N 250; diamonds, N = 350.
--
TABLE 2 Transition Temperatures for (TeF6)N Clusters of Various Sies As Determined by Various Methods T(melt), K
T(mono-bcc), K methodlN
Ua 6svcb bindiv' 4 d
ave
0
50
100
160
200
a
0.111 0.08
0.07 10
0.05 0.05
0.01
60
100
150
250
350
100
150
250
350
83 81
93 97
106 105
110 111
82
95
106
111
148 147 146 144 146
72 168 165 164 167
183 180 177 176 179
186 184 187 190 187
From caloric curve. From Lindemann index averaged over entire cluster. Point at which half the molecules have 6 greater than 0.09. Point at which half the molecules have coefficients of diffusion greater than 0.03 A2/ps. (I
T (K)
0
100
150
200
T (K) b Figure 5. Translational diffusion coefficients (a) and Lindemann indices 6 (b) averaged over the molecules in individual shells of a 150-molecule cluster of TeF6 as a function of temperature in during heating: filled circles, shell 1 (innermost); triangles, shell 2; solid squares, shell 3; diamonds, shell 4; solid triangles, shell 5 .
We then define the melting point to be the temperature at which the cluster is half melted. This requires a criterion for whether molecules in a given shell are in a frozen or molten state. For the threshold value for S we simply chose the same as the one proposed above for the average value over an entire cluster. Several different threshold values of 0, were examined for suitability as a criterion for melting. It was found that the value of 0.03 AZ/ps
(or 3.0 X mz/s) gave results which were consistent with those of the other two criteria, Le., those for the caloric curves and for curves of the Lindemann index, according to Figures 1, 4, and 6b. The value we found is also in accord with the criterion proposed for CC4 clusters30and is of the same general order of magnitude as that implied by results of M D simulations of SFs clusters3*and NMR studies of bulk SF6.37938Figure 6 illustrates for all four clusters the fraction F(T)of a cluster that is melted as a function of temperature. Curves constructed from both the diffusion coefficient and 6( T ) for individual shells are plotted, and characteristic melting points are estimated from them. Melting points as defined by thevariouscriteria and temperatum of the transition from monoclinic to bcc are summarized in Table 2. The average transition temperatures for monoclinic to bcc and for melting are plotted for N-molecule clusters as functions of N-Il3in Figure 7. Linear least-squaresfits of the points illustrated, including the bulk melting point, are shown. Implications of classical capillary models to the effect that transition temperatures should decrease linearly as the reciprocal of the particle diameter increases tend to be borne out. Next are considered the relative stabilities of several of the efficiently-packed solid phases. For this study, clusters of 128 molecules were carved from lattices of each of the following crystal
13554
The Journal of Physical Chemistry, Vol. 97, No. 51. 1993 250 I
i
190
8 160
0
0.1
0.06
0.15
0.2
0.25
““3
Figure 7, Transition temperatures of TeF6 clusters as a function of N-1/3: circles, melting temperatures; triangles, temperatures for the monoclinic to bcc solid-state phase transition. Lines correspond to least-squaresfits of the points displayed, including the bulk melting point. The corresponding solid-state temperature for the bulk is unknown (see text). -18
1
I
-20 -21 -22 -23 -
-19
1 0
8
3
-24
-25ko
6b
sb
100
120
140
i o
T (K)
Figure 8. Variation of configurational energy per TeF6 molecule in 128molecule clusters undergoing heating, starting with four different initial
structures: upper four curves, simulations excluding electrostatic interactions; lower four curves, simulations including them (see text). The structures began as monoclinic ( C 2 / m ,solid, filled circle),rhombohedral (R3, solid, triangle), orthorhombic (Pnma, dashed, solid square), and second monoclinic (P2l/c, dashed, diamond), and all ultimately transformed to body-centered cubic. structures: orthorhombic (Pnma, Z = 4, the stable lowtemperature phase of bulk TeFs),’ monoclinic ( C 2 / m , 2 = 3 , ‘monol”, the phase formed when bcc is cooled rhombohedral ( R 3 , Z = 3, a phase found for bulk wc16),39and a second monoclinic ( P 2 , / c ,2 = 2, “mono2”). According to the present model potential function incorporating ionic character in Te-F bonds, the latter structure is the most densely packed of all of the 14 structures investigated so far.35 Configurational energies of clusters prepared to have the above structures are plotted at various stages of heating, starting at 50 K,in Figure 8. Results are shown both for runs neglecting effects of partial atomic charges and for runs including them in the intermolecular interactions. All clusters were at least metastable at low temperatures, and all transformed into bcc when they were warmed. All except for mono2 with partial charges transformed directly into bcc, but this variant of monoclinic underwent a transition to rhombohedral before the bcc phase formed. Implications of the above results are discussed in the next section. Discussion The spontaneous transition that occurs as all bcc clusters are cooled (Figure 2) is to the monoclinic phase (monol). Exactly corresponding transitions are also found for SF6 and SeF6 both in M D simulations and in electron diffraction experiments on large clusters. In the sulfur and selenium cases, the phase change is believed to be thermodynamically reversible.5~~~ In the case of TeF6, however, the equilibrium change is not to monoclinic but
Xu and Bartell to orthorhombic.s As discussed elsewhere: the nonequilibrium transition observed experimentally and in MD runs in rapidly Cooled clusters of TeF6 is seen on the short time scale of observations because the transformation is kinetically favored. The quasispherical molecules need to do little more than reorient to achieve the monoclinic form, and only one-third of them are required to r e ~ r i e n t . ~A~ ?more ~ ~ major, and hence slower, reorganizationis involved in the change to orthorhombic. Images of Te atoms in a 150-moleculecluster illustrated in Figure 1 show that the bcc (framea) and monoclinic(frame b) arecloselysimilar in structure,although the warmer bcc plastically crystallinephase is appreciably more disordered. In the previous paper of this series,34 the similarity in organization of the two phases was also corroborated by the similarities in the pair-correlation functions g ( r ) and angular distribution functions P(0) of the two phases. Figures 1-6 show that the melting transition is spread over a range of temperatures and that it occurs first at the surface as clusters are warmed, as it is believed to, for matter in generaL42 In fact, all of the phase changes induced on warming, including the various solid-state transitions observable in Figures 2 and 4, take place over a fairly wide temperature range and begin at the surface. Some of the lack of sharpness may be associated with the very fast rate of heating (2 X 10” K/s), but it seems to be an intrinsic characteristic of very small particles. As will be discussed below, the classical capillary model of clusters that is customarily invoked to account for the lowering of transition temperatures of small particles also predicts a diffuseness of the t r a n s i t i ~ n . ~The ~ . predicted ~~ diffuseness, however, is much more limited than that actually found in simulations. The fact that transitions are always initiated at the surface is no doubt due to the fact that surface molecules are more loosely bound and readier to adopt the higher-temperaturephase possessing larger-amplitude vibrations. In the view of some, the surface is said to catalyze the transition, and such a catalysis would account for the reproducibility of the transitions in clusters being heated. Clusters being cooled, on the other hand, do not exhibit the same reproducibility. Transitions are initiated in the interiors of clusters by a stochastic nucleation process. This process, will be analyzed in more detail in a later paper, appears to be genuinely that of homogeneous nucleation, uncatalyzed by a surface (inasmuch as it is never initiated at a surface). In the transitions we have seen in cooling clusters, so far, we have invariably found clusters to be single crystals when the liquid freezes and when bcc goes to monoclinic. In related studies of the bw-monoclinic phase change for SF6, Pawley and co-workers reported that twinning or multiple twinning usually occurred upon cooling in their simulation^.*^^^^ This may be a consequenceof their faster cooling rates. Figure 7 demonstrates that both melting and solid-state transition temperatures decrease more or less linearly as the reciprocal of the cluster radius increases, as expected according to conventional, simplified capillary models. In these models it is assumed that interfacial tensions, molar transition entropies, and densities are independent of temperature and particle size. Clusters ofTeF6 reveal no size threshold below which this decrease stops. Although such a threshold was reported for gold clusters supported on tungsten, at about 250 atoms,44 our melting temperatures continue to drop. Melting points for clusters with a few as 12 molecules fall close to the line plotted in Figure 7.4s Since the first classical capillary theory of melting point depression was introduced over a century ago,46several variants have been pr0posed.43,~~~~ Attempts based on density functional theory to go beyond the classical theory in the treatment of the liquid-solid interface have achieved some success in modeling nucleation phenomena.50Not enough informationabout the TeF6 system is available, however,to warrant an analysis of the present results in terms of anything more complex than the simplest capillary model. The most elementary model applicable to both
The Journal of Physical Chemistry, Vol. 97,No. 51, 1993 13555
Solid-state Transitions of TeF6 Clusters melting and the solid-state transition is that of Reiss, Mirabel, and Whetten (RMW).43 It places a spherical solid core inside a larger liquid cluster that wets the solid (or, equivalently, a solid core of the lower-temperaturephase inside the higher-temperature solid phase), assigns interfacial free energies to both interfaces present, and neglects any effect of thediffuseness of the interfaces. For simplicity, the difference between the molar volumes Pof the phases is neglected, as are the variations of interfacial free energy ai,, and entropy of transition mfus/ Tmmwith temperature and particle size. In systems adhering to the above simplifications, the temperature at which a small solid particle has the same chemical potential as a large body of undercooled liquid engulfing it is identical to that of conventional capillary theory. When it is asked what the temperature is at which the free energy of a solid cluster of radius r is the same as that of a liquid cluster of the same size, however, that temperature T," is
v,
(3)
a value which is depressed by 50% more than that of a cluster possessing the same thermodynamic quantities according to the conventionalcapillary theory of melti ng. Although the interfacial free energy uls is not known for TeF6, it is possible to test the order-of-magnitude compliance with the classical or RMW theories by invoking an empirical relation proposed by Turnbullsl that makes (4)
with k~ in the vicinity of 0.32 for metalloids and nonmetals. The constant for metals is somewhat higher.sl,s2 Experimentalvalues determined in this laboratory3v4for nonmetals from nucleation rates have also yielded values for kT close to 0.32. Of course, because neither uls nor the other thermodynamic quantities can be expected to remain constant or to vary in step with each other over the rather large temperature range of the present simulations, it cannot be expected that k~ is really a constant of nature. Nevertheless, if eq 4 is substituted into eq 3, the very simple result
is obtained for a cluster of Nmolecules, implying a linear variation of T," with N-Il3. Moreover, a line drawn through the bulk melting point to cleave the MD points as evenly as possible has a slope corresponding to kT = 0.32. Although there is nothing rigorous about the result, particularly in view of the fact that our operational definition of melting point is rather arbitrary in the first place, the result nevertheless hints that there is a certain element of physical validity in the treatment. Note that if the conventional thermodynamic treatment had been applied, the factor of 3 in eq 3 would have been replaced by a factor of 2 and the line drawn through the data would have implied that kT = 0.48.
In addition to the favorable aspect of the RMW capillary model just demonstrated, it should be noted that the RMW model also predicts, qualitatively, exactly the tendency found in the MD simulations for a solid cluster to melt at the surface before the interior melts. Nevertheless, one should not expect too much of an attractively simple model introduced to be purely illustrative. As shown in detail elsewhere,30the model fails to reproduce the breadth of the temperature range over which melting takes place in simulationsfor small clusters, and the failure is severe. Perhaps the most crucial elements missing from the model are the effects of the diffuseness of the solid-liquid and liquid-vapor interfaces.
Next, the degree to which the size dependence of the solidstate transition temperature shown in Figure 7 resembles that for melting is considered. If the surface stress associated with the lattice misfit between the solid phases can be ignored, the thermodynamicsof the monoclinic-bcctransition exactly parallels that of melting. DibbleS offered evidence implying that, for crystals as small as the present clusters, the surface stress is negligible. Unfortunately, little reliable, systematic information existsabout magnitudesof interfacialfree energiesbetween liquids and solids. Even less exists for interfacesbetween two solid phases. The four points in Figure 7 suggest that the size-dependent depression of the freezing point, as originally noted elsewhere,53 closely resembles that for melting. If the bulk transition temperature were known for the monoclinic-bcc phase change, the degree of linearity of the transition temperature could be assessed with more confidence. Only the temperature for the orthorhombic-bcc transition has been reported, however, and the value,38200 K, does not seem to be known with precision. Because monoclinic is metastable with respect to orthorhombic, its transition temperature can be expected to lie lower. Figure 8 suggests that the lowering might be 15 deg or so, implying that the bulk transition temperaturemight, perhaps, be in the vicinity of 185 K. If the four MD temperatures are represented by a function linear in N-'I3 chose by least squares, the intercept for the bulk material is 170 f 22 K, assuming uncertainties in the individualcluster transition temperatures of 5 K. The agreement between the estimated experimental and extrapolated MD values for the bulk temperature is well within the fairly large uncertainties, particularly considering the fact that the MD potential function is not reliably calibrated against experiment in the first place, in view of the large uncertainity (10-20%) in the heat of sublimation of TeF6. If we were to adopt 170K (or other definite value) for the bulk temperature, the interfacial free energy could be estimated from a relation analogous to eq 3. Alternatively, if an equation of the form of eq 5 is applied, a value of k~ = 0.5 is derived. In experiments on solid-state nucleation rates on large clusters of SeF6 and tert-butyl chloride, a value of kT closer to 0.3 was found. If we were to impose the analog of eq 3, with a line whose slope corresponded to kT = 0.32 passing through the MD points, the intercept would have been close to 140 K. Such a low value may betray the need for partial charges in a realistic potential function (see Figure 8 and the discussion below) inasmuch as such charges are neglected in Figure 7. Although uncertainties are not small, it appears that from the data in Figure 7 as if the Turnbull relation applies, at least in order of magnitude, to interfaces between two solid phases of one-component systems as well as to solid-liquid interfaces, and that values of interfacial free energies derived from the size depressionof transition temperatures agree, in order of magnitude, with those determined from nucleation rates. In each case, the essence of the capillary model applied is identical, of course. And the present results, in their diffusenessof transition temperatures, confirm that the capillary model is seriously inadequate. Therefore, the generalized Turnbull relation, while interesting and suggestive, is far from secure. Relative stabilities of the five solid phases examined in this work can be inferred from the results in Figure 8. It is found that the relative stabilitiesdepend appreciably upon the model potential function adopted. For all potential functionsexamined,including several more than the two included in the present paper,35the bcc phase has a higher (less stabilizing) potential energy but is the most stable high-temperature phase. First let us consider the model neglecting partial atomic charges. Potential energies of all of the low-temperature phases are similar and almost within the statistical uncertainties. Of greater significancethan potential energies are the relative chemical potentials. To the extent that the statistical noise of the present runs can be neglected, this information can be inferred from the temperatures at which the
13556 The Journal of Physical Chemistry, Vol. 97, No. 51, 1993
lower-temperature phases transform to bcc, the less stable forms transformingat lower temperatures. It appears, then, that mono2 is theleast stableof thelow-temperature forms examinedalthough it may well turn out to be the stable high-pressure form.35 Rhombohedral and monol are intermediate and comparable, and orthorhombic is the most stable low-temperature structure, just as it is in the bulk. When electrostatic interactions are included, the potential energy of the monol form becomes distinctly less binding than that of rhombohedral and orthorhombic even though it is the monol form that is spontaneously generated when the bcc phase is cooled. The reason for this has already been discussed above. Themono2 structure, whilecompetitivewith theothersin potential energy, is decidedly less stable than the others when all are executing thermal motions, and it transforms to rhombohedral before the mono2-bcc transition temperature is reached. Orthorhombic is decisively the most stable low-temperature form. In Figure 8, one of the more striking consequences of electrostatic interactions is that, while they have only a modest effect on the energy of the bcc phase, they have a considerable effect on the stabilities of the more compact phases. This effect is correlated with an appreciableincreaseof transition enthalpiesand transition temperatures to the bcc phase. One cannot assess the virtue if a model potential function solely on the basis of its performance with the bcc phase. As shown e l s e ~ h e r e , treatments ~*~~ of structures and thermodynamic properties of condensed phases are significantly more realistic if electrostatic interactions are included, and results of model potential functions, at least for the heavier hexafluorides, based only on fluorins-fluorine interactions are quite mediocre. In this study it has been shown that small clusters can exhibit at least some of the same phase changes in molecular dynamics simulations that large clusters do in the laboratory, even in the caseof solid-statetransformations,despite the enormousdifference between time scales involved. When changes are induced, they take place first at the surface upon heating, and in the interior uponcooling. Both melting and solid-statetransition temperatures have been found to decrease with decrease in particle size. Structural and dynamic results of simulations on TeF6 clusters are qualitatively the same when electrostatic interactions are neglected as when they are included. All of the behavior encountered in simulations is in qualitative accord with implications of theconventionalcapillary representation. The capillary model assuming molecularly sharp interfaces between phases, however, predicts phase changes with temperature that are much too sharp. Further aspects of phasechanges, including molecular rotation and a more detailed examination of nucleation, will be presented in forthcoming papers in this series.
Acknowledgment. This research was supported by a grant from the National Science Foundation. We are indepted to Dr. W. Smith of the Daresbury Laboratory for the program MDMPOL. We gratefully acknowledge the assistance of Dr. F. Dulles in many aspects of computing and helpful input from Mr. K. Kinney based on his analysesof packing efficienciesin various crystalline phases of hexafluorides.
Xu and Bartell
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