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Measurement of velocities of clusters generated in supersonic flow
Journal
of
MOLECULAR STRUCTURE Journal of Molecular Structure 376 (1996) 59-63
Measurement
of velocities of clusters generated in supersonic flow’
James W. Hovick, Richard J. French, Lawrence S. Bartell* Department
of Chemistry.
University
of Michigan,
Ann Arbor. MI48109.
USA
Received 20 June 1995; accepted 10 July lYY5
Abstract An apparatus is described for measuring the speeds of molecular clusters in a flight tube under conditions of gas density that would overwhelm the system in conventional measurements of gas velocities. Although the apparatus incorporates no new design principles it is effective in practice. It provides the information needed in timing the evolution of the extremely fast phase changes that occur in highly supercooled clusters. The state of clusters is monitored as a function of their distance and, hence, time-of-flight along their path. Illustrative examples of velocity measurements are given.
1. Introduction Cluster research is a fertile and rapidly expanding field of pursuit in chemistry, physics, and materials science. One of the promising applications being explored is a study of the dynamics of homogeneous nucleation in freezing and certain solid-state phase transitions. In the past, research in this fundamental area has been severely hindered by experimental and theoretical difficulties. The discovery [l-7] that clusters can serve as realistic models of bulk systems in transition, while retaining the convenient measurement techniques available for clusters, has revitalized research on nucleation. The advantages and limitations of clusters in investigations of nucleation were recently reviewed in detail elsewhere [8]. Under the conditions of extremely high super* Corresponding
author.
I Dedicated to Professor James E. Boggs on the occasion of his 75th birthday.
cooling that can be attained in cluster systems, transformations take place in microseconds while clusters are monitored by electron diffraction [2] or coherent Raman spectroscopy [l]. Because transition times of clusters traveling at supersonic speeds are most conveniently measured by timesof-flight, it is necessary to know the cluster speeds to convert flight distances to times. Many time-offlight measurements of velocity distributions in atomic and molecular beams have been carried out, and their systematic properties are quite well established [9-151. Although cluster velocities have been determined under milder conditions than those used in this laboratory [16,17], less is known about velocities of heavy particles than about gas velocities. In a seminal paper [18], Schwartz and Andres demonstrated how to calculate the speeds to which small particles are accelerated in supersonic flow under certain conditions. At high gas densities the particles flow without slipping, but as the expansion rarefies and gas flow continues to accelerate, clusters begin to
0022-2860/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved SSDI 0022-2860(95)09042-S
60
J. W. Hovick et al./Journai
of Molecular
lag behind and ultimately reach a terminal velocity. This happens well before the acceleration of the surrounding gas ceases. Unfortunately, in its present form the theory of Schwartz and Andres does not apply to conditions of flow as complex as those encountered in studies of clusters in this laboratory. Therefore, it was necessary to develop a suitable experimental technique for measuring cluster speeds. Such a technique is described in the following.
2. Experimental section The electron diffraction unit with its supersonic system for generating clusters has been described in detail elsewhere [ 19,201. A flight tube was added to determine the speeds of clusters entering the diffraction chamber. Attached to a port opposite the nozzle, the 3” tube is isolated from the diffraction chamber except for an orifice 2 mm wide and 8 mm high to admit the supersonic jet. A 300 1 s-’ diffusion pump maintains a background pressure of 10-3-10-4 mbar during experiments. In order to detect clusters, a fast ionization gauge (Beam Dynamics FIG-l) is mounted in the flight tube 441 mm away from a chopper in the diffraction chamber. The chopper is located between the nozzle and the flight tube at a distance of 5 mm from the nozzle exit and 5-10 mm from the entrance to the flight tube. Signals from the fast ionization gauge (FIG) are displayed on an oscilloscope. A light emitting diode (LED) offset from the axis of the supersonic jet senses the chopper slits as they pass and triggers the horizontal sweep of the oscilloscope. The accompanying pulse on the oscilloscope screen serves as a reference point in determinations of the zero point of time. To cancel effects of the uncertainty in the position of the LED in computations of flight times, measurements are made with chopper rotation in both clockwise and counterclockwise directions. The chopper itself is a disk 16 cm in diameter notched by two slits 180” apart, each 2 mm wide and 8 mm deep. The disk is attached directly to the shaft of a reversible 115 VAC motor operated at 165 Hz. Rotation frequencies are measured accurately with a Hewlett Packard 5232A
Structure
376 (1996)
59 63
electronic counter. With the small loads imposed on the motor, no precautions are required to prevent overheating. The fast ionization gauge is similar to a BayardAlpert gauge, with a heated filament 22 mm in length serving as a source of electrons. Electrons accelerated by 160 V ionize gas molecules and clusters passing through the ionization volume of approximately 0.33 cm3. The resultant ions are detected by a central collector, 22 mm in length. It can be appreciated that gas expanding through our miniature Lava1 nozzles (throat diameters 0.1-0.2 mm), having initial pressures typically of several atmospheres, is far too dense when it encounters the chopper to qualify as a molecular beam. Therefore, the gas pulses transmitted by the chopper and by the entrance to the flight tube undergo severe skimmer interactions, and the profiles of the gas pulses registered by the FIG are badly distorted. Accordingly, even though both gas and cluster signals in a pulse are visible on the oscilloscope screen, the gas pulse is of little use. Fortunately, it is the speed of the clusters, not the gas speed, that is needed for the kinetic analyses.
3. Results Initial experiments with the flight tube were performed on systems, namely neat helium and neon, that do not produce clusters under our usual expansion conditions. These experiments confirmed the poor skimming at the entrance of the flight tube. The peak reading of the FIG implied a faster gas velocity than expected from kinetic theory. Presumably, this was due to a distortion of the density/time profile attributable to the decline in skimming quality with time. Earlier elements of a gas pulse encounter a clean orifice, but subsequent elements are progressively more attenuated by scattering from the build up of gas. Because clusters are tens of thousands times more massive than gas molecules, they are scarcely perturbed as they coast through regions of gas dense enough to scatter gas molecules. Nucleation experiments have been carried out in this laboratory on a variety of cluster systems,
J. W. Hovick Ed al./Journal
of Molecular
Structure
376 (1996)
59-63
61
Table 1 Expansion conditions for cluster formation and associated data from oscilloscope traces Subject + carrier gas
Subject mole fraction
P, (bar)
r,” (cm)”
rP (cm?
I, (cm)’
fb (cm?
2s (ms)’
SF, + Ar SF6 + Ar SF6 + Ar SF6 + Ar NH, NH3 + Ne CC& + Ne (CH,)$CI + Ne CH,CCI? + Ne Hz0 + Ne’ H,O + Nep
0.04 0.04 0.04 0.04 1.0 0.125 0.125 0.22 0.088 0.2 0.4
1.4 2.8 4.1 5.5 3.1 3.1 1.2 1.9 1.9 4.4 4.4
5.55 7.36 7.54 7.60 4.94 3.24 1.36 7.06 6.25 5.00 5.00
6.85 7.98 8.55 5.98 5.97 6.03 8.16 7.61 8.63 5.10 5.00
2.51 4.31 4.58 2.08 4.00 1.95 4.94 5.03 4.30 3.15 3.60
1.18 2.35 3.00 3.17 2.34 2.05 3.00 2.85 3.86 1.60 1.60
0.871 0.X68 0.851 0.833 0.457 0.527 0.758 0.679 0.672 0.535 0.480
’ Location of cluster signal for clockwise chopper rotation; time scale 0.2 ms cm-‘. b Location of cluster signal for counterclockwise chopper rotation. ’ Location of the LED signal for clockwise chopper rotation. d Location of the LED signal for counterclockwise chopper rotation. ’ Calculated from Eq. Cl9 of Ref. [21]. f Stagnalion temperature, 95-C [7]. s Stagnation temperature, 120°C [7].
including those listed in Table 1. This Table records expansion conditions and elapsed times between the LED signal and the arrival of clusters, for both clockwise and counterclockwise rotations. Table 2 tabulates the corresponding cluster speeds derived from the times in Table 1 by a straightforward analysis outlined elsewhere [21]. A characteristic oscilloscope trace indicating the
LED pulse, the carrier gas signal, and the later signal from the slower clusters is shown in Fig. 1 for clusters of SF6 seeded into argon. The 32 ps risetime of the cluster signal is ten-fold longer than the risetime of the FIG and is a consequence of the finite width of the chopper slit. Random cluster velocities associated with the thermal distribution in a gas flow (assumed to be about 50 K for
Table 2 Cluster velocities derived from data of Table 1 Subject
Subject mole fraction
StagnatIon temperature (“)
Carrier gas
pt (bar)
Cluster velocity (ms ‘)
SF, + Ar SF6 + Ar SF6 + Ar SF, + Ar NH3 NH2 + Ne CC14 + Ne
0.04 0.04 0.04 0.04 1.0 0.125 0.125 0.22 0.088 0.2 0.4
23 23 23 23 23 23 23 23 23 95 120
Ar Ar Ar Ar
1.4 2.8 4.1 5.5 3.1 3.1 1.2 1.9 1.9 4.4 4.4
507 f 6 508 + 6 518 zt 6 530 f 6 965 f 12 837 f 10 582f7 650 f 8 651 f 8 825 * 10 919f 11
(CH&CCl
+ Ne
CHQ& + Ne Hz0 + Nea Hz0 + Ne” a Data from Ref. [7].
Ne Ne
Ne Ne Ne Ne
62
J. W. Hovick et al.~Journal of Molecular Strucmv
376 (1996) 59-63
sharply, flight times of the clusters can be measured with reasonable accuracy. Some substances such as ammonia form clusters readily, even in the absence of a carrier gas [6]. Fig. 2 shows the appearance of the oscilloscope trace for clusters of neat ammonia. It differs from Fig. 1, mainly in the absence of a signal due to gas.
4. Discussion
Fig. 1. Oscilloscope trace of time-of-flight distribution for SF, clusters in Ar carrier. Subject mole fraction, 0.04; stagnation pressure, 4.1 bar; time scale, 0.2 ms per division.
the sake of illustration) would imply a spread in arrival times of only 3 11s. Although the cluster pulse peaks sharply, its slow rate of decay clearly does not correspond to the true density profile of a cluster packet. The long tail of the cluster signal is due to molecules produced by the disintegration of clusters on impact with the FIG and other surfaces. The fragments subsequently scatter from various parts of the apparatus before they are ionized. Nevertheless, because the cluster beam peaks
Fig. 2. Oscilloscope trace of time-of-flight clusters formed from neat NHj. Stagnation time scale, 0.2 ms per division.
distribution for pressure, 3.1 bar;
Results confirm that clusters reach their terminal velocity long before the surrounding gas does. Although a classic paper [22] on nonequilibrium flows (which is excellent in other respects) suggested that no such slippage of clusters relative to the gas flow should occur, experiments and the theoretical work of Schwartz and Andres [18] prove otherwise. Consistent with expectations based on crude extrapolations of the SchwartzAndres theory, our clusters are observed to lag markedly behind the gas that surroundeds them during their formation by condensation. Alternatively, computations [23,24] treating the condensation, evaporation, thermal accommodation, and gas dynamics of supersonic flow show that the gas velocity at the exit of the Lava1 nozzle and beyond markedly exceeds the measured velocity of the clusters. Therefore, even though the expanding gas beyond the nozzle is scattered badly by the chopper and flight tube orifice, the gas at these junctures is too rarefied to have a significant effect on cluster trajectories. This argument is confirmed quantitatively by the work of De Martin0 et al. [25] who investigated the scattering of clusters by a buffer gas. Clusters as heavy as those generated in our nucleation studies, i.e. made up of about lo4 molecules, suffer little interference in their passage through an assembly of molecules an order of magnitude more numerous than encountered by our clusters before they arrive at the FIG. Terminal velocities of clusters are reached when the expanding flow becomes so rarefied that collisions and the concomitant momentum exchanges are too infrequent to keep accelerating the massive clusters. Although cross sections for collisions of gas molecules with clusters greatly exceed those
J. W. Hovick et al./Journal
of Molecular
for collisions of molecules with other molecules, masses of clusters increase with the cube of cluster radii whereas cross sections only increase with the square. Therefore, beyond an effective “quitting surface” the expanding gas is unable to influence the velocities of clusters, even though it continues to undergo collisional acceleration. Of course, the cessation of the acceleration of clusters is not a sudden event at a mathematical surface, but is the result of a gradual tapering off centered at the quitting surface. Treatments of the gas dynamics of nonequilibrium nozzle flow yield the thermal and velocity profiles of gases before, during, and after the condensation of clusters [23,24]. This information, together with experimental measurements of cluster terminal velocity, makes it possible to locate the effective position of the quitting surface for clusters. This has proven to be well inside the exit of our Lava1 nozzles in most cases examined to date. Some of the systematic effects of flow conditions on cluster velocities can be seen in Table 2. Pressure effects are evident in the case of SF6 seeded into argon at a concentration of 4 mol%. As the initial pressure is increased, collisions persist until later stages of the expansion where the flow velocity is higher, i.e. an increase in the pressure shifts the quitting surface outward. Temperature effects are shown in the example of water clusters, where the faster velocity of the clusters formed in a warmer flow reflects the greater enthalpy available for conversion into mass flow. More complex is the case of ammonia, for which effects due to a change in the heat capacity, the mean molecular weight, and thermal accommodation act in contrary directions. The Schwartz-Andres theory of slippage of heavy particles in gas flow was developed for preexisting particles in free jet expansions. It would be pleasing if the theory were extended to preexisting particles in nozzle flow or, even better, to clusters formed during the course of nozzle flow. A reliable theory could save appreciable experimental labor. Until such a theory is available, it is reassuring to find that cluster velocities can be measured in flight tubes even under conditions of fairly high gas densities, entirely unsuitable for conventional measurements of gas velocity.
Structure
376 (1996)
59-63
63
Acknowledgments This research was supported by a grant from the National Science Foundation. We thank Mr. Paul Lennon for his expert help in carrying out the experiments. We gratefully acknowledge the award of a Gomberg Fellowship to J.W.H.
References [l] R.D. Beck, M.F. Hineman and J.W. Nibler, J. Chem. Phys., 92 (1990) 7068. [2] L.S. Bartell and T.S. Dibble, J. Am. Chem. Sot., I12 (1990) 890. [3] L.S. Bartell and T.S. Dibble, J. Phys. Chem., 95 (1991) 1159. [4] T.S. Dibble and L.S. Bartell, J. Phys. Chem., 96 (1992) 2317. [5] T.S. Dibble and L.S. Bartell, J. Phys. Chem., 96 (1992) 8603. [6] I. Huang and L.S. Bartell, J. Phys. Chem., 98 (1994) 4543. [7] I. Huang and L.S. Bartell, J. Phys. Chem., 99 (1994) 3924. [8] L.S. Bartell, J. Phys. Chem., 99 (1995) 1080. [9] E.W. Becker and W. Henkes. Z. Phys., 146 (1956) 320. [lo] K. Bier and O.F. Hagena, 2. Angew. Phys., 14 (1962) 658. [1 l] J.B. Andersen and J.B. Fenn, Phys. Fluids, 8 (1965) 780. [12] P.B. Scott, P.H. Bauer, H.Y. Wachman and L. Trilling, in C.L. Brundin (Ed.), Rarefied Gas Dynamics. Academic Press, New York, 1967, Vol. 2, p. 1353. [13] O.F. Hagena and A.K. Varma, Rev. SC!. Instrum., 39 (1968) 47. [14] J.A. Alcalay and E.L. Knuth, Rev. Sci. Instrum., 40 (1969) 438. [15] H. Haberland, U. Buck and M. Tolle, Rev. Sci. Instrum.. 56 (1985) 1712. [16] E.W. Becker, K. Bier and W. Henkes, Z. Phys.. 146 (1956) 333. [17] I. Bauchertand and O.F. Hagena, 2. Naturforsch., Teil A, 22 (1965) 1135. [IR] M.H. Schwartz and R.P. Andres, J. Aerosol. Sci., 7 (1976) 281. [19] L.S. Bartell, R.K. Hccnan and M. Nagahima, J. Chem. Phys., 78 (1983) 236. [20] L.S. Bartell and R.J. French, Rev. Sci. Instrum., 60 (1989) 1223. [21] J.W. Hovick, Ph.D. Thesis, University of Michigan, 1994. [22] P.P.Wegener, in P.P. Wegener (Ed.). Nonequilibrium Flows, Marcel Dekker, New York, 1969. [23] L.S. Bartell, J. Phys. Chem., 94 (1990) 5102. [24] L.S. Bartell and R.A. Machonkin, J. Phys. Chem., 94 (1990) 6468. [25] A. De Martino, M. Benslimanc, M. Chatelet, C. Crazes, F. Pradere and H. Vach, Z. Phys. D, 27 (1993) 185.
of
MOLECULAR STRUCTURE Journal of Molecular Structure 376 (1996) 59-63
Measurement
of velocities of clusters generated in supersonic flow’
James W. Hovick, Richard J. French, Lawrence S. Bartell* Department
of Chemistry.
University
of Michigan,
Ann Arbor. MI48109.
USA
Received 20 June 1995; accepted 10 July lYY5
Abstract An apparatus is described for measuring the speeds of molecular clusters in a flight tube under conditions of gas density that would overwhelm the system in conventional measurements of gas velocities. Although the apparatus incorporates no new design principles it is effective in practice. It provides the information needed in timing the evolution of the extremely fast phase changes that occur in highly supercooled clusters. The state of clusters is monitored as a function of their distance and, hence, time-of-flight along their path. Illustrative examples of velocity measurements are given.
1. Introduction Cluster research is a fertile and rapidly expanding field of pursuit in chemistry, physics, and materials science. One of the promising applications being explored is a study of the dynamics of homogeneous nucleation in freezing and certain solid-state phase transitions. In the past, research in this fundamental area has been severely hindered by experimental and theoretical difficulties. The discovery [l-7] that clusters can serve as realistic models of bulk systems in transition, while retaining the convenient measurement techniques available for clusters, has revitalized research on nucleation. The advantages and limitations of clusters in investigations of nucleation were recently reviewed in detail elsewhere [8]. Under the conditions of extremely high super* Corresponding
author.
I Dedicated to Professor James E. Boggs on the occasion of his 75th birthday.
cooling that can be attained in cluster systems, transformations take place in microseconds while clusters are monitored by electron diffraction [2] or coherent Raman spectroscopy [l]. Because transition times of clusters traveling at supersonic speeds are most conveniently measured by timesof-flight, it is necessary to know the cluster speeds to convert flight distances to times. Many time-offlight measurements of velocity distributions in atomic and molecular beams have been carried out, and their systematic properties are quite well established [9-151. Although cluster velocities have been determined under milder conditions than those used in this laboratory [16,17], less is known about velocities of heavy particles than about gas velocities. In a seminal paper [18], Schwartz and Andres demonstrated how to calculate the speeds to which small particles are accelerated in supersonic flow under certain conditions. At high gas densities the particles flow without slipping, but as the expansion rarefies and gas flow continues to accelerate, clusters begin to
0022-2860/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved SSDI 0022-2860(95)09042-S
60
J. W. Hovick et al./Journai
of Molecular
lag behind and ultimately reach a terminal velocity. This happens well before the acceleration of the surrounding gas ceases. Unfortunately, in its present form the theory of Schwartz and Andres does not apply to conditions of flow as complex as those encountered in studies of clusters in this laboratory. Therefore, it was necessary to develop a suitable experimental technique for measuring cluster speeds. Such a technique is described in the following.
2. Experimental section The electron diffraction unit with its supersonic system for generating clusters has been described in detail elsewhere [ 19,201. A flight tube was added to determine the speeds of clusters entering the diffraction chamber. Attached to a port opposite the nozzle, the 3” tube is isolated from the diffraction chamber except for an orifice 2 mm wide and 8 mm high to admit the supersonic jet. A 300 1 s-’ diffusion pump maintains a background pressure of 10-3-10-4 mbar during experiments. In order to detect clusters, a fast ionization gauge (Beam Dynamics FIG-l) is mounted in the flight tube 441 mm away from a chopper in the diffraction chamber. The chopper is located between the nozzle and the flight tube at a distance of 5 mm from the nozzle exit and 5-10 mm from the entrance to the flight tube. Signals from the fast ionization gauge (FIG) are displayed on an oscilloscope. A light emitting diode (LED) offset from the axis of the supersonic jet senses the chopper slits as they pass and triggers the horizontal sweep of the oscilloscope. The accompanying pulse on the oscilloscope screen serves as a reference point in determinations of the zero point of time. To cancel effects of the uncertainty in the position of the LED in computations of flight times, measurements are made with chopper rotation in both clockwise and counterclockwise directions. The chopper itself is a disk 16 cm in diameter notched by two slits 180” apart, each 2 mm wide and 8 mm deep. The disk is attached directly to the shaft of a reversible 115 VAC motor operated at 165 Hz. Rotation frequencies are measured accurately with a Hewlett Packard 5232A
Structure
376 (1996)
59 63
electronic counter. With the small loads imposed on the motor, no precautions are required to prevent overheating. The fast ionization gauge is similar to a BayardAlpert gauge, with a heated filament 22 mm in length serving as a source of electrons. Electrons accelerated by 160 V ionize gas molecules and clusters passing through the ionization volume of approximately 0.33 cm3. The resultant ions are detected by a central collector, 22 mm in length. It can be appreciated that gas expanding through our miniature Lava1 nozzles (throat diameters 0.1-0.2 mm), having initial pressures typically of several atmospheres, is far too dense when it encounters the chopper to qualify as a molecular beam. Therefore, the gas pulses transmitted by the chopper and by the entrance to the flight tube undergo severe skimmer interactions, and the profiles of the gas pulses registered by the FIG are badly distorted. Accordingly, even though both gas and cluster signals in a pulse are visible on the oscilloscope screen, the gas pulse is of little use. Fortunately, it is the speed of the clusters, not the gas speed, that is needed for the kinetic analyses.
3. Results Initial experiments with the flight tube were performed on systems, namely neat helium and neon, that do not produce clusters under our usual expansion conditions. These experiments confirmed the poor skimming at the entrance of the flight tube. The peak reading of the FIG implied a faster gas velocity than expected from kinetic theory. Presumably, this was due to a distortion of the density/time profile attributable to the decline in skimming quality with time. Earlier elements of a gas pulse encounter a clean orifice, but subsequent elements are progressively more attenuated by scattering from the build up of gas. Because clusters are tens of thousands times more massive than gas molecules, they are scarcely perturbed as they coast through regions of gas dense enough to scatter gas molecules. Nucleation experiments have been carried out in this laboratory on a variety of cluster systems,
J. W. Hovick Ed al./Journal
of Molecular
Structure
376 (1996)
59-63
61
Table 1 Expansion conditions for cluster formation and associated data from oscilloscope traces Subject + carrier gas
Subject mole fraction
P, (bar)
r,” (cm)”
rP (cm?
I, (cm)’
fb (cm?
2s (ms)’
SF, + Ar SF6 + Ar SF6 + Ar SF6 + Ar NH, NH3 + Ne CC& + Ne (CH,)$CI + Ne CH,CCI? + Ne Hz0 + Ne’ H,O + Nep
0.04 0.04 0.04 0.04 1.0 0.125 0.125 0.22 0.088 0.2 0.4
1.4 2.8 4.1 5.5 3.1 3.1 1.2 1.9 1.9 4.4 4.4
5.55 7.36 7.54 7.60 4.94 3.24 1.36 7.06 6.25 5.00 5.00
6.85 7.98 8.55 5.98 5.97 6.03 8.16 7.61 8.63 5.10 5.00
2.51 4.31 4.58 2.08 4.00 1.95 4.94 5.03 4.30 3.15 3.60
1.18 2.35 3.00 3.17 2.34 2.05 3.00 2.85 3.86 1.60 1.60
0.871 0.X68 0.851 0.833 0.457 0.527 0.758 0.679 0.672 0.535 0.480
’ Location of cluster signal for clockwise chopper rotation; time scale 0.2 ms cm-‘. b Location of cluster signal for counterclockwise chopper rotation. ’ Location of the LED signal for clockwise chopper rotation. d Location of the LED signal for counterclockwise chopper rotation. ’ Calculated from Eq. Cl9 of Ref. [21]. f Stagnalion temperature, 95-C [7]. s Stagnation temperature, 120°C [7].
including those listed in Table 1. This Table records expansion conditions and elapsed times between the LED signal and the arrival of clusters, for both clockwise and counterclockwise rotations. Table 2 tabulates the corresponding cluster speeds derived from the times in Table 1 by a straightforward analysis outlined elsewhere [21]. A characteristic oscilloscope trace indicating the
LED pulse, the carrier gas signal, and the later signal from the slower clusters is shown in Fig. 1 for clusters of SF6 seeded into argon. The 32 ps risetime of the cluster signal is ten-fold longer than the risetime of the FIG and is a consequence of the finite width of the chopper slit. Random cluster velocities associated with the thermal distribution in a gas flow (assumed to be about 50 K for
Table 2 Cluster velocities derived from data of Table 1 Subject
Subject mole fraction
StagnatIon temperature (“)
Carrier gas
pt (bar)
Cluster velocity (ms ‘)
SF, + Ar SF6 + Ar SF6 + Ar SF, + Ar NH3 NH2 + Ne CC14 + Ne
0.04 0.04 0.04 0.04 1.0 0.125 0.125 0.22 0.088 0.2 0.4
23 23 23 23 23 23 23 23 23 95 120
Ar Ar Ar Ar
1.4 2.8 4.1 5.5 3.1 3.1 1.2 1.9 1.9 4.4 4.4
507 f 6 508 + 6 518 zt 6 530 f 6 965 f 12 837 f 10 582f7 650 f 8 651 f 8 825 * 10 919f 11
(CH&CCl
+ Ne
CHQ& + Ne Hz0 + Nea Hz0 + Ne” a Data from Ref. [7].
Ne Ne
Ne Ne Ne Ne
62
J. W. Hovick et al.~Journal of Molecular Strucmv
376 (1996) 59-63
sharply, flight times of the clusters can be measured with reasonable accuracy. Some substances such as ammonia form clusters readily, even in the absence of a carrier gas [6]. Fig. 2 shows the appearance of the oscilloscope trace for clusters of neat ammonia. It differs from Fig. 1, mainly in the absence of a signal due to gas.
4. Discussion
Fig. 1. Oscilloscope trace of time-of-flight distribution for SF, clusters in Ar carrier. Subject mole fraction, 0.04; stagnation pressure, 4.1 bar; time scale, 0.2 ms per division.
the sake of illustration) would imply a spread in arrival times of only 3 11s. Although the cluster pulse peaks sharply, its slow rate of decay clearly does not correspond to the true density profile of a cluster packet. The long tail of the cluster signal is due to molecules produced by the disintegration of clusters on impact with the FIG and other surfaces. The fragments subsequently scatter from various parts of the apparatus before they are ionized. Nevertheless, because the cluster beam peaks
Fig. 2. Oscilloscope trace of time-of-flight clusters formed from neat NHj. Stagnation time scale, 0.2 ms per division.
distribution for pressure, 3.1 bar;
Results confirm that clusters reach their terminal velocity long before the surrounding gas does. Although a classic paper [22] on nonequilibrium flows (which is excellent in other respects) suggested that no such slippage of clusters relative to the gas flow should occur, experiments and the theoretical work of Schwartz and Andres [18] prove otherwise. Consistent with expectations based on crude extrapolations of the SchwartzAndres theory, our clusters are observed to lag markedly behind the gas that surroundeds them during their formation by condensation. Alternatively, computations [23,24] treating the condensation, evaporation, thermal accommodation, and gas dynamics of supersonic flow show that the gas velocity at the exit of the Lava1 nozzle and beyond markedly exceeds the measured velocity of the clusters. Therefore, even though the expanding gas beyond the nozzle is scattered badly by the chopper and flight tube orifice, the gas at these junctures is too rarefied to have a significant effect on cluster trajectories. This argument is confirmed quantitatively by the work of De Martin0 et al. [25] who investigated the scattering of clusters by a buffer gas. Clusters as heavy as those generated in our nucleation studies, i.e. made up of about lo4 molecules, suffer little interference in their passage through an assembly of molecules an order of magnitude more numerous than encountered by our clusters before they arrive at the FIG. Terminal velocities of clusters are reached when the expanding flow becomes so rarefied that collisions and the concomitant momentum exchanges are too infrequent to keep accelerating the massive clusters. Although cross sections for collisions of gas molecules with clusters greatly exceed those
J. W. Hovick et al./Journal
of Molecular
for collisions of molecules with other molecules, masses of clusters increase with the cube of cluster radii whereas cross sections only increase with the square. Therefore, beyond an effective “quitting surface” the expanding gas is unable to influence the velocities of clusters, even though it continues to undergo collisional acceleration. Of course, the cessation of the acceleration of clusters is not a sudden event at a mathematical surface, but is the result of a gradual tapering off centered at the quitting surface. Treatments of the gas dynamics of nonequilibrium nozzle flow yield the thermal and velocity profiles of gases before, during, and after the condensation of clusters [23,24]. This information, together with experimental measurements of cluster terminal velocity, makes it possible to locate the effective position of the quitting surface for clusters. This has proven to be well inside the exit of our Lava1 nozzles in most cases examined to date. Some of the systematic effects of flow conditions on cluster velocities can be seen in Table 2. Pressure effects are evident in the case of SF6 seeded into argon at a concentration of 4 mol%. As the initial pressure is increased, collisions persist until later stages of the expansion where the flow velocity is higher, i.e. an increase in the pressure shifts the quitting surface outward. Temperature effects are shown in the example of water clusters, where the faster velocity of the clusters formed in a warmer flow reflects the greater enthalpy available for conversion into mass flow. More complex is the case of ammonia, for which effects due to a change in the heat capacity, the mean molecular weight, and thermal accommodation act in contrary directions. The Schwartz-Andres theory of slippage of heavy particles in gas flow was developed for preexisting particles in free jet expansions. It would be pleasing if the theory were extended to preexisting particles in nozzle flow or, even better, to clusters formed during the course of nozzle flow. A reliable theory could save appreciable experimental labor. Until such a theory is available, it is reassuring to find that cluster velocities can be measured in flight tubes even under conditions of fairly high gas densities, entirely unsuitable for conventional measurements of gas velocity.
Structure
376 (1996)
59-63
63
Acknowledgments This research was supported by a grant from the National Science Foundation. We thank Mr. Paul Lennon for his expert help in carrying out the experiments. We gratefully acknowledge the award of a Gomberg Fellowship to J.W.H.
References [l] R.D. Beck, M.F. Hineman and J.W. Nibler, J. Chem. Phys., 92 (1990) 7068. [2] L.S. Bartell and T.S. Dibble, J. Am. Chem. Sot., I12 (1990) 890. [3] L.S. Bartell and T.S. Dibble, J. Phys. Chem., 95 (1991) 1159. [4] T.S. Dibble and L.S. Bartell, J. Phys. Chem., 96 (1992) 2317. [5] T.S. Dibble and L.S. Bartell, J. Phys. Chem., 96 (1992) 8603. [6] I. Huang and L.S. Bartell, J. Phys. Chem., 98 (1994) 4543. [7] I. Huang and L.S. Bartell, J. Phys. Chem., 99 (1994) 3924. [8] L.S. Bartell, J. Phys. Chem., 99 (1995) 1080. [9] E.W. Becker and W. Henkes. Z. Phys., 146 (1956) 320. [lo] K. Bier and O.F. Hagena, 2. Angew. Phys., 14 (1962) 658. [1 l] J.B. Andersen and J.B. Fenn, Phys. Fluids, 8 (1965) 780. [12] P.B. Scott, P.H. Bauer, H.Y. Wachman and L. Trilling, in C.L. Brundin (Ed.), Rarefied Gas Dynamics. Academic Press, New York, 1967, Vol. 2, p. 1353. [13] O.F. Hagena and A.K. Varma, Rev. SC!. Instrum., 39 (1968) 47. [14] J.A. Alcalay and E.L. Knuth, Rev. Sci. Instrum., 40 (1969) 438. [15] H. Haberland, U. Buck and M. Tolle, Rev. Sci. Instrum.. 56 (1985) 1712. [16] E.W. Becker, K. Bier and W. Henkes, Z. Phys.. 146 (1956) 333. [17] I. Bauchertand and O.F. Hagena, 2. Naturforsch., Teil A, 22 (1965) 1135. [IR] M.H. Schwartz and R.P. Andres, J. Aerosol. Sci., 7 (1976) 281. [19] L.S. Bartell, R.K. Hccnan and M. Nagahima, J. Chem. Phys., 78 (1983) 236. [20] L.S. Bartell and R.J. French, Rev. Sci. Instrum., 60 (1989) 1223. [21] J.W. Hovick, Ph.D. Thesis, University of Michigan, 1994. [22] P.P.Wegener, in P.P. Wegener (Ed.). Nonequilibrium Flows, Marcel Dekker, New York, 1969. [23] L.S. Bartell, J. Phys. Chem., 94 (1990) 5102. [24] L.S. Bartell and R.A. Machonkin, J. Phys. Chem., 94 (1990) 6468. [25] A. De Martino, M. Benslimanc, M. Chatelet, C. Crazes, F. Pradere and H. Vach, Z. Phys. D, 27 (1993) 185.