Rotational relaxation time in free jets of He + N2 mixtures

21 July 1995

CHEMICAL PHYSICS LETTERS

ELSEVIER

Chemical Physics Letters 241 (1995) 209-214

Rotational relaxation time in free jets of He

+ N2

mixtures

Andrey E. Belikov *, Ravel G. Sharafutdinov Institute of Thermophysics, Novosibirsk 630090, Russian Federation Received 28 March 1995

Abstract

Non-equilibrium rotational energy distributions of N 2 in free jets of He + N2( < 5%) were measured by the electron beam technique. The rotational temperature was derived from the spectrum of the first negative system of N f induced by high energy (E ~ 10 keV) electrons. The experimental data were compared with kinetic calculations of rotational energy in free jets based on the linear relaxation equation: d ER/d t = - ( E R - E t ) / ~ . The rotational relaxation times I- were derived from available transport collision integrals, evaluated with three different potential energy surfaces: HFD1, B'IT and M3SV. The HFD1 surface leads to an overestimated rate of rotational relaxation. Two other surfaces agree satisfactorily with experiment, similarly as observed by Dickinson et al. for transport coefficients. To decide between BT1~ and M3SV, more precise measurements should be carried out over a wide temperature range.

1. Introduction

The data for rotational relaxation rate coefficients in mixtures of N 2 + He are presently severely limited. The rotational relaxation time ( r 2) in nitrogenhelium collisions has been measured in a sound absorption experiment [1] at a temperature of 304 K, [1 69+°-91~ (nT)2 = ~l. -0.551 (101° c m -3 s ) ,

which corresponds to the rotational energy relaxation cross section, O ' 2 = ( n T 2 ( V ) ) -1 = ( 4 . 4 1 _ +12. 5. 140) (ilk2).

Here n is the number density of He, ( v ) = (8kBT/~rlx) 1/2, Ix is the reduced mass. Comparison with the results of classical theories [2,3] has been inadequate: 0-2 = 0.60 ,~2 [2]; 15.6 ~2

* Corresponding author. Elsevier Science B.V. SSD1 0 0 0 9 - 2 6 1 4 ( 9 5 ) 0 0 6 1 7 - 6

[3]. However, classical trajectory (CT) calculations [4], using an interatomic potential of Morse form, are in reasonably good agreement with data derived from acoustical absorption measurements [1,4]: 0"2 = 6.84 .~2 at T = 308 K. In more recent works the temperature dependences of 0"2(T) were calculated. CT calculations of transport collision integrals from 77 to 1100 K were performed [5,6] using three potential energy surfaces: HFD1 [7], B T r [8], M3SV [9] (we use here the authors' abbreviations). Close-coupling (CC) calculations of the collision integrals for the intermolecular potential HFD1 and comparison with approximate methods were carried out in Ref. [10] for the same region of temperature. From comparisons with measurements of diffusion, viscosity, thermal conductivity, rotational relaxation, depolarized Rayleigh light scattering and magnetic field effects on diffusion, viscosity and thermal conductivity it follows [6] that none of the current potential energy surfaces is yet

210

A.E. Belikov, R.G. Sharafutdinov / Chemical Physics Letters 241 (1995) 209-214

ture of E R from E t was controlled by a variation of the stagnation parameters and distance from the gas source. The measured data of E R were compared with the results of kinetic calculations in free jets using Eq. (1) and theoretical values of (n'r 2) = f ( T ) [6,10] for three potential energy surfaces.

o< ~

2

..

...

x x

o

~10 I o~ o r..)

2. E x p e r i m e n t a l 2 ....

i

10

~ ' ' ~ ..... 10 2 ~ ' ' " 1 0 3

TEMPERATURE, K Fig. 1. Integral effective cross-sections of the rotational energy relaxation of N2 in He. (~) Experiment [1]; (O) calculated [5,10] with HFD1 [7]; ([2) calculated [6] with BTI" [8]; (zx) calculated [6] with M3SV [9]. Best fits by power law and extrapolation are shown by solid and dashed lines. able to fit all the available transport and relaxation data. However, BTT and M3SV potentials are closer to the measurements than HFD1, and BTF is closer than M3SV. At the same time the M3SV surface yields cross sections in satisfactory agreement [11] with crossed-beam inelastic differential cross-section measurements [12]. All the potentials (HFD1, B T r , M3SV) give results consistent with the only available measurement for the rotational relaxation cross section [1], though differing substantially among themselves. Both the theoretical and experimental rotational relaxation cross sections from Refs. [1,5,6,10] are presented in Fig. 1. In this work the rotational energy relaxation of N 2 molecules in He are reported for the temperature region T < 300 K. The experimental method used is similar to one that was applied to rotational relaxation studies on the N2-Ar and N2-N 2 systems [13]. Non-equilibrium rotational energy distributions have been produced and measured in free jets of He + N 2 mixtures (a mole fraction of N 2 < 5%). An analysis of the experimental data has been performed based on the relaxation equation, dER/dt

= - (E R- Et)/r,

(1)

where E R and E t are the non-equilibrium and equilibrium values of the rotational energy. The depar-

The experimental apparatus, methods and techniques have been described in detail previously [13], and only the essential features are listed here. Conic sonic nozzles of diameters d = 2.1 and 6.3 mm served as the source for the supersonic gas jet. The thickness of the nozzle edge was much less than the diameter of the orifices. The background gas pressure was less then 5 × 10 -3 Torr under all stagnation pressures from 3 to 150 Torr. Because of this, the background gas had no effect on the results of the measurements at the axis of the jet. The electron beam (with diameter of = 2 mm), optical axis of monochromator and jet axis were at right-angles to each other. The distance nozzle-beam X was measured accurate to 0.1 mm. The sizes of the detected region were about 0.2 mm along the jet and 1.5 mm along the electron beam axis. The reversal dispersion of the monochromator and instrumental function were 6 A / m m and 0.5 .~. The nozzle was held at a constant temperature (TO= 293 K), controlled by a thermocouple.

3. C a l c u l a t i o n s o f

ER

For the rotational energy relaxation of N 2 in the He + N 2 mixture on the axis of the free jet Eq. (1) may be rearranged to give u

dE R

( n 1 + n2)

dx

ER

-

-

Et

(n'r)rai x '

(2)

where E t = kBTt; u, I t , n 1, n 2 are the equilibrium values of flow rate, temperature and number densities of N 2 and He at distance X from the nozzle; x = X/d;

(n'r)milx = o t ( n ' r ) l 1W (1 - o t ) ( n ' r ) 2 1

(3)

A.E. Belikov, R.G. Sharafutdinov / Chemical Physics Letters 241 (1995) 209-214

a is the mole fraction of N2; indexes 1 and 2 refer to the relaxation in pure N: and He. It is suggested that the mixture species have the same gas-dynamic parameters, which correspond to isoentropic calculations with the ratio of heat capacities: ")/mix

24+5 7 2 4 + 3 ' Yr% = 5 ' YHe

5 3

(4)

This assumption is justified [14] for used Knudsen numbers in source and distances from the nozzle. The values of (n-r) 1 for the temperature range T < 300 K were taken from Ref. [15],

(nr)l = 1.48 × lOl°(T/90) °'68-°'2° log(T/90) (cm -3 s).

(5)

Because the collision integrals were presented in Refs. [5,6,10] for only T > 70 K, an extrapolation in the low-temperature region was necessary to compare with experimental data. Approximations of the data by the power law o ' = A T s and extrapolations are shown in Fig. 1 by solid and dashed lines, respectively. The fitting coefficients for the different potentials are HFDI: A = 84.8, B = - 0.46, BTT: A = 38.9, B = - 0 . 3 9 , M3SV: A = 73.8, B = - 0 . 5 6 , where T is in K and o- in ~2. It should be pointed out that since the values of r 1 and T2 are of the same order [1,13] Eqs. (2) and (3) can be simplified for mixtures with a low mole fraction of N 2 .

4. M e a s u r e m e n t s o f E R

The rotational level populations of the N 2 molecules were derived from the rotational line intensities of the first negative system (FNS) of nitrogen (N~- B 2 ~ - , v ' = 0 , j ' ~ N ~ - X 2 ~ - , v " = 0, j") excited by high energy ( E -- 10 keV) electrons (N 2 X 1 •g + , v, j ~ N~- B zXff, v' = 0, j') [16]. The line intensities of the R branch are

I/otNj, j ' / ( 2 j ' + 1).

(6)

The relative level populations Nj and Ny of the

211

states N2(X) and N~'(B) are related by the rotational transition probabilities ~j, [17],

Nj, = EPjj, Nj, J

E N j = E N j , = 1. J j'

(7)

To measure the rotational energy distribution function by this method the excitation of the radiating state is bound to be produced by direct electron impact. In pure nitrogen and in a mixture of N e + Ar it was shown [13] that this is indeed so for these gases. However, in a mixture of He + N2(< 5%) other competitive ways of excitation come into existence. Overpopulation of the upper rotational levels of the N~(B) state and increasing its decay time in He + N 2 electron-beam plasma were reported in Ref. [18]. Penning ionization, He(2 1S, 2 3S) + N2(X ) ~ N ~ ( B ) ,

(8)

should be excluded, since in reaction (8) the essential rotational heating was not observed [19-21]. In addition, Penning ionization (8) produces vibrational level populations in agreement with Franck-Condon factors [19-21], whereas under the operating conditions of our experiments a strong overpopulation of the upper (v' > 2) vibrational levels was evident. A strong rotational heating was observed on ionization of N 2 in charge transfer reactions with atomic and molecular helium ions [20]. The rates of these reactions are substantially higher than the rate of Penning ionization [22], and at a pressure of ~ 1 Torr the relative contribution of atomic He + ions to the excitation of N~-(B, v ' = 0) by charge transfer reactions is only ~ 3% [20]. Be that as it may, the implementation of the electron beam technique to measure the rotational level populations in He + N2(< 5%) mixtures calls for taking into account or eliminating the additional (in relation to electron impact) ways of producing N~-(B). Since it is beyond the scope of this Letter to review the excitation processes, we tried to obtain the results under conditions which were free from undesirable effects. The rotational line intensities in the spectrum of FNS in the He + N 2 mixture clearly show (Fig. 2) a bimodal character of rotational energy distribution in N~-(B 2£ u, v ' = 0). The populations of the lower rotational levels correspond to a temperature (T L) close to the equilibrium temperature in this flow

212

A.E. Belikov, R.G. Sharafutdinov/ Chemical Physics Letters 241 (1995) 209-214

region. The populations of the upper levels follow a Boltzmann distribution as well, but with the considerably higher temperature of T n ~ 580 K, which depends only slightly on the conditions. Furthermore, it turns out that each relative population of the upper rotational levels depends only on the local number density of He (when n 2 >> n 1) independently of the distance from the gas source, nozzle diameter and stagnation pressure. This means that the excitation of the upper levels is predominantly due to some unified process through excited particles of helium (He + and He~- probably). On the contrary, the lower level populations behave in agreement with the concept of rotational relaxation. It is known [23], that the product of stagnation pressure and nozzle diameter (po d) is a parameter of rotational relaxation when the nozzle temperature is kept constant. In Fig. 3 the intensities of the lower rotational lines are plotted versus P0 d. The measurements were performed at a distance x / d = 6.3 for two nozzle diameters d = 2.1 and 6.3 mm. The good agreement between these data indicates that the additional excitation of these levels by excited particles of He is negligible in relation to direct excitation by electrons. Thus we can use these

/

0

i~o [] 0 /x 0

O [3

OOOOO13.0 ooooo 7.5 a~sa 3.4

O

t,

,_I -2

28 28 30

O []

zx

[]

O O

A

O D

A

d=2.1 mrn 1

0

j=3 -1.0

•,-~

j=5 ~"~--~-o- ....... o

-2.0

j=9

~4 -a.O

O

x/d=&3 -4.0 0

~ - ~ r ~ " 50

pod,

o~o d=2.1 ====:~ d=6.3 ~ r ~ " 100 150 200

Torr

mm

Fig. 3. Relative intensities of the rotational lines of FNS N~ (R branch, (00) band) as a function of po d, /sum= Y'-lj. Data are connected by lines for the sake of clarity.

data to measure the rotational level populations in the ground state of N 2. Because only a few lower levels of N2(X) are accessible to analysis or because it is the distinctive property of rotational relaxation in He + N 2, disruption of the Boltzmann distribution of the rotational level populations is not observed. Thus, the population temperature of these levels will be identified with the rotational energy expressed in K.

5. Results

545 572 580

O

[]

•

o ..... !::1

n m [ 1 0 t n e m -3] TL,K TH,K

O

-1

0.0

100

..... / ............................. 200 300 400

500

j'(j'+l) Fig. 2. Distribution of rotational line intensities of FNS N~ (R branch, (00) band) in electron beam excited mixtures of He + N2. j' is a rotational quantum number of N~ (B). TL and TH are the population temperatures of the lower and upper levels. The local number density of N2 is 4.2X 1015 cm-3. The equilibrium temperature is 26 K.

and discussion

The non-equilibrium rotational energy of N 2 molecules in free jets of He + N 2 is shown in Fig. 4 as a function of relaxation parameter (p0 d). Experimental data (points) were obtained at distances from the nozzle of x / d = 3.15 and 6.30 with two nozzle diameters d = 2.1 and 6.3 mm. The rotational energy was derived from the temperature of the lower level populations of the ground molecule state. Rotational level populations in N2(X 1•;, U-~-0) were obtained from line intensities in the R branch, (00) band of FNS N~- using Eqs. (6) and (7). Good agreement between data, measured with the different nozzle diameters, justified the experimental technique. The results of kinetic calculations with theoretical relaxation rate coefficients are represented by lines

A.E. Belikov, R.G. Sharafutdinov / Chemical Physics Letters 241 (1995) 209-214

55

,

45

al

25

E-

15

,

,

,

,

, , i

°X

35 1:~

,

,

and does not give inelastic differential cross sections and transport coefficients in good agreement with experiment. It can be seen from Fig. 4 that the efficiency of the rotational relaxation is overestimated as well.

,

x/d=3.15

"',, "'"". . . .

213

0

Tt=I

Acknowledgement

40

,

zo

This work was supported by the Russian Fundamental Science Foundation (Grant No. 93-02-14461). The authors thank Dr. M.L. Strekalov and Dr. A.V. Storozhev for useful discussions.

x/d=6.3 30 20 10 0

10

""

\

Tt I

~

References ' ' ~ .....

~

6

100 pod, T o r r m m

Fig. 4. The rotational temperature as a function of po d. Experiments: (&) d = 6 . 3 ram; (zx) d = 2 . 1 mm. Calculations with: HFD1 [7] (dotted lines), M3SV [9] (short dashed), BTT [8] (solid). Equilibrium temperatures are shown by long dashed lines.

in Fig. 4. Theoretical relaxation times were derived from collision integrals, calculated in Refs. [5,6,10] for three different potential energy surfaces [7-9], and were extrapolated in the low temperature region (Fig. 1). The calculations with potential BTI" [8] show the best correlation with experimental data. There are no statistically significant discrepancies between our measurements and calculations with the M3SV potential [9], which have favourably described [11] the crossed-beam state-to-state differential cross-section data [12]. Good agreement with rotational relaxation data is observed for both the seemingly different BTI" and M3SV potential energy surfaces. However, the relaxation rates are close to each other in the temperature region where the rotational energy begins to depart from its equilibrium value in our free jets (see Fig. 1). To decide between the BTI" and M3SV surfaces, more precise measurements should be carried out over a wide range of temperature, where these surfaces produce different relaxation times. Previous comparisons [5,6,24] have shown that the HFD1 potential [7] is too anisotropic

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