J. Phys. Chem. 1988, 92, 7241-1246 figures also show the calculated concentrations of OzF, 02F2,and
7241
Acknowledgment. This work was performed under the auspices of the United States Department of Energy. We appreciate the support from the DOE Division of Nuclear Materials and the FOOF/Superacids Project of the Los Alamos National Laboratory. We also thank Dr. L. B. Asprey for preparation of the high-purity fluorine and our many colleagues for helpful discussions of the research. Registry No. 02,7782-44-7; F, 14762-94-8; 0 2 F , 15499-23-7; 02F2,
F. Temperature Dependence Table I11 summarizes the room-temperature rate constants described above (center column), The last column is our estimate of the temperature dependence of these and several other rate constants. These estimates will be useful for estimating synthesis or decomposition rates for the oxygen fluorides.
7783-44-0.
Capillarity Theory for the “Coexistence” of Liquid and Solid Clusters H. Reiss,* P. Mirabel,t and R. L. Whetten Department of Chemistry and Biochemistry, University of California, Los Angeles, Los Angeles. California 90024 (Received: September 20, 1988)
The apparent coexistence of liquid and solid clusters of argon, observed in argon jets, is discussed in terms of a simple theory based on capillarity phenomena. A ”Coexistence” range is easily demonstrated,limited from above by the true melting temperature of the cluster (Le., where the vapor pressures of solid and liquid clusters are equal) and from below by the temperature of “critical supercooling” for the nucleation of crystals in the melt. However, from the practical point of view the coexistence range is determined by the probabilities of observing either solid clusters (upper limit) or liquid clusters (lower limit). Coexistence is demonstrated to be not a thermodynamic phenomenon but one dependent on the existence of a free-energy barrier that lowers the rate of transition between solid and liquid and vice versa. One sort of cluster is in metastable equilibrium while the other is stable. Definition of the cluster “melting point” as the temperature for which the chances of observing liquid and solid clusters are equal is convenient only when the clusters are prevented from exchanging material with their surroundings, and, indeed, the vapor pressure of the liquid cluster exceeds that of the solid at this temperature (which must be lower than the true depressed melting temperature). Substitution of macroscopic thermodynamic parameters (density, surface tension, and heat of fusion) into the simple theory yields results in fairly good agreement with those obtained from molecular theory and simulation. Even the transition rate is consistent with the simple theory, but “magic numbers” cannot be derived unless the above-mentioned thermodynamic parameters are allowed to depend on both temperature and cluster size.
clusters of molecular size. In fact, if a solid cluster “melts” from the surface, inward, the molten layer should be all “transition zone” and never “bulk”, so that the capillarity approximation should be seriously unphysical. Nevertheless, we shall use it (just as it is used in nucleation theory), but our goal need not be a fully quantitative description of cluster behavior. Rather, we aim at a qualitative explanation of the phenomenon and fit the derived formulas to the experimental data. In this approach we treat quantities like surface tension u and volume per molecule v as parameters that can be adjusted to achieve a best fit. Under
Introduction The coexistence of liquid and solid clusters has been observed in “reality” in freely expanding jets or molecular beams and “computationally” in computer simulations.’ Berry2 has offered an explanation for the phenomenon based on quantum statistical arguments. However, the systems are for the most part classical, and a more straightforward classical analysis should be possible. We offer such an analysis based on the “capillarity approximation” familiar to nucleation theorye3 In this approximation, clusters of almost molecular size are treated as macroscopic drops or crystals, in the sense that they are assumed to have macroscopic volume and surface properties, Le., surface tensions and densities, identical with those of the same substance in bulk. Although both of these properties, especially surface tension, lose much of their thermodynamic character as the clusters become very small, experience shows that the concepts can be extrapolated almost down to the molecular level without the introduction of catastrophic error, insofar as the forms and trends of phenomena are concerned. Even great quantitative error does not seem to be introduced by such e~trapolation.~ It i s easy to understand why this is so. Surface energy expresses merely the unsaturated “bonding” at a surface, and this feature certainly retains its meaning at the molecular level. On the other hand, we know that an interface, especially one involving a liquid phase, is not a mathematical plane of zero thickness, but really a transition zone in which the properties of one phase continuously change into those of the other over a distance of several molecular diameters.5 This gives rise to a dependence on curvature of surface tension itself, for example, so that the macroscopic quantity should not be applicable to ‘Permanent address: Strasbourg, France.
(1) (a) Whetten, R. L.; Hahn, M. Y. Phys. Rev. Lett. 1988,61, 1190. (b) Bosiger, J.; Lentwyler, S. Ibid. 1988, 59, 1895. (c) Valente, E. J.; Bartell, L. S. J . Chem. Phys. 1984,80, 1458. (d) Jellinek, J.; Beck, T.; Berry, R. S. J. Chem. Phys. 1986,84,2783. (e) Honeycutt, J. D.; Andersen, H. C. J. Phys. Chem. 1987, 91, 4950. (2) Berry, R. S.; Jellinek, J.; J. Natanson, G. Phys. Reu. 1984, A30, 919. For a review, see: Berry, R. S.; Beck, T. L.; Davis, H. L. Adu. Chem. Phys. 1988, 90(2), 75-138. (3) Volmer, M. Kinetic der Phasenbildung Steinkopf Dresden, 1939 (translated into English by the Intelligence Department Air Material Command, US.Air Force, as document AT1 No. 81935). Frenkel, J. Kinetic Theory of Liquids; Oxford University Press: New York, 1946; Chapter 7; J. Chem. Phys. 1939, I, 200,538. Abraham, F. F. Homogeneous Nucleation Theory; Academic Press: New York, 1974. Zettlemoyer, A. C. Nucleation, Dekker: New York, 1969. Nucleation Phenomena; Zettlemoyer, A. C., Ed.; Elsevier Scientific: New York, 1977. Reiss, H. Ind. Eng. Chem. 1952, 44, 1284; Precipitation, Faraday Discuss. Chem. Sac. 1976, 61. Hirth, J. P.; Pound, G. M. In Progress in Materials Science; Chalmers, B., Ed.; Pergamon: London, 1963; Vol. 11. Becker, R.; DBring, W. Ann Phys. 1935, 24, 719. Zeldovich, J. B. Acta Physicochem. USSR 1943, 18. For application to small systems see: Hill, T. L.; Thermodynamics of Small Systems; Benjamin: New York, 1963; Chapter 5. (4) Langmuir, I. Chem. Reu. 1933, 13, 147. Frenkel, J. Kinetic Theory of Liquids; Clarendon Press: Oxford, 1946; p 344. ( 5 ) Rowlinson, J. S.; Widom, B. Molecular Theory of Capillarity; Clarendon Press. Oxford, 1982; Chapter 2.
Institut de Chimie, Universite Louis Pasteur,
0022-3654188 , ,12092-7241$01.50/0 I
0 1988 American Chemical Societv -
7242 The Journal of Physical Chemistry, Vol. 92, No. 26, 1988
Reiss et al. uls that of the interface between solid and liquid. ul and cisare related by Young’s equation5 ulS = us -
61 COS
8
(4)
where 8 is the contact angle between liquid and solid, and us is the surface tension of the solid. Although a liquid need not “wet” its solid, it is not a bad approximation to set 8 = 0, so that eq 4 becomes bls
Figure 1. Partly melted cluster of radius R. Solid core (shaded) of radius r. Relevant surface tensions uI and uIsare shown.
We use this relation later. In eq 3 , 47r(R3- $ ) / 3 u is the number of molecules in the liquid layer and 4ar3/3u,the number in the solid core. By setting dC/dr equal to zero, one can demonstrate that G possesses an extremum at r = r * = - 2UlSU
favorable circumstances, when that best fit is achieved, u and u will not differ too much from the measured bulk values. Coexistence Theory A cluster will be modeled as a sphere of radius R . Indeed we can consider it to be a “composite” sphere (see Figure 1) having a spherical solid core of radius r < R surrounded by a spherical molten layer of thickness R - r. For convenience we refer to this configuration as model A. Alternatively we could consider the core of radius r to be molten and the outer layer of thickness R - r to be solid. This configuration will be called model B. It should be noted, however, that the strict sphericity of boundaries implies that we are averaging over “fluctuated” paths of transition, many of which involve nonspherical configurations. (The same “order of averaging” problem appears in conventional nucleation theory.) Model B is almost certain to be of higher energy than model A since the liquid will usually have a molar volume larger than that of the solid, and therefore the outer solid layer will be strained. The additional elastic energy will therefore render model B less stable than model A. In rare cases, e.g., water, germanium, etc., where the system contracts upon melting, model B may be relevant, but for most cases it will probably not represent an intermediate stage of a cluster in the process of melting. Thus we will focus on model A. The most complete model should have the volume per molecule u1in the liquid different from the corresponding quantity us in the solid. Then both R and r would vary as the melting process takes place, the relation between them being
where n is the total number of molecules in the cluster. However, once having chosen model A, we can develop most of the argument by choosing UI
= us = u
1
47r(R3 - r 3 ) 3u 1.1
+
{
$}kS
(6)
PI - Ps
Furthermore (7)
so that the extremum at r* is a maximum. Rearranging eq 6, we obtain
and adding 2ulu/R to both sides of eq 8 gives 2u,u 2U& 20,u pl+-=p R S r* R
+-
+-
(9 1
The left side of this equation is the pressure-adjusted chemical potential in the liquid layer, 2 q / R being the amount by which the pressure in the liquid exceeds the pressure outside of the cluster. Similarly, (2ak/r*)+ (2ul/R)is the amount by which the pressure in the solid core exceeds that outside of the cluster, so that the right side of eq 9 is the pressure-adjusted chemical potential in the solid. Thus eq 9, equivalent to eq 6, simply indicates the equality of chemical potentials at the maximum, showing that, at r*, the cluster is in unstable equilibrium. At r = 0 (fully molten cluster) or at r = R (fully solid cluster) it is in either metastable or stable equilibrium, depending upon which minimum is “local” and which is “global”. Assuming that A, the heat of fusion per molecule, is to a first approximation independent of temperature, the Gibbs-Helmholtz relation of thermodynamics requires where To is the bulk melting temperature. Substituting this equation into eq 6 gives
(2)
and assuming that u is constant, Le., that the cluster is incompressible. This offers the advantage of having to deal with much less algebra, because R remains constant as r varies. On the other hand the full argument, with us # u,, can be unfolded if one so desires. Specializing to eq 2 and adopting the capillarity approximation, the free energy of the cluster may be expressed as G=
(5)
= us- u1
+ 4rR2ul+ 47rr2ulS( 3 )
In this equation p,and ps are the chemical potentials per molecule in the bulk liquid and solid, respectively, at the pressure outside of the ~ l u s t e rwhile , ~ ~ ~uIis the surface tension of the liquid and (6) Reiss, H.; Wilson, I. B. J . Colloid Sci. 1948, 3, 551. (7) Reiss, H. Methods of Thermodynumics; Blaisdell Publishing Co.: New York, 1965; Chapter 1 I .
Since r* > 0, this equation shows that for G to have a maximum, T must lie below the melting temperature To. When a maximum exists, liquid and solid clusters can coexist on opposite sides (see Figure 2 ) of it, one in metastable equilibrium while the other is in stable equilibrium. Thus eq 11 shows that this range of coexistence lies below the melting temperature. Some words of qualification are however necessary. Without a maximum the solid and liquid states of the cluster may be continually visited. If, for example, the solid cluster has a lower free energy than the liquid one, the rate of transition (in the absence of a barrier) from liquid to solid will be very high so that the lifetime of the liquid cluster will be so small that the cluster may not be separately observed. Carrying the argument further, if the free energies of the two states are comparable, the lifetimes in both states may be so short that only some average over both states will be observed. Thus the “states” may coexist without the presence of barrier but the “separate clusters” will not coexist.
The Journal of Physical Chemistry, Vol. 92, No. 26, 1988 7243
“Coexistence” of Liquid and Solid Clusters
clusters. The melting point, Le., the temperature at which fully solid and fully molten clusters can “truly” coexist in equilibrium without spontaneous transfer of material, is determined by Ps = PI
(17)
However, this equilibrium is unstable, since the slightest transfer of material from molten to solid cluster increases pIand decreases p, and vice versa. To distinguish this melting point from To, the bulk melting point, we designate it as Tb. Substituting eq 15 and 16 into eq 17 gives ~uI,v h(T0 - T’o) (18) M l - C L s = R - -
TO
where in the last step we have used eq 10. But, as we have seen, when r* = R and the cluster is fully solid, the maximum on the path between solid and melt vanishes. The temperature T’at which this occurs may be obtained from eq 11 by replacing r* by R . Then we get 0
r
Figure 2. Schematics for the free energy G(r) of the cluster for four temperatures, T’,, > T , > T”,, > T2.The curves for the melting temperature T $ and for the coexistence melting temperature T’$ are included. The free energy barrier vanishes at Tb. Locations of barrier tops are indicated by r*.
But this is just the last equation, in eq 18, if T’is identified with Tb. Thus the maximum vanishes at the melting point. Note, however, that from eq 13 and 14
By the “coexistence range“ we shall mean the range of coexistence of the clusters. We can rearrange eq 11 to the form
This equation shows that as T i s raised toward To,r* increases, becoming infinite a t T = TO. When r* 1 R , however, the path from melt ( r = 0) to solid ( r = R ) exhibits no maximum. We explore this condition a bit further. When the cluster is fully solid ( r = R ) , eq 3 with the help of eq 5 yields 4rR3 3v
-p,
G, =
+ 4rR2u,
where eq 5 and 10 have been used. Even at Tb, Le., at the melting point, the path from fully melted to fully solid cluster is uphill free-energy-wise (unless ais = 0). This can be proved by setting T, in eq 20, equal to T b and using the last equation in eq 18 to eliminate a,, in eq 20. We then find that G,(T‘,) - GI(T‘,) = 2rR3X(To- Tb)/vTo > 0. Nevertheless, at Tb, the path exhibits no maximum. Replacing R’, in eq 19, by T b , we find
and when it is fully molten ( r = 0), eq 3 gives
GI =
(
T ) p l
+ 4rR201
Since, except for r = r*, the chemical potentials of liquid and solid phases in the cluster are not equal, it is not convenient to define the chemical potential of a partially molten cluster. The chemical potential of either the solid or fully melted cluster may, however, be defined in the usual manner’ as
2 usv
2(Ul
= k + y = P L S +
+ %)V
(15)
for the solid (where p is the external pressure and where eq 1 and 5 have been used, while the bar over ps indicates the (internal) pressure adjusted chemical potential) and
Thus, the smaller the cluster, Le., the smaller the value of R , the lower will be the melting temperature Tb. As we have discussed, for T # Tb, solid and molten clusters can only “coexist” if the path from one to the other exhibits a free-energy barrier, Le., it must possess a free-energy maximum. Since, as we have seen, this can be true only for T < Tb, the coexistence range must lie below T‘o. Equation 21 shows that To - Tb, the melting point depression due to surface effects, is (2ukvTo)/RX).Very small clusters will exhibit large melting point depressions, and the coexistence range may lie far below the bulk melting temperature To. If the free-energy barrier is low enough, clusters will fluctuate and make transitions between solid and molten states at a finite rate. If the barrier is too high, the lifetime of either state will be too long for transitions to be observed. In the coexistence range, the ratio of the average number of solid to the number of liquid clusters, due to the chemical-like equilibrium that is established by the fluctuation process, will be given by
2ap = P I + R
for the liquid. The definitions of ps and pI clearly require the relaxation of the condition n = constant, so that material can be exchanged between the cluster and the surrounding phase, and in particular between (different) fully solid and fully molten (8) Reference 7, pp 163-165.
where eq 20 has been used. In the next section we analyze the cluster transition rates between solid and liquid states. Practically speaking, the limits of the ”coexistence” range may be determined, not so much by quantities such as Tb, but rather
7244
The Journal of Physical Chemistry, Vol. 92, No. 26, 1988
(for the upper limit) by the temperature at which N,/Nl becomes so small that solid clusters are unlikely to be observed and, similarly, for the lower limit, by the temperature at which N J N , becomes sufficiently large. At Tb and above, the absence of the free-energy barrier makes the observation of the solid cluster impossible, i.e., its lifetime will be too short. Furthermore, since, at T i and above, G, > GI,the system will appear molten. In other words, it melts at T’,,. It has been somewhat conventional to define the cluster melting point as the temperature? within the coexistence range, at which C, = GI, so that NJN, = 1. We might call this the “coexistence” melting temperature and denote it by T’b. Using eq 20 in G, = G,, we solve for T’b and find
Comparison of eq 21 and 23 shows that T’’,, < T b (24) so that T’b lies in the coexistence range. The emergence of T’b in the cluster literature is one example of the poor communication between the “cluster” community and workers in the fields of both capillarity and nucleation phenomena. A vast literature has been compiled in these latter fields during almost a century, and many subtle (almost paradoxical) problems have been discussed and resolved. The distinction between T b and T’b is one of these subtle problems and, in particular, the forms of the p’s derive from the fact that G, and GI, unlike the case of bulk systems, are not homogeneous functions of the mole numbers in the first degree.1° T b is the true melting point, since at this temperature there is no driving force for the spontaneous transfer of material between liquid and solid clusters of the same size. In fact the vapor pressures of the two kinds of clusters are equal. This is by no means the case at T’b, where the vapor pressure of the liquid cluster exceeds that of the solid cluster (of the same size). To give significance to T’b, it is necessary to constrain the clusters so that no exchange of material with the surroundings is possible. When the clusters are formed in the jet, no such constraint is actually present, but the temperature is often low enough so that rates of evaporation are small and the exchange of material between clusters is at least slow. This constraint is also employed in computer simulations that have been performed on the process of clustering, since the possibility of “evaporation” (or condensation) is disallowed on the basis that it involves a very long time relaxation process in comparison to the time for the internal equilibration of a cluster. Nevertheless, it is a process that must be considered if paradoxes such as that involving T b and T’b are to be avoided. Without allowing for exchange of material with the environment, the cluster is simply addressed as a large molecule rather than a thermodynamic phase, capable of equilibrium with other phases.
Transition Rates The free energy barrier on the path from solid to liquid is simply AGSl = G(r*) - G, (25) and, for the path from liquid to solid, is AGI, = C ( r * ) - GI
(26)
Setting r in eq 3 equal to r*, given by eq 6, and substituting the result, together with eq 13 and 14, into eq 25 and 26 yield 16~~~~1,3 AGI, =
3 ( ~ 1- P,)’
(27)
In the coexistence range, below Ttt0,the molten cluster is meta(9) Natanson, G.; Amar, F.; Berry, R. S. J. Chem. Phys. 1983, 78, 399. (10) Reiss, H. Methods of Thermodynamics; Blaisdell Publishing Co.: New York, 1965; p 164, eq 11.35.
Reiss et al. stable, Le., it represents a supercooled liquid, and therefore it is not surprising that AG1,, the free energy barrier to solidification, proves to be the standard barrier for the “nucleation” of the solid phase in the supercooled melt. Indeed, eq 27 shows that AGh is this standard barrier.” Since the solid cluster is the stable phase in the coexistence range below T’b, AGd must exceed AGk. Indeed, comparison of eq 27 and 28 shows that AG,, exceeds AG,, by the amount
-
-
Note that as R 0 0 , eq 28 requires AGsl m. Thus, for a bulk solid, the barrier against fluctuation to liquid is infinite, and for the bulk, no such fluctuation will ever be observed (below the melting point). This of course is the proper result. On the other hand, for a metastable supercooled liquid, fluctuation to the solid can occur at a finite rate, Le., conventional nucleation can occur. Correspondingly, AGl, is independent of R. The rates of transition can be estimated (very crudely) from conventional transition-state theory employing the mean thermal vibration frequency k T / h (where h is Planck‘s constant) as a preexponential factor. Then we find for the frequency of transition from liquid to solid
and for the frequency from solid to liquid
Note that both AGI, and A&, specified by eq 21 and 28, can be expressed as functions of temperature by using eq 10. Also note that the transition state at the top of the barrier corresponds to a “saddle point”-alternative paths between liquid and solid, e.g., a nonspherical solid core, must involve larger surface to volume ratios and therefore higher free energies. The above development suggests that the extreme limits of the “coexistence range” are determined by Tb, the depressed melting point, on the upper side, and by Tdt, the temperature of “critical supercooling” for nucleation, on the lower side. Furthermore, it appears as though “coexistence” is not coexistence at all, since in coexistence range, one cluster is metastable while the other is stable. However, because the cluster is so small, the barrier, eq 28, from solid to liquid is small enough so that fluctuations back to liquid remain possible. It should be noted that things like “magic numbers”12 cannot be treated directly by using the present model. This phenomenon resides in the actual dependences of u and X on cluster size while we have taken these parameters to be constant. Thus magic numbers are only “implicit” in the crude theory based on capillarity. Finally we comment on model B, corresponding to a situation such that vI < us. The entire development can be repeated for this model, and it can easily be shown that the “coexistence” range will then lie above the bulk melting temperature. For reasons of brevity, we do not present this development here. Some Numbers As emphasized above, the present theory should not be regarded as a quantitative predictive tool, but rather as a qualitative explanation of the coexistence range phenomenon. Even within the limitations of the capillarity model, the theory is not complete. For example, no provision has been made for (1) the change of volume upon melting; (2) the dependences of heat of fusion, density, and surface tension on both temperature and cluster size; (3) the inclusion of the contact angle 8. In spite of these various, serious omissions it is of some interest to see how well the theory ( 1 1 ) Frenkel, J. Kinetic Theory of Liquids; Clarendon Press: Oxford, 1946; p 415. (12) Knight, W. D.; Clemenger, K.; de Heer, W. A.; Saunders, W. A,; Chou, M. Y.; Cohen, M. L. Phys. Reu. Lett. 1984, 52, 2141.
"Coexistence" of Liquid and Solid Clusters TABLE I: Ratio of Probabilities of Solid and Liquid Clusters
The Journal of Physical Chemistry, Vol. 92, No. 26, 1988 7245 TABLE II: Conversion Factors for Reduced (Lennard-Jones) Units
T 'b
60.1 52.0 48.2 48.0 45.0 40.0
1.23 X 10" 2.2 x 10-3 1.o 4.5 3.7 x 103 1.8 x 109
agrees with (computer) experiment. (In the Appendix, a brief summary is given of data extracted from simulations.) For this purpose we examine the case of the 55-atom icosahedral argon cluster, for which considerable information is available from molecular dynamics s t ~ d i e s . l * ' ~ These . ' ~ studies indicateI3 that the coexistence range for the 55-atom cluster lies between 32 and 41 K and that the transition frequency vlS for the temperature, or N , = Nl, is 3 X lo8 s-l. The values 36.5 K, at which G, = GI, of the parameters that we employ in applying the present theory are To= 83.8 K h = 2.02 X erg v = 3.75 x cm3 (32) These are simply handbook values, and h and u correspond to T = To. R is obtained from R = (3nu/4i~)'/~ (33) where n = 5 5 . We also need a value for uls. For this purpose we substitute eq 10 into eq 27, and the result into eq 30. We then solve the resulting equation for uls, obtaining ulS= {-[(3kTAZ(To - T ) z / 1 6 ~ T o 2 v 2In ) ] ( h ~ ~ , / k T ) ] ' / (34) ~
Choosing vlS= 3 X lo8 s-l and T = 36.5 K, all the remaining quantities on the right are known (see eq 32), and we find uls = 6.02 dyn cm-' (35) The value of uI at To = 83.8 K is known to bel5 ul = 13 dyn cm-'
(36) and will exceed this value at 32.5 K. Thus the value of ulsin eq 35 is not unreasonable in the light of eq 4 and 5. Substituting the values from eq 32, 33, and 35 into eq 21 gives for the depressed melting point T'o = 60.1 K (37) and substitution of the same values into eq 22 allows us to prepare Table I. From the practical point of view the observable coexistence range lies, according to this table, between 45 and 52 K. Outside these limits, the chance of Occurrence of more than one type of cluster is too small for observation by computer simulation. The extent of this coexistence range is comparable to that required by computer simulation (32-41 K), but it lies about 13 K higher than the range obtained by that method. When the simplicity of the theory is taken into account, this is not a bad result. Furthermore, not only the details of the range but also the transition frequency are consistent with the corresponding quantities derived from simulation, since vls was used in order to provide a value for uls which, in turn, proved to be reasonable. We also list, in Table I, the value of T'b, the "coexistence" melting point specified by eq 23. As required by theory, at 48.2 K it is lower than the real melting point, T',, = 60.1 K, and lies approximately at the midpoint of the coexistence range. Notice that, at the true melting point Tb, Table I indicates N,/Nl = lo4, so that with the constraint against material exchange in force, the probability of observing a solid cluster is still miniscule. On the other hand, with T reduced only infinitesimally below Tb, and with the constraint removed, all the material in a set of liquid (13) Whetten, R.L., unpublished work. Hahn, M. Y.,unpublished work. (14) Briant, C. L.; Burton, J. J. J . Chem. Phys. 1975, 63, 2045. (15) Stansfield, D. Proc. Phys. SOC.1958, 72, 854. Sprow, F. B.; Prausnitz, J. M. Trans. Faraday SOC.1966, 62, 1097.
erg x 10'5 Ne
cm
5.8 19.5 27.6 39.0
Ar Kr Xe
x IO*
2.74 3.35 3.57 3.88
t*,
m,
0,
€9
Tb
g
x
s x 1OI2
s*, h
1.22 1.14 1.49 1.71
6.74 21.5 39 64
3.32 6.64 13.9 22
TABLE 111: Two-State Ranees for Lennard-Jones Clusters ~
Ar
reduced
size
Tmin
13 55 147
0.21 0.26 0.32
Tm* 0.30 0.34 0.38
Tmim 26 32 39
K
Tman 37 41 46
K
TABLE I V Internal Energy E ( T ) in the Two-State Region N = 13 Arl3, erg K N = 55 ArS5,erg K N 147 AT1,,, erg K
E(T), solid -45.4 45.OT (-7.58 + 0.06263) X -285 + 207T (-47.6 0.288T) X -890 + 539T (-149 0.750T) X
+
+ +
E ( T ) , liquid -47.8 68.OT (-7.98 0.0946T) X -296 303T (-49.4 0.4223) X lo-" -920 + 808T (-154 1 . 1 2 0 X lo-"
+ + + + +
TABLE V Estimated X and A S of Fusion/Melting N 13 55 147
reduced ifus asrus T, 4.20 14.7 0.285 18.3 61 0.30 66.8 190 0.35
Ar, erg K 1 0 1 3 ~ 1 0 1 3 ~ s T,, K 0.70 3.06 11.0
0.021 0.085 0.26
34 36 42
TABLE VI: Expressions for CI - C. in Reduced Units N 4' term 4" term 13 55 147
-4.2AT -18.3AT -66.8 AT
-6.6(An2/P -28.8( An2J T' -94.2(An2/T'
clusters will transfer through the vapor phase to solid ones, so that larger (solid) clusters will result. Some of these may "flucturate" back to the liquid state, but again material will be transferred to the solid clusters which, in turn, will grow. As the process is repeated, the magnitude of R increases until N,/Nl becomes much larger than unity so that, in fact, all of the material will have frozen. In conclusion, all of the qualitative features of the cluster coexistence phenomenon are reproduced by the simple capillarity theory, and the quantitative features, derived via computer simulation, are matched with an error of no more than about 30%. Such agreement is sufficient to establish capillarity phenomena as the qualitative underlying basis for the coexistence range. Matters such as the unique stabilities at the icosahedral (magic) numbers, 13, 55, 147, etc., can only be injected into the theory by allowing u, u, A, etc., to depend on both temperature and cluster size. However, these dependences are not available from macroscopic considerations. Acknowledgment. This work was performed under NSF Grant No. DMR 8421383.
Appendix. Data from Simulations Summary of Raw Data and Sources. The solid-liquid transition in molecular clusters has been repeatedly studied by computer simulations in the past 15 years; see ref 1 and 2 for reviews. Some relevant features of this work are as follows: (a) Both Monte Carlo (MC) and molecular dynamics (MD) methods were used extensivelv in earlv work. More recentlv. so-called isothermal molecula; dynamfcs (IMD) has also bein applied to this problem.~e (b) The molecules (A) constituting the cluster ( A N )are invariably represented by the Lennard-Jones (6-1 2) potential. Much
1246 The Journal of Physical Chemistry, Vol. 92, No. 26, 1988
work is reported in the reduced units t = 1, u = 1, M A = 1, where the symbols have their usual meanings. The time unit is then t* = a ( M / ~ ) l / ~Table . I1 gives these units for the rare gases. The reduced temperature is often taken to be elkB. (c) Since 1983, most attention has been directed to the highly symmetrical clusters (Mackay icosahedra), N = 13, 55, 147, partly because of their clear two-state behavior in the melting region. This highly complete and reproduced data set will form the basis for the analysis below. For each of the selected clusters (13, 55, 147), simulations data are reduced to produce the several desired quantities (or sets of quantities) of interest. These are (a) The “coexistence” or twostate range, specified by Tminand T,,,, the lowest and highest temperature at which coexistence can be observed. These are taken from M D and IMD data for 13 and 55 and from hysteresis (supercooling and -heating) in IMD data for 55 and 147. (Sources are ref 7d,e and our own simulations.) (b) U and E vs T data for each state across the coexistence region. These are obtained as in (a) from detailed results of those simulations and are used to compute the free-energy differences AG( T ) across this region. (c) Two-state equilibrium constants for the solid-liquid equilibrium can be independently obtained by observing long trajectories (done for A13). For ASsand the fit in Honeycutt and Andersen’s paperle is used to compute these values, which in principle are equivalent to the GI - G, values. (d) Transition frequencies are determined from the times between transitions in very long trajectories (MD or IMD). Typically, these are known only at one or two temperatures, although some estimates can be performed from published simulations. Further work is needed here for the N = 55 and 147 clusters. The major source is our unpublished data. Two-State (Coexistence) Ranges. Table I11 contains the coexistence ranges (T,,, and T,,,) of the three clusters in both reduced units and units appropriate to argon (€/kB = 120 K). Average Potential and Internal Energy in the Coexistence Range. In principle, the potential energy curves U( 7) for the solid and liquid clusters are nonlinear functions of T. However, simulations data show them to be highly linear over the two-state range and into the normal (liquid or solid) range. Therefore only the formulas for the lines are given here. From M C simulations, the functions E(T) are obtained from V(T)by adding the kinetic energy 3/2NkBT. However, in IMD simulations the energy E(T) is computed directly. Table IV presents these data. Two additional quantities that can be obtained from the raw E( 7‘) curves are the latent heat of fusion (A = E, - E,) and the entropy of melting ( A S = A/T,). The midpoint of the two-state E,, and El (see Table V). region was used for T , (=Yo), GI- G, in the Melting Region. Next the free energy difference GI - G, in the melting region is computed across the two-state range. However, this requires an additional assumption, because the E( curves do not extend to a common reference state. The additional assumption used is that AG = 0 at some point, T, =
Reiss et al. T”,,, inside the two-state region; this is the temperature at which the two states are observed with equal probability. Then the difference G, - G, is computed for all other temperatures by the method outlined below. Start with the Gibbs-Helmholtz equation:
[a(AG/T)/ar], = -AH/P
(‘41)
where AG = G - G O , AH = H - Ho. Since at T,, AGl = AG, by definition, we take our reference condition to be at T,: Go = G,(T,) = G,(T,) = 0, without loss of generality. Then HIo - H,’ = AHf,,, = A. With these conventions and definitions, one can rewrite ( A l ) to obtain GI( T ) - G,( 7‘) = -T
dT’
(A2)
Because HI(T ) and H,( T ) are approximately linear functions of T, this can be expressed in terms of parametrized data: G , ( T ) - G,(T) = T S T T ‘ 2 { - A C ( T ’ - T,) - if,,,) dT’
(A29
Tm
where AC
Cl - C,. This expression can be integrated to yield
G,(T) - GAT) = -TAC In (T/ T,)
+ AC( T - T,) + ifus( 1 - T / T,)
(A3)
For T > T,, the first and last terms are negative; however, all three terms are typically similar in magnitude. To simplify this form for comparison with simulated data, define T* = T / T , and gather like terms to obtain
GI - G, = ACT*T,( 1 - In T* - T*-’ 1 + Afus(1 - T*)
(-44)
It is of interest to examine two special cases: (i) AC = 0. The two caloric curves are parallel lines and GI - G, sz -AI,,(T* - 1 ) (A4’) is a linear function of temperature. (ii) Afus = 0. To see the relation between the two remaining terms, expand In T*-’about T*-l = 1 . In T*-’ = -1
+ T*-l - (T*-’ - 1)2 + ...
to see that all constant and linear terms in the parentheses of (A4) vanish, leading to second order:
- 1)’/ T* (A4”) so that a quadratic dependence on AT T* - 1 is found. GI - G, = -ACT,( T*
Table VI gives the important quantities for the calculation of GI - G, from simulations data using the AT T* - 1 shorthand. Equilibrium populations in the two-state region and transition rates will be presented in a separate paper. Registry No. Ar, 7440-37-1.
