Vibrational CARS measurements in a near-atmospheric pressure plasma jet in nitrogen: II. Analysis
VVibrational CARS measurements in anear-atmospheric pressure plasma jet in nitrogen:II. Analysis
J Kuhfeld, D Luggenh¨olscher, U Czarnetzki
Ruhr University Bochum, Institute for Plasma and Atomic Physics, GermanyE-mail: [email protected]
Abstract.
The understanding of the ro-vibrational dynamics in molecular (near)-atmospheric pressure plasmas is essential to investigate the influence of vibrationalexcited molecules on the discharge properties. In a companion paper [1], results ofro-vibrational coherent anti-Stokes Raman scattering (CARS) measurements for ananosecond pulsed plasma jet consisting of two conducting molybdenum electrodeswith a gap of 1 mm in nitrogen at 200 mbar are presented. Here, those results arediscussed and compared to theoretical predictions based on rate coefficients for therelevant processes found in the literature. It is found, that during the discharge themeasured vibrational excitation agrees well with predictions obtained from the ratesfor resonant electron collisions calculated by Laporta et al [2]. The predictions arebased on the electric field during the discharge, measured by EFISH [1, 3] and theelectron density which is deduced from the field and mobility data calculated withBolsig+ [4]. In the afterglow a simple kinetic simulation for the vibrational subsystemof nitrogen is performed and it is found, that the populations of vibrational excitedstates develop according to vibrational-vibrational transfer on timescales of a few µ s,while the development on timescales of some hundred µ s is determined by the losses atthe walls. No significant influence of electronically excited states on the populationsof the vibrational states visible in the CARS measurements ( v (cid:46)
7) was observed.
Submitted to:
J. Phys. D: Appl. Phys. a r X i v : . [ phy s i c s . p l a s m - ph ] F e b ARS measurements in nitrogen
1. Introduction
In recent years, (near-) atmospheric pressure plasmas have gained a lot of interest inthe plasma community due to their numerous possible applications. Those plasmasoften employ relatively complex gas mixtures including molecular gases. Depending onthe discharge conditions a significant amount of the energy input can be stored in thevibrational excitation of molecules, which can potentially enhance chemical reactionsleading to possible use cases in plasma chemistry [5] and plasma catalysis [6, 7]. Toinvestigate the influence of the vibrational excitation it is critical to know the vibrationaldistribution function. Besides tunable diode laser absorption spectroscopy (TDLAS),Fourier transform infrared (FTIR) spectroscopy and spontaneous Raman scattering,coherent anti-Stokes Raman scattering (CARS) is a popular technique to measure thevibrational excitation in gaseous media which was already employed in plasmas in thepast [8]. Nonetheless, CARS has a drawback in the sense that the measured signal doesdepend on the population differences between vibrational states, i.e. the populationscannot be determined directly from the measured CARS spectra. This is not a majorproblem only for equilibrium systems where the vibrational population densities followa Boltzmann distribution. With the knowledge of the distribution function theoreticalCARS spectra can be calculated and fitted to measured ones with the temperatureas fitting parameter [9]. The use of a simple Boltzmann distribution function is notpossible in non-equilibrium systems as low-temperature plasmas. Here, several differentevaluation methods were used in the past. One approach is to estimate the populationdensity of one state and use this as starting point to calculate the others with thepopulation differences obtained from the CARS spectra. This is done for example in anearly work related to CARS in plasmas by Shaub et al [10] where the densities of the firsttwo vibrational states were estimated by assuming a Boltzmann distribution for those.In other works [11, 12] it is assumed that the upper state for the highest detectabletransition is approximately zero. The latter approximation is certainly reasonablefor vibrational temperatures close to room temperature where the ratio between twoneighboring states is about N v +1 N v ≈ exp (cid:16) − (cid:126) ω e k B T vib (cid:17) ≈ . × − with ω e ≈ − and T vib = 300 K. This changes drastically for higher temperatures, e.g. T vib = 5000 Kwhere N v +1 N v ≈ .
5. For this reason in our companion paper [1] a different approach isused. Similar to CARS measurements in thermal equilibrium a distribution functionis assumed for fitting theoretical spectra to measured ones, like it was done already byMessina et al [13] in a plasma burner. They assume that both the vibrational and therotational decree of freedom are Boltzmann distributed, but have different temperatures.This works reasonably well for their measurements as they measure only up to thethird vibrational state to the limited spectral range. In our measurements in [1] andprevious CARS measurements in different plasma sources it can be seen that usuallythe higher vibrational states are overpopulated compared to a Boltzmann distributiondetermined by the first two or three vibrational states. For this reason a distributionfunction is chosen which includes two vibrational temperatures and one rotational, either
ARS measurements in nitrogen −
50 0 50 100 150 200 250 300 3500 . . . . . . . Time / ns V o l t ag e / k V C u rr e n t / A Figure 1: Voltages (solid) and currents (dashed) at the cathode used for themeasurements in [1].called vibrational two-temperature distribution or simply two-temperature distributionfunction (TTDF) in the following. As the TTDF is motivated by the underlying plasmaphysics in the discharge, a detailed derivation was omitted in [1] where the focus was onthe diagnostic method. In the present paper we motivate the use of the TTDF basedon the excitation processes in the plasma in section 3.1. Additionally, simple modelsare derived for the parameters in the TTDF connecting them to the plasma parameters.Finally, the population difference results from [1] in the afterglow where the TTDF isnot valid anymore are compared to a simple kinetic model for the vibrational subsystemin section 3.2.
2. Summary of the results in [1]
In [1] a ns-pulsed discharge with a parallel electrode configuration is studied. Thedischarge jet consists of two molybdenum electrodes with a length of 20 mm and athickness of 1 mm. The 1 mm gap between the electrodes is enclosed with two glassplates at the long edges of the electrodes. In the middle of one glass plate a hole
ARS measurements in nitrogen × N ( v, J ; T rot , T vib,c , T vib,h , R h ) = g ( J )e − E ( v,J ) − E ( v, kBTrot (1) × (cid:104) − R h Z c e − E ( v, − E (0 , kBTvib,c + R h Z h e − E ( v, − E (0 , kBTvib,h (cid:124) (cid:123)(cid:122) (cid:125) for v> (cid:105) is proposed which distinguishes between the rotational(-translational) temperature T rot and two vibrational temperatures - T vib,c for a vibrationally cold distribution and T vib,h for a smaller, vibrationally hot distribution. For nitrogen the degeneracy of therotational states is g ( J ) = (cid:40) J + 1) , if J even3(2 J + 1) , if J odd. (2)The partition function for the vib. cold molecules is given by Z c = (cid:88) v,J g ( J )e − E ( v,J ) − E ( v, kBTrot × e − E ( v, − E (0 , kBTvib,c (3)and for the vib. hot molecules by Z h = (cid:88) v> ,J g ( J )e − E ( v,J ) − E ( v, kBTrot × e − E ( v, − E (0 , kBTvib,h . (4) ARS measurements in nitrogen − − − v ˜ k ( v ) ˜ k ( v ; T vib,c = 1300 K , T e = 1 eV ) Boltzmann fit with T vib,h = 5450 K Figure 2: Sum of excitation rates into vibrational states v for an electron temperature of T e = 1 eV and a vibrational background with a (vib.) temperature of 1300 K (normalizedby (cid:80) v ˜ k ( v )). The solid line is a Boltzmann distribution with a temperature of 5450 K.A detailed motivation of this distribution function is given in section 3.1 where alsothe measurement results are presented in figures 6 and 2 together with predictions fromdata for vibrational excitation by resonant electron collisions. It should be noted, thathere and in the following particle densities are always normalized to one, i.e. they aredivided by the gas density.As (2) is motivated by the excitation processes in the discharge it is not adequateto describe the afterglow. Therefore, there only the population differences ∆ N v = N v − N v +1 are inferred from the CARS spectra and the individual population densitiesare obtained by extrapolating the number density of the upper state of the highestdetectable transition. For more details see [1]. Some results are shown in section 3.2where they are compared with a simple volume averaged model for the vibrationalsystem. ARS measurements in nitrogen
800 1 ,
000 1 ,
200 1 ,
400 1 ,
600 1 2 35 , , , T vib,c / K T e / eV T v i b , h / K , , , , , , Figure 3: Temperature of the newly excited states in dependence of electron andvibrational temperature before the discharge.
800 1 ,
000 1 ,
200 1 ,
400 1 ,
600 1 2 30 . . T vib,c / K T e / eV ˙ R h / n e / − c m s − . . . . . . Figure 4: Dependence of R h /n e on the temperature of the cold molecules and theelectrons. ARS measurements in nitrogen
3. Description of the vibrational dynamics
The two-temperature distribution function in equation 2 was already introduced in [1]but not further motivated. This shall be done here.The main concept of equation 2 is that the nitrogen molecules can be divided into twomostly independent populations N ( v, J ) = (5) N c ( v, J ; T rot , T vib,c , R h ) + N h ( v, J ; T rot , T vib,h , R h )where the population of the vibrationally cold molecules is characterized by the coldtemperature T vib,c and the population of the hot molecules by the hot temperature T vib,h . Both populations share the same rotational temperature T rot and the fraction ofhot and cold molecules compared to the total amount of nitrogen is given by R h and(1 − R h ) respectively.On time scales of the nanosecond pulse there is essentially no exchange of vibrationalexcitation among the nitrogen molecules as V-V and V-T collisions are important only onlonger time scales. The dominant process of vibrational excitation during the dischargeis most certainly excitation by resonant electron collisions [5]. This leads to the followinginterpretation of the two distributions: N c is the steady-state background distributionwhich comprises the bulk of the nitrogen molecules. During the discharge some moleculesare transferred from N c to N h by electron collisions. This means the total excitation rate˙ R h and the vibrational temperature of the newly excited molecules are solely defined bythe corresponding cross sections, the electron density and the electron temperature.For the analysis of the measurements in [1] the cross section and rate dataset calculatedby Laporta et al [2, 14, 15] - freely accessible in the Phys4Entry database [16] - isused. First the characteristics of the hot distribution are investigated. While in(2) a Boltzmann distribution is assumed, there is no obvious physical reason whichsuggests this choice. To see that the Boltzmann distribution is still a reasonable goodapproximation for the range of vibrational states visible in this work in the followingthe excitation process is investigated. To begin we assume, that the electron conditionsduring most of the discharge are essentially constant. This is motivated by the constantelectric field in [1] and the nearly constant current. So the rate equation for a vibrationalstate v ≥ N v = (cid:88) i
20 ns, i.e. during the plateau phase. In figure 6 R h calculated by integrating (10) is compared to the measured values of R h during thedischarge pulse. The initial value for the integration is chosen to be the average of thethree data points before the ignition of the discharge ( t < To analyze the dynamics of vibrationally excited states in the afterglow between twodischarge pulses a kinetic model is developed for vibrationally excited nitrogen up to v = 9.As the ionization degree is very low, superelastic collisions with the plasma electronsare neglected here. Furthermore, the influence of electronically excited molecules isignored and the dissociation degree is assumed to be small. These assumptions reducethe reaction set to V-V and V-T collisions among the nitrogen molecules.The rates for the V-V processN ( v + 1) + N ( w ) → N ( v ) + N ( w + 1) (13)are calculated from the rate for the processN (1) + N (0) → N (0) + N (1) (14) ARS measurements in nitrogen . . . . . . . Time / ns R h Figure 6: Fraction of the newly excited molecules during the discharge pulse. The solidlines are the theoretical rates calculated with the data from [2] for electron temperaturesand densities obtained by current and field measurements for 4 kV applied voltage(green) and 3 kV (blue). The shaded ribbons give the uncertainty due to the uncertaintyin the initial values.via the scaling law of the semiclassical forced harmonic oscillator (FHO) model [18, 19] k V V ( v + 1 , w → v, w + 1) = (15) k V V (1 , → , v + 1)( w + 1) 3 − e − λ − λ exp (cid:18) ∆ E k B T (cid:19) where k B is the Boltzmann constant and T the gas temperature. λ is given by [19] λ = 13 √ (cid:114) θT | ∆ E | ω (cid:126) (16)with the harmonic angular frequency ω and θ = π ω mα k B . ∆ E = E ( v +1)+ E ( w ) − E ( v ) − E ( w + 1) is the energy defect due to the anharmonicity of the vibrational potential, m isthe collisional reduced mass and α is the exponential repulsive potential parameter [19].For k V V (1 , → ,
1) the value 0 . × − cm s − is used [20].For completeness the V-T rates from the calculations provided by Billing and Fisher [20]for a (rotational-translational) gas temperature of 300 K are included. It should be noted ARS measurements in nitrogen × ×
10 mm. As first approximation the particle densities are assumed to behomogeneous in the whole volume. The influx of is considered via [21] dN dt (cid:12)(cid:12)(cid:12)(cid:12) in = N gas CQP atm
V P (17)where Q = 10 sccm is half of the set gas flow - as it is assumed that the total gas flowof 20 sccm is split equally to both sides of the jet. C = 1 . × − m / s / sccm is theconversion factor to convert sccm to m s − , P atm = 1 bar is the atmospheric pressureand P = 200 mbar is the pressure in the discharge chamber and N gas is the gas density.In (17) it is assumed, that the inflowing particles are all in the vibrational ground stateas at room temperature there is now significant vibrational excitation. The vibrationalexcited nitrogen molecules in state v exiting the jet are described by dN v dt (cid:12)(cid:12)(cid:12)(cid:12) out = − N v CQP atm
V P . (18)The loss of excited molecules by diffusion to the walls is given by [21, 22] ∂N v> ∂t (cid:12)(cid:12)(cid:12)(cid:12) W = − N v> (cid:18) Λ D v + 2(2 − γ v ) Vv th γ v A (cid:19) − (19) ≈ − v th γ v A − γ v ) V with the characteristic length scale for the given geometry Λ [22], the volume andsurface area V and A , the diffusion coefficient for the species D v and the correspondingdeactivation coefficient γ v giving the probability to lose a particle in state v when ithits the wall. γ v is not very well known for states v > v = 1 it is typically inthe order of about 4 × − to 3 × − for different materials [23]. In absence of betterknowledge the deactivation coefficient is chosen here to be γ = 1 × − for all states.This motivates the approximation in (20) which is consistent with the assumption offlat density profiles: the very low deactivation coefficient means that the loss of excitedparticles is not limited by the diffusion, but instead by the deactivation process once themolecules reach the walls. Finally, we assume that the deactivation happens directlyinto the ground state, so that the walls are effectively a source for v = 0: ∂N ∂t (cid:12)(cid:12)(cid:12)(cid:12) W = − (cid:88) v> ∂N v ∂t (cid:12)(cid:12)(cid:12)(cid:12) W (20)The initial conditions for the simulation are obtained from the two temperaturedistribution directly after the discharge pulse. In this way the initial population densitiesare extrapolated up to v = 9. The results of the simulation are compared to themeasurement results from [1] in figure 7. For clarity of presentation the results for theother two measurement conditions are shown in the appendix (see figures A1 and A2).Those show a similar good agreement. ARS measurements in nitrogen − − − − − . . . . . N / N g a s i n % − − − − − . . . . . N / N g a s − − − − − . . . . N / N g a s − − − − − . . . . . Time / s N / N g a s Figure 7: Relative population densities for the measurement 3 kV, 200 ns. Symbols arethe population densities obtained with the method described in [1] and lines are theresults from the kinetic model.
ARS measurements in nitrogen et al [11], Montello et al [12] used a pin-to-pin discharge with a 10 mm gap.Their estimated electric field reaches up to about 275 Td during the ignition phaseand stays at about 125 Td during the discharge plateau which is significantly higherthan the value obtained by E-FISH measurements [1] in the discharge investigatedhere. Additionally, their pulse is only 150 ns long which further increases the relativeimportance of the high electric field during the ignition. Therefore, it is very likely, thatin their discharges the density of electronically excited states relative to the vibrationallyexcited states is significantly higher.
4. Conclusion
In this paper, the vibrational two-temperature distribution function used in thecompanion paper [1] to evaluate CARS spectra during a ns-pulsed discharge wasmotivated and the different parameters were connected to the underlying physicalprocesses. It is found that under the investigated discharge conditions the directresonant excitation through electron collisions is the main path for production ofvibrationally excited nitrogen. The distribution function of the newly excited moleculesfollows therefore the shape of the corresponding excitation probabilities or rates. Inthe case of resonant excitation this shape closely resembles a Boltzmann distributionfor small v , leading to the two-temperature distribution function (2) consisting ofa Boltzmann distributed cold background and the also Boltzmann distributed newlyexcited hot molecules. For the case that the excitation rates do not follow a Boltzmanndistribution the hot part of the distribution function can be modified easily. If sufficientdata for the excitation process is known the parameter in the distribution function canbe estimated, and it is found that the estimates - using the rates reported by Laporta [2]in combination with current and field measurements - agree very well with the measuredvalues. This shows that the rather simple two-temperature distribution - and itsgeneralization by allowing non-Boltzmann distributions - provides a useful frameworkfor analyzing and potentially optimizing the vibrational excitation of molecules inplasmas where the resonant excitation by electron collisions is the dominant process.Furthermore, the fact that the current and electric field are nearly constant duringalmost the whole high voltage pulse for the discharge type investigated in this work,indicates a constant electron density during the majority of the discharge which iscreated essentially only during the ignition of the pulse. This promises an easy tool forestimating the amount of vibrational excitation a priori when one is able to predict thedensity and electric field for the given discharge. ARS measurements in nitrogen
Acknowledgements
This project is supported by the DFG (German Science Foundation) within theframework of the CRC (Collaborative Research Centre) 1316 ”Transient atmosphericplasmas - from plasmas to liquids to solids”.
Appendix A. Additional simulation results
For completeness here the results of the simulation are compared to the measurements”3 kV, 200 ns” and ”4 kV, 200 ns” from [1] in figures A1 and A2. The agreement is asgood as in figure 7, leading to the conclusion, that for all conditions investigated theprocesses included in the simulation are sufficient to explain the experimental results.
References [1] Kuhfeld J, Lepikhin N D, Luggenh¨olscher D and Czarnetzki U Vibrational CARS measurementsin a near-atmospheric pressure plasma jet in nitrogen: I. Measurement procedure and results[2] Laporta V, Little D A, Celiberto R and Tennyson J 2014 Electron-impact resonant vibrationalexcitation and dissociation processes involving vibrationally excited N2 molecules
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Journal of Physics D: Applied Physics Journal of Physics D: Applied Physics ARS measurements in nitrogen − − − − − . . . . . N / N g a s i n % − − − − − . . . . . . N / N g a s − − − − − . . . . . N / N g a s − − − − − . . . . . . . Time / s N / N g a s Figure A1: Relative population densities for the measurement 3 kV, 250 ns. Symbolsare the population densities obtained with the method described in [1] and lines are theresults from the kinetic model. [8] Lempert W R and Adamovich I V 2014 Coherent anti-Stokes Raman scattering and spontaneousRaman scattering diagnostics of nonequilibrium plasmas and flows
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