J. Phys. Chem. 1984, 88, 4513-4521
4513
An Analysis of HF Excitation and (HF), Dissociation in a Nozzle Beam by a Continuous Wave HF Laser E. L. Knuth* and M. Wilde Max-Planck Institut f u r Stromungsforschung, Gottingen, West Germany (Received: March 22, 1983)
Mass-spectrometer measurements of dimer-fragment signals in an HF nozzle beam in which some fraction of the monomers has been vibrationally excited by an antiparallel coaxial continuous wave (CW) HF laser are analyzed. Relative to an earlier more simplified analysis, the most important extensions of the present analysis are that (a) the excitation of monomers is taken to occur at a fixed value of the gas density and (b) the dimer concentration at this location contains contributions from both the equilibrium source state and dimer formation in the free jet. An expression relating (a) the ratio of the dimer signals for the laser on and off with (b) the ratio of the dimer concentration to the excited-monomer concentration at the monomer-excitation location and with (c) a measure of the number of collisions which a dimer molecule would experience with excited-monomer molecules is developed. This expression is fitted to measurements for the 1P2, 1P4,and 1P6 laser lines by selecting values of two fitting parameters, one related to the fractional excitation and the other related to the monomer-excitationlocation. It is concluded that, for the investigated experimental conditions, (a) the monomer-excitation location and the downstream end of the dimer-formation region are always downstream from the nozzle throat, and that the monomer-excitation location is always downstream from the dimer-formation region, and (b) approximately 2/3 of the dimers are formed during the expansion. Application of the present model to a typical set of experimental conditions indicates that about 29% of the laser-beam energy is utilized in exciting the HF monomers, that about 14%(a relatively large percentage) of the monomers are excited, and that the excited-monomer flux at the downstream end of the excitation region is approximately 5 X 1O”sr-l s-’. It is found that, if dimers cannot be avoided, then the terminal excited-monomer flux is maximized (for given species, temperature, and nozzle size) if the source. pressure is such that the dimer mole fraction at the end of the dimer formation region is about 3/8 the fractional excitation of the monomers at the excitation location.
Introduction In a recent publication, Ellenbroek et al.’ presented preliminary results and a simplified analysis of an antiparallel coaxial arrangement (Figure 1) for vibrational excitation of HF molecules in a nozzle beam by a C W HF laser. The described arrangement appears to have at least two major advantages, namely: (a) since the laser photons propagate upstream in the free jet until they meet a molecule in the appropriate internal-energy state, the strongest laser lines from a relatively powerful chemical laser can be utilized fully, and (b) by a judicious selection of the (dimensionless) product of source number density, collision cross-sectional area, and throat diameter, one can achieve extensive rotational cooling with negligible change in vibrational state. They monitored the vibrational excitation of the H F monomers indirectly by observing the dimer-fragment signal H2F+in a mass-spectrometer beam detector. Five different HF laser lines (1P2, 1P4, 1P6, 2P6, and 2P7) were used; independent measwements in a gas cell at 295 K and 382 mbar indicate that, for the 1P2 and 1P4 lines, absorption by the monomers is a t least two orders of magnitude greater than absorption by the polymers. Hence the H2F+-signal depletions observed for the 1P2 laser line for several different beam-source pressures were analyzed by using a simplified model in which (a) the laser beam excites only monomers, (b) collisions between excited monomers and dimers excite vibrational degrees of freedom of the dimers, and (c) the vibrationally excited dimers dissociate before arriving at the detector. The expression for dimer dissociation derived for this simplified model was fitted then to the data for the 1P2 laser line by selecting values for (a) the fraction of the monomers in the free jet excited by the laser beam and (b) the cross-sectional area for monomerdimer V-V energy transfer. However, although this simple model has helped to clarify the fate of the laser energy added to the nozzle beam, it is relatively crude. Crude model features include the following: 1. The excitation of monomers by the laser photons occurs at the nozzle throat, independent of source pressure. 2. The dimer concentration at the location where monomers are excited is the equilibrium concentration at the source conditions, Le., no dimers are formed during the gas expansion.
* Permanent address: Chemical Engineering Department, University of California at Los Angeles, Los Angeles, CA 90024. 0022-3654/84/2088-4513$01.50/0
3. The dimer mole fraction is small in comparison with unity. 4. The frequency with which excited monomers collide with dimers is a function of only the local concentrations, i.e., the product of the collision cross section and relative velocity is independent of temperature in the region of interest. 5. The molecular flow rate through the skimmer is 4% of the flow rate through the nozzle. In the present paper, these several model features are relaxed and/or extended in order to describe selected physical processes more precisely; the application of the model is extended to include the measurements for the 1P4 and 1P6 laser lines as well as those for the 1P2 line. Model and Analysis
With the objective of describing selected physical processes more precisely and extending the range of applicability of the model, consider a model with the following features: 1. The excitation of monomers by the laser photons occurs at an axial location corresponding to a fixed value of the local gas density at the free-jet centerline. (Hence, for a given source temperature, the distance from the source orifice to the monomer-excitation location increases monotonically as the source pressure is increased.) 2. The dimer concentration at the monomer-excitation location is the terminal dimer concentration in a free jet without laser excitation, Le., the dimer concentration contains contributions from equilibrium source conditions and from dimer formation in the free-jet expansion. (The assumption that the dimer formation terminates upstream from the monomer-excitation location will be examined later in a discussion of the results of the analysis.) 3. The dimer mole fraction is not necessarily small in comparison with unity. However, the mole fractions of higher polymers are small in comparison with unity. (Higher polymer mole fractions less than the dimer mole fraction are consistent with recent experimental and analytical results, which results modify some earlier data interpretations, particularly with regard to hexamer mole fractions.) (1) T. Ellenbroek, J. P. Toennies, J. Wanner, and M. Wilde, J . Chern. Phys., 75, 3414 (1981).
0 1984 American Chemical Society
Knuth and Wilde
4514 The Jourrtal of Physical C h e m i s t r y , Vol, 88, No. 20, 1984
where the subscript e refers to conditions immediately downstream from the monomer-excitation location. Now substitute for em. in eq 4 from eq 5 to obtain
+
HF-Laser
CaFp-Lens
=-a(,,)
2dCd Cd[(2Cm*
8 kT ' I 2 dz
- Cd)e + c d l
(6)
U
Integrating from a point immediately downstream from the monomer-excitation location (subscript e) to a point far downstream (subscript a),one obtains
10-5 m bar
(cd - 2cm*)e
- (Cd/2Cm*)e + (Cd/2Cmt)e(Cd-/Cde)
In
-
Nozzle, I D 100Urn
Figure 1. Schematic drawing of the nozzle-beam system used in ref 1 . The nozzle beam is directed upward; the laser beam is directed downward and focused onto the nozzle orifice.
Since the mass-spectrometer signal is related more directly to mole fractions than to mass fractions, convert to mole fractions with the result 1 - (Xd/Xm*)e
+ (Xd/Xm*)e(Ae/A-)(Xdm/Xde)
4. The frequency with which the excited monomers transfer (via appropriate collisions) their excitation energy to the dimers is given by
where u is the collision cross section for this excitation-energy transfer, nm*is the number density of the excited monomers, nd is the number density of the dimers, k is Boltzmann's constant, T i s temperature, and -1 = - 1 + - 1= - 3 p m 2m 2m where m is the mass of the monomer molecule. Since the present analysis focuses on the relative change in terminal dimer miile fraction due to the monomer excitation, the total molecular flow rate through the skimmer is not of interest here. For this model, downstream from the monomer-excitation location, the individual species continuity equations for excited monomers m* and for dimers d may be written dem* pu
pu
dCd
= -mZm*d
Consider now the interesting case in which the variation of the average mass per molecule A can be neglected, so that me i= m 5 mm. Then one may write 1 - (Xd/Xm*)e
In
1 - (xd/&n*)e
-k (Xd/Xm*)e(Xd-/Xde)
-
The right-hand side is a measure of the number of collisions with excited monomers which a dimer molecule would experience if it survived (as a dimer) the expansion from z = z, to z = m. More specifically, the integral is the number of collisions with excited monomers which a dimer molecule would experience if all other molecules were excited monomers whereas x,*, is the fraction of all molecules at z = ze which are excited monomers, so that the right-hand side is the number of collisions with excited monomers which a dimer would experience if xm*remained at xmlefor z > ze. Hence, for convenience write
= -2mZm*d
U
where p is the local mass density, u is the local flow velocity (mass-weighted average velocity), cm*and cd are the mass fractions of species m* and d, and z is the distance from the nozzle. (The use of mass density, mass-weighted average velocity, and mass fraction is preferred for flow systems with chemical reactions.) Substituting for Zm*d in eq 3a from eq 1, one may write
(4) where cmr = mn,./p and c d = 2mnd/p. A relation between em* and c d is obtained if one eliminates Zm*d from eq 3a and 3b. One obtains d(2cm. - cd) =o pu dz which implies 2cml - cd = constant For convenience, evaluate this constant immediately downstream from the monomer-excitation location. Then
If xd.. is the mole fraction of dimers far downstream when the laser is on and Xde is the mole fraction of dimers at the downstream end of the dimer-formation region, hence also the mole fraction far downstream when the laser is off, then one may write I * / I o = Xdm/Xde
(9)
where I* is the mass-spectrometer signal with the laser on and Io is the mass spectrometer signal with the laser off, so that one may write eq 7c in the convenient form
_I* -Io
[1 - (Xd/Xm*)el expbm*eNe[l -
(xd/xm*)el~
-
+ L1 - (Xd/Xmr)el
(10)
It is seen that, according to this model, the ratio of the dimer signals with the laser on and off is a function of (a) (Xd/Xm*)e, the ratio of the dimer mole fraction to excited-monomer mole fraction at z = ze, and (b) xmVeNe, a measure of the number of collisions which a dimer molecule would experience with excited-monomer molecules. The comparison of predictions of eq 10 with measured values requires additional considerations of the parameters xde, x,.,, and Ne. Consider first Xde, the dimer mole fraction. The minimum
The Journal of Physical Chemistry, Vol. 88, No. 20, 1984 4515
HF Excitation and (HF)2 Dissociation information necessary for an independent prediction would be values of the terminal dimer mole fraction in the free jet as a function of stagnation pressure Po for the existing source temperature and nozzle diameter, d. Relevant information is available in the analysis of dimer formation in monatomic free jets by Knuth? the relative ion signals as a function of stagnation pressure for HF free jets given in Figure 3 of Ellenbroek et al.,' and mass-spectrometer measurements for H F free jets by MullerS3 Combining eq 39-41 of ref 2, one obtains
where uA-A and cA-A are respectively the zero-potential radius and the potential well depth for collisions between monomers. Note that the right-hand side indicates a variation with pressure Po (implicit in n ), to the 513 power. Figure 3 of ref 1 indicates that, for stagnation pressures less than about 120 mbar, the m l e 21 (dimer) signal varies as Poto the 813 power whereas the m l e 20 (monomer) signal varies as Poto the first power so that the dimer fraction varies as Po to the 513 power. Hence, although eq 11 was derived for noble gases, one is notivated to apply it also to dimer formation in HF free jets. Dyke et al! report, on the basis of radio-frequency and microwave spectra, an F-F distance of 2.79 8. Ab initio calculations and experimental measurements indicate bonding energies from 5 to 7 kcal mol; a typical value is the 6 kcal/mol given by Smith.5 Keeping in mind the crudeness of using eq 11 for dimers formed from diatomic molecules, we used these values for aA-A and 'A-A. The data of ref 1 were taken for To= 295 K and d = 0.010 cm. Using these values in eq 11, one obtains
where Pois in mbar. Another estimate of the coefficient appearing here can be obtained from the mass-spectrometer measurements of Muller.3 For Po = 950 mbar and the same temperature and nozzle diameter as used in ref 1, Muller recorded mass-spectrometer signals for the several lowest H F polymers. With the species enrichments for the dimers and trimers assumed to be respectively two and three times that for the monomer, and the ionization cross sections for the dimers and trimers assumed to be respectively 1.6 and 2.4 that for the monomer, the measured mole fraction of dimers was estimated to be 0.47. (Cf. ref 6 for an analysis of enrichment factors for molecular beams formed from a free jet dominated by a diatomic species. The ratios of ionization cross sections were estimated from ratios of ionization cross sections for hydrocarbons of various sizes.) These values of Po and xd are consistent with xd/(l - xd)' = 1.8 x 10-5P05/3
(1 2b)
where Pois again in mbar. The agreement of the coefficients in eq 12a and 12b is believed to be fortuitous. In the analysis which follows later in this paper, Xde will be calculated from eq 12a, which is the same as 12b to the first significant digit in the coefficient of the stagnation-pressure term. Consider now xmae, the excited-monomer mole fraction immediately downstream from the excitation location. With our present knowledge of the interaction of laser beams with expanding gases, an a priori quantitative prediction of x,;, does not appear possible. Hence, the value of some suitable parameter which fixes xm., will be determined via a fitting of eq 10 to the available data. It is to be anticipated that the value of this parameter would depend strongly on the laser line and the laser power used but (2) E. L. Knuth, J. Chem. Phys., 66, 3515 (1977). (3) Personal communication from K. Muller, Max-Planck Institut fur Stromungsforschung, Gottingen, 1982. (4) T. R. Dyke, B. J. Howard, and W. Klemperer, J . Chem. Phys., 56, 2442 (1972). ( 5 ) D. F. Smith, Molecular Dynamics, 3, 473 (1959). (6) S. Yoon and E. L.Knuth, "Rarefied Gas Dynamics", R. Campargue, Ed., Commissariat a l'Energie Atomique, Paris, 1979.
could be chosen to minimize dependence on dimer concentration at the excitation location. Hence introduce the parameter f via &*e
= f(1 - xde)
(13)
where f is the fraction of the monomer at z = z, which is excited with a specified laser line at a specified laser power. (Note that, for the present model, 1 - xde is the monomer mole fraction at z = z,.) In the analysis which follows later, the value off (and one other fitting parameter) will be determined for a given laser setting by fitting eq 10 to data taken at several stagnation pressures. Consider finally Ne, the number of collisions with excited monomers which a dimer molecule would experience in traveling from z = z, to z = m if all other molecules were excited monomers. This number depends on both the source conditions and the excitation location. Recall that, in the present model, the excitation location corresponds to a fixed value of the local gas density at the free-jet centerline. However, once again, with our present knowledge, an a priori quantitative prediction of this location (hence a quantitative prediction of Ne) does not appear possible. Hence the form of the dependence of Ne on stagnation pressure is predicted on the basis of known propertiks of free-jet expansions, and a suitable parameter is introduced, the value of which will be determined also via a fitting of eq 10 to the available data. It is to be anticipated that the value of this parameter also would depend on the laser line and the laser power used. Recall the definition of Ne, namely U
The dependence of Ne on stagnation pressure is contained in n and in the dependence of z,, the excitation location, on stagnation pressure. The indicated integration is simplified if the integrand is written as a function of the local Mach number Ma. Then, given Mu as a function of z, the integral can be evaluated numerically. The number density n is related to the Mach number Ma via the isentropic-process thermally perfect gas relation
where y is the specific heat ratio. The flow velocity u is related to Ma via the Mach-number definition
where m is the mean mass per molecule. The cross section u is generally a function of temperature, but this dependence is not known for the energy transfer of interest here. However, information obtained from vibrational relaxations of excited monomers in the presence of dimers indicates that the value of this cross section is relatively large-perhaps of the same order of magnitude as gas-kinetic cross sections. Hence, if we take into account the known general temperature dependence of total collision cross sections at low temperature^,^ the crude approximation
is made. (Fortunately, the temperature dependence of u is almost negligible in comparison with the isentropic-process temperature dependence of n.) Combining eq 8, 14-16, and 2, setting y = 7 1 5 , and approximating m by m, one obtains
Now the dependence of Ne on stagnation pressure is contained in 4and in the dependence of z, on stagnation pressure. The ratio (7) J. 0. Hirschfelder, C. F. Curtiss, and R. B. Bird, "Molecular Theory of Gases and Liquids", Wiley, New York, 1954.
4516
The Journal of Physical Chemistry, Vol. 88, No. 20, 1984
Knuth and Wilde
TABLE I: Values o f Parameters f a n d N,' Selected in Fitting the Present Model t o Measurements laser line
f
1P2
0.14
1P4 1P6
0.15 0.07
1.0
Ne' 50 50
80
z,/d is related to no via eq 14 applied to the excitation location, i.e., via
and the dependence of Ma, on z , / d . A simplified form of eq 17 is obtained for Ma, >> 1, in which case (cf. ref 8) Ma, = 3 . 6 4 ( ~ , / d ) ~ / ~
(19)
---
so that Ne = 0.1 9(n,/no)'7/30noood Numerical evaluations of eq 17 were made from the data of Ashkenas and Sherman (ref 9, Figure 2 ) for z / d < 3 and results of method-of-characteristics calculations for z / d > 3 (ref 10). For 1 < z e / d < 10, the results are fitted better by Ne = 0.20(ne/no)17~30nouod
I
I
I
I I I l l 1
1
I
I I I I I L
-
--
-
-
-
-
(20)
For constant values of To and d , the dependence on Po may be written Ne = N,'(P0/1013)13/30
(21)
where N,' is the value of Ne for Po = 1013 mbar ( 1 atm) and Po is in mbar. The parameter N,' is the second of the two fitting parameters which will be determined for a given laser setting by fitting eq 10 to the data.
Fit of Model to Data Although Ellenbroek et al.' measured the mass-spectrometer m / e 21 signal with the laser on and off for five laser lines (1P2, 1P4, 1P6, 2P6, and 2P7; cf. their Figure 6 ) , they fit their model to the data for only the 1P2 line (cf. their Figure 8 ) . Here, the refined model developed in the present paper is fitted to the data for the 1P2, 1P4, and 1P6 lines. The present model is not applied to the 2P6 and 2P7 lines since, for these lines, none of the laser beam is absorbed by the monomers. Already for the 1P6 lines, absorption by dimers may be significant (cf. Figure 4 of ref 1 ) . The model developed here, namely, eq 10, was fitted to the data for each of the three forementioned laser lines by appropriate choices of the fitting parameters f and N i . The selected values off and N,' are given in Table I. Comparisons of predicted values with measured values are included in Figure 2, where the signal ratio Z,/Zo for m / e 21 ( H F dimers) is plotted as a function of stagnation pressure Po, and in Figure 3, where Z./Zo is plotted as a function of the excited-monomer mole fraction divided by the dimer mole fraction immediately downstream from the excitation location. Discussion As the first step in the examination of the fit of the present model to the data, compare the deduced results with the considered model features. Consider first the relative locations of (a) the monomer-excitation region and (b) the end of the dimer-formation region. One can characterize a given location conveniently by using either a measure of the number of collisions downstream (8) E. L. Knuth, Department of Engineering 64-53, University of -Report . California, Los Angeli, 1964. (9) H. Ashkenas and F. S. Sherman, "Rarefied Gas Dynamics", J. H. deleeuw, Ed., Academic Press, New York, 1966. (10) F. S. Sherman, Technical Report: Fluid Mechanics 6-90-63-61, Lockheed Missiles and Space Co., Sunnyvale, CA, May 23, 1963.
. e
I
0
E
0.1
1 EXCITATION/DItIERIZATION
10
RATIO, ( x , * / x ~ ) ~
Figure 3. Same as Figure 2 except that the dimensionless parameter ( x , . / x ~ ) ~appearing in eq 10 is used for the abscissa.
from this location or the distance from the nozzle to this location. The number of collisionsdownstream from the monomer-excitation location has been characterized by Ne =
lern..( 8 -) kT ; dz -
'I2
= P
= N,'(P0/1013)13/30
(21)
The number of collisions downstream from the nozzle can be characterized similarly by U
(
~ ) l / l a u o d ~ m
The weakest factor in this evaluation of N* is the value of uo,the cross section for transfer of vibrational energy from the excited monomers to the dimers. Considering the integral elastic cross sections for HF-HF collisions calculated by Gianturco et ale," the integral cross sections for HF dimer-monomer cross sections estimated by Muller3 from H-HF, H-Xe, and HF-Xe measurements, and the product of collision cross section and probability for the subject energy transfer obtained by Ellenbroek et al.,' the ~
(1 1) F. H. Gianturco, U. T. Lamanna, and F. Battaglia, Inr. J . Quanrum Chem., 19, 217 (1981).
HF Excitation and (HF), Dissociation
The Journal of Physical Chemistry, Vol. 88, No. 20, 1984 4517
TABLE 11: Relative Locations of Key Processes in the Free Jet line 1P2 1P4 1P6
P,, mbar 173 666 184 660 160 653
Ne 23 42 24 42 36 66
authors are motivated to use K and d = 0.01 cm
go
N* 310 1192 329 1181 286 1169
Nf 283 807 294 800 265 791
10
T 1/3 L =(+,
= 100 A2. Then, for To = 295
00
y =
1
N* = 1.79P0
715
(23)
where Po is in mbar. Finally, a measure of the location of the downstream end of the dimer-formation region is provided by the sudden-freeze model, according to which the terminal dimer concentration (in the case of no dimer destruction due to laser action) is the equilibrium dimer concentration at the sudden-freeze location. Combine eq 12a with the definition of the equilibrium constant
10-1
to obtain
P 2.0 X 10-5P02/3= ,Kp(T)
(25) 10-3
To the extent that the expansion is adiabatic and reversible, one may relate the pressure ratio PIPoto the temperature ratio TITo by using the isentropic relation and write, for y = 7/5 2.0
x
10-5~02/3 = (T/T~)~.~K,(T)
(26)
The dependence of Kp+onT has been studied by several investigator~.'*-'~The relation given by Franck and Meyer,13 namely
0
To - 19.9)
RTO Tf
(28)
where T f is the temperature at the sudden-freeze location. For specified values of Po and To,eq 28 yields implicitly T f / T o a, measure of the location of the downstream end of the dimerformation region. Predictions of eq 21, 23, and 28 for the minimum and maximum values of Poinvestigated by Ellenbroek et al. are given in Table 11. It is found that both the monomerexcitation location and the downstream end of the dimer-formation region are always downstream from the nozzle throat (Le., Ne and Nf C N*) and that the monomer-excitation location is always downstream from the dimer-formation region (Le., Ne < Nf). The conclusion that the monomer-excitation location is always downstream from the dimer-formation region (for the experimental conditions of ref 1) is particularly significant since, in this case, the monomer excitation does not interfere with the dimer formation. Incidentally, in order to position the excitation location at the throat (Ne = P), Po = 2 mbar is required for the 1P2 and 1P4 lines and Po = 4 mbar for the 1P6 line. Subsequent to eq 8, a specific axial location in the free jet has been characterized by the number of collisions which a dimer (12) G. Briegleb and W. Strohmeier, 2. Elektrochem., 57, 668 (1953). (13) E. W. Franck and F. Meyer, Z . Elektrochem., 63, 571 (1959). (14) G. T. Armstrong and R. S. Jessup, J . Res. Natl. Bur. Stand. U.S.A., Sect. A , 64, 49 (1960).
6
8
10
12
Figure 4. The number of collisions N experienced by a dimer molecule downstream from a given location, normalized by the number of collisions N* experienced downstream from the nozzle location, as a function of dimensionless axial distance z / d . TABLE 111: HF-Dimer Mole Fractions in the Source and at the End of Dimer-Formation Regiona
Po,mbar
where K p ( T ) has units mbar-' and RT has units cal/mol, is preferred since its temperature dependence agrees best with recent predictions and measurements of the HF-dimer bond energy. Combining eq 26 and 27, one may write
4
2
A X I A L DISTANCE, x l d
K p ( T ) = exp( RT 6700 - 19.9)
2.0 x 10-5po2/3 = (;)"exp( -
-2
xdo
Xde
160 666
a
0.032 0.080 0.11 0.38 Source conditions: T o = 295 K , d = 0.010 cm.
molecule would experience if all other molecules were excited monomers and if it survived (as a dimer) the expansion from this specific axial location to z = m. One could use alternatively the distance z from the nozzle throat. Note that N / N * is related to the dimensionless distance z / d by
The resulting dependence of N / N * on z l d is shown in Figure 4. It is seen that most of the collisions occur close to the nozzle; e.g., 91% of the collisions have occurred already at z / d = 1. In the present model, the dimer concentration at the monomer-excitation location contains contributions from both the equilibrium source conditions and the dimer formation in the free jet. The relative sizes of these two contributions are apparent from the comparison of the predictions of eq 12a with those of 24 and 27 given in Table 111. It is seen that approximately 213 of the dimers at the downstream end of the dimer-formation region were formed during the expansion. Note also that, for the highest pressures, the dimer mole fraction is not small in comparison with unity. The terminal values of the excited-monomer mole fraction, xm*-, predicted by the present analysis are of interest for two reasons: (a) For a viable model, the predicted terminal values must not exceed physically reasonable upper limits, and (b) a potential user
4518
The Journal of Physical Chemistry, Vol. 88, No. 20, 1984
TABLE IV: Terminal Values of the Excited-Monomer Mole Fraction Predicted for Experimental Conditions of Ref 1 Po, mbar
150 200 300 400 500
1P2 Line 0.068 0.039
1P4 Line 0.075 0.045 0.008
0.007
0.001 0.000
0.001
0.000
1P6 Line 0.016 0.005 0.000 0.000 0.000
of such a laser-excited molecular beam might be interested in those operating conditions which maximize xmern.Apply eq 5 to the state far downstream (subscript a),convert to mole fractions and eliminate Xd../Xde and x,,, by using eq 9 and 13 to obtain
(
xm*..= f - f+-
"I, e.)"
Knuth and Wilde TABLE V: Extent of Excitation Region Compared with Extents of Related Regions line Po, mbar ANe Ne Nf-Ne 1P4 184 29 24 260 660 3 42 758 1P6 160 98 36 229 653 8 66 725 An estimate of the axial dimension of the excitation region, and a comparison of this dimension with other axial dimensions, is of interest. In order to make this comparison as meaningful and clear as possible, the excitation-region axial dimension will be characterized by the number of collisions a dimer molecule would experience while passing through this region. For the case in which the mass fraction of excited monomers is fixed (at the physically maximum possible fraction), eq 6 may be written in the convenient form
*-,.(
-)
Pa 8 k T ' I 2 dz dCd _ Typical values of x , . ~ predicted with the present model for the U cd m * P experimental conditions of ref 1 are given in Table IV. (Keep in mind that the denominator in x,* is based on the sum of all To the extent that the average molecular weight can be approxmonomers and dimers, not on only the monomers in a given imated by the monomer molecular weight, one may write internal-energy state.) It is seen that all of these mole fractions are smaller than the fractional populations (0.27,0.12, and 0.019) dXd/Xd = -X,r d N (32) of the 1P2, 1P4, and 1P6 states in H F a t 300 K (cf. Table I of where ref 1). Equation 30 and Table IV indicate that experimental conditions yielding smaller values of dimer mole fraction, Xde, yield larger values of terminal excited-monomer mole fraction; holding (33) other conditions constant, a reduction in dimer mole fraction leads to a reduction in excited-monomer deexcitation via collisions with Since x , ~is constant in this case, one may integrate eq 32 to obtain dimers. The values of the fractional monomer excitation, f, and the (Xde - AXde)/Xde = -Xm*'ANe (34) excited-monomer mole fraction immediately downstream from where Xde is the dimer mole fraction at the end of the dimerthe excitation region, xmee,predicted by the present analysis are formation region (hence also the dimer mole fraction at the benot to be taken too literally. According to this model, the axial ginning of the monomer-excitation region), Xde is the dimer dimension of the excitation region is sufficiently small that esmole fraction at the end of the excitation region, x,*' is the sentially all monomer-dimer V-V energy transfer occurs down(constant) excited monomer mole fraction in the excitation region, stream from this region. Physically, if the laser intensity is so and meis the number of collisions with excited monomers which high relative to the flux of molecules in the appropriate interwould be experienced by a dimer molecule while passing through nal-energy state that the excitation of molecules in this state is the excitation region if all other molecules were excited monomers. saturated at a given location, the remainder of the laser beam (Note that x,*'AN, is the actual number of collisions with excited continues upstream, thereby extending the excitation region. For monomers which would be experienced by a dimer molecule while sufficiently high laser intensities, this excitation region is so broad passing through this region.) Since each laser photon entering that some monomers are excited (by laser photons), deexcited (by this region ultimately leads to the dissociation of a dimer, the collisions with dimers), and excited a second time (by laser reduction in the dimer mole fraction in this region equals the photons) before leaving this region. In the present model, such difference between the calculated values of x,,,.~and its physically repeated excitations are reflected in unrealistically high values maximum possible value, i.e. of xmre For the experimental conditions of ref 1, eq 13 predicts AXde = Xmee - X,t' values of x,., from 0.09 to 0.13 for the 1P2 line, from 0.09 to (35) 0.14 for the 1P4 line, and from 0.04 to 0.06 for the 1P6 lines. The so that values for the 1P4 and 1P6 lines are considered to be unrealistically large in comparison with the fractional populations given in the preceding paragraph and are interpreted as indicating repeated excitations of monomers by a laser intensity high in comparison where x,*' is of the order of magnitude of one-half the fractional with fluxes of molecules in these states in the beam. The values for the 1P2 line are from 2.4 to 3.4 times as large as the value population of the associated internal-energy state in the source. Hence xmee- x,,' ranges from 0.03 to 0.08 for the 1P4 line and 0.038 determined by Ellenbroek et al.' using the simplified analysis. from 0.03 to 0.05 for the 1P6 line. The corresponding values of Several significant differences between the earlier simplified AN, are compared in Table V with the values of Ne (a measure analysis' and the present extended analysis have been noted; they of the extent of the region downstream from the excitation region) are summarized briefly at this point before continuing this disand the values of Nf - Ne (a measure of the extent of the region cussion. In the earlier model, the monomer excitation was taken between the downstream end of the dimer-formation region and to occur always at the throat; in the present model, it is found the downstream end of the excitation region). It is seen that, for to occur always significantly downstream from the throat (cf. the laser power used, the excitation region never overlaps the Table 11). In the earlier model, the dimer concentration was taken dimer-formation region (Le,, ANe < Nf - Ne),that the thickness to be the equilibrium concentration in the source; in the present of the excitation region is relatively small for the larger values model, it is found that approximately 2/3 of the dimers are formed of Po, but that this thickness is relatively large for the 1P6 line in the free jet (cf. Table 111). In the earlier model, the dimer mole if Po is small. fraction was taken to be small in comparison with unity; in the The population of internal-energy states of the unexcited mopresent model, it is found to have values up to 0.38 (cf. Table 111). Largely as a result of these model differences, the values of x ~ * ~nomers in the excitation region is somewhere between that at temperature Toand that at the temperature Te at the downstream deduced with the present model are about three times as large end of the excitation region. Tois known from the experimental as those deduced earlier.
The Journal of Physical Chemistry, Vol. 88, No. 20, 1984 4519
H F Excitation and (HF)2 Dissociation conditions, and for the measurements of ref 1 was 295 K. T, may be estimated by approximating the expansion from To to T, by an isentropic process. Then, substituting for n,/no from eq 20 and for Ne from eq 21, one obtains (for y = 7/5, To = 295 K, and d = 0.010 cm) T e / T o = ( n e / n o ) 2 / 5= 0.20 N,”2111P0-2/5
I
I
I
I
I
I
(37)
where Po is in mbar. Hence, for the ranges of Po investigated in ref 1, T, ranges from 69 to 119 K for the 1P2 line, from 70 to 116 K for the 1P4 line, and from 97 to 171 K for the 1P6 line. If the internal-energy-state populations are closer to those for these estimated values of T , than to that of To,then the extents of the excitation region for the 1P4 and 1P6 lines (but not for the 1P2 line) are greater than estimated in the preceding paragraph. The temperature decrease due to expansion considered in the preceding paragraph is offset partly by the temperature increase due to dimer formation. Consider the changes in dimer mole fraction indicated in Table 111. If all the heat released due to dimer formation went into heating of the gas (and none into increasing the directed kinetic energy, Le., into increasing the hydrodynamic velocity), then (with the approximation m = m) one may relate the temperature increase to the dimer-mole fraction increase by
AH
AT = - X(:de
- Xdo)
(38)
CP
6700 cal 2 mol K AT=mol 7( 1.986 cal)(xde -
(39)
For the changes in dimer mole fraction indicated in Table 111, one finds that AT ranges from 47 to 266 K. How much of this heat goes into random vs. directed kinetic energy depends on the location at which this heat is added; the earlier in the expansion this heat is added, the larger the fraction going into directed kinetic energy. Recall that, in eq 10, the product x,&’, appeared as a measure of the number of collisions which a dimer molecule would experience with excited-monomer molecules. The product may be written in the convenient form
&,,*,Ne= f(1 - X,jc)N~(Po/1013)o’43
(40)
For the experimental conditions of ref 1, its value ranges from 3.0 to 3.6 for the 1P2 line, from 3.2 to 3.9 for the 1P4 line, and from 2.3 to 2.9 for the 1P6 line. Note that this product is the number of energy-transferring collisions which a dimer molecule would experience with excited-monomer molecules if the mole fraction of excited monomers were constant at x,., for z > z,; since x,. decreases for z > z,, the number of collisions is less than yield this product. If ( X , * / X d ) , 5 1, then smaller values of larger values of Z./Zo (cf. Figure 3). As indicated in Figure 3 and by eq 10, (X,*/?& = 1 divides the dependence of Z*/Io on operating conditions into two regimes. dominates over For (x,./x~)~ < 1, the dependence on (&*/&)e the dependence on xmlcNewith For ( X , * / X d ) e with
%
1
- (X,t/Xd),
for
(Xm*/Xd),
< 0.5
(41)
> 1, the dependence on xmleNeis more important I./& = e-xm,Jve for
(X,*/Xd),
>> 1
(42)
At (x,*/x~)~= 1 , eq 10 becomes I*/Io = 1/(1
+ x,*JV,)
for
(&t/Xd)e
=1
(43)
Since xmeeis related to f by eq 13, the condition ( x , * / x ~ ) ~= 1 occurs at xde/(l
4w
0
I 0
I
I
I
I 5
I
I
I
I
I 10
EXCITATION LOCATION, m
where -AH is the heat released per mole of dimer formed and Cp is an average heat capacity. Using the value of AH given by Franck and MeyerI3 and the approximation zp = (7/2)R, one obtains
z*/zo
I
mn
d = 0.14
-Xde)
=f
(44)
Figure 5. Percent attenuation of SF6SF5+(dimer) signal for SF, nozzle beam intersected by a transverse IR CO, laser beam as a function of the location of the intersection point relative to the nozzle. Data from ref 15.
For the experimental conditions of ref 1, eq 44 is satisfied for Po = 220,230, and 138 mbar respectively for the 1P2, 1P4, and 1P6 lines. Hence, most of the measurements of ref 1 were made in the regime corresponding to ( X , r / X d ) e C 1 (cf. Figure 3), i.e., under conditions such that sufficient dimers were present to relax essentially all excited monomers back to the ground vibrational state (cf. Table IV). Measurements somewhat similar to those reported in ref 1 and analyzed here have been made by Rechsteiner et al.I5 They used a perpendicular arrangement for vibrational excitation of SF6 molecules in a nozzle beam by an IR CO, laser. For the conditions of their measurements, they observed polymers containing as many as 100 SF6molecules. The measured attenuation of the SF6SF5+ (dimer) signal as a function of the distance from the nozzle to the laser beam is shown in Figure 5. The experience gained with the model used in the present paper suggests a model in which the attenuation of the SF6SF5’ signal is due to the dissociation of dimers following collisions with excited monomers downstream from the excitation location; as the laser-intersection point is moved downstream, the number of collisions available for dimer dissociation decreases and the percent attenuation decreases. For a graphical comparison of the predictions of the present model with the measurements of Rechsteiner, the predicted laser-induced decrease in the H F dimer signal is plotted in Figure 6 as a function of the distance from the nozzle to the excitation region. [The distance z,/d was obtained by calculating the ratio Ne/N* for a given source pressure with eq 21 and 23, and then reading z,/d from Figure 4. The decrease in dimer signal was calculated for the same pressure with eq lo]. The observed similarities in the curves of Figures 5 and 6 are taken as supporting the suggestion of dimer dissociation by collisions with excited monomers downstream from the monomer-excitation location. Finally, of considerable interest to potential users of laser-excited nozzle beams is the flux of excited monomers immediately downstream from the excitation region: (a) A comparison of this flux with the laser power provides a measure of the efficiency with which the laser beam is used. (b) The magnitude of this flux is an upper limit to the useable excited-monomer flux which could be achieved if deexcitation via collisions could be eliminated. This (15) R. Rechsteiner, R. Monot, L. Waste, J. M. Zellweger, and H. van den Bergh, Ber. Friihjahrstagung Schweizerischen Phys. Gesell., 54,282 (1981).
4520
Knuth and Wilde
The Journal of Physical Chemistry, Vol. 88, No. 20, 1984 I
I
I
I
f = 0.14 N; = 50 To = 295 K d = 0.01 cm
Y
1 .n
I
I
2.0
EXCITATION LOCATION, z,/d
Figure 6. Attenuation of m / e 21 (dimer) signal for HF nozzle beam excited by an antiparallel coaxial CW HF laser beam as a function of the location of the excitation region. Predicted from the model of the present paper.
SOURCE PRESSURE,
Po
(mbar)
Figure 7. Mole fractions of HF dimers and excited monomers at excitation region and far downstream as functions of source pressure. Predicted from eq 10, 12a, 13, and 30.
flux F, of excited monomers immediately downstream from the excitation region is given by ne
F, = n,u,x,., = neu&l - Xde) = -nouJ(l no
- xde) (45)
where no/ne is given by eq 18. If the expansion is adiabatic and Ma, 5 3, then uc = (2E,T0)1/2
(46)
Hence, for y = 715 and To= 295 K
nou, = 2.25
X 1Oz1P0cm-2 s-'
(47)
where Po is in mbar. The ratio no/neis evaluated by calculating (for a given value of Po) the ratio Ne/N* with eq 21 and 22, then reading z,/d from Figure 4, then extracting Ma, by using ref 9 and 10, and finally calculating no/ne with eq 18. Values off are given in Table I, values of Xde are calculated with eq 12a. One finds that F, ranges from 3 X lo2' to 4 X 1O2I cm-2 s-l for the experimental conditions of ref 1. The focusing of the laser was such that its diameter at the excitation region is believed to be of the order of 0.02 cm. For this diameter, a typical value of Fe (namely, 3.8 X lo2' cm-2 s-' for the 1P2 line with Po= 173 mbar) represents a utilized laser power of 0.092 W, which represents 29% of the laser power (0.32 W). The 61% difference between 0.092 and 0.32 W is attributed to some combination of losses in delivery of the laser power and uncertainties in the calculation of the utilized power. For the potential user, a flux per unit solid angle is perhaps most easy to assess; for the forementioned typical case, 3.8 X lo2' cm-2 s-l corresponds to 5 X 1017 sr-' s-'. Potential users of laser-excited nozzle beams sometimes are interested in maximizing the absolute value of the terminal excited-monomer flux in the presence of dimers. The pressure dependence of the flux per unit solid angle is contained in PG,.,. Substituting from eq 30 for xmSa one obtains
For the present purpose, one may neglect f and Z./Io in comparison with unity and write
To the same approximation, eq 12a may be written
0
200
400 SOURCE PRESSURE,
600
800
inno
Po (mbar)
Figure 8. Fluxes of HF dimers and excited monomers at excitation region and far downstream as functions of source pressure. Conditions same as for Figure 7.
Setting the differential of Pflrnern with respect to Po equal to zero, one finds that the flux is maximized if Xde
%
%f
(51)
Le., if the dimer mole fraction at the end of the dimer-formation region is about 3/8 the fractional excitation of the monomers at the excitation location. For the experimental conditions of ref 1, this condition is met most closely by the measurements at Po = 173 mbar for the 1P2 line, for which Xde = 0.64f. Using the value of xmr, given by eq 30, one calculates a flux of 2 X 1017 to 3/af(by reducing Po appropriately) would s i 1s-'. Reducing yield a higher flux. Reducing Xde below 3/af(by reducing Poto a still lower value) would reduce the deexcitation of excited monomers via collisions with dimers, but would also reduce the absolute value of the excited-monomer flux. The variations of the several mole fractions and fluxes with stagnation pressure are shown in Figures 7 and 8. The flux of excited monomers (per steradian) increases as source pressure increases even though the laser power is fixed due to the fact that the distance to the excitation region increases (and the subtended angle decreases) as the source pressure increases. Consistent with eq 51, the excited-monomer flux Fm*-peaks (Figure 8) at the value
H F Excitation and (HF), Dissociation of Po for which Xde = 3f/8 = 0.05 (Figure 7). Figures 7 and 8 emphasize the important role of dimers in the deexcitation of the excited monomers. The possibility of reducing Xde by either increasing the source temperature or decreasing the nozzle diameter [cf. eq l l ] provides incentive for further development of this source.
Addendum A reviewer of the manuscript suggested that a more detailed critical comparison be made between the model used in ref 1 and the model used in the present paper. Hence, additional comments are provided here, keeping in mind that the simplified analysis of ref 1 represents a pioneering effort, that the present analysis stands on the shoulders of that first analysis, and that it is frequently easier to improve on an existing analysis than to create the pioneering analysis. The advantage of the present paper is not that it reduces significantly the differences between model predictions and measured values; since each model contains two fitting parameters, reasonably satisfactory fits can be obtained from either model. The advantage of the present paper is rather that several model features are in significantly better agreement with known properties of molecular collisions, free jets, and dimer formations in free jets, so that the level of confidence in conclusions drawn from this model is expected to be higher. The most important differences in these two models include the following: 1. In the model of ref 1, the only dimers present in the free jet are those already present in the stagnation chamber. In the present model, additional dimers are formed in the free jet. At a source pressure of 666 mbar, the present model predicts a dimer mole fraction at the dimer-formation freezing point 3.5 times that in the stagnation chamber. 2. In the model of ref 1, the laser excitation of monomers occurs at the source orifice. In the present model, this excitation occurs a t an axial location corresponding to a fixed value of the local gas density. At a source pressure of 666 mbar, the present model predicts that the number of collisions available for monomer deexcitation is only 0.035 that available if excitation occurs at the source orifice. 3. In the model of ref 1, the product of collision cross section and V-V transfer probability is 480 Az.(The stated cross section is 400 A,, an inadvertant error of a factor of 4 was made in the definition of the cross section, and a V-V transfer probability of 0.3 was selected.) In the present model, a V-V transfer cross
The Journal of Physical Chemistry, Vol. 88, No. 20, 1984 4521 section of 100(T0/T)1/3 AZis used. At the source orifice, the ratio of these two cross sections is 0.22. 4. In the model of ref 1, the number density in the free jet varies according to
n (-4 n*
-
0.04 (6)l/2(i 0 . 6 ~ / d ) ~
+
(Cf. eq 4-7 of ref 1 with u = u-.) In the present model, the local number density is based on the local Mach number obtained from the data of ref 9 for z / d < 3 and from the calculations of ref 10 for z / d > 3 . It is seen that the number density used in ref 1 is too small by a factor of 0.016 at the source orifice ( z / d = 0). For large values of z / d , the above approximation predicts number densities 9.8 times as large as given by the method of characteristics (ref 10) and confirmed by measurements. 5. In the model of ref 1, the approximation
(LT)”’
t
(E5)I’Z
$ = ($
is made. In the present model, the variation of T with axial distance is taken into account. At the source orifice, the exact function is 2.4 times as large as the approximate function. It is seen that the difference in the two models are indeed quantitatively significant. A pragmatic consequence of these differences is in the values off, the fraction of the monomer excited by the laser, deduced from these two models. Using the first model, one finds f = 0.038 for the 1P2 line; using the present model, one finds f = 0.14.Since the fractional excitation which can be obtained is an important factor in such applications of lasers, the relatively large ratio (3.7) of these deduced values of f is believed to justify this second-generation model and analysis. Acknowledgment. E.L.K. received support from the Alexander von Humboldt Foundation while a guest of the Max-Planck Institut fur Stromungforschung, Gottingen, W. Germany, and on sabbatical leave from the University of California at Los Angeles. Stimulating discussions with Professor J. P. Toennies, J. Wanner, and K. Muller contributed to the research described here. Mrs. P. Gilbert, Chemical Engineering Department, UCLA converted bits and pieces of handwritten material into a coherent typed manuscript. Registry No. Hydrogen fluoride, 7664-39-3; hydrogen fluoride dimer, 30664-12-1.
