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
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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 N2 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.
0022-3654/87/2091-2479$01.50/0
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 N2 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
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Wegener
The Journal of Physical Chemistry, Vol. 91, No. IO, 1987 100
I
I
I
T = 6 0 K / p = 8 0 torr
AG kT
50
0
.,
0 -0.001t% -0.3 ...: 4 1 . I 20 30 40
I
.
I
,
I
60T(K)70 74 Figure 1. Experimental results of condensatiori of pure nitrogen in the
50
supersaturated state in small supersonic nozzles: A, ref 2; 0, ref 3 and 4; ref 5; 0 , ref 6; 0 , ref 7; 0 , ref 8.
Zeldovich fact~r’~-a measure of the departure from the equilibrium state-(e.g. eq 38 of ref 14) is given by
In eq 1 and 2, AG*/kT refers to the critical Gibbs free enthalpy or energy of formation of a cluster of critical size containing n* molecules as formed from the vapor phase. The surface area of this cluster is given by
O* = ( 3 6 ? r ~ ~ ) ’ / ~ ( n * ) ~ / ~
(3)
with u the molecular volume. Finally
6 = p/(2?rmkT)’/’
(4)
is the rate at which molecules of mass m impinge on unit surface at the state p and T. In eq 1 we assume implicitly that the condensation coefficient equals one, an assumption that has been shown to be reasonable.16 The usual ideas of the theory of nucleation such as the establishment of Szilard’s steady state, the presence of a carrier gas to equilibrate the temperature, and the assumption of sphericity of the cluster recur. The value of the energy of cluster formation for a standard pressure but varying n and T is given by AGo(n,po,T)from the original printout of the calculations described in the preceding paper.’ Now AG(n,T) at any pressure is found from13
AG = AGO
+ (1 - n)kT In @/PO)
(5)
Comparison with Experiment A calculation of A G / k T = f ( n ) by eq 5 for three states of p and T is shown in Figure 2. We note in contrast to similar curves computed for the liquid drop theory that stepwise irregularities are present. This result is to be expected and it agrees with the previous argon work.13 For each state a maximum energy difference arises for a certain value of n defining the energy of formation and number of molecules for the critical cluster. For the lowest curve in Figure 2 no single maximum exists, rather two such states at n* = 22 or 60 are seen. This effect turns out to be insignificant for the calculation of the nucleation rate within the usual uncertainties of the theory. With this information in hand for any state of interest within the range of the Pal and Hoare calculations, the work proceeds as follows. The nozzle expansions (15) Zeldovich, Y.B. Zh. Eksp. Theor. Fiz. 1942, 12, 169. (16) Wegener, P. P.; Wu, B. J. C., ref 9, pp 325-417.
0 50 n 100 Figure 2. Energy of formation of microclusters of nitrogen as a function of the number of malecules in the cluster. Energy values taken from Pal and Hoare.’ Three states are chosen on an isentropic expansion starting at the melting point (Figure 3) and approaching at decreasing temperature the Wilson zone of Figure 1.
start at effectively zero flow speed in a reservoir at conditions in the “superheat” region. The initial pressure and temperature at this state are denoted po and To, respectively, and isentropic expansions are computed from Poisson’s law
p / p o = ( T / To)Y/(Y-’) (6) Until condensation occurs the gas can be treated as thermally and calorically perfect with the ratio of the specific heats y = 7/5 = 1.4 = constant. In Figure 3 four isentropes covering the experimental range are shown together with the Wilson zone of Figure 1. Since our interest centers on the coexistence region the curves start at the vapor pressure curve. It has been established in experimental gas.dynamicsthat eq 6 is indeed applicable provided nozzle calibrations are performed.16 It is not clear if all data in Figure 1 were similarly handled indicating one contribution to the scatter. The expansion proceeds from right to left in Figure 3, and for all states on the isentrope conditions for a critical cluster can be determined as shown, e.g., in Figure 2. In this example the isentrope starting on the melting point is singled out and three successive states approaching the Wilson zone are depicted. If all goes well, the theory must indicate an extreme rise of nucleation rate on the isentrope where it approaches the Wilson zone. For example, for Figure 2 we find an increase of nucleation rate by about 20 orders of magnitude between the conditions at the top and bottom curves. Such an increase leads to the catastrophic collapse of the supersaturated state well-known since Wilson’s time. It is unimportant to which exact value of J we assign the actual onset of nucleation, rather its sudden increase is significant. In -~ Figure 3 we indicate the locations of J = loo and lo6 ( ~ m s-l) on the sample isentropes by closed and open symbols. In this calculation we have at first set 2 = 1. The agreement of predicted and observed condensation-except at higher temperatures-is gratifying; however, it will have to be qualified in several respects. The cluster calculationsof the preceding paper are not expected to apply above about 45 K, a temperature that is 1 / 3 of the well temperature of the interaction potential. Hence anharmonicities appear in the cluster which are not considered in the theory. Nevertheless-except for the uppermost expansion-theory and experiment agree reasonably well even above 45 K. Setting 2 = 1 is permissible since the Zeldovich factor computed with eq 2 for the isentropes shown gives 0.03 < 2 < 0.1 near the Wilson zone. This effect can be ignored in view of the fact that J is normally not known within two or more orders of magnitude. Next we ask if the steady state of nucleation can be achieved in these rapid expansions. Estimates of the relaxation time needed to establish this state have often been made (e.g. ref 17 and 18) and (17) Probstein, R. F. J. Chem. Phys. 1951, 29, 619. (18) Kantrowitz, A. J. Chem. Phys. 1951, 19, 1007.
The Journal of Physical Chemistry, Vol. 91, No. 10, 1987 2481
Nucleation of Nitrogen
10
5~10~
IO2
P (torr)
P (torr)
”
W I L SON
”
Z0NE
10’
IO’
loo
100 Figure 3.. A p T diagram showing ‘four sample isentropic expansions, the vapor pressure curve,of N2,and the Wilson zone of Figure 1. Solid and open circles on the isentropes denote calculated nucleation rates obtained with the theory of nucleation for cluster energies of formation by Pal and Hoare’ (0,J = 10’; 0, J = lo6 cm-3 s-I).
a particularly-simple result is given by 1/r = 6.42?PO*
(7)
where 7 is the required relaxation time.I9 With the terms in eq 7 given previously, we can now estimate 7. For the three states of Figure 2 we find the order s, while at the lowest temperatures and pressures the relaxation time increases to about 10-7s. Considering that the rapid rise of J with temperature takes several microseconds, we can safely assume that the steady state of nucleation prevails. Finally, a comparison of our results with those of the classical theory is indicated. We recall that, in argon with the carrier gas helium, the liquid drop theory based on an estimated surface free energy of solid argon yielded a nucleation onset at much higher temperatures and pressures than observed or predicted by the cluster form~1ation.l~ Previous comparisons of pure N2 condensation with the classical theoryI6gave relatively small discrepancies. Here the correction for the preexponential factor proposed by Feder et al.B to handle in part the effects of the absence of a carrier gas was applied. Pure vapors condense at lower temperatures than those in carrier gases (see later) as demonstrated, e.g., for steam.21 Estimating a “surface tension” for N2 (eq 14 of ref 13), we obtain c = 0.01 5 N/m. With this value, the uncorrected classical theory does not fail as badly as before for argon. However, at the higher temperatures the rise of nucleation rates is predicted ahead of the cluster result while for the lowest isentrope in Figure 3, this condition is reversed. In both instances-in opposite directions-the temperature of condensation is off by about 4 to 5 deg, a difference that is substantially larger than the measuring accuracy.
Carrier Gas Effects The theory in all its variants requires the knowledge of the temperature of the condensing vapor. Normally, a large amount of a noncondensing carrier gas is present to ensure this fact. To (19) Kanne-Dannetschek, I.; Stauffer, D. J. Aerosol Sci. 1981, 12, 105. (20) Feder, J.; Russel, K. C.; Lothe, J.; Pound, G. M. Adu. Phys. 1966, 15, 111. (21) Barschdorff, D.; Dunning, W. J.; Wegener, P. P.; Wu, B. J. C:Nature (London), Phys. Sci. 1972, 240, 166.
I T (K) Figure 4. Experimental results on the condensation of N in helium
30
LO
50
obtained at Yale in a cryogenic shock.tube with supersonic outflow.23 Onset of nucleation shown for four different mole fractions of N2in He: W, 50%; 0 , 20%; A, 10%; A, 5%. The calculated nucleation rates of Figure 3 and the Wilson zone of Figure 1 are repeated. \
understand the importance of this effect in relation to the results of the previous section, experiments with varying mole fractions of N2 in helium were performed at our laboratory. A return to an earlier technique using a shock tube with supersonic outflow22 was indicated. The expansion originated at a low temperature (200 K) in order to keep the Mach number at condensation at reasonable levels. The onset of condensation was determined in a novel manner reversing the usual procedure.I6 Rayleigh light scattering was applied and the results of this are shown in Figure 4. Also shown is the Wilson zone and the calculations of Figure 3. Four different mole fractions of N2 in He were investigated and the 50% values are close to the Wilson zone. The lower values deviate increasingly from the pure vapor data as expected. In pure vapors the condensing vapor itself must remove the heat of formation leading to delayed condensation with respect to vapors in carrier gases.21 The theory shows better agreement at the lower temperatures with those new experimental results.
Conclusions In conclusion we find that the classical theory of nucleation gives an excellent qualitative representation of the behavior of condensing N2 in the supersatured state as found before for many vapors (e.g. note comparisons in ref 16). Yet quantitative agreement within the usual uncertaintiesof nucleation experiments fails for the liquid drop approach. However, if free energies for the formation of molecular microclusters of Lennard-Jones nitrogen are inserted in the rate equation, satisfactory agreement is obtained in the temperature range for which the statistical mechanics approach is valid. Acknowledgment. This work is dedicated to the memory of my friend and colleague Gilbert D. Stein. The cooperation of Drs. M. R. Hoare and P. Pal over the years was in part made possible by a grant from the Thermodynamics and Mass Transfer Program of the National Science Foundation. The author is grateful to the Institute for Advanced Study Berlin where the paper was written. Registry No. N2, 7727-37-9. (22) Wegener, P. P.; Lundquist, G. J. Appl. Phys. 1951, 22, 233. (23) Zahoransky, R. A. ZFW 1986, 10, 34.
