J . Phys. Chem. 1987, 91, 2474-2479
2474
b
5
G -2
'S I
I
m.0
I
1 WA
I
1
w. 4
1
t
)
1 U.4
I
I
m.2
I
e
-
!
!
!
--
-
c [IO' C"]
Figure 6. Several atomic Mo emission bands in solid neon following 4d55p 7F"' 4d55s'S excitation near 27 250 cm-'. (a) 5G2 'S emission; it exhibits a long rise time and a >30-ms decay time; (b) 5P3 'S emission; long rise time, decay time T = 33 f 5 ms; (c) SD,, 7S emission; no rise-time,T = 30 f 5 ns.
that, in solid Ne, the a3P state which is at 20607 cm-' drops below the 5Dstate and that the short lifetime is due fast nonradiative relaxation into this state. The long rise times observed on the *P and SGemission may then be due to population via this 3P state. The absence of direct observable emission from the intermediate 3Pstate would then have to be attributed to the strongly forbidden nature of the 3P 'S emission. It should, however, be noted that, if this interpretation is accepted, the relatively high intensity of the 5Demission is rather surprising.
-
Conclusions and Summary Pulsed laser vaporization combined with matrix isolation is found to be an extremely versatile technique for studies of a wide variety of transient species. It is applicable to studies of most metals, including highly refractory species such as molybdenum or tungsten. It should be particularly well suited for mixed, heteronuclear studies using alloys or, as in our gas-phase studies, pellets pressed from a mixture of powders in case of metal combinations which do not alloy readily. One can also use the technique to generate and investigate metal oxides, nitrides, carbides, and similar species, either by directly vaporizing the corresponding solids or by adding suitable reactants to the matrix gas and forming them by gas-phase reactions. While in the present studies the characterization of the samples was accomplished by laser-induced fluorescence and, in some cases,
by electronic absorption spectroscopy, in principle it should be easy to extend this work into the infrared. Infrared emission spectroscopy should be quite useful for transition-metal clusters, many of which will possess low-lying electronic states. While using conventional matrix isolation techniques deposition times of several hours are often required, most samples studied here were deposited in 1-3 min. This suggests that, with increased deposition times, it should be possible to stabilize high enough quantities of infrared-active species for direct absorption studies. In many cases, the distribution of clusters is not a smooth function of the number of atoms, but one or several particularly stable species dominate. Thus in carbon vapor most of the large cluster^^*^^^ seem to converge to Ca. Similarly, in laser-vaporized InP, 3-4 specific clusters were found to dominate the distribution." Again the technique described here should be extremely useful in providing information about the geometric, electronic, and vibrational structures of these interesting species. Registry No. Ca2, 12595-85-6; Pb2, 12596-92-8;PbO, 1317-36-8; Mo2, 12596-54-2; Ne, 7440-01-9. (52) Trevor, D. J.; Whetten, R. L.; Cox, P. M.; Caldor, A. J . Am. Chem. SOC.1985, 107, 518.
(53) Kroto, H. W.; Heath, J. R.; OBrien, S. C.; Curl. R. F.: Smallev. R. E. Nature (London) 1985. (54) Bondybey, V. E.; Reents, W. P.;Mandich, M. L J . Chem. Phys., submitted for publication.
Thermodynamic Propertles and Homogeneous Nucleation of Molecular Clusters of Nltrogen P. Pal* and M. R. Hoare Royal Holloway and Bedford New College, University of London, Egham Hill, Egham, Surrey TW20 OEX, England (Received: June 17, 1986) A theoretical model for the dynamics of nitrogen microclusters, based on the assumed free rotation of N2 molecules within
a solidlike harmonic vibrating structure, is described. This approach allows the known morphology of rare-gas-type atomic clusters to be adopted to account for the statistical thermodynamicsof nonpolar diatomic species with considerablesimplification. As judged by the prediction of thermodynamic parameters related to homogeneous nucleation, the approximation would appear to be justified in a favorable temperature range.
Introduction
In recent years atomistic theories of the critical nucleus have effectively supplanted the classical liquid-drop model for the investigation of thermodynamic properties and hence nucleation 0022-3654/87/2091-2474$01.50/0
rates in the microcluster size range of some two to several hundred atoms. The great advantage of such method is, of course, that no knowledge of macroscopic mechanical or thermodynamic properties need be assumed, so that rate information is, in principle, obtainable ab initio from the characteristics of the force field alone. 0 1987 American Chemical Society
The Journal of Physical Chemistry, Vol. 91, No. 10, 1987 2475
Homogeneous Nucleation of Nitrogen Clusters TABLE I: Components of Degrees of Freedom of a Nitrogen Cluster Containing n (Diatomic) Molecules under Three Different Models free rotor solidlike Reed model translation overall rotation intermolecular vibration intramolecular vibration internal rotation
3 3 3n - 6
3 3
n 2n
6n - 6 0 0
6n
6n
W
a
3 3 3n - 6
(neglected) (neglected)
3n
In fact it is nearly 35 years since Reed’ took the first tentative steps in this direction in his paper on N 2 condensation, and some 15 years since Burton2 and the present authors3s4made the first realistic computation of microcluster statics of rare-gas atoms which paved the way to the calculation of genuinely “atomistic” nucleation rates. By a fortunate coincidence, this period also saw the development of experimental methods of sufficient refinement to allow comparisons of measured and computed results of nucleation, at least for rare-gas-type systems. In due course, the authors’ results were shown to be in broad agreement with the supersonic nozzle and shock tube rate measurements of Wegener and co-workers.5 In this paper we return to the system first studied in Reed’s 1952 paper, namely nitrogen. We discuss possible models for the dynamics of N 2 clusters and justify the choice of a free internal-rotation approximation for the calculation of the free energy of formation and hence nucleation rate. The results obtained are offered for comparison alongside the shock-tube measurements of Wegener et al. described in the following paper, where a detailed account of experimental evaluation will be given.
The Model Although a full dynamic treatment of nitrogen microclusters would be exceedingly difficult, on account of the lack of spherical symmetry in the intermolecular potential, considerable simplification seems justifiable in the energy range of interest for the calculation of nucleation rates. Thus while, at first sight, the low-energy region would seem to be characterized by librational motions of the individual molecules about an extraordinary complex variety of equilibrium configurations, and the medium-energy region might properly be considered as that in which these go over to a regime of restricted rotor dynamics, in practice there seems to be a strong case for assuming internal rotations to be free, with each effectively spherical N 2 molecule interacting as would an atom, under the appropriate rotationally averaged central potential.6 This is because we know,’ on the one hand from scattering experiments, that the anisotropic component in a diatomic force field, at energies well below the dissociation energy, is much less than the unfortunate image of a “dumbbell” tends to suggest (an egglike ellipsoid would be nearer the truth) while, on the other hand, we know from calorimetric and spectroscopic evidence on the solid N2 that free rotation sets in, even in the bulk crystal lattice, although the vibrational degree of freedom remains in the ground state. Thus, a fortiori, the assumption of free (1) Reed, Jr., S. J. J. Chem. Phys. 1952, 20, 208. (2) Burton, J. J. J. Chem. Phys. 1970, 52, 345. (3) Hoare, M. R.; Pal, P. Adu. Phys. 1971, 20, 161. (4) Hoare, M. R.; Pal, P. Adv. Phys. 1975, 24, 645. (5) Hoare, M. R.; Pal, P.; Wegener, P. P. J. Colloid Interjace Sci. 1980, 75, 126. ( 6 ) Rowlinson, J. S. The Fluid Stare; Pergamon: London, 1963. (7) Preliminary work on the statics of diatomic microclusters under nonspherical potentials (e.g. the “atom-atom” potential with two Lennard-Jones sources in a rigid “dumbbell” relationship) shows the variety of available equilibrium configurations to be enormous and virtually unclassifiable, even at sizes as small as n = 6 or 7. Whereas with atomic clusters a good guess at a deep stable minimum can be made by maximizing nearest-neighbor contacts, the complexities of angular relationships between neighboring diatomic molecules rule out any such approach. We have been virtually limited to the examination of accidentally discovered minima in stereo computer graphics; in view of the uncertainty in the chosen potential any attempt to classify these or analyze the librational motion about them seems pointless. (Hoare, M.R.; Andrew Barker, J., unpublished work.)
b
C
d Figure 1. A few small nitrogen clusters relaxed under the Lennard-Jones potential function (eq 1). Each sphere represents a nitrogen molecule enveloping a core, containing two atoms, of dumbbell shape with random orientation. The midpoint of separation of the two atoms coincides with the center of the sphere, and all L-J distances ri, are measured with reference to these centers: (a) n = 2, linear dimer (CJ; (b) n = 3, triangular trimer (C3J;(c) n = 4, tetrahedron (Td);(d) n = 5 , tetrahedral ( e ) n = 7,pentagonal bipyramid ( & h ) . bipyramid, (&A);
rotation in the much less restricted microcluster seems well justified. The question arises as to how we should assign a representative Lennard-Jones potential to the N 2 molecule. The method of Kihara* provides a suitable rationale in terms of differential geometry of spherocylindrical or ellipsoidal hard-core molecules, the details of which need not concern us here. For consistency with the experimental work of Wegener’s group, we adopted the potential
in which rij = Iri - rjl is the distance between the centers of mass of the ith and j t h molecules. These mass centers are assumed to be located at the midpoint of the line joining the atom pair of each molecule which corresponds to the Kihara “core length” of 1.094 8, for nitrogen. The parameters e / k and u take the values of 95.05 respe~tively.~It is obvious that the distance of K and 3.698 %., closest approach for a pair of molecules under Lennard-Jones potential is well over three times the molecular “core length” and this effectively means that free rotation of each molecule is possible even in large clusters. Thus in the spirit of the above approximation, we may take it that, as far as potential energy is concerned, the N, microcluster behaves statically as though composed of central-force units interacting under the potential (1) while dynamically the internal motions are those of the harmonic oscillations of each pair of atoms of a nitrogen molecule with the additional contribution of its internal rotation. These internal vibrations and internal rotations contribute one and two degrees of freedom respectively to every constituent molecule in the cluster. The whole cluster is considered in the appropriate harmonic-oscillator/rigid-rotor (HORR) approximation. In fact, although the intramolecular oscillations at (8) Kihara, T. J . Phys. SOC.Jpn. 1951, 6, 289.
(9) Hirschfelder, J.; Curtiss, C.; Bird, R. Molecular Theory of Gases and Liquids; Wiley: New York, 1954.
2416
Pal and Hoare
The Journal of Physical Chemistry, Vol. 91 No. 10, 1987
TABLE II: Contributions to the Helmholtz Free Energy of n-Meric Clusters by Translational, Rotational, Vibrational, and Internal Motions at Three Different Temperatures free-energy components, )(IO2' J
n
translation
rotation
3 5 IO 13 19 24 30 55 65 70 80 90 100 110
783.79 794.42 808.77 814.16 826.86 831.55 837.49 844.11 847.56 849.08 851.84 854.19 856.40 858.47
83.10 101.04 129.20 126.72 182.21 189.94 200.57 154.05 216.99 219.48 224.17 228.18 231.77 235.08
zero Doint
internal motions
1.17 5.93 15.04 16.42 3 1.06 87.79 162.47 66.25 93.45 96.07 130.03 156.95 163.99 202.09
317.9 971.5 3 088.0 4885.0 8 066.0 10 770.0 14410.0 31 420.0 37 680.0 41 380.0 47 910.0 54 920.0 62 630.0 69610.0
5 1.76 86.27 172.55 224.45 327.98 414.25 517.78 949.30 1121.99 1208.68 1380.40 1552.95 1725.50 1898.05
vibration
30°K
~~~~~
T = 10K
3 5 10 13 19 24 30 40 55 65 70 80 90 100 110
2149.7 2415.6 2494.6 2510.7 2534.4 2548.9 2562.6 2580.8 2600.2 2610.6 2614.3 2623.9 2630.9 2637.5 2643.3
317.6 371.4 455.9 375.6 448.5 614.9 638.1 670.0 530.5 719.3 726.7 740.4 752.8 763.6 773.1
T=30K 44.72 149.50 377.70 481.2 773.1 1085.4 1505.7 2251.5 1979.9 2442.5 2587.0 3092.2 3544.5 3840.5 4360.6
317.9 971.5 3 088.0 4 885.0 8 066.0 10 770.0 14410.0 20 270.0 3 1 420.0 37 680.0 41 380.0 47910.0 54 920.0 62 630.0 69610.0
291.5 485.6 972.7 1264.7 1848.2 2334.8 2918.3 3891 .O 5350.4 6323.6 6808.1 7791.3 8754.5 9727.6 10700.8
3 5
3667.4 3714.6 3779.2 3804.1 3839.5 3861.2 3881.7 3908.4 3938.3 3953.8 3960.6 3973.0 3984.2 3994.2 4002.8
514.3 593.2 721.8 601.3 710.6 960.3 995.1 1042.9 833.0 1 1 16.8 1127.4 1148.5 1167.2 1183.3 1197.6
T=45K 108.7 359.6 913.7 1185.8 1874.1 2561.7 3464.3 5029.0 5006.7 6093.1 6485.1 7640.5 8696.5 9485.4 10659.4
317.9 971.5 3 088.0 4 885.0 8 066.0 I O 770.0 14410.0 20 270.0 31 420.0 37 680.0 41 380.0 47910.0 54 920.0 62 630.0 69610.0
508.7 855.4 1710.7 2224.4 3276.7 4106.6 51 32.8 6845.4 9410.8 1 1 119.1 11976.3 13690.8 15399.0 17 107.3 18821.7
IO 13 19 24 30 40 55 65 70 80 90 100 110
the N-N bond frequencies are easy to include, they will scarcely be excited at the temperatures of interest to us here and could almost be neglected. We may summarize by giving the breakdown of degrees of freedom in Table I, where comparisons with a rigid solidlike model and the primitive model of Reed are also shown. The immediate advantage of proceeding this way is that we can take over all our data on the statics of argon-like clusters as the starting point for dynamic calculations on nitrogen, or indeed any other nonpolar diatomic molecule. The taxonomy of minimal structures in soft atomic packings is a complex subject in its own right, made particularly interesting by the dominant Occurrence of noncrystalline fivefold symmetric motifs of icosahedral type. Certainly beyond n = 5 5 , icosahedral arrangement appears to predominate the growth sequence. Here we can only refer the interested reader to the extensive earlier literature in this fieldlo where exhaustive accounts of the actual geometry of the minimal cluster structures, we have assumed in this work, will be found. It is to be mentioned that structures of the presumed minimal
20°K
10°K I
Figure 2. Vibrational entropy S / k per molecule of nitrogen clusters as a function of size at different temperatures.
clusters are determined by exploration of available local minima in the total potential energy function n-l
n
v(n)= C C cXrl,r,)
(2)
I'lJ=1+1
with 4(rl,rJ)as given by (1). A few illustrative minimal structures are shown in Figure 1. Thermodynamic Functions
Under the assumptions described above, the computation of thermodynamic functions for nitrogen microclusters proceeds according to the following time-honored algorithm: (a) Investigate the available static equilibrium structures by optimization of the total potential energy function (2) and select the available absolute minimum configuration as thermodynamically representative for the particular cluster size n. (b) Having selected the representative structure for given n, carry out a vibrational analysis in the harmonic oscillator approximation and obtain the vibrational partition function, qvlb.for the cluster as a whole. (c) From the geometry of the minimal configuration, obtain the principal moments of inertia ZA, IB, and 1, and hence the rotational partition function, qrot,for the cluster in the rigid-rotor approximation. The appropriate symmetry factor must be determined and included. The translational partition function, qtran,is determined by straightforward calculation. Departing from the procedure for monatomic systems, the internal partition must be determined for diatomic constituents. (d) Obtain the partition functions q,ntrot and qIntnb for internal rotation and internal vibration, respectively, from the geometry and spectroscopic data of the N, molecule. (e) The total partition function, Q(n), for the cluster is assembled as the product of these components, which then leads to the required thermodynamic functions, most importantly the free energy of formation AG(n,T,P). The whole sequence must be repeated for each cluster size n over the whole size range in which the critical nucleus is expected to lie. Further details of the above scheme, together with a justification of the approximation involved, will be found in ref 4 and 10. The key equation, giving the free energy of formation AG(n,T,P) for the cluster of n molecules reads AG(n,T,P) = -kT[ln Q ( n ) - n In Q(1) - (1 - n) In (P/Po)] (3)
Here Porefers to the standard atmospheric pressure and Q( 1) and Q(n) denote the total partition functions for the single molecule (monomer) and n cluster, respectively. The subpartition functions are given by the well-known formulae: (4)
(10) Hoare, M.
R. Ado. Chem. Phys.
1979, 40, 49
The Journal of Physical Chemistry, Vol. 91, No. 10, 1987 2411
Homogeneous Nucleation of Nitrogen Clusters
TABLE 111: Standard Cibbs Free Energy of Formation of n-Meric Clusters at Different Temperatures AG(n.T,Pd X 1021 J n 5K 10 K 15 K 20 K 25 K 30 K 35 K -3.01 -2.8 1 -2.63 -2.47 -2.36 -2.26 3 -2.19 -8.07 -5.31 -7.09 -6.17 -4.49 5 -9.03 -3.74 7 8 12 13 15 20 24 30 36 40 50 55 60 70 80 90 100 110
-16.48 -19.92 -38.81 -45.63 -53.77 -80.63 -101.3 -135.9 -168.8 -191.3 -246.9 -297.4 -324.1 -392.5 -454.6 -521.4 -595.2 -661.8
-14.66 -17.81 -34.81 -4 1.07 -48.78 -73.24 -92.36 -124.4 -154.5 -175.8 -227.0 -273.8 -298.5 -362.4 -420.1 -482.4 -551.6 -613.8
-12.78 -15.65 -30.68 -36.34 -43.65 -65.57 -83.07 -1 12.6 -140.2 -159.9 -206.4 -248.9 -271.5 -330.6 -383.7 -44 1.4 -505.6 -563.2
-10.97 -13.55 -26.62 -3 1.68 -38.62 -58.01 -73.91 -101.1 -126.0 -144.3 -186.1 -223.9 -244.5 -298.7 -347.4 -400.2 -459.4 -512.5
Here translational motions of the n clusters are contained within unit volume.
where I*, ZB,and IC are the principal moments of inertia of the cluster as a whole and 5 is its symmetry factor 3n-6 qvib
=
[
exp(-hvi/2kT) (1
i=l
- exp(-hvi/kT))
where 4 refer to the normal-mode frequencies of the cluster in free rotation. The vibrational analysis incorporates all intermolecular interactions of a cluster which is accomplished by forming the force-constant matrix with respect to 0.5n(n - 1) bond lengths. This matrix is then transformed with respect to Cartesian coordinates into a 3n X 3n matrix from which the eigenvalues are extracted. This Cartesian coordinate method, due to Gwinn,]’ differs from our earlier treatment used for the monatomic (argon) clusters. The eigenvalue matrix of order 3n yields 3n - 6 positive eigenvalues and 6 zero values. Comparison of these normal modes shows detectable differences from the nearest-neighbor results in which more low-order frequencies tend to be excited. q i n t vib
=
1”
[
=
” (5) a21kT
The “core length” and reduced mass of the nitrogen molecule have been used to calculate the moment of inertia I appearing in eq 8. Finally these subpartition functions determine Q(n) which is given by Q(n) = q i n t vibqint rotqtranqrotqvib exP(-v(n) /kT) (9) with V(n), the binding energy of the minimum configuration relative to the energy at infinitite separation of all molecules of N2. Although our primary interest is in the free energy of formation of N2clusters as a route to the nucleation rate, it is also instructive to analyze the simpler Helmholtz free energy F = -kT In Q(n,T) (1 1) Gwinn, W. J. Chem. Phys. 1971, 55, 477. (12) Herzberg, G. Molecular Spectra and Molecular Structure,
Van Nostrand: New York, 1955.
-7.57 -9.63 -18.92 -22.78 -29.08 -43.52 -56.34 -78.75 -98.71 -1 14.3 -146.8 -175.3 -192.0 -236.6 -276.2 -320.0 -369.2 -413.5
-5.99 -7.81 -15.27 -18.55 -24.57 -36.62 -47.95 -68.12 -85.67 -99.90 -128.0 -151.8 -166.7 -206.6 -241.9 -281.2 -325.6 -365.4
45 K -2.12 -2.36 -3.03 -4.40 -8.34 -10.48 -16.02 -23.43 -3 1.93 -47.79 -60.70 -72.44 -9 1.94 -106.5 -1 17.8 -148.6 -182.0 -206.2 -241.2 -272.7
for its dependence on temperature and size. The proportional contributions of several components in the total Helmholtz free energy for a number of clusters at three different temperatures have been tabulated in Table 11. These figures show the relatively dramatic rise of the vibrational component with increase in temperature, in contrast to the effectively classical behavior of translation and rotation. They also illustrate the importance of zero-point energy. Thus to take a specific example, we see that for the n = 100 cluster, the translational contribution rises from only 1.3% at 10 K to 4.2% at 45 K, while in the same range that of overall rotation rises from 0.3%to 1.2%. In contrast to this, the vibrational contribution rises from 2.6% to some 10% and internal rotation from 2.6% to 18%. In the temperature quoted, the contribution of N 2 intraatomic vibration is totally negligible The dominant role of zero-point energy stands out clearly in the table. Although the free-energy function as a whole is of primary importance in nucleation rate calculations, it is of interest to examine in addition the vibrational entropy function for clusters, since it is the entropic component in the free energy which seems likely to influence the size dependence of stability at least as much as the energy term. The vibrational entropy function is given by S/k =
hvi/kT ’E6[(exp(hvi/kT) - In (1 - exp(-hvi/kT)) - 1) i=l
(7)
1 - exp(-hvo*c/kT) Here the frequency vo, denotes the fundamental N-N (interatomic) bond stretching frequency of the molecule acting as a simple oscillator with zero dipole moment.12 We have used the value v, = 2 3 3 0 . 7 ~(c = velocity of light). q i n t rot
-9.23 -11.55 -22.70 -27.16 -33.76 -50.65 -64.99 -89.72 -112.1 -129.0 -166.2 -199.3 -218.0 -267.3 -311.4 -359.7 -413.8 -462.5
40 K -2.15 -3.03 -4.41 -6.07 -1 1.74 -14.45 -20.22 -29.93 -39.82 -57.81 -73.01 -85.97 -109.7 -128.9 -141.9 -177.2 -208.3 -243.3 -282.9 -318.5
1
Figure 2 shows the variation of this function with cluster size at different temperatures. These curves have ,the remarkable property that the “structure” appearing in the range n < 55 is virtually eliminated as size increases beyond this value. Thus, while there might be minor influences on stability in the lower size range due to the emergence of particularly stable and symmetric minima at, for example, n = 13 (the single-shell icosahedron), n = 19 (the double icosahedron), and n = 55 (the two-shell “Mackay” ic~sahedron’~), it would seem that there is no such tendency and certainly no Occurrence of structural ”magic numbers” at 60 < n < 110.
Free Energy and the Critical Nucleus The key function in the atomistic equivalent of the Gibbs theory of nucleation is the Gibbs free energy of formation of clusters and its dependence on size. This is the function which expresses the compensation of “bulk” and “surface” behavior as well as, more subtly perhaps, the opposing “energetic” and “entropic” forces at work in determining cluster stability and ultimately concentrations in the supercooled gas. The first requirement of an atomistic theory is that it mimics the Gibbs liquid-drop model in producing
Vol. I; (13) Mackay, A. L. Acta Crystallogr. 1961, 15, 916.
Pal and Hoare
2478 The Journal of Physical Chemistry, Vol. 91, No. 10, 1987
$1
8
250-
70
200 -
C
0 150-
a, 37
L
Figure 3. Variation of total Gibbs free energy of formation AG(n,T,P ) / k T ( X lo2’J) with the size of clusters at different temperatures and pressures: (A) T = 30 K, P = 1 Torr; (B) T = 35 K, P = 2.7 Torr; (C) T = 35 K , P = 2 Torr; (D)T = 45 K,P = 13 Torr; (E)T = 50 K, P = 25 Torr.
a,
Q
.-
a,
0
100-
3 C
cc
a maximum in the free energy of formation curve as a function of size at realistic temperatures and pressures. This maximum is then identified with the critical nucleus, whose concentration in the supercooled gas is used to derive a nucleation rate, with additional kinetic assumptions. It is by no means a foregone conclusion that any atomistic model will produce such a maximum at realistic pressures; in fact, the projected results of the primitive model of Reed do not, which is due to the neglect of internal motions. Standard Gibbs free energy which is involved in forming an n-meric cluster from n molecules at standard atmospheric pressue has been tabulated in Table I11 for a number of clusters in the temperature range 5 K < T < 45 K. Figure 3 shows the curves of free energy of formation AG(n,T,P) for nitrogen clusters as a function of size for several combinations of temperature and pressure, all in the realistic parameter range for cryogenic shock-tube experiments. These are calculated from the functions sampled in Table 111 with the addition of pressure-dependent terms in eq 3. It is immediately clear that the required maxima are present well within the range accessible to both experiments and theoretical model. It is true that the structure in the curves, particularly at low temperature and pressure, makes the identification of the critical nucleus n* slightly less clear-cut than in a continuum model.
0
? 50
-
square-root term in the preexponential factor is almost irrelevant to the task of determining the temperature and pressure conditions under which the rate J changes suddenly from a microscopic value to an appreciable level. Rates of nucleation according to eq 11 are shown in Figure 4. The sudden switch from negligible condensation to an observable rate is clearly obvious and the critical pressure for this is readily identified. An alternative form of rate equation, due to Katz16 and ReissI7 (see also ref 5 ) takes account of the equilibrium concentration of all clusters of size n < n*, since in an atomistic model condensation can occur on other subcritical nuclei. When Nucleation Rates The Volmer-Weber-Becker-Doring-Frenkel-Zeldovi~h~~~~~ this is allowed for, we obtain for rate (to within an unessential factor): theory of nucleation, as adapted to the atomistic model, leads to several alternative prescriptions for the calculation of nucleation rates. Since what is required is qualitative predictions of catastrophic breakdown of the supersaturated phase rather than detailed values for the condensation flux, most of the refinements Here the summation is carried out up to n = n*, although the flux that have been suggested from time to time are inconsequential rate remains insensitive beyond n*. This is the form which is in practical terms. Only two will be considered here, for a fuller actually used in the following paper, where a full comparison of account see ref 4, 5 , and 10. our calculations with the results of shock-tube condensation exFollowing our previous work on argon, it will be convenient to periments is carried out in terms of “Wilson line” describing the use the rate equation: locus in P and T of points where onset of rapid nucleation has been observed. J(n*.T.P) =
Here S(n*) denotes the surface area of the cluster of critical size and C( 1) is the equilibrium concentration of monomer molecules per unit volume. We emphasize that the accuracy of S(n*) and (14) Volmer, M.; Weber, A. Z. Phys. Chem. 1926, 119, 277. (15) Lothe, J.; Pound, G. M. Nucleation, Zettlemoyer, A,, Ed.; Dekker: New York, 1969.
Discussion Although the model can be viewed as a suitable analogue of the liquid-drop model in the small size range and a predictor of thermodynamic conditions for experiments under which condensation can be attained, this does not warrant the conclusion that it is altogether adequate. It is a somewhat fortunate coincidence that the experimental conditions in, for example, a cryogenic shock (16) Katz, J. L. J . Slat. Phys. 1970, 2, 137. (17) Reiss, H. J . Stat. Phys. 1970, 2 ( 1 ) , 83.
J. Phys. Chem. 1987, 91, 2479-2481 tube are conducive to satisfying the rather generous assumptions that have none into the mechanical Dicture. At temperatures near and beyond ‘45 K, the simple harmonic approximation for vibrational partition function is not tenable because of the emergence of anharmonic effects. At very low temperatures, the assumption of free internal rotation is likely to break down. If the motion were then to become dominated by librational modes, these would not only be very difficult to describe in terms of appropriate force constants, but would also
2479
introduce an additional zero-point energy contribution, unaccounted for in the present work.
Acknowledgment. The authors are particularly grateful to Professor P. P. Wegener and his colleagues of the Department of Applied Mechanics of Yale University for their hospitality and financial support under a grant from the National Science Foundation, which made this study possible. Registry No. Nz, 7727-37-9.
Nucleation of Nitrogen: Experiment and Theory Peter P. Wegener Applied Mechanics, Yale University, New Haven, Connecticut 06520 (Received: June 17, 1986) Experimental results on the condensation of pure nitrogen in the supersaturated state are collected. New experiments on pure N2 and N2 in He are added. A comparison of experiment and the classical theory of nucleation using the liquid drop approach and recent work by Pal and Hoare computing energies of formation of microclusters of Lennard-Jones N2 is made. The new results give satisfactory agreement with experiment in their range of applicability.
Introduction Aside from the intrinsic interest, homogeneous nucleation of nitrogen has recently taken on practical importance in conjunction with cryogenic transonic wind tunnels. To achieve the high Reynolds numbers required, such wind tunnels operate with pure nitrogen near saturation. Much experimental work on nitrogen condensation in the supersaturated state is available; however, detailed comparisons with theory are lacking. This gap will be filled in this study. In addition to new experiments, calculations with the classical theory of nucleation will be given. However, in the low temperature range of interest, the liquid drop approach to the determination of the energy of formation of droplets from the vapor will be replaced by new energy computations for microclusters based on the work by Pal and Hoare given in the preceding paper in this issue.l Experimental Results Approximately 20 experimental studies of the condensation of nitrogen in the absence of a noncondensing carrier gas have come to the attention of the author. Of these results a group of nine sources published in seven papers was selected as shown in Figure 1 with the references identified in the caption. Not considered are results that may have been affected by heterogeneous nucleation or whose cooling rates (free jets) were too high to be compatible with the theory of nucleation, etc. The results shown were obtained in small supersonic nozzles with steady flow at cooling rates of the order of lo6 OC/s. Different grades of N 2 purity were used, and with two exception^^,^ the onset of condensation in the supersaturated state was detected by static pressure measurements. The experiments cover a remarkable time span from 1952 to 1984. They include recent results from our laboratory6 using precooled N2 and Rayleigh light scattering to find condensation at a low Mach number ( M 2.5). In Figure 1 increased scatter at the higher temperatures is evident. New unpublished results from the Jiaotong University, Xi’an7 were obtained by attaching a nozzle directly to a vaporizer fed by liquid ( I ) Pal, P.; Hoare, M. R. J. Phys. Chem. paper preceding in this issue. (2) Faro, I.; Small, T. R.; Hill, F. K. J . Appl. Phys. 1952, 23, 40. (3) Arthur, P. D.; Nagamatsu, H. T. Heat Transfer and Fluid Mechanics Insfitute;University of California: Los Angeles, 1952; p 125. (4) Willmarth, W. W.; Nagamatsu, H. T. J. Appl. Phys. 1952, 23, 1098. (5) Goglia, G. L.; Van Wylen, G. J. J . Heat Transfer 1961, 83, 27. (6) Wegener, P. P.; Wu, B. J. C.; Stein, G. D., unpublished. (7) Guo Youyi; Ji Guanghua; Wang Qun; et al. private communication.
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nitrogen, and in the Gottingen works bottled N2 of extremely high purity was used. Both sets of experiments exhibit a higher supersaturation than previous work in the same range.5 Not shown-and to be discussed at the end of the paper-are new results from our laboratory with N 2 in He at varying mole fractions. In spite of the difficulties of all these experiments, the time span, and the various techniques involved, the combined results permit the determination of a “Wilson zonen of the states where condensation is to be expected. Here we apply an analogy to the Wilson line well-known for steam expansions. Theoretical predictions of the condensation of N 2 can now be matched to the experimental evidence.
Theory of Nucleation The theory of nucleation has been reviewed on many occasions (e.g., ref 9-1 1). Aside from possible fundamental thermodynamic problems, the liquid drop approach to the calculation of the energy of formation of droplets (or clusters) from the vapor phase is expected to fail at low temperatures for very small clusters. Moreover if clusters are expected to be solid in the temperature range below the melting point, property values such as surface free energy are not available. For these reasons calculations of the energy of cluster formation for argon were recently extended by Pal and Hoare to nitrogen1 assuming this substance to be a Lennard-Jones gas. The extensive literature preceding these calculations has been reviewed by Abraham” and Hoare.I2 The previous joint work comparing theory and experiment for argon13 proved to be successful of predicting the onset of nucleation at states where the liquid drop assumption failed. To compute J, the nucleation rate per unit time and volume, we take J = ZPO*C, exp(-AG*/kT)
(1)
(e.g. eq 20 of ref 14). Here C, is the number density and the (8) Koppenwallner, G., private communication. Based on work by Wagner, B.; Diiker, M. 53rd AGARD FM Panel Symposium, 1983. (9) Zettlemoyer, A. C., Ed. Nucleotion; Dekker: New York, 1969. (10) Zettlemoyer, A. C. Ado. Colloid Interface Sci. 1977, 7. (11) Abraham, F. F. Ado. Theor. Chem. 1974. (12) Hoare, M. F. Ado. Chem. Phys. 1979, 40. (1 3) Hoare, M. R.; Pal, P.; Wegener, P.P. J . Colloid Interface Sci. 1980, 75, 126. (14) Lothe, J.; Pound, G. M., ref 9, pp 109-149.
0 1987 American Chemical Society
