J. Phys. Chem. 1995, 99, 1080-1089
1080
FEATURE ARTICLE Nucleation Rates in Freezing and Solid-state Transitions. Molecular Clusters as Model Systems Lawrence S. Bartell Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109 Received: July 12, 1994; In Final Form: November 10, 1994@
Molecular clusters are sometimes seen to undergo the same phase changes observed in bulk systems, but at much deeper supercoolings and at enormously higher rates. Under other conditions they can be seen to transform spontaneously to crystalline structures not found in the bulk, for reasons that are beginning to be understood. Nucleation rates in clusters have been measured experimentally in supersonic beams by electron diffraction and computationally by molecular dynamics techniques. The two approaches are proving to be complementary and effective research tools. Despite the fact that a large proportion of the molecules in clusters lie in surface layers, compensating characteristics make clusters attractive objects of inquiry. Although surface layers are poorly representative of ordinary phases, cluster cores have been shown by experiment and computation to have properties close to those of bulk matter. Advantages and limitations of clusters in research on the dynamics of phase changes are reviewed, and illustrative examples are presented.
Introduction Clusters have become extremely popular subjects for research, in large measure because they occupy a regime intermediate between the molecular and the bulk. Almost all of the research has been carried out on aggregates of relatively few atoms or molecules, with atomic clusters pred~minating.'-~Of particular interest has been the question of how properties of clusters evolve toward those of bulk matter as their size increases. The thrust of the present paper is entirely different. The clusters to be considered are large and composed of polyatomic molecules. They are deliberately made large enough for the properties of their cores to be good approximations of those of bulk systems. The advantages offered by such clusters over macroscopic systems in research on liquids and solids is one of the main themes of the present paper. Clusters serve as particularly convenient models of condensed matter in studies of homogeneous nucleation in phase changes and yield intimate views of the molecular motions involved. These advantages, which will be outlined in more detail presently, are not possessed to the same extent by atomic clusters. Although atomic clusters are endowed with their own unique elements of interest, they differ markedly from molecular clusters in the rate at which their structures approach those of the bulk. As a rule they must be much larger. Structures and properties of small clusters of metallic atoms tend to be governed by the electronic shell structures of the aggregates (quantum confinement Structures of small van der Waals clusters of atoms are directed by the drive to achieve a maximum packing efficiency. It turns out that polyicosahedral (amorphous) arrays provide a denser packing of limited numbers of soft spheres than does the cubic closest packing enjoyed by the bulk crystals. Bulk packing arrangements do not begin to propagate until an amorphous core of perhaps 1000 atoms has formed.6 On the other hand, in those cases we have examined, solid clusters containing only a few dozen polyatomic molecules are able to organize spontaneously 'Abstract published in Advance ACS Abstracts, January 1, 1995.
0022-365419512099- 1080$09.0010
into lattices with very nearly the same structure and intermolecular spacings as those of the corresponding macroscopic crystals.' We emphasize, then, that molecular clusters differ essentially from atomic clusters in the size dependence of their properties. In the remainder of this paper we shall consider only molecular clusters. When investigations of large molecular clusters were first begun at the University of Michigan, it was not anticipated how useful clusters could be in research on phase changes. To the structural chemists involved the challenge was to find whether an alternative experimental approach might offer useful new information about the liquid phase-the motivation being that, among the common forms of matter, liquids are surely the least understood. Although the traditional tools for studying liquid structure are X-ray and neutron diffraction, electron diffraction appeared to offer certain advantages, in principle. The best electron diffraction patterns possess signal-to-noise ratios more than an order of magnitude higher than any reported for X rays or neutrons, and electrons can be easily focused to yield sharper resolution of detail than is customarily attained in conventional studies of liquids. Although liquid diffraction patterns tend to be diffuse, the sharpness of the first peak in intensity gives information about the range of order. The greatest advantage of electrons in gas-phase experiments, however, is a serious impediment in research on liquids. Electrons are scattered perhaps 100 million times more strongly by molecules than are X-rays or neutrons. Therefore, it is a simple matter to record electron diffraction patterns of gas molecules. It is quite a different matter to obtain diffraction patterns from condensed phases that are free from excessive absorption and multiple scattering and, thereby, readily interpretable. Only if the condensed sample is exceedingly thin, say of the order of 100 A, can the desired conditions be met. It seemed that it might be possible to achieve such thin samples of liquids by condensing vapor in supersonic flow to generate large liquid clusters. As early as a quarter of a century ago the Orsay electron diffraction group successfully produced large clusters by the
0 1995 American Chemical Society
Feature Article supersonic t e c h n i q ~ eand ~ , ~verified that electron diffraction was an ideal technique for studying them. Being physicists, the Orsay scientists concentrated upon clusters of rare gas atoms and of a few particularly simple substances such as water and carbon dioxide. Several years later gas dynamicists at Northwestern University followed a similar course.l0 Only solid clusters were observed by the two groups. Nevertheless, when vapors of such polyatomic molecules as benzene or butane were examined at Michigan, liquid clusters were found immediately.11.12 The anticipated superior signal-to-noise ratios and resolution of diffraction detail were realized at once. Moreover, it was soon recognized that liquid clusters could be studied at highly supercooled temperatures which gave correspondingly sharper pair correlation functions and more discriminating tests of potential f ~ n c t i 0 n s . l ~It proved to be simple to generate clusters 100 A in diameter, and clusters referred to in experimental studies in the remainder of this paper will typically be of that size. What diverted the research program from continuing to concentrate upon the structures of liquids were observations that, for some of the substances investigated, as many as two or three different cluster structures could be produced, depending upon the conditions of the supersonic f l ~ w . ’ ~ ,It’ ~ seemed astonishing that one could, as it were, control how molecules packed together by the way one passed them through a miniature nozzle. Therefore, a program was undertaken to learn how such a phenomenon could occur. It became apparent that the various liquidlike and solid structures of the large clusters produced corresponded unmistakably to phases of macroscopic systems. Eventually, it was recognized that the thermal history of the clusters as they formed depended upon conditions of the expansion, and variations in the conditions produced different phases.l63l7 What was surprising was the speed with which some of the clusters transformed from one phase to another. Sometimes it was found that transitions were controlled by kinetic factors rather than by thermodynamics so that phases could be produced which had never been observed in macroscopic ~ y s t e m s . ~ J It * even proved to be feasible to observe some of the same phase changes in molecular dynamics simulations that occurred in supersonic jets, thus making it possible to follow the course of the transformations in molecular detai1.7,18-21 It is an account of the readily observable phase changes in clusters and what can be learned from them that constitutes the principal theme of the present paper.
Significance of Research on Phase Changes Phase changes are so commonplace and simple in comparison with many of the chemical transformations carried out by synthetic chemists that it is easy to take them for granted. They have been recognized and pondered about since ancient times. Nevertheless, a recent perspective in the journal Sciencezz lamented our lack of understanding of one of the most familiar phase transitions of all, that of freezing. Although the thermodynamics of phase changes has been a well developed discipline for a century?3 the kinetic aspects have been troublesome to study. When David Turnbull began his seminal research on the freezing of metals a half-century ago, it was commonly believed that metals could not be ~ u p e r c o o l e d . ~The ~ notion that nucleation was involved in freezing was considered dubious because it was supposed that the local environment of atoms in a melt was too much like that in a crystal for such an idea to have much relevance. What had obscured fundamental aspects of the dynamics of freezing in prior observations was the rapid heterogeneous nucleation of the process induced by the inevitable traces of foreign matter found in bulk liquids. Turnbull showed that when liquids were divided into droplets so fine
J. Phys. Chem., Vol. 99, No. 4, I995 1081 that only a small percentage contained foreign particles catalyzing the transition, he could indeed substantially supercool liquid metals and other material^.^^-^^ Moreover, he developed a promising theory of homogeneous nucleation in freezing and showed that it successfully accounted for his experiments.28 Although Turnbull showed the way progress could be made in the elucidation of phase changes in condensed matter, it remained extremely difficult to carry out definitive, wellcharacterized experiments. Indeed, data from Turnbull’s classic 1952 study of the freezing of mercuryz9 are still being cited as the only thoroughly reliable results available to test new treatments of the temperature dependence of nucleation rates.30 This very fact gives a correct impression about the scarcity of trustworthy data on the kinetics of nucleation. One problem remaining even when liquids were divided into very small droplets was that the droplets had to be somehow kept apart. Droplets were commonly suspended in an emulsion or surrounded by an oxide layer which, one hoped, played no active role in the transformation itself. Quite apart from the experimental obstacles impeding research on nucleation are difficulties in formulating a working theory of nucleation rates. The method advanced by Turnbull and coworkersz8 was based on a capillary model. Growing evidence confirms that the capillary or “classical” model is excellent qualitatively but deficient quantitatively. It assumes bulklike properties for molecular aggregates as small as the “critical nucleus,” the structural fluctuation believed to initiate the phase change, and neglects any effects of the thickness of an interface between phases. Treatments based on density functional t h e ~ r y ~ show l - ~ ~considerable promise but have not yet been implemented into treatments suitable for routine analyses of experimental data. Imperfect though it be, at least the capillary theory is simple to apply, and it does furnish a means to predict with useful accuracy a rate at one temperature given the rate determined at a different temperature. Presumably the availability of new experimental data on the kinetics of phase changes would provide renewed impetus to the development of effective and practicable nucleation theories. Because nucleation is believed to lie at the heart of many important processes in catalysis, metallurgy, materials engineering, meteorology, earth sciences, and science and technology in general, we are obliged to seek alternative ways to investigate it. The next few sections review the advantages and limitations of the cluster approach and sketch practical features of experiments and results. Readers who seek to gain a quick impression of the sorts of things that can be studied without consideration of the technical details involved should skip directly to the illustrative examples at the end of the article. The reader should bear in mind that the cluster techniques described herein are in their early stages of development. We hope that the obvious imperfections and unfinished aspects of experiment and interpretationwill motivate those with an interest to make significant improvements.
Advantages and Limitations of Clusters as Model Systems Advantages. Just because clusters closely simulate bulk matter in some respects and are convenient to produce is, of course, insufficient reason for expending resources on their study. What is positive about clusters is that they offer an alternative avenue in the search for ways to learn about matter in transition. As reviewed above, the conventional approaches to research on nucleation have proven to be troublesome, partly because of the difficulty of adequately freeing the systems of heterophase contaminants. When it was found that nucleation
Bartell
1082 J. Phys. Chem., Vol. 99, No. 4, 1995 rates could be measured quite readily in large molecular clusters that were in contact only with their own vapor or, in some cases, with rarefied noble gases,35 it seemed fitting to explore the matter further. Already proven to be fruitful among the experimental techniques are electron d i f f r a ~ t i o n ~and ~ -coher~~ ent Raman s ~ a t t e r i n g . Computational ~~,~~ techniques have also yielded valuable insights. Monte Carlo (MC)42-44and molecular dynamics (MD) s i m ~ l a t i o n s ~ ~45-49 ~ * - have ~ ~ - given significant insights into the behavior of polyatomic molecules in clusters as they evolve from one structure to another. Besides making it possible to work with systems free from the influence of contamination, there are several other advantages associated with clusters. As a consequence of the small volumes and rapid cooling rates that can be imposed, exceptionally deep supercoolings can be attained. In many cases this supercooling leads to extraordinarily high nucleation rates, rates which can easily be followed by the techniques applicable to clusters. This allows nucleation to be examined in heretofore unstudied regimes and sometimes leads to the formation of phases never before seen in the bulk.@ In addition, both the experimental and computational techniques are as applicable to certain solid-state transformations as to freezing. They have yielded the first homogeneous nucleation rates known to the writer for solid-state transitions in pure, one component systems. But if it is really information about the bulk dynamics that is sought, why not carry out simulations on bulk systems-finite systems with periodic boundary conditions-rather than on clusters? The reason is that periodic boundary conditions can interfere severely with nucleation and growth of crystal phases unless the systems are enormous. This is particularly true when phases of low symmetry are formed. Free boundaries, namely those of clusters, appear to exert no such interference. Especially important among the results yielded by the molecular dynamics simulations are the direct views of the cooperative motions of the molecules. Critical nuclei, entities which until recently had been more the imagined objects of theorists than the concrete objects of observation, can be watched as they form,20.48.49 Limitations. As felicitous as the above advantages of clusters may be, they do not come without penalty. It is not easily possible to control over wide ranges the temperatures of freely coasting clusters. Generally temperatures are in the vicinity of the so-called “evaporative cooling temperat~re,”~O-~~ a quantity to be discussed in more detail later. In addition, with present experimental arrangements, the time scale of experiments is limited to the range from a microsecond to perhaps 100 ps. If it were expedient to do it, this range could be extended appreciably in either direction. A consequence of the thermal and temporal limitations is that only a fraction of possible transitions are accessible by the current experimental techniques. Even more limited are the time scales of MD simulations feasible with current hardware, although the temperature can, of course, be controlled at will. How large nucleation rates must be if transitions in clusters are to be observable can be inferred directly from the characteristic cluster sizes and time scales of the method of investigation. From the definition of nucleation rate J ,
J = (1/V) dN*ldt
(1)
or the number N* of critical nuclei produced per unit volume per unit time, it is elementary to estimate what magnitude J must have if a critical nucleus is likely to form in the volume of a cluster in the time available in the experiment or simulation. In electron diffraction experiments with clusters of 100 8, and times of 10-100 pus, J must be of the order of -1028-1029
m-3 s-’. In MD simulations with cluster diameters typically about 30 8, and times of perhaps tens of picoseconds at a given temperature, Jmust be m-3 s-l. When it is recalled that nucleation rates observed in conventional experiments on the freezing of small droplets range from perhaps lo7 to 10l6 m-3 s-l, or some 15-30 orders of magnitude lower than rates corresponding to the supersonic jet or the MD techniques, it is clear that it cannot be taken for granted that the transitions observed by the new methods can be considered typical. Nevertheless, as will be seen, it has been possible to study some of the same transitions by molecular dynamics simulation^^^^^^^^ as were observed by electron d i f f r a ~ t i o ndespite , ~ ~ the 7 order of magnitude difference in rates. Unfortunately, electron diffraction has monitored only one transition, so far (the freezing of ~ a t e r ) , that ~ ~ ,had ~ ~ been investigated by conventional ~~,~~ methods (at rates 20 orders of magnitude s l o ~ e r ) . Although the overlap of substances studied by the conventional method and by the cluster method has been regrettably small, the two methods gave closely corresponding results in the one overlapping case, as we shall see. Most of the nucleation rates measured to date have been on materials so volatile that their vapors can be seeded into supersonic flows at room temperature. This situation is changing rapidly, however, as sample systems capable of operating at high temperatures are being devised.58 Results for nonvolatile systems are not yet available for presentation. Another concem about the cluster method is that not everyone agrees that matter confined to spaces as small as cluster volumes is genuinely representative of bulk matter.59-62 These reservations are particularly strongly expressed in the case of supercooled water whose unusual properties are at the heart of an active c o n t r ~ v e r s y . ~Even ~ - ~ ~in more pedestrian systems, the smallness of cluster volumes raises questions. The fraction of molecules in a cluster which are on the surface is very high. A simple estimate of the surface fraction F for a cluster of N molecules is
where (3)
-
-
For clusters of the sizes typically encountered in electron diffraction ( N lo4- lo3) and in MD simulations ( N 50050), these fractions are 21-41% and 48-80%, respectively. Surface layers are not adequately handled by the conventional capillary theory of nucleation. Quite apart from that problem, when the old phase wets the new phase being formed (a circumstance occurring in almost all of the systems examined to date at Michigan), nucleation in cooling clusters always occurs in the interior and never in the surface layer. Therefore, in transitions occurring in MD simulations of clusters of a few hundred molecules, only a small fraction of the volume of the cluster can be considered to be effective in nucleation.
Characteristic Temperatures When clusters produced are warmer than their characteristic evaporative cooling temperatures, they cool at initial rates as high as lo7 Ws as they evaporate. During this period a rate would have to be extraordinarily high to be observed. Such cooling, of course, rapidly reduces the vapor pressure until the evaporation rate becomes so low that the temperature-time profile flattens. (Strictly, the temperature continues to fall, but its decrease is roughly linear in the logarithm of the evaporation time.) For -100 A clusters the effective flattening occurs, on
J. Phys. Chem., Vol. 99, No. 4, 1995 1083
Feature Article a 10-50 ms time scale, at a sufficiently well-defined temperature that it has been named the “evaporative cooling temperature” Tevp,proposed51 to occur at
Tevp= Q.Q4AEv,JR
(4)
for substances obeying Trouton’s rule. An alternative rule36is sometimes used, especially for associated liquids not obeying is the temperature Trouton’s rule and for solids, namely, that Tevp at which the vapor pressure is -0.4 Pa. Although these two rules afford a quick, approximate idea of cluster temperatures, a more accurate evaluation can be obtained by calculating the temperature profile via kinetic t h e 0 1 - y ~(or ~ ~the ~ equivalent Mots statistical thermodynamic treatment).52 It is possible to cool clusters significantly below their evaporative cooling temperature if they are generated in a high concentration of carrier gas. The rate at which such clusters continue to cool is so low, however, that a transition which has not already happened is unlikely to happen during the usual observation time. It is the manipulation of the temperature profile by adjustment of the flow conditions that allows one to cause a transition to occur in a convenient spatial region of the supersonic jet.
Some Aspects of Classical Nucleation Theory and Its Parameters Nucleation Theory for Condensed Phases. As initially formulated for freezing by Turnbull and Fisher 28 and Buckle,67 the steady-state rate of formation J of critical nuclei per unit volume per unit time was expressed as
J = A exp( -AG*/kT)
(5)
where AG* is the free energy barrier to the formation from the liquid of the critical nucleus believed to initiate the transition. The preexponential factor in the Tumbull-Fisher theory is
A = N,(kT/h) exp(-dkr) where N, and E represent respectively the number of molecules per unit volume and the activation energy for the jumps of molecules across the boundary from the old phase to the new. A current variant using viscous flow to model the molecular jumps across the solid-liquid interface in the classical theory of homogeneous nucleation gives36
A = 2(as,kT)”2/(v,5’37)
(7)
where a,]is the interfacial free energy per unit area of the boundary between solid and liquid, vm is the molecular volume l/Nv, and q(T) is the liquid viscosity. A related formulation can be applied to solid-state transitions. For those so far encountered in clusters, the molecular jumps to the new phase correspond principally to reorientations instead of to translations, which is why the transitions are facile enough to be seen by our supersonic technique in the first place. For such cases the preexponential factor takes the
A = 2(aijkT)1’2~,-U3[~0 exp(-~/kr)] where the rotational jump frequency [YOexp(-~/kr)] can be estimated from spectroscopic information or rotational diffusion coefficients derived from molecular dynamics simulations. The free energy barrier, AG*, to the formation of a spherical nucleus is given by
AG* = 16nai,3/[3(AGv
+ w’I21
(9)
as shown by Gibbs over a century ago?3 where AG, < 0 is the Gibbs free energy of the transition from the old to new phase per unit volume at the standard pressure and supercooled temperature calculable from the relation AGv(T) = -(l/VJTrAS(T)
dT
In the above expression V is the molar volume, and AS( r ) is the molar entropy change at temperature T estimated with the aid of an extrapolation of the difference between the heat capacities of the two phases. The term w’may include the work of elastic deformation of a nucleus growing in a solid phase or the work of changing the surface area of an outer liquid phase i due to the volume change accompanying the formation of the nucleus of phase j . Some evidence exists that the contribution to AG* from the misfit between the old solid phase and the new may play a minor role in clusters.38 The contribution w’ for a nucleus growing in a spherical liquid of radius ro is (2aI/ro)(el where 2a,/rois the Laplace pressure exerted by the outer phase on the inner phase and the e’s are densities of the phases.68 It is in the exponent AG*/kTrather than in the factor A that a, can exert an ovemding effect on the nucleation rate. For example, a 20% change in usl for CC4 changes J for freezing at 175 K by 10 orders of magnitude.36 A similar change in ul, for very small clusters of tert-butyl chloride at 100 K changes J for the phase I11 to phase IV transition by only an order of magnitude, however. From the above relations it is apparent that the parameter a, can be derived from the observed nucleation rate if the temperature of nucleation and the other physical properties in the equations are known (or can be plausibly estimated). For sake of example we shall adopt these relations for the time being but will indicate the desirability of alternative treatments and suggest simple modifications when we look at specific examples. Remarks about the Interfacial Free Energy. A presentation of rates of phase transitions, the principal observable of the present work, is of marginal interest, in itself. In the limited numbers of studies of nucleation rates it has been customary to flesh out accounts of results by interpreting rates in terms of the classical (capillary) theory of nucleation sketched in the previous section. The key parameter in this theory is the “interfacial free energy” a,. Derivable from the nucleation rate, a, plays a role somewhat analogous to that of the activation energy in the kinetics of chemical reactions. Although its name is suggestive of a thermodynamic variable, it is a kinetic parameter, of course. Its most important role, perhaps, is to facilitate the estimation of nucleation rates at greater or smaller degrees of supercooling from a given measured rate. Such an undertaking has proven to be helpful in this laboratory in forecasting whether or not transitions seen to occur on the microsecond time scale of electron diffraction experiments might be reproduced at greater supercoolings on the picosecond time scale of molecular dynamics simulations. To what extent ulJ reflects the true thermodynamic variable has not been determined very precisely. Obvious difficulties associated with measurements of the work required to increase the interfacial area between a solid and another phase without performing other work (such as that of elastic or plastic deformation) have made direct determinations of the thermodynamic variable an elusive goal. Alternative measurements in terms of equilibrium forces have been carried Additional factors blur the interpretation of alJ. Because it is possible for two phases to coexist at equilibrium at ambient pressures at only a single temperature,
eJ)/e,,
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1084 J. Phys. Chem., Vol. 99, No. 4, 1995
cases studied to date, the growth of the solid phase is so rapid in comparison with the rate of appearance of nuclei (see below) that the nucleation rate is derivable quite directly from F(t) once the size of the clusters has been found. This observable can be deduced from the breadths of the Debye-Schemer diffraction rings via the relation74
A
D = 0.553/W6
I
1
I
I
2
I
I
3
I
I
4
I
(12)
where D is the diameter of the cluster (assumed to be spherical) and We is the full width at half-maximum of the Bragg peaks expressed in terms of the conventional X-ray variable (sin 8)/
A.
What are not supplied with adequate sensitivity by the diffraction pattems are the temperatures of the liquid clusters. Figure 1. Electron diffraction pattems of freezing ammonia clusters Even though the lattice constants of the crystalline clusters may at various times of flight @s) beyond the nozzle: (a) 24.6, (b) 27.2, establish their temperatures if prior measurements of lattice (c) 29.7, (d) 32.3, re) 34.9, (f) 37.4, (g) 45.0, and (h) 50.3. From ref constants at several temperatures are available, this information 39. does not disclose the liquid temperature. For one thing, the heat of crystallization evolved has a substantial effect on the some theorists consider that ozjhas a meaning only at that single temperature. For another, what determines the cooling rate in temperature and not at the deep supercooling which may be the first place, and the “evaporative cooling temperature” at encountered in nucleation experiment^.^^ Worse, as already which cluster properties are commonly observed, is the evaporamentioned, it is clear that the classical theory of nucleation has tion rate, and evaporation rapidly dissipates the heat evolved only qualitative validity, and therefore, the kinetic parameter as the clusters freeze.39 Investigations of the freezing of liquid oil must, to some extent, be a bit of a fiction. Be that as it clusters by coherent Raman scattering are able to yield apmay, we will interpret results in terms of conventional capillary proximate cluster temperatures in at least some case^.^^^^^ A theory for the remainder of the paper because it does provide a combination of diffraction and spectroscopic measurements means of comparing results of enormously disparate nucleation might be advantageous. Fortunately, a large fraction of clusters rates on a common and intuitive basis. Furthermore, we will investigated in this laboratory can be shown to leave the Lava1 follow T u r n b ~ l and l ~ ~Spaepen’l in assuming that oljtends to nozzle in which they are generated at temperatures appreciably increase in temperature for liquid-solid interfaces because of above the evaporative cooling temperature. For these clusters the excess order imposed on the liquid in contact with the solid it is a simple matter to compute the rate of cooling and, hence, at the interface. For purposes of illustration we will assume the temperature profile, probably to within a degree or two at that us[increases as 2” with n assumed to lie in the range 0.3a given time of flight. 0.4, the magnitude found in Turnbull’s careful experiment^.^^ The Freezing of Ammonia. What is remarkable about the One way of assessing the values of interfacial free energies results observed in the supersonic jets is how fast the freezing derived from rates of nucleation is to see whether they are more or less consistent with an empirical relation found by T ~ m b u l l . * ~ of the highly supercooled clusters can be. For ammonia clusters, whose diameters are about 99 A when they begin to freeze at Turnbull noted that os[as determined from freezing rates tends the nucleation rate is 1.2 x 10 30 m-3 s-l. 120 K (Figure to be a fairly universal fraction, k ~ of, the enthalpy of fusion Although the nucleation rate for ammonia had not been per unit area, or measured previously to the writer’s knowledge, rates for the somewhat similar and highly studied case of water were lower, and lower by 16-20 orders of m a g n i t ~ d e ! ~ ~Is. it~ ’reasonable to expect this enormous enhancement of rates characteristic of where AHf,, and V represent respectively the molar enthalpy clusters (of a few thousand molecules) over that of “bulk” of fusion and molar volume, and N A is Avogadro’s number. ammonia (drops of over 1 p m , containing over 1Olomolecules) For metals kr is about 0.45, and for semimetals and nonmetals to have any relevance in research pertaining to ordinary systems? it is often approximately 0.32. We shall see that an analogous One way to assess this is to examine the interfacial free energy relation appears to apply to solid-state transitions. implied by the nucleation rate. The result for ammonia clusters is 23 mJlm2, a result that is comparable to the value of -29 Illustrative Examples d l m 2estimated from Turnbull’s empirical relation between the kinetic parameter osl and the heat of fusion for nonmetallic Experimental Information about Phase Changes. The freezing of water, ammonia, and carbon tetrachloride provides substances. The comparison would have been even closer if the 120 K result had been adjusted to higher temperatures by instructive examples to begin with. Electron diffraction pattems the amplification factor F referred to above. The term w’ in of ammonia39 at increasing times of flight are shown in Figure 1. The diffraction pattems yield several crucial pieces of eq 9 facilitates nucleation but changes the derived a,l by less than 2%. information. They establish at once what solid phase is produced, which knowledge had seldom been available in prior Growth Rates of Solid Phase and Consequences. In the studies of nucleation rates. In the case of ammonia the calculation of a nucleation rate for the freezing of ammonia, it crystalline phase formed is the stable cubic phase of bulk was supposed that the freezing was mononuclear; Le., it was ammonia. For CC4, whose phase behavior is c ~ m p l e x ,and ~ ~ , ~ ~assumed that each cluster froze immediately after a critical for water, the crystalline phases generated are not those formed nucleus appeared in it and before another nucleus could form. The validity of the assumption can be tested by Kashchiev’s when the bulk liquid is frozen, as will be discussed presently. In addition, the time-dependent diffraction pattems are readily criterion75 for mononuclear freezing. That is to say, if the linear growth rate G of the solid into the liquid obeys the inequality analyzed in terms of the fraction F(t) of clusters frozen. In all
S (25
Feature Article
J. Phys. Chem., Vol. 99,No. 4, 1995 1085
35 where V is the volume of the liquid sample and a is the ratio of the volume in which the transformation is detected to the sample volume, then freezing is mononuclear. In the case of ammonia the mean cluster volume and nucleation rate require, for a set to unity, that G be much larger than 4 “/s. Because no information about the growth rate is available, it is necessary to estimate it. One choice, consistent with the preexponential factor A of eq 7, is to use the Wilson-Frenkel (WF) theory of diffusion-limited g r o ~ t h . ~ Application ~,’~ requires an estimate of the viscosity of the liquid in the highly supercooled regime, an estimate that can only be made rather crudely (see ref 39), but the real problem lies deeper. It tums out that the velocity computed on the basis of WF theory is marginal in satisfying eq 8, despite indirect evidence that the freezing of ammonia and, indeed all other liquid clusters studied to date, is in fact mononuclear. A possible explanation lies in a heuristic treatment by Burke, Broughton, and Gilmer (BBG).77 Their treatment strongly suggests that the Wilson-Frenkel approach, while giving the correct order of magnitude of the growth rate of crystals near the melting point, underestimates by many orders of magnitude the rate at large supercoolings. Therefore, it is quite certain that the Kashchiev criterion is satisfied in the freezing of ammonia.78 What is more important is the implication that eq 7 is inadequate. Presumably the growth rate of nuclei, once formed, should depend upon the rate of jumping of molecules from the old phase across the boundary into the new phase in much the same way as it does in the original rate of formation of critical nuclei. It is plausible, then, to modify the preexponential factor A of eq 5 in such a way as to embody the difference between the WF and BBG jump frequencies. One way of doing it gives the expression
A = (5f/av, 213 )(3~r,Jm)”~[1- exp(-AG,,/RT)]
(14)
in whichfis the fraction of jumps that are successful (commonly taken as ~ 0 . 2 7 )a, is ~ ~a characteristic intermolecular spacing, and m is the molecular mass. In a fundamental analysis of crystal growth rates Oxtoby and H a r r ~ w e l lcorroborate ~~ the BBG conclusion that the WF theory tends to underestimate the growth rate and sometimes to underestimate it severely. They also find that the BBG approach tends to exaggerate it. In the examples tabulated by Oxtoby and Harrowell, then, the correct magnitude tends to be intermediate. To provide a little more perspective about effects of theoretical uncertainties in the illustrative examples given, we shall plot results based on both eq 7 and eq 14. Somewhat better results than those given by either alternative alone might well be approximately the geometric means of the separate quantities (arithmetic means of the logarithms). In Figure 2 are plotted nucleation rates for ammonia that might be expected at different degrees of supercooling as projected from the single experimental nucleation rate at 120 K. Plainly, experimental data at other temperatures would help considerably in discriminating between alternative formulations of rate theory. One of the weaknesses of the supersonic technique is the difficulty in studying rates over a wide range of temperatures. In any event, the projections of Figure 2 indicate that the freezing of ammonia is not a favorable case for simulation by molecular dynamics techniques with today’s computers. If eq 14 is adopted instead of eq 7 in the analysis of the nucleation rate, it will, of course, influence the estimate of the interfacial free energy parameter, raising its value by perhaps 15% in the examples in this section. This increase will not
30
25 20 15
10
5
0 80
100
120
T,
140
160
180
K
Figure 2. Temperature dependence of nucleation rate (m-3 s-l) for the freezing of ammonia according to classical nucleation theory with prefactor A of eq 7 (long dashes) and eq 14 (short dashes), adjusting a,, at 120 K to make the curves pass through the experimental point at 120 K, and taking a,l to be proportional to P 3 .
improve the slightly low estimate for ammonia referred to above in the comparison with the value based on the empirical Turnbull relation because the information incorporated in the Tumbull database was derived using a preexponential A more closely related to eq 7 than to eq 14. Freezing of CCh and HzO. The cases of carbon tetrachloride and water are similar to that of ammonia in that clusters of all of the substances freeze at about the same rate at their evaporative cooling temperatures. As mentioned above, one difference among the three examples is that only ammonia freezes in clusters to the same phase as it normally does in the bulk. When bulk carbon tetrachloride is cooled below 245 K, it solidifies to a metastable cubic form (Ia) which only slowly transforms to the stable rhombohedral form (Ib) unless the system is cooled below 234 K.73 At temperatures colder than 234 K the transition to Ib is rapid. Below 225 K a monoclinic form (11) is the stable structure.73 Large molecular clusters, cooled by evaporation to 175 K, freeze neither to Ia, the form kinetically preferred near the freezing point, nor to the monoclinic form that is stable at 175 K. Instead, they freeze directly to the rhombohedral form Ib.36 In accordance with Tumbull’s empirical relation between interfacial free energy and the heat of transformation, the barrier to the formation of monoclinic nuclei is expected to be higher than that for rhombohedral nuclei. Therefore, the fact that monoclinic crystals were not formed is readily understood from the kinetics. It is less clear why the cubic form was not encountered. Only the nucleation rate at 175 K was determined for carbon tetrachloride. Projections of the rate to other temperatures (adopting the tentative rule that oSlit proportional to predicted that MD simulations would be unlikely to lead to crystallization as a sample is cooled on the MD time scale.. In fact, a liquid cluster of 250 CC4 molecules cooled in a subsequent MD simulation failed to crystallize and solidified, instead, to an amorphous mass.34 An interfacial free energy of 5.46 d / m 2 was founds0 from the nucleation rate (via eq 7) in approximate agreement with the 5.0 d / m 2 value predicted from the Tumbull relation, indicating that the cluster is behaving as would be expected of a normal macroscopic system of a nonmetallic substance, despite its small size and great supercooling. Water is a particularly interesting case. The unresolved c o n t r o ~ e r s i e over s ~ ~the ~ ~interpretation ~~~ of its properties when it is highly supercooled are too complex to be reviewed here. It is clear, however, that 75 A molecular clusters cooled to 200 K by evaporation manage to avoid the extreme anomalies
Bartell
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35 30
25
-
20-
I
0.5
I
I
I
1.5
I
3.5
2.5
10
I
I
I
4.5
S (A") Figure 3. Electron diffraction patterns of freezing water clusters at various times of flight (us) beyond the nozzle: (a) 10.9, (b) 14.5, (c) 18.2, (d) 21.8, (e) 25.5, and (fl 29.1. From ref 54. 260
250
I
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-
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-
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-
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I
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"
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28 26
200 190 -20
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Figure 5. Experimental nucleation rates (m-? s-I) for the freezing of water: clusters, circles; water-in-oil emulsions, squares, ref 56; and solid diamonds, ref 57. Calculated temperature dependence according to eq 5 with prefactor of eq 7 (long dashes, us)a P 3 )or of eq 14 (short dashes, us[ PO9).
36" 34
240 Y
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-
"
_ _ - - - - ._' -.
-
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-
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20
30
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t, microsec Figure 4. Temperature-time profile computed for 75 A water clusters
as they cool by evaporation after leaving the nozzle. Arrows indicate the times at which the patterns of Figure 3 were recorded. From ref 54. proposed to occur in bulk water in the vicinity of 226 K if nucleation could be avoided.54 Whether this immunity is because their small size and large surface-to-volume ratio disqualify them from consideration as a bulk system or because the hypothetical anomalies are not real is unresolved. In any event, liquid clusters of water can be continuously monitored by diffraction as they cool to 200 K and below, and the only noteworthy phenomenon seen is that of freezing, at about 200 K, to cubic ice. Figures 3 and 4 show the time evolution of diffraction patterns and of the temperature of water clusters. The type of ice nucleated, cubic, has been produced by several other techniques at low temperatures.81,82It retains its cubic structure for extended periods of time unless warmed above 160 K when it changes sluggishly to the more stable hexagonal ice.52 Quite apart from the widespread intrinsic interest in water, the system is of special concern because it is the only one so far to have been studied by both the conventional and the cluster techniques. Figure 5 compares our nucleation rates for clusters generated from runs with water vapor at 95 and 120 "C (seeded into 3.5 atm of neon carrier gas) with those from droplets dispersed in an e m u l ~ i o n .Despite ~ ~ ~ ~ the ~ 15-20 order of magnitude difference in nucleation rates, the cluster and droplet results appear to be logically connected, as suggested by the dotted and dashed lines passed through the cluster results and projected to lower supercoolings using the p 39 law (for eq 14) or 10.3(for eq 7 , the difference between the two exponents being smaller than the uncertainty). This conclusion is more easily recognized in Figure 6 plotting the interfacial free energies derived from the nucleation rates. For uniformity and simplicity,
201 180
. ' 200
*
'
'
220
I
240
'
I
260
a
I 280
T, K Figure 6. Interfacial free energies for the boundary between ice and
water. Values derived from nucleation rates to cubic ice, prefactor of eq 7: clusters, circles; water-in-oil emulsions, squares, ref 56, and solid diamonds, ref 57; dashed line, calculated with us[= P,?.Value at 273 K with error bar computed from equilibrium contact angles between water and two crystals of hexagonal ice sharing a grain boundary (ref 69). all of the kinetic parameters in the figure were recalculated directly from the reported nucleation rates and put on the same basis, using the expression for A in eq 7 . In the prior work on droplets, the authors had adopted various and appreciably different conventions, especially in their simplified calculations of the free energy of freezing. Details of the calculations will be reported elsewhere.55 What is of particular interest is the fact that the kinetic interfacial free energy parameters for the clusters and for the droplets are in excellent agreement with each other but in considerable disagreement with the thermodynamic value derived from equilibrium contact angles between the liquid and two crystals whose grain boundary adjoined the It is the thermodynamic value that closely agrees with the Tumbull relation. The interpretation of these facts is quite straightforward. The ice in contact with the liquid is the normal, thermodynamically stable, hexagonal ice. The ice to which the liquid clusters froze was readily identified as cubic ice. Although the solid to which the liquid droplets of Butorin and Skripov5' and of Wood and WaltonS6 crystallized was not determined, the present results leave no doubt that it was also cubic ice. Despite the claim of Wood and Walton that their interfacial free energy extrapolated to that of hexagonal ice in water at the freezing point, their extrapolation was surely incorrect. The temperature dependence of the cited extrapolation was much too steep, and the real dependence was buried
Feature Article in the experimental As confirmed by the low interfacial free energy derived from the nucleation rates, the reason that cubic ice forms at low temperatures instead of the stable hexagonal ice is kinetic. The low interfacial free energy is associated with a much lower barrier to the formation of critical nuclei. Transitions in the Solid State. Those transformations from one crystalline form to another which have been followed by electron diffraction in supersonic jets to date have all been nonreconstructive. That is, the principal reorganization taking place in the transformation is one of molecular reorientation, not translation, with the reorientation rate being govemed by rotational diffusion. Examples studied in supersonic jets include tert-butyl chloride3Eand several hexafluoride^.^^^^^ Available evidence suggests that the temperature dependence of rotational diffusion in solids remains in conformity with an Arrhenius rate law to much lower temperatures than does translation diffusion in liquids. Hence, no marked falloff of rates appears to occur in cooled solids in the way it does for liquids when the glass transition is approached. As a consequence, calculations based upon the classical theory of nucleation suggested to us that extremely high nucleation rates might be attained at large supercoolings for both the hexafluorides and the alkyl chloride. As it turned out, the calculations were correct in their projection that rates would become high enough for the transitions to be seen in MD simulations. Results for both experiment and simulation are presented below. tert-Butyl Chloride. Clusters of tert-butyl chloride generated at comparatively high mole fractions in neon carrier gas (Xsubject > 0.1 at 4 bar) emerged from the Lava1 nozzle as crystals of phase 111, a form stable in the bulk over the range from 183 to 218 K. As they cooled by evaporation they transformed to phase IV at about 156 K. Phase I11 was known to be tetragonal and plastically crystalline, with rotational disorder of molecules about the unique axis. The structure of phase IV was unknown even though neutron diffraction powder patterns of quite high resolution existed for the phaseE3in addition to the more diffuse electron diffraction patterns (of considerably smaller crystals).38 It was clear that phase IV was of lower crystallographic symmetry, but a successful indexing of the diffraction lines had not been found. Nevertheless, the experimental nucleation rates for the clusters did provide some clues about the colder phase. Calculations based on nucleation theory and known properties of the tetragonal phase ruled out all conceivable transition mechanisms except that from orientational disorder to order if the observed rate was to be achieved. Translational mechanisms were simply too slow. It was gratifying to find in MD simulations that a system of 188 molecules not only existed stably in the tetragonal phase over a suitable temperature range, it spontaneously transformed to a lower temperature, ordered phase when sufficiently supercooled.21 Upon warming and recooling, the cluster could be repeatedly cycled back and forth between the two phases. Figure 7 illustrates the transformation. Moreover, the ordered phase (monoclinic, P21/m) proved to be the phase found in the bulk, for diffraction patterns computed from the MD molecular packing in the core correctly reproduced the neutron powder patterns found earlier. This illustrates another application of clusters, namely as aid in solving crystallographic problems. From the nucleation rates of 6.4 x loz7mF3s-l in supersonic at 156 K and 1.4 x in MD s i m ~ l a t i o n sat~ 90 ~ K were deduced interfacial free energies of 3.3 and 3.4 rdlm2, respectively, ignoring the term w’ in eq 9. In the latter term the strain energy and the possible effect of the contraction of the cluster’s external surface tend to oppose each other, but
J. Phys. Chem., Vol. 99, No. 4, 1995 1087
Figure 7. Images of a crystalline cluster of 188 molecules of fer& butyl chloride at various stages of cooling, looking down the 3-fold molecular axis: top, orientationally disordered tetragonal phase, at 130 K; center, nucleus of monoclinic phase growing in tetragonal phase at 80 K; bottom, ordered monoclinic phase at 50 K after transformation.
Surface molecules tend to be disordered at all temperatures. neither effect has been estimated reliably in small clusters. The values of a,, for tert-butyl chloride are of interest, in part, because the MD and experimental results agree approximately and, in part, because they are close to the value predicted from an extension of Turnbull’s relation (between as]and the enthalpy of fusion) to the corresponding relation (between assand the enthalpy of transition from IV to 111). If the same factor of 0.32 is used for the change from tetragonal to monoclinic as has been suggested for the freezing of nonmetals, an interfacial free energy of 3.4 d l m 2 is obtained. Therefore, Turnbull’s relation appears to have an even greater generality than was originally proposed.
Hexafluorides Clusters of the hexafluorides AF6 (A = S, Se, Te, Mo, and W) have been studied extensively, both in supersonic flow and in simulation. All adopt the bcc structure when they freeze, and all have been seen in the monoclinic phase (C2/m, Z = 3) when cooled. What is most interesting about the hexafluorides
Phys. Chem., Vol. 99, No. 4, 1995 -18
-25
Bartell 1
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'
40
60
80
100
120
140
I 160
T (K) Configurational energies (in kJ/mol of TeF6) of 128-molecule clusters undergoing heating in MD runs. Computations started with four different initial solid structures, all of which eventually transformed spontaneously to bcc when warmed (see ref 19 for computational details). Upper four curves, simulations excluding electrostatic interactions; lower four curves, simulations including them. The four starting structures were monoclinic (P21/c, dashed, diamonds), a more stable monoclinic (C2/m, solid, filled circles), rhombohedral ( R i , solid, triangles), and orthorhombic (Pnma, dashed, filled squares). In all simulations carried out on clusters of a given size the orthorhombic required the highest temperature to induce the transformation to bcc and, hence, is the structure of lowest free energy even though the other structures have comparable configurational energies. From ref 19. is that the Te, Mo, and W compounds do not crystallize in the monoclinic phase in the bulk,'* even though they readily do in clusters. As will be shown, this difference is not due to an intrinsic difference between clusters and the bulk. The coldest known phase of the heavier hexafluorides is orthorhombic, a phase also seen in their clusters if they are nucleated and grown in sufficiently cold conditions of flow (low subject mole fractions and high enough carrier pressure). Monoclinic clusters are never seen to transform to orthorhombic on the MD or supersonic time scales. Molecular dynamics computations show beyond doubt that the monoclinic phase is generated spontaneously when the bcc phase is cooled quickly because it is kinetically, not thermodynamically, favored. Even though the monoclinic and a half-dozen other packings of the hexafluorides, including the orthorhombic, are of comparable configurational energies, the orthorhombic phase has a substantially lower free energy as can be inferred from Figure 8.19 The key to understanding the facile transition between bcc and monoclinic, as shown by Raynerd et aLg5 and Pawley and Dove,*6 lies in the close similarities of structures of the two phases. The principal rearrangement required for the transition (via a trigonal form very like the monoclinic) is for one-third of the molecules to rotate 60" about a 3-fold axis while the other two thirds move very little. The much more extensive rearrangement required to change from bcc to orthorhombic can easily take place on the long time scales of conventional crystallographic and thermodynamic investigations but not on those of typical cluster studies. Interfacial free energies between the bcc and monoclinic phases computed from nucleation rates are more uncertain, both in MD and in supersonic studies, because of difficulties in establishing the degree of supercooling with precision. Neither the bulk transition temperatures nor the heats of transition are known, although both can be estimated approximately. Results from experiment and computation indicate that the generalized Tumbull relation connecting the interfacial free energy with the heat of transition holds, at least approximately.
Concluding Remarks As model systems for investigating condensed matter in transition, clusters are beginning to prove their worth. They offer an alternative approach to conventional studies of the dynamics of phase changes. Novel regimes of rate and supercooling are encountered, and certain difficulties encumbering the older techniques are avoided. Although limitations in the current experimental and computational cluster techniques prevent them from being universally applicable, they are well suited to many systems. Despite the fact that nucleation rates in clusters tend to be astronomically higher than in conventional studies, the enhancement appears to be attributable to the much greater supercooling incurred rather than to intrinsic differences in mechanism. Examples do occur, however, of phase changes enabled for kinetic reasons outstripping those favored by thermodynamics. Although analyses of nucleation rates in terms of the classical capillary theory of homogeneous nucleation yield helpful and qualitatively plausible results, MD simulations of cluster behavior detect many of the same deficiencies of the capillary theory that are implied by recent formulations of density functional theory. Research on nucleation rates in clusters can be expected to evolve in several directions. In experiments to date, nucleation rates have been measured for only a small number of systems, partly due to the extreme tedium of finding optimal experimental conditions and converting photographically recorded intensities to digital transcriptions. Moreover, rates have so far been measured at only a single temperature per system. We are designing an apparatus, modeled after one described by Ewbank et al.,87for recording intensities digitally, in real time. With such a recording device it should become feasible to study many more systems and over an appreciable range of cluster sizes and temperatures of transition. The information derivable may help to establish the temperature dependence of the interfacial free energy parameter associated with the formation of critical nuclei. Very little is known about this quantity. Revealing though the existing molecular dynamics simulations of phase changes have been, they have not yet provided definitive results for such important quantities as the sizes of critical nuclei. The reason is less because of inherent deficiencies in the simulations than in the primitive state of diagnoses of the results. It has become evident that a prime inadequacy of classical nucleation theory is its failure to take into account the transition layer between the two phases. Until a rational way of treating this far from negligible layer is devised, it will not be possible to apportion the molecules in the interface between the old and the new phases. Clearly, what is needed is an improved and generally applicable theory of nucleation. Such a theory would be a major help in diagnosing results of both experiments and simulations. Conversely, further research on clusters could provide valuable tests of the consistency of proposed theories of homogeneous nucleation. It is not too much to hope that the new approach to nucleation opened up by clusters will stimulate further advances in theory.
Acknowledgment. This research was supported by a grant from the National Science Foundation. I thank Drs. J. Huang and J. Chen for permission to present portions of their unpublished results and Dr. J. Hovick for helpful remarks. References and Notes ( 1 ) Jena, P., Rao. B. K., Khanna, S . N., Eds. Physics and Chemistty of Small Clusters; Plenum: New York, 1987. (2) Echt, 0..Recknagel, E.. Eds. Proceedings of the Fi'h Intemational Symposium on Small Particles and Inorganic Clusters. In Z. Phys. D 1991, 19&20.
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