The Effect of Nozzle Geometry on Cluster Formation in
Molecular Beam Sources
J. T. McDaniels*, R. E. Continetti**, D. R. Miller*
Department of Mechanical And Aerospace Engineering* and Department of Chemistry and Biochemistry**,
UC San Diego, La Jolla, California, 92093
Abstract. We have investigated cluster formation in small diverging nozzles with throat diameters down to 70 microns,
150 half-angle divergence sections, and an L/D of five throat diameters. These nozzles are suitable for molecular beam
experiments and we present data for the beam speed ratio and the cluster formation for argon expansions comparing the
diverging nozzle source with both sharp-edged orifice and capillary free-jet sources of similar throat diameters. We
compare performance using the parameters PoD, Po2D, and PoD2, and at two source temperatures, 300K and 250K, to
qualitatively assess the effects of binary and ternary collisions, modest cooling, and to compare at the same source flow
rate. Values of PoD were varied from three to fifteen torr-cm. We find that all three of the nozzle sources give essentially
the same beam characteristics for source-skimmer interference and for terminal speed ratio of the argon monomer at the
same PoD. However, as expected, the diverging source provides substantially larger clusters at higher beam
concentrations. Clusters up to eleven argon atoms are identified in the diverging nozzles. The advantage of the capillary
source compared to the sharp-edged orifice reported earlier by Murphy and Miller [1] is confirmed for these small
sources. The two-step dimer formation kinetics of Knuth [2] are combined with a simple quasi-one dimensional nozzlejet isentropic expansion, to explain the results. The calculations do remarkably well in predicting the general quantitative
amounts of dimer, but do not predict as much enhancement as observed for the diverging nozzle.
INTRODUCTION AND BACKGROUND
The formation of clusters in free-jet expansions has important applications in molecular beam chemistry and thin
film, nano-scale structures. The free-jet expansion from a simple orifice is sufficiently rapid and reaches such low
densities that normally the source is used to generate small clusters with only a few molecular units. Murphy and
Miller [1] showed that a capillary tube source could improve the cluster formation rate, at the same flow rate,
because of the exit conditions near the sonic surface before the free-jet expansion. Abraham et al.[3,4] showed that
adding a long diverging section to the supersonic nozzle could generate copious amounts of very large clusters in the
tens of nanometer size range. The latter authors utilized very long nozzles (200-500 throat diameters) with small
divergence angles (~50 half-angle) in order to slow the rate of expansion and permit significant cluster generation
and growth. They also presented pressure tap and pitot tube data which showed that the viscous boundary layer
effects in their diverging nozzles were substantial, effecting the entire nozzle flow. These viscous effects are the
reason why early molecular beam researchers eliminated the supersonic diverging section and utilized simple
orifices and the subsequent free-jet expansion to obtain optimal expansion characteristics. Abraham et al. did not
report the effect of the boundary layers in their diverging nozzles on the molecular beam properties. We wish to
generate molecular beams with clusters of size and density in between the two types of sources. To avoid the
boundary layer effects we have studied cluster formation in supersonic sources with 150 half-angle divergence
sections and an L/D of only five throat diameters. Existing correlations [5] suggest that the viscous boundary layers
in such geometries should have little effect on the flow.
There have been many studies of clustering and dimer formation in free jet expansions and many proposed
scaling rules for predicting the onset of clustering and nucleation; discussed, for example, by Hagena [6,7]. For the
small clusters classical nucleation rates are not appropriate because equilibrium conditions do not prevail and
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
670
important parameters such as surface tension are not known for small clusters. Clustering and growth kinetics are
difficult to model, especially for the dimer formation step, because of the need for multiple or three-body collisions,
energy transport and relaxation of internal degrees of freedom, and the lack of knowledge of intermolecular potential
parameters for the clusters. Even the modest two step dimer theories, which we use below, indicate that there are no
simple scaling rules which might quantitatively predict clustering distributions. Our intent here is to show
experimentally that small nozzles with short diverging sections can be very useful molecular beam sources of large
clusters when compared with other geometries, and that they do not compromise other beam properties such as
terminal speed ratio. Although we are interested in other species for our research program, we report data for argon
because of the rich literature available for comparison, and because argon simplifies the gas dynamic and kinetic
analysis, avoiding difficulties associated with internal states. In the sections below we first briefly describe the
apparatus, the mass spectrometer results, and then we use the two-step dimer formation mechanism of Knuth [2],
combined with a simple quasi-one dimensional nozzle-jet isentropic expansion, to examine the results.
EXPERIMENTAL APPARATUS AND PROCEDURES
The molecular beam apparatus is a standard design with a nozzle chamber, connected to a mechanical chopper
chamber by a skimmer (0.023 cm diameter), and then a detection chamber; the base pressures in torr were 10-6, 10-7,
and 10-8 respectively for this study. Under normal operating conditions the nozzle source chamber could reach 10-4
torr. The apparatus was built to study gas-surface interactions but for this study the surface manipulator in the
detection chamber was removed and direct line-of-sight quadrupole mass spectrometry was used to monitor the
beam composition and to make standard time-of-flight velocity distribution measurements. The details of the
apparatus are given in McDaniels [8].
The three nozzle geometries used in this study are shown in Figure 1. The sharp edged orfice (SEO) and the
diverging nozzle (DN) were machined directly from brass stock. The capillary nozzle (CAP) was stainless steel
hypodermic tubing fixed into a brass disc. The SEO had a length-to-diameter ratio of one. The DN had a length to
diameter ratio of five with a 150 half angle. We have studied two SEO diameters, 0.011cm and 0.014cm, and two
DN throat diameters, 0.007cm and 0.014 cm. The CAP nozzle was 0.01 cm internal diameter and had a length-todiameter ratio of 100. All of the nozzles were interchangeable on a brass nozzle block, cooled by circulating fluid
through the block and monitored with a thermocouple attached near the nozzle exit. The thermocouple was also
calibrated by time-of-flight energy balance on the terminal argon beam, and the source stagnation temperatures
reported below are felt to be accurate to within two degrees Kelvin. The diameters of all of the nozzles, except the
capillary were calculated from measured flow rates of argon analyzed by assuming sonic conditions at the throat and
an ideal gas expansion. In the capillary nozzle the stagnation pressure drops along the length of the capillary due to
friction. An effective stagnation pressure for the CAP is defined as the stagnation pressure that would produce the
same measured flow rate in an SEO nozzle of the same geometric diameter. Murphy and Miller [1] have previously
shown that this effective stagnation pressure very reasonably characterizes the collisions in the supersonic free jet
expansion, which is confirmed by our results below.
FIGURE 1. Nozzle Geometries. L/D is the axial nozzle length in throat diameters
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3
1x
10x
100x
(N=1+,2++)
2.5
(N=2+,4++)
2
signal
(N=4+,8++)
1.5
(N=5+,10++)
1
(N=6+,12++)
(N=3+,6++)
(N=7+,14++)
0.5
(N=7++)
(N=3++)
(N=1++)
(N=5++)
(N=8+,16++)
(N=9++)
(N=11++)
0
0
20
40
60
80
100
120
140
160
180
200
220
240
260
280
300
320
340
mass/charge
FIGURE 2. Mass Spectrum for Argon Expansion from Diverging Nozzle at To = 249K ; PoD/(To/300K) = 13 torr-cm
Experiments were run at ~300K (warm) and ~ 250K (cold) source temperatures, at several noD, the product of
source number density and nozzle diameter. This product is proportional to the number of binary collisions a
molecular species will experience in the supersonic expansion. We compare and report results as PoD/(To/300)
because most experiments in the literature are near 300K and pressure rather the density is most often used for the
correlations. The factor (To/300) is then a modest correction for our cold runs to account for the difference between
pressure and density. We report cluster data below as functions of PoD, Po2D, and PoD2. For the capillary, the
effective Po discussed above in used. The first parameter is proportional to binary collisions, the second to three
body collisions, and the last to the mass through-put which the system vacuum pumps must handle – a pragmatic
parameter for design of a cluster beam source. As we noted above there is no simple scaling rule for dimer or cluster
formation and the literature reports a range of PnD where n varies around the value of two and, as expected, the
temperature scaling is more complicated, [7]. Nevertheless, these parameters provide some basis for normalizing the
experiments in order to attempt to isolate the effects of nozzle geometry. Obviously the source temperature is a
primary variable that experimentalists can use to adjust cluster formation. We have purposely restricted this study to
only modest cooling in order to more clearly determine the effect of nozzle geometry. As the data show, however,
even this modest cooling has a substantial effect.
The mechanical chopper had both narrow slots for time-of-flight velocity distribution measurements and wide,
equal on-off, square wave slots for frequency modulated mass spectrometer measurements of species concentration.
The time-of-flight distance was L = 0.47m and a typical chopper gated pulse was 20 microseconds. The mass
spectrometer species measurements were made using a lock-in amplifier and sweeping the mass spectrum slowly
from 10 to 350 amu at fixed source conditions, or by sitting on one cluster mass and varying the source pressure at
constant temperature. Figure 2 is an example of a sweep mass spectrum of a beam extracted from a diverging nozzle
at 249K and PoD ~ 11 torr-cm. The argon clusters are clearly resolved and N=11 clusters are identified by the
doubly ionized mass peak. As a prelude to the results below, in a similar expansion from an SEO nozzle we only
resolve N =4 clusters.
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RESULTS AND DISCUSSION
To characterize the behavior of the diverging nozzles compared to the orifice and capillary nozzles we made
measurements of the skimmer interference and the terminal speed ratio, V/√(2kT/m), the ratio of mean velocity to
the thermal spread in velocities, related to the translational energy beam resolution [9]. We found little difference in
the nozzle-skimmer interference behavior, the beam intensities for these small nozzles peaked near 100 nozzle
diameters (x/D). We could interchange sources and only a modest tuning of the nozzle-skimmer distance was
necessary to optimize the beam intensity. This result was expected since the same flow rates were the same and most
of the skimmer interference in this regime is due to jet molecules scattering back into the jet from the skimmer
surface [9].
A more sensitive measure of the beam quality is the terminal speed ratio, obtained from the time-of-flight
measurements on the beams extracted by the skimmer [9]. Figure 3 shows actual time-of-flight data. The inset gives
the speed ratio, deconvoluted for the chopper gating function, and beam temperature which is related to the width of
the time or velocity distribution. As is clear from the data, we consistently found that the three sources provide the
same beam velocity distribution when compared at the same PoD. Again, for the capillary, the effective Po discussed
above in used. These results suggest that the viscous boundary layers in these short diverging nozzles have little
effect on the terminal translation properties and the subsequent molecular beams extracted from the expansions.
1
SEO
0.8
DN
CAP
PoD
14
14
16
SR
21
20
21
T(K)
1.7
1.9
1.7
S/Smax
0.6
0.4
0.2
0
0.8
0.9
1
1.1
1.2
t/to
FIGURE 3. Argon time-of-flight signal for three nozzle geometries at similar source conditions, To = 296K . The data are
normalized by the maximum signal and the mean time of flight.
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SEO-140w
7
DN-140w
CAP-100w
Dimer/Monomer(%)
6
SEO-140c
DN-140c
5
CAP-100c
4
3
2
1
0
0.0E+00
5.0E+03
1.0E+04
1.5E+04
2.0E+04
2.5E+04
Po^2*D/(To/300K)^2 (torr^2-cm)
FIGURE 4. Argon Dimer-to-Monomer Intensity versus Po2D/(To/300K)2 at To = 250 K ( c ) and To = 295 K ( w )
It is not possible to make quantitative measurements of cluster concentrations because we are not able to
calibrate the mass spectrometer for cluster fragmentation or detector sensitivity. We present our data for argon
clusters normalized to the monomer signals. This normalization should account for run-to-run time variations in
detection efficiency and provide a reasonable basis for comparing between the three nozzle sources. The formation
of dimers is compared in figure 4 as a function of Po2D for all three sources at the warm and cold temperatures. The
discreet data are connected by lines only to guide the eye and do not represent any data fit or calculated dependence.
The figure inset indicates the type of nozzle, the diameter in microns, and whether the source was warm or cold; e.g.
DN-140C is a diverging nozzle with a throat diameter of 140 microns, and with the source temperature within two
degrees of 250K. The data show the improved dimer formation as the nozzle geometry is changed from the SEO to
the CAP to the DN nozzle, and from warm to cold sources. Further, as expected, the data are not adequately scaled
by the binary collision parameter no2 D, or Po2 D/To.
0.75
DN-070c
Quadrimer/Monomer (%)
DN-140c
CAP-100C
0.50
SEO-110c
SEO-140c
0.25
0.00
0.00
0.05
0.10
0.15
0.20
PoD^2/(To/300K) (torr-cm^2)
FIGURE 5. Argon Quadrimer-to-Monomer Intensity versus PoD2 /(To/300K) at To = 250 K
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0.25
1.0E-03
DN-70C
DN-140C
7.5E-04
Octomer/Monomer(%)
CAP-100C
5.0E-04
2.5E-04
0.0E+00
0.000
0.020
0.040
0.060
0.080
0.100
0.120
0.140
0.160
0.180
0.200
PoD^2/(To/300K) (torr-cm^2)
FIGURE 6. Argon Octomer-to-Monomer Intensity versus PoD2 /(To/300K) at To = 250 K
Figure 5 shows data for the quadrimer (Ar)4 cluster formation, again normalized by the monomer intensity and
expressed as a percent. Since an acceptable scaling parameter is not known, we have plotted the cluster formation as
a function of PoD2, related to the mass flux from the nozzle and therefore to the pumping speed requirement of the
vacuum system. This data show that the diverging nozzle is much better than either the SEO or CAP nozzles for
producing beams of these larger clusters in a given vacuum system with fixed pumping speed. In fact, we can barely
detect the quadrimer from the SEO in this system at these temperatures. Note that the effect of the modest source
cooling is much more substantial for the DN than the SEO, which indicates an interplay between increased collision
frequency and the lower thermal energies, i.e. cooling doesn’t help as much if there are not sufficient collisions to
take advantage of it or the rate of cooling is too rapid.
Figure 6 shows similar data for the octomer (Ar)8 cluster formation, comparing the DN to only the CAP because
we could not resolve the octomer for the SEO at these source conditions. The advantage of the diverging nozzle is
seen to be even greater for these larger clusters. This a general result which we have observed, that the effectiveness
of the DN compared to either the SEO or CAP is enhanced as we go to larger clusters beyond N = 4. However, for
smaller clusters our data suggest that the CAP is a good choice and much easier to fabricate than the diverging
nozzle.
The primary reason for the increased clustering as we go from SEO to CAP to DN is that cooling due to
expansion occurs at higher density and the rate of expansion is slower permitting more clustering to occur,
especially important near the sonic point of the expansions. This effect was examined by Murphy and Miller
[1,10,11] in more detail when they compared the SEO to the CAP. Following their approach, we have analyzed the
SEO and DN expansions by the simplest of schemes [8]. We first solve for the isentropic flow properties T(x) and
P(x) using the well known method of characteristic solutions for the free jet [9] and standard quasi-one dimensional
calculations for the DN, given the area A(x), found in any compressible flow text. We then assume that we can
decouple the clustering kinetics for the small concentrations encountered in this study, assuming the clustering and
associated heat of formation have little effect on the expansion characteristics and thermal bath properties. For the
dimer kinetics we assume the well known two step mechanism in which the first step is a binary collision formation
of a virtual orbiting diatom pair, A*2 , which is then stabilized as a dimer A2 in subsequent binary collision. The
sequential binary collsions have been shown [1,2,10,11] to provide more quantitative predictions for argon dimer
formation in free jet expansions than an alternative single three body collision mechanism. Following Knuth, this
mechanism leads to the simple kinetic expression for dimer formation:
675
dnA 2
dt
[
3
= kf K1 nA − nA 2 nA K2
]
(1)
where nA and nA2 are the argon monomer and dimer concentrations respectively and the constants are functions of T
and the standard Lennard-Jones molecular parameters σ, ε [2,8] . We have integrated this kinetic equation along the
centerline of the expansion, starting in the subsonic section of the nozzles, to model the dimer formation. A typical
example of the dimer kinetics calculation is shown in Figure 7. The dimer formation rate is frozen after two nozzle
diameters for the SEO free-jet expansion, while the dimers are still forming in the DN. We terminated the
calculations at five diameters to coincide with the exit of the DN.
To examine the formation of the larger clusters we have used a simple kinetic growth model starting from the
dimer population given by the two step mechanism above:
d ni
= (ci −1 ni −1 − ei ni ) − (ci ni − ei +1 ni +1 )
dt
(2)
where ni is the cluster concentration for i > 2 and c and e are condensation and evaporation coefficients. The
coupled equations assume growth of the dimer by monomer addition and loss. For this growth model we set the rate
of monomer evaporation equal to the rate of monomer condensation from an equilibrium vapor. The coefficients c
and e are then estimated from the free molecular mass flux rates, assuming the clusters to be spherical liquid drops:
c = α 4πr 2 P
e = α 4πr 2 Pds
(2π m kT )
(3)
(2π m kTd )
(4)
where α is the accommodation coefficient and Psd is the drop saturated vapor pressure and Td the drop temperature.
We used the Kelvin relation to correct the vapor pressure for cluster drop size:
Pds = P s (∞ ) exp[2δ ( ρ R Td r )]
(5)
0.015
SEO
DN
XAr
2
0.010
0.005
0.000
0
1
2
3
4
5
x/D
FIGURE 7. Theoretical argon dimer mole fractions vs x/D for the sharp-edged orifice and diverging nozzle expansions. Source
conditions were PoD = 11 torr-cm and To = 250K.
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where δ is the drop surface tension, corrected for cluster size according to Hirschfelder et. al. [12], ρ the drop density
and r is the drop radius. The coupled equations are again integrated to x/D =5. This simple growth model is not
expected to be quantitative for such small clusters. The clusters are not liquid drops, the accommodation coefficients
and drop temperatures are not known, and just as in equilibrium cluster models the surface tension is poorly
understood. For the results below we assume that the drop temperature is the expansion bath temperature and α = 1.
Table 1 is an example of a comparison between the above kinetic model predictions and experiments for cold
expansions at To = 250K and a PoD/(T0/300K) ~ 13 torr-cm. The table gives the ratio, in percent, for the cluster-tomonomer intensity up to N=5 argon atoms. The kinetic model value is that computed at x/D =5, while the
experimental value is that measured by the mass spectrometer. While the predicted dimer concentrations are
remarkably close to the observed values, the enhancement of the DN compared to the SEO is underestimated.
Experimentally we measured an enhancement of about 3.6 while the model predicts an enhancement of about 2.
The kinetic growth model used to predict N>2 is not nearly as rigorous as the two step dimer model and the results
shown in Table 1 indicate this. Although the kinetic model supports the advantage of the DN over the SEO nozzle, it
substantially overestimates the amount of larger cluster growth, even considering the fact that the experimental data
likely underestimate the larger cluster concentrations because of ionization fragmentation in the mass spectrometer.
This discrepancy could be reduced by parametrically setting the cluster drop temperature to be higher than the
surrounding bath temperature or by adjusting surface tension or accommodation coefficients, but the simplicity of
the model does not warrant such a parametric study.
TABLE 1. The Effect of Nozzle Geometry on Argon Cluster Formation at PoD/( To/300K) = 13 torr-cm
Nozzle
Ar2/Ar1 %
Ar3/Ar1%
Ar4/Ar1%
SEO (experimental)
1.99
0.26
0.04
SEO (kinetic model)
1.18
0.84
0.65
DN (experimental)
7.20
2.20
0.60
DN (kinetic model)
2.34
1.65
1.28
Ar5/Ar1%
0.00
0.53
0.33
1.05
ACKNOWLEDGMENTS
Support for J. McDaniels was made possible by a Dreyfus Teacher-Scholar award to R.Continetti .
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