Capturing molecular multimode relaxation processes in excitable gases based on decomposition of acoustic relaxation spectra

Existing two-frequency reconstructive methods can only capture primary (single) molecular relaxation processes in excitable gases. In this paper, we present a reconstructive method based on the novel decomposition of frequency-dependent acoustic relaxation spectra to capture the entire molecular multimode relaxation process. This decomposition of acoustic relaxation spectra is developed from the frequency-dependent effective specific heat, indicating that a multi-relaxation process is the sum of the interior single-relaxation processes. Based on this decomposition, we can reconstruct the entire multi-relaxation process by capturing the relaxation times and relaxation strengths of N interior single-relaxation processes, using the measurements of acoustic absorption and sound speed at 2N frequencies. Experimental data for the gas mixtures CO2–N2 and CO2–O2 validate our decomposition and reconstruction approach.


Introduction
In the past several decades, the study of molecular relaxation in gases has substantially promoted the development of many fields, such as anomalous absorption in polyatomic gases [1][2][3][4], molecular lasers [5,6], thermal phonons [7], quantum computation [8][9][10], plasma discharges [11,12] and so on. The frequency dependencies of acoustic absorption and sound speed both are determined by molecular relaxation in excitable gases [2,3]. Moreover, air-coupled acoustic transducers, such as parking sensors, are cheap and robust [13] and thus acoustic measurement is a very promising, efficient and convenient tool for exciting, capturing and exploring relaxation in gases. In practice, however, it is quite difficult to measure the entire molecular relaxation process by changing the frequency of transducers over a sufficiently wide range, since commercially available transducers have fixed resonance frequencies [13,14]. Since relaxation time varies inversely with gas pres sure, the traditional approach is to make the measurements at a handful of frequencies while varying gas pressure over a wide range [15][16][17][18][19][20]. This provides a broad range of frequency-pressure ratios ( f/p) to cover the entire relaxation process. However, the necessary pressures are often so small that the acoustic signals are swamped in noise or so large that the non-ideality of the gas needs to be considered [21]. Furthermore, the requirement to make many measurements at different pressures is time consuming. Thus, it is desirable to develop an efficient acoustic approach to measure molecular relaxation in gases.
To address the drawbacks due to varying gas pressure, Petculescu and Lueptow (PL) presented a two-frequency reconstructive algorithm to synthesize primary (single) molecular relaxation processes at a single pressure [21]. However, the algorithm utilizes the frequency dependence of the effective specific heat of gases for the reconstruction, which requires the gas density to be measured with the necessary pre-processing of the gases. Unlike the effective specific heat approach, the acoustic relaxation spectra of gases can be obtained by only measuring acoustic absorption and sound speed, which avoids the complexity of detecting the gas density. Based on this fact, we recently proposed a method to capture the primary relaxation processes by reconstructing the acoustic relaxation spectra [22]. However, compared with the traditional measurements with varying gas pressure, these fast reconstructive algorithms cannot capture the entire molecular multimode relaxation process for systems having more than one dominant interior single-relaxation process, which results in the loss of significant relaxation information for various applications, such acoustic gas sensing [20,[23][24][25][26].
In this paper, we decompose the acoustic relaxation spectrum of a molecular multimode relaxation process into spectra of interior single-relaxation processes. Based on this decomposition, we can reconstruct entire multi-relaxation processes by capturing the relaxation times and relaxation strengths of N interior single-relaxation processes using measurements at 2N operating frequencies. Experimental data validate our decomposition and reconstruction approach. The present paper is organized as follows. In section 2 we introduce the decomposition of the acoustic spectra of multimode relaxation processes. In section 3 we demonstrate the reconstruction of the entire molecular multi-relaxation process in gases. Section 4 concludes the paper.

General expression of acoustic relaxation spectra
The molecular collisional relaxation processes resulting from acoustic propagation in excitable gases (diatomic or polyatomic gases and their mixtures) can be expressed as an effective acoustic wave number ( ) ω k which depends on the frequency-dependent phase speed ( ) ω c and the molecular relaxation absorption coefficient ( ) α ω r [21]: where ω π = f 2 is the acoustic angular frequency and = − i 1. The graphical representation of the relaxation processes is the acoustic relaxation spectra ( ) µ ω , which represent the frequency dependence of the dimensionless relaxation absorption coefficient α λ r , where λ is the wavelength of sound. Thus ( ) µ ω can be represented in terms of the real and imaginary parts of ( ) ω k : can also be expressed in terms of the frequency-dependent effective isochoric molar specific heat of gases ( ) ω C V eff [21]: where ρ 0 and p 0 are the equilibrium gas density and pressure, respectively, and R = 8.31 J mol −1 K −1 is the universal gas constant. From equations (2) and (3), we can see that the effective specific heat determines the relaxation processes and the acoustic spectra of gases. Therefore, the acoustic relaxation spectra ( ) µ ω can be expressed by using the effective specific heat is also a complex number and can be written as To find the dependence of ( ) µ ω upon ( ) ω x and ( ) ω y , we square equation (4) for ( ( )) ω k 2 and expand two items on the right of the square of equation (4) respectively as i .

Acoustic spectra of single-relaxation processes
To describe the relation of multi-relaxation processes to singlerelaxation processes, we first deduce the acoustic spectra of single-relaxation processes. For a gas with a single vibrational mode, the effective specific heat of the gas is [2,21] ( ) where τ is the relaxation time characterizing the single-relaxation process [27,28], ∞ C V is the external specific heat from the translational and rotational degrees of freedom of molecules [29] and C V vib is the vibrational specific heat, which is the internal specific heat C V int of most molecules (except H 2 ) around room temperature [4].
According to equation (9), C V vib is determined directly by the characteristic frequency of vibrational mode υ. Table 1 provides the vibrational frequencies expressed by the spectroscopic convention in terms of the inverse wavelength and the results of C V vib for some gases at T = 300 K. Comparing C V vib in

Using equation (8) and setting
, the real and imaginary parts of Substituting (10) into µ A in (7), it is evident that Thus we can set ωτ = 1 in µ A to obtain the general expression of acoustic spectra for singlerelaxation processes: Equation (11) is deduced from the effective specific heat, and a thermodynamics approach confirms this equation [30]. In equation (11), A C s V vib is independent of ω and determines the amplitude of ( ) µ ω s , while ( ) ωτ ωτ + 1 2 is a bell-shaped function.

Acoustic spectra of multimode relaxation processes
Now we consider the acoustic spectra of multimode relaxation processes. For a gas mixture with W kinds of molecules consisting of N types of vibrational modes ( ⩾ N W), the effective specific heat of the multi-relaxation process is the sum of inter ior single-relaxation processes [28,31]: C l is the external specific heat of gas molecule l, a l is the mole fraction of molecule l and C i int and τ i are the internal specific heat and the relaxation time of the ith singlerelaxation process, respectively. In [28], we proved that the number of decomposed single-relaxation processes equals the number N of all vibrational modes of the component gas molecules. Using equation (12) and the real and imaginary parts of [28] and > ∑ ∞ C a C j j V vib (here a j is the mole fraction of vibrational mode j; all a j in the lth gas molecule are equal to a l ), so we have Following the approach used to obtain equation (11), we substitute (13) into (7) and set all ωτ = 1 i in µ A to gain the general expression of acoustic spectra for multi-relaxation processes: (14) and (11), we can deduce the relationship between the acoustic spectra of multi-relaxation processes and those of single-relaxation processes.

Decomposition of acoustic spectra of multi-relaxation processes
According to equation (12), the internal specific heat of a multi-relaxation process can be regarded as the sum of all   [19] 1554 υ = 2.67 × 10 −1 Cl 2 [3] 577 υ = 4.55 CH 4 [19] 2915 (8), we define the effective specific heats of these single-relaxation processes as (15) and similar to equation (11), the acoustic spectra of these single-relaxation processes are obtained as Comparing (16) with (14), the only difference between A i and Therefore, the acoustic spectrum of a multi-relaxation process is essentially the sum of the spectra of interior singlerelaxation processes: In other words, we decompose the entire spectrum of a multi-relaxation process into N spectra of single-relaxation processes.
One point should be noted in this decomposition. For A m in equation (14) and A i in equation (16) independent of the value of ω (from 0 to ∞), ≅ A A i m , so equation (17) can be obtained. The reason we select ωτ = 1 is to make the peak values of the acoustic relaxation spectra more accurate.

Results of the decomposition
The binary gas mixture 80% CO 2 -20% N 2 and the ternary mixture 40% CH 4 -10% Cl 2 -50% N 2 are used to illustrate the decomposition of acoustic relaxation spectra. According to equation (12), the number of decomposed single-relaxation processes equals the number of all vibrational modes of the component gas molecules [28]. Thus, the multi-relaxation spectra of CO 2 -N 2 and CH 4 -Cl 2 -N 2 are decomposed into N = 4 and N = 6 spectra of interior single-relaxation processes, respectively.
As shown in figure 1(a), 20% CO 2 -80% N 2 has only one significant decomposed single-relaxation spectrum (i = 1) which is almost equal to the multi-relaxation spectrum in figure 1(b) and three very small-amplitude decomposed spectra (i = 2, 3, 4) which can be ignored. For 20% CO 2 -80% N 2 , figure 1(b) shows that the sum curve (dashed) of decomposed single-relaxation spectra overlaps with the multi-relaxation spectrum (solid) by our general expression and matches the experimental data (circles) [19] very well.
Considering now the N = 6 single-relaxation processes for 40% CH 4 -10% Cl 2 -50% N 2 , figure 1(c) shows only two dominant decomposed single-relaxation spectra (dotted) and the sum curve (dashed) of single-relaxation spectra overlaps with the multi-relaxation spectrum (solid) again. Therefore, our decomposition approach is confirmed and clearly illustrates the constructive characteristics of molecular multimode relaxation processes in gases.
According to the results of the decomposition, the multirelaxation processes in gases are generally composed of one or two dominant single-relaxation processes. This explains why most multi-relaxation spectra generally have only one or two significant peaks in the relaxation (moderate) frequency range, as shown in figures 1(b) and (c). Based on this decomposition, we can reconstruct the entire molecular multi-relaxation process.

Method of reconstruction
According to equation (11), when ωτ = 1 (i.e. the acoustic period is commensurate with the relaxation time), the acoustic loss due to a molecular single-relaxation process is a maximum A C /2 s V vib . Thus the strength of molecular relaxation is manifested by the peaks of the relaxation spectra. So we define the peak amplitude ε = AC /2 V vib as the 'relaxation strength' corresponding to the relaxation time τ. Thus equation (17) can be rewritten as According to equation (18), ε i and τ i define all interior singlerelaxation processes in an entire multi-relaxation process.  (12). (b) Spectra and the sum of the decomposed spectra in (a) for 80% CO 2 -20% N 2 (circles, experimental data [19]). (c) Spectra, decomposition and the sum of decomposed spectra of 40% CH 4 -10% Cl 2 -50% N 2 for the two dominant singlerelaxation processes. The solid line represents the theoretical total multi-relaxation spectra; the dotted line is the decomposed singlerelaxation spectra; and the dashed line is the sum of the decomposed spectra.
Therefore, the reconstruction of molecular relaxation processes captures ε i and τ i from the measured acoustic absorption and sound speed. In excitable gases, the acoustic absorption α is the sum of the relaxation contribution α r and the classical contribution α c . α r is related to the energy exchange between molecular internal and external degrees of freedom in the pairs of collisional molecules, while α c is associated with transport phenomena, i.e. heat conduction, viscosity and diffusion. Since α c is generally very small compared with α r at the range of f/p < 10 6 Hz atm -1 , the classical contribution to the acoustic absorption can be omitted in the relaxation (moderate) frequency range without loss of generality [21]. In addition, the non-ideal behavior of gases is considered to be negligible in this work, and the correction [32][33][34] of our reconstructed results at high pressure and low temperature will be considered in future work.
With the measurements of acoustic absorption and sound speed at two frequencies, two spectral values, ( ) µ ω s 1 and ( ) µ ω s 2 , can be calculated from equation (2). Using equation (11) and the two spectral values, we can capture the relaxation strength ε and the relaxation time τ of a single-relaxation process as  (19) is our previous two-frequency reconstruction algorithm for capturing the primary relaxation processes [22]. According to equation (18), the spectrum of the entire multirelaxation process could have several peaks and cannot be replaced by a primary relaxation process completely. Solving 2N equations in the form of equation (18), which are evaluated with the values of ( ) µ ω m at 2N frequencies, we can obtain ε i and τ i for N decomposed single-relaxation processes and then reconstruct the entire molecular multi-relaxation process.
Obviously, equation (19) is the analytical solution of equation (18) when N = 1. Therefore, our previous two-frequency algorithm is a special case of the reconstruction method based on the decomposition approach described in this paper. In other words, our decomposition develops the reconstruction method from the primary (single) relaxation processes to the entire molecular multimode relaxation processes.

Results of reconstruction
Since gases and mixtures generally result from only one or two dominant single-relaxation processes [15][16][17][18][19][20], we now demonstrate how the method can use the measurements at four frequencies to reconstruct the entire molecular multirelaxation process, based on equation (18) with N = 2.
For 5% CH 4 -95% N 2 , figure 2(b) shows the reconstructed primary relaxation spectrum (dashed) by the two-frequency algorithm of PL using at 92 and 215 kHz (the same as figure 3 in [21]) and the reconstructed entire relaxation spectrum (dashed) using our approach at 0.1, 0.7, 92 and 215 kHz. Compared with the entire theoretical curve (solid), our method reconstructs the entire relaxation process more effectively. The two-frequency algorithm misses the secondary relaxation process, having a peak to the left of the primary relaxation peak. Otherwise, the reconstructed spectrum of the primary relaxation process by PL is similar to our reconstructed spectrum of a single-relaxation process. Therefore, Figure 2. Reconstruction of the molecular relaxation processes of gases: (a) 40% CH 4 -10% Cl 2 -50% N 2 , (b) 5% CH 4 -95% N 2 . Solid lines are the theoretical multi-relaxation spectra and decomposed single-relaxation spectra; bold dots are the spectral values at the measured frequency for the reconstruction; dashed lines are the reconstructed results of multi-relaxation spectra and decomposed single-relaxation spectra.
our decomposition approach provides a method for capturing the molecular relaxation processes from single relaxation to multi-relaxation in gases.

Error analysis of the reconstruction
In the reconstruction procedures, the selected frequencies and the measurement spectral values both have determinative influences on the reconstructed results. Considering the measurement errors in these two aspects, the mixture 40% CH 4 -10% Cl 2 -N 2 is used in our simulations. The measured frequencies of 40 kHz, 125 kHz, 215 kHz and 1 MHz are labeled as f 1 , f 2 , f 3 and f 4 , respectively.
Firstly, we fix the measured frequencies and observe the influence of the measurement errors of relaxation spectral values at each frequency on the reconstructed results. According to equation (2), the relaxation spectra are decided by the measurements of acoustic absorption and sound speed of the gases. Since the sound speed can be obtained accurately, we choose the acoustic absorption with relative errors of ±5% to reconstruct the entire multimode relaxation process. Table 2 shows the errors of the reconstructed relaxation strengths and relaxation times of the decomposed singlerelaxation spectra.
From table 2, the measurement errors at f 1 mainly cause the errors in reconstructed decomposed single-relaxation spectrum 1 at low frequency, while the measurement errors at f 4 mainly cause the deviations of single-relaxation spectrum 2 at high frequency. Spectra 1 and 2 are labeled in figure 2. Since f 2 and f 3 are located in the ranges of both decomposed singlerelaxation spectra 1 and 2, their measurement errors influence the reconstructions of spectra 1 and 2 and cause larger errors than f 1 and f 4 .
According to the reconstructed results in figure 2, f 1 and f 2 should be located in the relaxation range of the decomposed single-relaxation spectrum 1 while f 3 and f 4 should be situated in the decomposed single-relaxation process 2. Thus, the selected f 1 is supposed to be less than the effective relaxation frequency of low-frequency decomposed spectrum 1 while f 4 is larger than the effective relaxation frequency of high-frequency decomposed spectrum 2. f 2 and f 3 should be evenly distributed between f 1 and f 4 .
According to the simulations, when f 1 and f 4 cover the entire molecular multi-relaxation process, the frequency changes of f 2 and f 3 do not significantly influence the reconstructed errors resulting from their measurement errors. Once one of the decomposed single-relaxation processes is not covered by the range between f 1 and f 4 , the single-relaxation spectrum would be missed in the reconstruction. Moreover, the values of f 1 and f 4 are difficult to determine for unknown gases. Thus, one needs to study the effects of different values of f 1 and f 4 on the reconstruction errors. Since the frequency changes of f 1 and f 4 would display similar simulation results, we choose f 1 with measurement errors to illustrate the effects.
In simulation, f 1 varies from the lower value of 10 kHz to the higher one of 50 kHz. Then we assume the error distribution of measured spectral values µ 1 is uniform within ±10%, which is probably a worst case (the error distribution is more likely to have a Gaussian distribution). Figure 3 depicts the dependence of the errors of the reconstructed results, including the relaxation strengths ε 1 , ε 2 and relaxation times τ 1 , τ 2 , on the selected frequencies and measured spectral values.
From figure 3, we can first see that the measurement errors at f 1 mainly cause the errors of ε 1 and τ 1 , as in table 2. The closer f 1 approaches to the effective relaxation frequency of the decomposed spectrum 1 (45 kHz), smaller errors in the reconstructed results and a more accurate spectrum can be obtained. It is also observed that the errors in the reconstructed relaxation results are approximately linear depending on the errors of the measured spectral values. The linear dependence reveals that the errors of the reconstructed results could be effectively reduced with averaging of multiple measurements. Thus, our proposed method could guarantee robust measurement results.

An application example from experimental data
To reconstruct the entire molecular multi-relaxation process in practice, we recommend that the range of working frequency covers all dominant single-relaxation processes of the tested gases. In other words, the frequency range should cover the peaks of the acoustic relaxation spectra. In order to capture all the dominant interior single-relaxation processes, four or more working frequencies should be distributed over the relaxation frequency range. Repeating measurements and averaging the reconstructed results is an effective and convenient approach to eliminate the deviations in reconstruction results.
The gas mixture 90% CO 2 -10% O 2 at T = 600 K and T = 450 K [17] is used to illustrate the reconstruction from experimental data based on our recommendations. As shown in figure 4(a), we select the four experimental data points (bold dots) to reconstruct the entire molecular multi-relaxation process of 10% CO 2 -90% O 2 at T = 600 K. The discrepancy between the reconstructed curve (dashed) and the theoretical spectrum (solid) [17] can primarily be attributed to the amplitude measurements of acoustic spectra of the analyzed gases, as shown in figure 4(a). In figure 4( Repeating measurements and averaging the reconstructed relaxation strengths and relaxation times is an effective approach to reduce the errors of reconstruction from experimental data. According to the recommended rule of working  The capturing of acoustic relaxation from experimental data. Solid line, the theoretical multi-relaxation spectra; dashed line, the reconstructed multi-relaxation spectra from selected frequencies points (bold dots) with amplitude errors in (a) and the reconstructed acoustic spectra resulting from averaged relaxation strengths and relaxation times in (b); squares and diamonds, the experimental data for 10% CO 2 -90% O 2 at T = 600 K and T = 450 K respectively [17]. frequency selection, we randomly select six groups of four experimental data points for 10% CO 2 -90% O 2 at T = 600 K. The averaged results from the six captured groups of relaxation strengths and relaxation times are 3.31 × 10 −2 , 4.95 × 10 −2 , 2.25 × 10 −6 and 3.10 × 10 −5 s respectively, and the relative discrepancies to the theoretical values are only 0.3%, 0.2%, 1.4% and 6.2%.
As shown in figure 4(b), the two reconstructed acoustic spectra (dashed) by the averaged relaxation strengths and relaxation times nearly overlap with the theoretical spectra (solid) of 10% CO 2 -90% O 2 at T = 600 K and T = 450 K. Therefore, the experimental data validate the reconstruction method based on the decomposition of entire molecular multi-relaxation processes in this paper. Given this case, our decomposition and reconstruction of molecular multi-relaxation spectra is robust when used within a wide temperature range.

Conclusion
In summary, we find that the frequency-dependent acoustic spectrum of a multi-relaxation process in gas is the sum of the spectra of decomposed single-relaxation processes. Based on this decomposition, we propose a method to capture the relaxation strengths and relaxation times of N decomposed single-relaxation processes using sound speed and absorption measurements at 2N frequencies. This method can reconstruct the entire molecular multimode relaxation process in excitable gases. Compared with existing measurements and reconstruction algorithms, the reconstruction method based on the decomposition in this paper not only obtains the entire molecular multi-relaxation processes at a single pressure but avoids troublesome measurements of gas density. Therefore, the method provides a guide for an effective and simple design for instruments and sensors to measure molecular relaxation in gases, which would promote various applications.