J. Phys. Chem. 1992,96,8801-8808 (9) Blom, R.; Faegri, K., Jr.; Volden, H. V. Orgatwmerallics 1990,9,372. (10) Kozen, A.; Nojima, S.;Tenmyo, J.; Asahi, H. J. Appl. Phys. 1986, 59, 1156. (11) Lewis, C. R.; Dietze, W. T.; Ludowise, M. J. J . Electron. Mater. 1983. -.__ 12. 507.(12) Allen, K. A.; Gowenlock, B. G.; Lindsell. W. E. J . Polym. Sci., Polym. Chem. Ed. 1974,12,929, 1131. (13) Ellis. A. M.; Robles, E. S.J.; Miller, T. A. J . Chem. Phys. 1991, 94, I - - . - -
I /JL.
(14) Robles, E. S.J.; Ellis, A. M.; Miller, T. A. J . Phys. Chem. 1992, 96, 3247. (15) Robles, E. S. J.; Ellis, A. M.; Miller, T. A. J. Phys. Chem. 1992, 96, 3258. (16) Robles, E. S.J.; Ellis, A. M.; Miller, T. A. J . Chem. Soc., Faraday Trans. 1992,88, 1927. (17) Robles, E. S.J.; Ellis, A. M.; Miller, T. A. J. Am. Chem. SOC.1992, 114, 7171. (1 8) Moore, C. E. Atomic Energy Levels as Derived from the Analyses of Oprical Spectra; Natl. Bur. Stand. Circ. No. 467; US GPO: Washington, DC, 1952. (19) Floss, F. P.; Traeger, J. C. J . Am. Chem. Soc. 1975, 97, 1579. (20) Cotton, F. A.; Reynolds, L. T. J . Am. Chem. Soc. 1958, 80, 269. (21) Ford, W. T. J . Organomet. Chem. 1971, 32, 27. (22) Parris, G. E.; Ashby, E. C. J . Organomet. Chem. 1974, 72, 1. (23) Hull, H. S.;Reid, A. F.;Turnbull, A. G. Inorg. Chem. 1967,6,805. (24) Begun, G. M.; Compton, R. N. J . Chem. Phys. 1973, 58, 2271. (25) Kharasch, M. S.; Reinmuth, 0. Grignard Reactions of Nonmetallic Substances; Prentice Hall: New York, 1954. (26) Nenitzescu. C. D. Bull. Soc. Chim. Romania 1930, !I, 130. (27) Whitesides, G. M.; Nordlander, J. E.; Roberts, J. D. Discuss. Faraday Soc. 1962, 34, 185.
8801
(28) Reinecke, M. G.; Johnson, H. W., Jr.; Sebastian, J. F. J . Am. Chem. SOC.1963,85, 2859. (29) Bak, B.; Christensen, D.; Hansen, L.; Rastrup-Andersen, J. J . Chem. Phys. 1956, 24, 720. (30) Gallinella, E.; Mirone, P. J . Lubelled Compd. 1971, 7, 183. (31) Crosswhite, H. M. J . Res. Natl. Bur. Stand. Sect. A 1975, 79A, 17. (32) H u h , K. P.; Herzberg, G. Molecular Spectra and Molecular Structure. IV. Constants of Diatomic Molecules; Van Nostrand Reinhold: New York, 1979. (33) Richardson, J. H.; Stephenson, L. M.; Brauman, J. I. J . Chem. Phys. 1973, 59, 5068. (34) Csicsery, S.M. J . Org. Chem. 1960, 25, 5 18. (35) Koopmans, T. Physica, Eindhouen 1933, I , 104. (36) Cotton, F. A. Chemical Applications of Group Theory; Wiley-Interscience: New York, 1971. (37) Yu,L.; Foster, S.C.; Williamson, J. M.; Heaven, M. C.; Miller, T. A. J. Phys. Chem. 1988, 92, 4263. (38) Friedman, L.; Irsa, A. P.; Wilkinson, G. J. Am. Chem. Soc. 1955,77, 3689. (39) Field, F. H.; Franklin, J. L. Elecrron Impact Phenomena and the Properties of Gaseous Ions; Academic Press: New York, 1957. (40) Gawboy, G.; Miller, T. A.; Shavitt, I. Unpublished results. (41) Engleman, R., Jr.; Damsay, D. A. Can. J . Phys. 1970, 48, 964. (42) Porter, G.; Ward, B. Proc. R . SOC.London A 1968, 303, 139. (43) Yu, L.; Williamson, J. M.; Miller, T. A. Chem. Phys. Lett. 1989, 162, 431. (44) Nelson, H. H.; Pasternack, L.; McDonald, J. R. Chem. Phys. 1983, 74, 227. (45) Garforth, F. M.; Ingold, C. K.; Poole, H. G. J. Chem. Soc. 1948,508. (46) Bailey, R. T.; Lippincott, E. R. Spectrochim. Acta 1965, 21, 389. (47) Fritz, H. P. Adv. Organomet. Chem. 1964, 1, 239.
Structure and Dynamics of Molecular Clusters. 2. Melting and Freezing of CCi4 Clusters Lawrence S. Bartell* and Jian Cben Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109 (Received: May 29, 1992; In Final Form: July 23, 1992)
Phase transitions of a 225-moleculecluster of carbon tetrachloride have been studied by a molecular dynamics simulation. A five-site model potential function was developed to reproduce the density and heat of vaporization of the bulk liquid. Computations began with orientationally disordered molecules distributed in fcc lattice sites of a nearly spherical cluster. The cluster was heated from a low temperature to 200 K in 10-deg step of 50 ps each and then cooled to 10 K. Translational and rotational transitions were monitored by following several indicators including the translational and rotational diffusion and rotational entropies of individual molecules. Melting began at the surface and propagated inward as the temperature increased. Solidification of the molten cluster proceeded from the center to the surface. At the high cooling rate of the simulation, however, molecules were unable to organize into a crystalline array and solidified into a glassy structure instead. Except for spatial order, the indicators of degree of liquefaction exhibited almost the same temperature dependence in the crystsl liquid as in the liquid glass transition, a behavior that could be rationalized on the basis of Lindemann’stheory of melting. Results were compared with predictions of an illustrative model due to Reiss, Mirabel, and Whetten. Qualitatively, the model included all of the features of the simulation. Quantitatively, the model grossly underestimated the range over which the melting transition took place.
-
-
Introduction
Clusters have become increasingly popular systems to investigate in studies designed to shed light on the nature of phase transitions. It is possible to follow certain aspects of transformations in clusters that are extremely difficult to monitor in bulk systems by existing techniques. Advantages of clusters in experimental studies have been discussed in detail el~ewhere.l-~A great many more investigations of transitions in clusters have been carried out by computer simulation than by experiment, however, partly because simulations are appreciably simpler to perform and partly because simulations reveal details of molecular behavior particularly directly. The great majority of the simulations performed to date have been on atomic clusters.e8 Because molecular clusters offer elements of interest that are not found in atomic clusters, and because the few experimental studies of phase transitions have
been on molecular clusters, it seemed worthwhile to initiate molecular dynamics studies of systems which had already been investigated experimentally. One such system is carbon tetrachloride, whose kinetics of freezing has been examined in some detai1.l Although carbon tetrachloride is an appropriate subject in several respects, beiig a quite well understood substance composed of highly symmetrical, fairly rigid molecules, it is not entirely favorable for molecular dynamics analyses of freezing and melting. For one thing, its crystallography is quite complex, as will be discussed. For another, the rate of freezing achieved in experiments on 100-Aclusters, namely = critical nuclei per second per cubic meter when supercooled by 80 K, is believed to represent very nearly the maximum rate theoretically attainable for carbon tetrach1oride.l Rapid though that rate be compared with rates
0022-3654/92/2096-880l S03.00/0 0 1992 American Chemical Society
8802 The Journal of Physical Chemistry, Vol. 96, No. 22, 1992
Bartell and Chen
TABLE I: Lennard-Jones Potential Parameters Used in the Simulation c, 0,
kJ/mol
A
c-c
CI-CI
C-CI
0.209 3.8
1.117 3.472
0.483 3.636
measured in bulk material, it is far too slow to be followed in a standard molecular dynamics simulation. Nevertheless, a molecular dynamics study of CCI, appeared to be an attractive choice. Even if melting of a crystal were to turn out not to be reversible upon cooling, the alternative process, freezing to a glass, would be interesting in its own right. Therefore, a molecular dynamics investigation of CC14 was initiated.
Procedure Choice of Startiug Structure. The aim of the simulation is to examine the molecular behavior when a cold, crystalline cluster of CCl, is first heated beyond its melting point and then cooled until it becomes solid again. What renders the starting point somewhat arbitrary is the complex crystal chemistry of the s u b stance.”’ When liquid CC14 is cooled under normal conditions it freezes to a metastable face-centered cubic solid (phase Ia) which, upon standing or further cooling, transforms to a rhombohedral structure. Additional phases are encountered at lower temperatures. When the liquid is cooled very rapidly, as it is when clusters are formed in a supersonicjet, it may freeze directly to the rhombohedral form.’ Unfortunately, the molecular packing in the rhombohedral form is not known. We chose, therefore, to begin with molecules arranged in the face-centered lattice. Even though it would be expected to be metastable, at least for large enough clusters, metastability on the laboratory time scale would not lead to decomposition of the structure on the time scale of a simulation. Somewhat more troublesome is the fact that the Td point group of the molecule is inconsistent with the site symmetry required for packing into an fcc lattice unless molecules are arranged with a distribution of disorder that averages to Oh symmetry. Therefore, it was decided to construct a cluster with molecules initially placed with random orientations into crystallographic sites and to let the assemblage relax freely. An initial lattice constant of 8.34 A was chosen. Computntional Decllls. Molecular dynamics simulations were performed on carbon tetrachloride clusters containing 225 molecules with a modified version of the program MDMPOL.lZ The initial configuration was constructed to be as spherical as possible consistent with the distribution of molecules described above, starting with a molecule at the center. The molecules were taken to be rigid tetrahedra with bond lengths of 1.769 A.13 Intermolecular interactions were constructed from pairwise additive Lennard-Jones atom-atom potential energy functions adjusted to yield the bulk densityI4 and heat of ~aporization’~ in a Monte Carlo simulation of the liquid phase. This simulation was carried out on a system of 128 molecules at 298 K and 1 atm pressure, with periodic boundary conditions imposed. Parameters for the potential functions are listed in Table 1. In the molecular dynamics runs, time steps were set to 5 fs, and an initial temperature of 5 K was chosen. To allow the cluster structure to begin to relax, the run started with 5000 time steps in a heat bath at 5 K, continued with loo00 time steps at constant energy, and then repeated 5000 time steps in a heat bath at 5 K and 10000 more at constant energy. Although it cannot be supposed that annealing for 0.15 ns at such a low temperature could possibly lead to an equilibrium codiguration, it is reasonable to expect annealing to continue during the subsequent process of heating and to be essentially complete by the time the melting transition is reached. A series of heating stages began at 10 K, each succeeding stage being nominally 10 K warmer than the previous one. Each stage consisted of 1000 time steps in a heat bath at constant temperature, followed by 9000 time steps at constant energy. Heating was continued to 200 K, well beyond the melting point of the cluster, after which the process was reversed until the cluster reached 10 K. In none of the runs did molecules evaporate from the cluster.
*.
.............................................
oh-*10 20 30 40 50 60 7b t (PS)
Figure 1. Time evolution of mean-square molecular displacements at 5 K (dots) and 200 K (triangles) averaged over all molecules in (rotationless) cluster. Various diagnostic tests were applied during the heating and cooling stages to monitor the molecular behavior. Melting and freezing were followed by observing the temperature dependences of the caloric curve, the Lindemann index 6, the translational and rotational coefficients of diffusion, the configurational entropy, and the center-of-mass pair correlation function. Additional insight was gained by viewing images of the molecular arrangement and Pawley-Fuchs projections of molecular orientation^.'^ Because many of the diagnoses could be applied to individual molecules, it was possible to locate the region of the cluster where phase changes began and to examine whether rotational and translational transitions took place simultaneously or not. Translational and rotational coefficients of diffusion could be determined for individual molecules with useful precision despite the property of random walks to the effect that the standard error of a mean-square displacement is comparable to the mean-square displacement,itself, irrespective of how many steps are taken. It was also possible to separate the mean-square amplitudes of vibration and libration of molecules from the diffusional displacements. During the first picosecond or so the statistical average of the quantity ([r(t) - r(0)l2) increases from zero to the mean-square vibrational amplitude of molecules in the cages of their neighbors. At low temperatures this displacement quickly reaches an asymptotic value. At higher temperatures the statistical average continues to rise with time, linearly, so that after a few picoseconds a function
([r(?) - r(t=0)l2) = const
+ 6D,t
(1)
develops whose slope of 6 0 , yields the coefficient of translational diffusion Dr.16 This behavior is illustrated in Figure 1. The angular mean-square displacement of bond directions is analogous, and the linear rise with time after the amplitude of libration has been established has a slope of 4D8,the factor of 4 corresponding to the two-dimensional character of the bond orientation. In the case of rotation some care must be exercised when displacements are large because distinguishable angular displacements of bonds are confined to the range imposed by the range of spherical coordinates. The mean-square angular displacement can easily approach the limit ( r 2- 4)/2 in simulations of modest length.” No serious problems arise, however, if the diffusion is calculated from averages taken over comparatively short segments of time. What makes it feasible to extract the diffusion coefficients for individual molecules despite the large inherent noise in individual trajectories is the Markovian nature of the diffusion process such that the statistical average ([r(t+t,)- r(ti)12)depends only upon the interval t and not upon t,. Therefore, amplitudes of vibration and diffusion parameters were extracted from the chaotic trajectories of individual molecules by averaging mean-square displacements over the different ti points available in the run.I7 This mode of averaging made it reasonable to recognize diffusion when the apparent coefficient of diffusion was smaller than the value of cm2 s-I suggested as a threshold criterion above the noise in MD runs by Angel1 et a1.I8 A typical plot of the mean-square amplitude, so averaged, for a single molecule in the cluster is plotted in Figure 2. Configurational entropy of rotation, averaged for each molecule over the orientational distribution established during the time steps
Structure and Dynamics of Molecular Clusters
80 r
The Journal of Physical Chemistry. Vol. 96.
-25
I
No.22, I992 8803 I
1
____.__.---.-.__.___...-~~~
0
10
20
30
40
time(ps) Figure 2. Time evolution of the mean-square displacement of an individual molecule located near the cluster surface. Cluster at 200 K. Dashed and dotted curves indicateranges of displacements reckoned from various initial times. Solid curve, average over the different initial times (see text). Error bars, characteristic standard deviations of the mean values.
Figure 3. Configurational energy per molecule as a function of temperature during the heating stage.
0.20 1
d
at a given temperature, provided an alternative index of degree of freedom of molecular motion. It was calculated, neglecting any coupling between orientation and other coordinates in phase space, via Boltzmann's H f u n c t i ~ n . ' The ~ resultant expression for a given molecule i is
where f is the distribution function in the Euler angles Bi, q$, and lli of molecule i. If the distribution is comparatively narrow it is well represented by a product of Gaussian functions whose variances are readily derived from the histograms of the angles. For broader distributions we regarded 4 4 , and as independent and regarded f as a product of Fourier series in 8, 4, and +, adopting the method of moment estimation (MOME)20to deduce the coefficients of the trigonometric functions. The coefficients were taken to be a, =
COS ne)
6, = 2(sin ne)
(3) (4)
etc., and the series was truncated when the coefficients fell beneath 5% of the first term in the series. At low temperatures where librational amplitudes were small this method yielded entropies in good agreement with those calculated assuming that the distributions were Gaussian. At higher temperatures, values were insensitive to the truncation point. For thoroughly molten samples and long enough run times the distributions are uniform and the corresponding molecular entropy has the limiting value of
S = k In ( 8 d ) = 4.3689k (5) At lower temperatures where rotational diffusion is a more severely restricted, activated process, a true equilibrium distribution may not be attained during the time of a given run. In such a case, the configurational entropy derived by the above method is in some measure a kinetic rather than a thermodynamic parameter. The other indexes of phase and molecular behavior examined to characterize events during the heating and cooling of the cluster are too familiar to document here. An analysis of the melting and freezing of the carbon tetrachloride cluster is presented in the next section. Shown in Figure 3 is the increase in configurational energy of the cluster as its temperature is increased. Although the change in slope of such a curve when a cluster begins to melt is often taken as evidence for a phase change, in the case of our CC14 cluster the change is too subtle to provide a very delicate measure of the transformation. A somewhat more conspicuous change occurs in Figure 4 which plots the Lindemann index, 6(r), corresponding to the ratio of the root-mean-square amplitude of vibration between centers-of-mass of adjacent molecules to the mean distance between themn5v2' When this ratio exceeds 0.1 for clusters of
Figure 4. Temperature dependence of the Lindemann index 6 during stages of heating (circles) and cooling (solid points). For atomic.clusters melting is considered to occur at 6 = 0.1. For clusters of polyatomic molecules with 6 based on distances between centers of mass of adjacent molecules, melting begins at a value appreciably smaller than 0.1. The lack of a pronounced break is indicative of a protracted process of melting over a range of temperaturts. Lennard-Jones spheres the clusters are generally considered to have For clusters of polyatomic molecules, the ratio corresponding to melting is believed to be somewhat less than 0.1.'' A more fundamental gauge of melting and freezing is desirable for quantitative studies. Several such gauges are available. They include the molecular cotfficients of diffusion D, and De and the configurational entropy. Before discussing these indexes of change, however, it is worthwhile to present illustrations of the molecular behavior which graphically depict the significant changes and which have immediate intuitive appeal. These are images of the clusters correspondingto projections of the centers-of-mass of individual molecules presented in Figure 5 . Changes suffered by the 225-molecule cluster of CC14during the heating and the subsequent cooling stages are shown. Vibrational and diffusional displacements of molecules are indicated by the arrays of dots corresponding to instantaneous coordinates plotted for the various molecules at regular intervals of time. At low temperatures the cluster can be seen to be largely crystalline, at least in translational order. As the temperature increases, amplitudes of vibration increase and, ultimately, the cumulative disorder betrays the onset of melting. It is apparent that melting begins at the surface and progresses inward as the cluster gets warmer. When the molten cluster is cooled the disorder remains but molecular motions become progressively more quiescent. To what extent the cluster can be considered to solidify when cooled, and to what extent changes in molecular rotations during phase transitions run parallel with those of molecular translations, remain to be analyzed. Particularly vivid characterizationsof the melting and freezing processes in the present research are offered by the coefficients of translational and rotational diffusion 0, and De. Their variations with temperature and with location within the cluster are shown in Figures 6 and 7. All of the aspects of change subjectively evident in Figure 5 are reproduced in the diffusion coefficients but the latter give quantitive force to the analysis and help to resolve additional questions, as well. In order to diagnose the physical state of a particular region of a cluster it is expeditious to propose plausible threshold values of D, and DBfor molecules above which the molecules can be considered to be undergoing genuine diffusion. There is a certain
8804 The Journal of Physical Chemistry, Vol. 96, No. 22, 1992
Bartell and Chen
1Ol2
I
10"
1o1O
io9 lo8 10'
I
I
..... ..e
,'e.
,
I
-..
.
. + . I . , . .
.....$..,
....... ........* . .......
-* *
.....
e e . 9 . b . Z
-I
..... . . . . . . ... . . . . ... . ..... :..* 8
1
1.0
Oa8 0.6
l f -
i
fll
.
.. .. .. .. .. .. .. .. I ........ - . I ..... . I .
.
Figure 7. Coefficients of rotational diffusion for molecules at various distances from the cluster center at 10,80, 120, and 160 K. For other aspects of the figure, see caption to Figure 6.
% . . .
Figure 5. Images of 225-molecule cluster at various stages of heating (left-hand column) and cooling (right-hand column). Projections of centers of individual, diffusing molecules are represented by sequences of dots plotted at regular intervals. Temperatures of left-hand figures, starting at bottom, are 10, 80, 120, and 160 K. Heating continued to 200 K before cooling began. Right-hand figures, beginning with the top, correspond to temperatures of 160, 120,80, and 10 K. All frames show identical orientations of the rotationless cluster.
107 10-9
,
9
10"O 10"
Figure 6. Coefficients of translational diffusion for molecules at various distances from the cluster center plotted at four different temperatures, namely, 10, 80, 120, and 160 K. Upper frame, during heating stage. Lower frame, during cooling stage. Clearly evident is the trend for increased diffusion as the temperature is increased and as the molecular site gets closer to the surface. The large relative noise at the colder temperatures can be seen to be very small on an absolute scale.
0.8 . 0.6 . 0.4
.
Ld?kLA
OS2 0.0 0
50
100
150
200
T (K) Figure 8. Fraction of molecules in the 'translationally melted region" of the cluster as a function of temperature according to two different criteria of melting. Lower frame, threshold coefficient of diffusion D, = 3.3 X IO" cm2/s. Upper frame, D, = cmz/s. Circles, heating; solid points, cooling.
arbitrariness in choosing such a threshold and, for such small clusters, only experience will provide an optimum choice. Considerations of displacement and signal-to-noiseratios in the present work suggest that plausible threshold values for D, and De might be 3.3 x lod cm2/s and 8 X lo9 rad2/s, respectively. Expressed in more easily compared units, they are, respectively, 3.3 X 1O'O A2/s for centers of mass and 2.5 X 1Olo A2/s for the chlorine atoms at the ends of the 1.769-A bonds. The close similarity is probably not accidental but it was not deliberately imposed. In any event, the uncertainty in the best value to choose is substantial in each case. For sake of comparison, we also present results based on the larger threshold value for D, of 1O-j cm2/s. This value was chosen for illustrative purposes partly because of the possibility that clusters reasonably characterizedas crystalline might display quite appreciable selfdiffusion22and partly because the value cm2/s has been suggested as a practical threshold value for MD simulations.I8 Once threshold values are assigned to diffusion it is possible to apportion molecules to regions which can be considered to be melted rotationally and/or translationally and to calculate the fraction of molecules which can be assigned to translationally or
The Journal of Physical Chemistry, Vol. 96, No. 22, 1992 8805
Structure and Dynamics of Molecular Clusters 1.0,
d - - l
if
0.6 O**t
0.8
I
m
Figure 9. Fraction of molecules in the “rotationally melted region” of
the cluster as a function of temperature according to two different criteria of melting. Lower frame, threshold for configurationalentropy of rotation taken to be 4.1 5k. Upper frame, threshold coefficient of diffusion taken to be De = 8 X lo9 rad2/s. Circles, heating; solid points, cooling.
r----l
2solll!Esd
4m0 3.0 n
L
W
M
bilities as low as those in a crystal is a point for others to debate. For simplicity we shall call it a glass. Simulation times were too brief to characterizethe relaxation mechanisms at issue. A feature which is considered by some to characterize a monatomic glass is a split in the second peak of the pair correlation function.23Such a split is displayed in the center-of-mass pair correlation functions for the quasispherical molecule reproduced in Figure 10. It develops as the temperature is lowered. A convenient distillation of the main results in Figures 6 and 7 can be made once the threshold values of 0,and Defor bonafide diffusion are adopted. They provide criteria to identify whether any given molecule can be considered to be in a translationally or rotationally “melted” region of the cluster. Curves representing the fractions of molecules so identified are given in Figures 8 and 9. Several results in Figures 8 and 9 attract notice. First, the cluster melts appreciably below the 246 K melting point of the fcc bulk as would be expected according several points of vie^.^^' The dependence of melting point on particle size is well established experimentally.26v28Next, it is plain that the cluster melts over a rather wide range of temperatures. Such behavior is well documented in simulations on clusters of Lennard-Jones Also of interest is the evidence that rotational melting appears to begin somewhat before translational melting, but the two phenomena overlap over most of the blurred range of the transition. An unequivocal conclusion about rotation vs translation is obscured by the effect of the uncertainty in the choice of threshold coefficient of diffusion. A comparison between the two parts of Figure 7 shows that the convention adopted for 0,has a not insignificant influence on the defined “melting temperature”. Quite striking is the modest hysteresis displayed in the curves despite the irreversible nature of the transition (crystal to liquid to glass). Consideratioas of Class Formation. The correspondence of the heating and cooling curves suggests a significant kinship between crystal and glass. One way to rationalize the relationship is to view freezing and melting from Lindemann’s point of view. He hypothesized2’ that, when the thermal amplitude of vibration of adjacent atoms relative to one another in a solid exceeds a certain fraction 6 of the mean distance between them, the rigidity of the assemblage can no longer be sustained and melting takes place. Many molecular dynamics and Monte Carlo studies of clusters corroborate this suggestion, for small rigid molecules as well as for atoms, although the critical value of 6 seems to be appreciably less than was supposed by Lindemann in 1910. From this standpoint the crucial requirement for solidity is the smallness of amplitudes of the thermal motions and not the degree of order in the structure. If this idea is tenable, the similarity between the heating and cooling curves in Figure 8 is unremarkable in view of the results in Figure 4. A certain degree of hysteresis is expected, in any event, because the heating rates and cooling rates are so extraordinarilyfast in comparison with rates characteristic of equilibrium processes in the laboratory that true equilibrium cannot be expected to be maintained. Comparison with Experimental Results. It is worthwhile to examine consequences of the published experimental investigation of the rate of freezing of CCl, clusters.’ It was concluded in this study that the maximum nucleation rate to be expected was in the vicinity of critical nuclei per cubic meter per second and that this maximum occurred at a temperature of about 165 K. At higher temperatures the thermodynamic drive to freeze is smaller and at lower temperatures the molecular motions become excessively sluggish. If the above nucleation rate applies to the present cluster whose volume is only 3.2 X m3, the number of nuclei to be expected per second would be about 3 X 10’. Restated, this corresponds on average to 6 X 1O’O simulation time steps for a critical nucleus to form. Therefore, even allowing for a substantial error in the estimate of maximum nucleation rate, it cannot be expected that a 225-molecule liquid cluster of CCl, will be observed to freeze to a crystal in a molecular dynamics simulation. This limitation, of course, does not prevent the solidification to a glass and, indeed, the way to form a glass deliberately is to cool a liquid so rapidly that the region of high nu-
1.o
0.0
10
20
30
40
R (A> Figure 10. Pair-correlation functions for molecular centers in the 225-
molecule cluster (ca. 39 A diameter) at various temperatures during cooling of the melt. From top to bottom 190, 130, 70,and 40 K. As solidification proceeds the second peak splits (see text). Ordinate applies only to the lowest, fully frozen case.
rotationally molten regions. Plots of the latter two quantities are given in Figures 8 and 9. Finally, pair correlation functions for centers of mass of the molecules were calculated for molten clusters and for once-molten clusters cooled well below the observed melting point. Results of the calculations are shown in Figure 10. Such pair correlation functions for atomic systems are considered by some authors to provide evidence about glass formation23but such a view is disputed by other^.'^^^^
Discussion General Features of the Simulation. In order to interpret the events occurring during the processes of melting and freezing it is, of course, essential to have a reasonable criterion to determine when the phase changes are taking place. What distinguishes liquids from solids most definitively is self-diffusion, the quantity whose behavior is set forth in some detail in Figure 6. The trends exhibited in these figures reveal many features of significance in the transitions and characterize quantitatively what is qualitatively apparent to the eye in the images of the cluster in Figure 5. Especially noteworthy as the cluster is heated is the fact that its outer portions begin to melt substantially sooner than its interior. Figures 6-9 also resolve the uncertainty about whether solidification occurs when the liquid cluster is cooled without crystallizing. The retum of self-diffusion coefficients to the low values originally possessed by the cold crystal leaves no doubt that freezing has taken place. The kinetic information which could have forecast the failure to recrystallize will be discussed presently. Whether the resultant amorphous aggregate should be considered to be a glass or a highly undercooled liquid with molecular mo-
8806 The Journal of Physical Chemistry, Vol. 96, No. 22, 1992
cleation rate is traversed too rapidly for nucleation to occur. The present investigation also raises questions about what fraction of the total volume of a cluster should be considered effective for nucleation to take place in. Clearly, the outermost layer of molecules, which compfiscs over half of the bulk of a 225-molecule cluster, has a greater propensity for becoming liquid than does the interior. Therefore, the volume in which a nucleus is likely to form is appreciably less than the total cluster volume. The fact that liquid clusters of CC14are unsuitable candidates for MD simulations of freezing to a crystalline phase does not mean that such transformations for polyatomic systems are beyond the capability of molecular dynamics, in general. For example, in MD simulations the freezing of TeFs to a bcc structure has been seen,29and transitions of bcc hexafluorides to lower temperature crystalline phases have been reported several time~.2’~593’ Compuiroa with RMW ModeL It is of interest to fmd whether the present results can be interpreted naturally in terms of an attractive model of clusters proposed by Reiss, Mirabel, and Whetten (RMW).31 This model is a natural extension of the popular capillary models of critical nuclei incorporated in classical nucleation theory for amdewtion from vapor and for the freezing of liquids. It was originally introduced for illustrative, rather than quantitative, purposes. What is especially appealing about the RMW model is the simple and natural way it treats aspects of cluster behavior found in the present simulations. These include the temperature range over which the transition is spread and the probability of fmding clusters consisting principally of a solid core surrounded by a liquid layer, and also of a mostly liquid cluster containing a small solid core. It is of interest to find how realistically the model applies to the present results. In the RMW model clusters are assumed to be spheres of radius R consisting of solid cores of radius r surrounded by liquid. What the RMW model provides is the free energy G(T) for all values of R / r ranging from zero (liquid) to unity (solid). The free energy of such clusters at any temperature can be expressed in terms of r, R, the chemical potentials of bulk solid and liquid, and uls, the liquid-solid interfacial free energy per unit area. Model results of immediate concem are the ‘thermodynamic” freezing point depression from the bulk freezing point To, due to the small drop size, of (6) To - To’ = (2~I,/R)(UTO/A) where D and A are the volume and heat of fusion per molecule, and T,’is the ‘thermodynamic” melting point at which solid and liquid forms of the clusters have the same vapor pressure. For any temperature T a t or below T i , the free energy G of a cluster with N molecules can be expresssd in terms of G,, the free energy a fully frozen cluster, and the fraction F of the cluster that is liquid, as G(T) = G, nA(T4 - T)F/To
+
+ 4?rR2u1,[2F/3+ (1 - q2j3 - 11 (7 )
or, if F is small, G(T) G,
+ NA(T,’ - T)F/To - 4*R2~1,F2/9
(8)
Therefore, according to the model, at the thermodynamic freezing point the free energy is at a maximum when the cluster is completely solid, and a cluster maintained at this temperature will spontaneously melt. This conclusion applies to clusters not exchanging matter by evaporation and condensation (e.g., to clusters in simulations such as the present one in which evaporation is too rare to have occurred). At temperatures lower than TO’ the free energy of the solid (F = 0) becomes progressively more favorable with respect to that of the liquid ( F = l), and the free energies of solid and liquid clusters become q u a l at a temperature designated TON by RMW, a temperature at which the freezing point lowering is 50% greater than the thermodynamic freezing point lowering. Below T i there is always a free energy barrier between liquid and solid but it is very low for s 1 just below T i and for 1 s far below T/. At T,,” itself the barrier is
-
-
Bartell and Chen AG*(T,,”) = 16*R2u1,/27
(9)
and substantial. According to RMW the relative abundances of liquid and solid clusters of radius R in equilibrium at a given temperature (not equilibrating through evaporation and condensation) can be expressed via the free energies of eq 7 by
N l / N s = exd-[G(F=l) - G(F=O)]/kT)
(10)
The above equation turns out to be quite misleading in its implied proportion of liquidlike and solidlike clusters because it neglects the extreme skewness of the barrier inherent in eq 7. What is more informative than eq 10’s relative numbers of the purely liquid and purely solid clusters is the ratio of populations of clusters that are mostly liquid (perhaps enveloping a small bit of solid) and clusters that are mostly solid (but coated with a film of liquid). By its very construction the RMW model quite reasonably implies a continuous range of ’’isomerization” between the extremes. If the free energy of any intermediate ‘‘isomer” given by eq 7 relative to that for F = 0 or 1 is not large compared with kT, there should be an appreciable probability of finding clusters with F values neither zero nor unity. That is, if the population in the range from F to F d F is taken to be dN(F) = K exp[-AG(F)/kT] d F (1 1)
+
where K is a normahtion constant, integrated relative populations of liquidlike and solidlike clusters can be found by integrating eq 11 on each side of the barrier. At T,,” it can be shown that N , / N , = (~kT,,”/108AG*)~/~
(12)
provided that the barrier AG* is large in comparison with kT,”, as it is in the case of CCl,. Here we emphasize that the ratio N l / N sof eq 12 has a different meaning from that of eq 10. It refers to the integrated populations N l and N, of liquidlike and solidlike particles, not to the more restricted and less utilitarian populations of completely liquid and completely solid particles. Calculating the ratio as a function of T makes it possible to estimate the temperature range over which the transition extends, for example, the range over which N J N , increases from, say, 0.1 to 0.9. In addition, eq 11 offers a way of estimating the probability of finding a cluster with a solid core and a liquid surface layer of any given thickness (R - r ) . Because kinetic aspects of the changes on heating or cooling a cluster as rapidly as was done in the MD study are far from trivial, the rate of crossing the barrier between solid and liquid forms needs to be considered. RMW propose a frequency of crossing of v = kT/h exp(-AG*/kT)
(13)
The number of time steps to cross the barrier, then, is presumably the order of magnitude of 1 / v ~ ,where is the length of a timestep ( 5 fs in the present work). Having referred to some key properties of the RMW model, not all of which have been presented before, we ask whether the illustrative model can give a quantitative account of the results of the present simulations. Fortunately, plausible values are available for all of the parameters required to test the model. These are discussed in the Appendix. Insertion of the parameters for a 225-molecule cluster of C C 4 into the RMW relations yields a 47 K thermodynamic lowering (To - To’) of the freezing point from the bulk freezing point of 246 to 199 K, and a 71 K lowering (To - T,,”) to 176 K to make the free energies of the solid and liquid clusters the same. Neither of these values is close to the 115 K midpoint of the transition seen in the warming stages plotted in the lower frame of Figure 8. In the MD simulations, melting begins at about 70 K and finishes by 160 K. By comparison, a solid RMW cluster upon being warmed would have been unable to surmount the barrier AG* and melt until being heated appreciably above TO’’ = 176 K, or 100 K above the MD onset of melting. This is because eq 13 suggests that some lo9 time steps would be required at T,”, or 100000 times more time steps than are taken in the MD simulation during the warming of a cluster by 100 K. Increasing
Structure and Dynamics of Molecular Clusters the temperature above T,,” rapidly increases the frequency of barrier crossing. Here it is well to point out that at T,,” the number of clusters in solidlike configurations does not equal the number of liquidlike clusters at equilibrium but exceeds them 20-fold according to eq 12. This asymmetry arises because the slope of the free energy curve is finite and approximately constant near F = 0 (solid) but is infinite at F = 1 (liquid). Of course this suppresses the probability of finding liquid clusters with small solid cores. Although solid clusters with liquid surfaces are less forbidden, quantitative calculations do not support the idea that solid cores are likely to be found surrounded by the thick, liquified regions so prominent in the MD clusters. Below T,,” the slope of G(F) is still so steep in the RMW model that fewer than half of the clusters would be expected to have a liquid surface containing as much as 3% of the bulk of the cluster. Of course, it is asking too much of the RMW model to reproduce accurately the exact behavior of a surface in a 225-molecule cluster whose outermost layer constitutes over half of the mass of the cluster. In summary, a literal application of the RMW relations to a solid cluster of CCl, 40 A in diameter, heated at a rate of 2 X 10” K s-I, indicates that few vestiges of melting will occur until the cluster liquifies rather sharply at a temperature between T,,” and TO’,near 186 K. Melting in the molecular dynamics simulation, on the other hand, is a protracted affair beginning at about 70 K. It is the broad range of temperature over which melting takes place in the MD run that we consider to be severely at odds with the RMW model. We attach less importance to the absolute value of the temperature at which the cluster is, say, half melted, for this hinges upon a somewhat arbitrary convention, and the bulk melting point corresponding to our model potential is not known.
The Journal of Physical Chemistry, Vol. 96, No. 22, 1992 8807 gross properties observed in the MD runs, it implies a much more
abrupt transition than was seen. It would be worthwhile if a more faithful model could be devised that is comparably accessible and as intuitively plausible. Such a model would requirea more explicit handling of the transition layer between phases. As such it would presumably also include the consequences of the Tolman parameter a well-known but controversial parameter which is s u p posed to characterize the thickness of the transition layer and to govern the interfacial free energy of very small particles. Whether such a model can be realized without a considerable increase in complexity is a question for the future.
Acknowledgment. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for the support of this research. We thank Messrs. S.Xu and F. Dulles for their expert assistance in computations.
Appendix Estimation of Parameters for Eqs 6-13. The most elusive parameter, the liquidsolid interfacial free energy uls, can be estimated with some confidence by applying Tumbull’s empirical relation3’ which holds that is proportional to the feat of fusion. In fact, because the value of uls has been determined for the interface between rhombohedral CCI, and the liquid (which, at 5.5 ergs/cm2, is consistent with the Tumbull r e l a t i ~ n )the , ~ ~value for the fcc/liquid interface can be taken as the value for the rhombohedral multiplied by the known ratio of the heats of fusion of the fcc and rhombohedral phase^.^^.^ The result is 3,g5 ergs/cm2. In the spirit of the model which neglects the differences in molar volumes and heat capacities between the liquid and solid phases, we disregarded the temperature dependence of ul,. To determine R and u we adopted the fcc molar volume of ref 9. In fairness it should be pointed out that To,the bulk melting point corresponding to the present model potential parameters, is not known and might deviate appreciably from the experimental melting point. The usually cited experimental melting point of 250 K is for the rhombohedral phase; that for the fcc phase is 5 K lowerag
Concluding Remarks Clusters composed of polyatomic molecules have already been shown to behave differently from atomic clusters in several respects. In addition to the obvious rotational degrees of freedom not found in atomic clusters, which introduce new elements of interest, there are pronounced structural differences. Van der Waals clusters of atoms are amorphous until they grow to comReferences and Notes paratively large sizes (consisting of approximately lo00 a t o n ~ s ) . ~ z ~ ~ ( I ) Bartell, L. S.;Dibble, T. S.J . Phys. Chem. 1991, 95, 1159. By contrast, clusters of polyatomic molecules are able to form (2) Bartell, L. S.; Dibble, T. S.;Hovick, J. W.; Xu,S.In The Physics and crystalline aggregates characteristic of the bulk when they contain Chemistry of Finite Systems: From Clusters to Crystals; Jena, P.. Rao. B. as few as several dozen m o l e ~ u l e s , I ~even , ~ ~when - ~ ~ the molecules K., Khanna, S. N., Eds.; Kluwer Academic Publishers: Dordrecht, 1992; Vol. 1, p 71. are quasispherical. In the present study of a 225-molecule cluster (3) Dibble, T. S.;Bartell, L. S.J . Phys. Chem., submitted for publication. of CCl, the size was more than ample for the cluster to exhibit (4) See, for example: Honeycutt, J. D.; Andersen, H. E.J . Phys. Chem. bulklike properties, including distinct phase changes. 1987, 91, 4950, and references cited therein. The melting transition began at the cluster surface, a phe( 5 ) Etters, R. D.; Kaelberer, J. B. J . Chem. Phys. 1977,66, 3233; Phys. Rev. A 1975, 11, 1068. nomenon often observed in prior simulations of atomic clusters. (6) Davis, H. L.; Jellinek, J.; Berry, R. S.J . Chem. Phys. 1987,86,6456. Rotational melting did not occur markedly before translational (7) Beck, T. L.; Jellinek, J.; Berry, R. S.J . Chem. Phys. 1987, 87, 545. melting in the particular instance of CCl, even though it has been (8) Quirke, N. Mol. Simul. 1988, I , 249. (9) Rudman, R.; Post, B. Mol. Crysr. 1968, 5, 95. observed to do so in the case of certain more spherical molecules (10) Cohen, S . ; Powers, R.;Rudman, R. Acra Crysrallogr. 1979, 835, such as hexafluoride^.^^^^^ Also unlike the melting transition of 1670. hexafluoride clusters, the melting of the C C 4 cluster was not (11) Bean, V. E.; Wood, S.D. J . Chem. Phys. 1980, 72, 5838. reversible on the time scale of the simulations. When the cluster (12) Smith, W.; Fincham, D. Program MDMPOL, CCP5 Program Liwas cooled, it froze to a glassy mass,consistent with extrapolations brary, SERC Daresbury Laboratory, Daresbury, U.K. (13) Bartell, L. S.;Brockway, L. 0.;Schwendeman, R.H. J . Chem. Phys. of experimental results for the freezing of C C 4 clusters in su1955, 23, 1854. personic jets.’ What is noteworthy about the crystal-to-liquid(14) Yaws, G . L. Physical Properties, A Guide; McGraw-Hill: New York, to-glass process is F( T), the fraction of the cluster that is liquid 1977. at temperature T. Except for a normal hysteresis associated with (15) Fuchs, A. H.; Pawley, G. S.J. Phys. Fr. 1988, 49, 41. (16) Einstein, A. Investigations on the Theory of the Brownian Morion; the high heating/mling rates of the simulation, this fraction was Dutton: New York, 1926. the same for the crystal-liquid and the glass-liquid transitions (17) Bartell, L. S.; Dulles, F. J.; Chuko, B. J . Phys. Chem. 1991,95,6481. according to the numerical indicators of F( 7‘). This independence (18) Angell, C. A.; Clarke, J. H. R.; Woodcock, L. V. Adu. Chem. Phys. of the nature of the solid is in harmony with Lindemann’s theory 1981, 48, 408. (19) Isihara, A. Srarisrical Physics; Academic Press: New York, 1971; of melting2’ P 5. Finally, the temperature range over which the melting trans(20) Silverman, B. W. Density Estimation for Srarisrics and Dara Analformation took place in the simulation of the 225-molecule cluster ysis; Chapman & Hill: New York, 1986; pp 23-25. (21) Lindemann, F. A. Phys. Z 1910,11, 609; Engineering 1912,94,515. of CCl, was compared with that predicted by a model formulated (22) Rousseau, B.; Boutin, A.; Craven, C. J.; Fuchs, A. H. Mol. Phys., by Reiss, Mirabel, and Whetten.” The RMW model is such an submitted for publication. attractively simple conception that it is regrettable it does not seem (23) See, for example: Kondo, T.; Tsumuraya, K.; Watanabe, M.S.J . to model cluster behavior more closely. Even though it is superior Chem. Phys. 1990, 93, 5182. to earlier treatment^^^+*^ in qualitatively incorporating all of the (24) Barrat, J.-L.; Klein, M. L. Annu. Rev. Phys. Chem. 1991, 42, 23.
8808
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Stable Conformations of Benzylamine and N,N-Dlmethylbenzyiamlne S.Li, E. R. Bernstein,* and J. I. Seeman Department of Chemistry, Colorado State University, Fort Collins, Colorado 80523 (Received: June 1 , 1992)
Mass-resolved excitation spectra (MRES) of deuterated and nondeuterated benzylamine, N,N-dimethylbenzylamine,and a number of their 4-ethyl-, 3-methyl-, and 3-fluoro-substituted derivatives are studied. The equilibrium conformations of these compounds are determined. In particular, the general torsion angle T~ (C,,rtb-C,w-C,-N) is verified as near 90' and the angle 72 is determined for both benzylamine [72 = (Cip
