Bulk viscosity of CO$_2$ from Rayleigh-Brillouin lightscattering spectroscopy at 532 nm

TL;DR

This study measures the bulk viscosity of CO₂ using Rayleigh-Brillouin light scattering at 532 nm across various pressures and temperatures, revealing it is primarily influenced by rotational relaxation modes rather than vibrational ones.

Contribution

It provides the first detailed measurement of CO₂'s bulk viscosity via light scattering, comparing two models and showing minimal dependence on pressure and temperature.

Findings

Bulk viscosity to shear viscosity ratio is approximately 0.41.

Bulk viscosity is four orders of magnitude smaller than ultrasound-based estimates.

Rotational modes dominate relaxation processes in light scattering of CO₂.

Abstract

Rayleigh-Brillouin scattering spectra of CO$_2$ were measured at pressures ranging from 0.5 to 4~bar, and temperatures from 257 to 355~K using green laser light (wavelength 532~nm, scattering angle of 55.7$^\circ$). These spectra were compared to two lineshape models, which take the bulk viscosity as a parameter. One model applies to the kinetic regime, i.e. low pressures, while the second model uses the continuum, hydrodynamic approach and takes the rotational relaxation time as a parameter, which translates into the bulk viscosity. We do not find a significant dependence of the bulk viscosity with pressure or temperature. At pressures where both models apply we find a consistent value of the ratio of bulk viscosity over shear viscosity $\eta_b/\eta_s = 0.41 \pm 0.10$. This value is four orders of magnitude smaller than the common value that is based on the damping of ultrasound, and signifies that in light scattering only relaxation of rotational modes matters, while vibrational modes remain 'frozen'.

Tables

Table 1. Table 1: Data sets for RB-scattering measurements in CO 2 gas recorded under conditions as indicated. The uniformity parameter is y 𝑦 y . For values of the temperature-dependent η s subscript 𝜂 s \eta_{\rm s} and λ th subscript 𝜆 th \lambda_{\rm th} we use Boushehri et al. ( 1987 ) and Eq. ( 3 ), respectively. The bulk viscosity and the ratios η b / η s subscript 𝜂 b subscript 𝜂 s \eta_{\rm b}/\eta_{\rm s} are derived in a fit to the experimental data. The bulk viscosity η b T subscript superscript 𝜂 𝑇 b \eta^{T}_{\rm b} is based on the Tenti-S6 model, while η b H subscript superscript 𝜂 𝐻 b \eta^{H}_{\rm b} is based on the Hammond–Wiggins hydrodynamic model. The parameter C int = 2 / 2 ​ R subscript 𝐶 int 2 2 𝑅 C_{\rm int}=2/2\>R , for all cases is the heat capacity of rotational motion.

Data set	$p$	$T$	${\eta }_s$($\times {10}^{-5}$)	${\lambda }_{\mathrm{th}}$($\times {10}^{-3}$)	${\eta }_b^T$($\times {10}^{-5}$)	${\eta }_b^T/{\eta }_s$	${\eta }_b^H$($\times {10}^{-5}$)	${\eta }_b^H/{\eta }_s$	$y$
Unit	bar	K	Pa$\cdot$s	W/m$\cdot$K	Pa$\cdot$s		Pa$\cdot$s
0.5 bar	0.500	273.2	1.37	13.4	0.64	0.47			1.03
	0.500	293.2	1.47	14.3	0.72	0.49			0.93
	0.505	313.2	1.56	15.3	0.91	0.58			0.85
	0.508	333.2	1.66	16.2	1.65	0.99			0.78
	0.503	353.2	1.75	17.1	1.79	1.02			0.71
1 bar	1.033	258.1	1.30	12.7	0.50	0.38	0.30	0.23	2.32
	1.038	274.3	1.38	13.4	0.46	0.33	0.24	0.18	2.13
	1.011	293.2	1.47	14.3	0.54	0.37	0.28	0.19	1.88
	1.055	312.9	1.56	15.2	0.69	0.45	0.31	0.20	1.79
	1.048	330.8	1.65	16.1	0.77	0.47	0.33	0.20	1.62
	1.028	353.2	1.75	17.1	0.67	0.38	0.34	0.19	1.46
2 bar	2.012	257.4	1.29	12.6	0.47	0.37	0.49	0.38	4.54
	2.037	274.5	1.38	13.4	0.46	0.34	0.45	0.33	4.18
	2.000	293.2	1.47	14.3	0.40	0.27	0.37	0.25	3.71
	2.047	312.9	1.56	15.2	0.60	0.38	0.47	0.30	3.47
	2.050	331.8	1.65	16.1	0.70	0.42	0.51	0.31	3.15
	2.042	354.8	1.76	17.1	0.64	0.36	0.40	0.23	2.89
3 bar	3.012	257.1	1.29	12.6	0.66	0.51	0.73	0.57	6.80
	2.996	273.2	1.37	13.4	0.65	0.47	0.63	0.46	6.18
	3.037	295.7	1.48	14.4	0.66	0.45	0.68	0.46	5.59
	3.050	313.7	1.56	15.3	0.77	0.49	0.75	0.48	5.15
	3.064	332.4	1.65	16.1	0.98	0.59	0.89	0.54	4.76
	3.021	354.4	1.76	17.1	0.75	0.43	0.61	0.35	4.28
4 bar	4.026	258.1	1.29	12.6	0.98	0.76	1.07	0.83	9.04
	4.052	274.9	1.34	13.5	0.70	0.51	0.79	0.57	8.29
	4.048	295.2	1.48	14.4	0.61	0.41	0.70	0.47	7.46
	4.041	313.1	1.56	15.3	0.73	0.47	0.78	0.50	6.84
	4.042	332.7	1.65	16.2	0.86	0.52	0.86	0.52	6.27
	4.000	353.5	1.75	17.1	0.94	0.54	0.89	0.51	5.68

Equations

$${\eta }_{\mathrm{b}}=p\frac{2}{{\left(3+{\sum }_i{N}_i\right)}^2}\sum _i{N}_i{\tau }_i$$

$${\eta }_{\mathrm{b}}=\frac{4}{25}p{\tau }_{\mathrm{rot}}$$

$$k_{\mathrm{sc}}=\frac{2\pi }{{\lambda }_i}2n\sin (\theta /2)$$

$$y=\frac{p}{k_{\mathrm{sc}}{v}_0{\eta }_{\mathrm{s}}}$$

$$\lambda_{\rm th}=\mbox{$\frac{5}{2}$}\eta_{\rm s}\>C_{\rm t}+\rho\>D\>C_{\rm int}$$

$${\chi }^2=\frac{1}{N}\sum _{i=1}^N\frac{[I_e(f_i)-I_m(f_i){]}^2}{{\delta }^2(f_i)}$$

$${\eta }_{\mathrm{b}}/{\eta }_{\mathrm{s}}\propto {\left[1+\frac{{\pi }^{3/2}}{2}{\left(\frac{T^{\ast }}{T}\right)}^{1/2}+\left(\frac{{\pi }^2}{4}+\pi \right)\frac{T^{\ast }}{T}\right]}^{-1}$$

Full text

I Introduction

The light scattering properties of carbon dioxide remain of interest, both from a fundamental perspective studying the relaxation in molecular gases and for determining their thermodynamic properties, as well as from an applied perspective. Details of Rayleigh-Brillouin (RB) phenomena, scattering spectral profiles of CO2 gas at differing pressures and temperatures Lao et al. (1976); Pan et al. (2005); Gu et al. (2014) as well as its cross section Sneep and Ubachs (2005), are of relevance for current and future remote sensing exploration of the planetary atmospheres where CO2 is the main constituent, either under high-pressure and high-temperature conditions as on Venus Rasool and De Bergh (1970) or under low pressure and low-temperature conditions as on Mars Aharonson et al. (2004). The fact that carbon dioxide is the prime greenhouse gas has spurred large-scale activity in the transformation of this gaseous species Sakakura et al. (2007); van Rooij et al. (2018), through catalytic hydrogenation Wang et al. (2011), electrochemical conversion into renewable energy Kauffman et al. (2015), as well as in plasma-driven dissociation for the synthesis of fuel from CO2 Kondratenko et al. (2013); Bongers et al. (2017). Apart from issues of capture, fixation and transformation of Teratons of carbon dioxide, its storage and transport, either in the liquid or gas phase, forms an important challenge Mikkelsen et al. (2010); Boot-Handford et al. (2014). For these purposes study of the transport coefficients of CO2 gas, such as thermal conductivity Uribe et al. (1990), heat capacity, and shear viscosity Trengove and Wakeham (1987); Bock et al. (2002), is of practical importance. Light scattering is an elegant way to determine the thermodynamic properties of a gas, because the Rayleigh scattering phenomenon resulting in the elastic peak is connected to entropy fluctuations Strutt (1899) (Lord Rayleigh), while the Brillouin-side peaks are associated with density fluctuations or sound Brillouin (1922); Mandelstam (1926). The macroscopic gas transport coefficients govern the scattering spectral profiles and can in turn be deduced from measurement of such profiles Hirschfelder et al. (1948). This holds for both spontaneous RB-scattering Letamendia et al. (1982); Marques Jr and Kremer (1993); Vieitez et al. (2010) as well as for coherent RB-scattering Pan et al. (2004); Meijer et al. (2010).

The bulk viscosity $\eta_{\rm b}$ Cramer (2012); Jaeger et al. (2018), is the most elusive transport coefficient. It is associated with the relaxation of internal degrees of freedom of the molecule, i.e. rotations and vibrations. The bulk viscosity is commonly measured from the damping of ultrasound at frequencies in the MHz domain Herzfeld and Litoviz (1959). It can also be retrieved from the light scattering spectrum of molecular gases. This was demonstrated by Pan et al. (2005) and recently for N2, O2 and air by Gu and Ubachs (2014), and for N2O gas by Wang et al. (2018). In light scattering, the frequencies $f_{\rm s}$ involved are those of sound with wavelengths comparable to that of light, three orders of magnitude larger than the frequencies used to measure $\eta_{\rm b}$ from ultrasound experiments. The bulk viscosity of CO2 from light scattering is found to be four orders of magnitude smaller than that from ultrasound experiments(Lao et al., 1976; Pan et al., 2005; Gu et al., 2014). A simple explanation is that at high (hypersound) frequencies the relaxation of vibrational modes of the CO2 molecule no longer plays a role, i.e. the vibrational energy stays frozen in.

The bulk viscosity can be expressed in terms of relaxation times of intra-molecular degrees of freedom,

[TABLE]

where $p$ is the pressure, $N_{i}$ is the number of degrees of freedom of mode $i$ (rotations, vibrations), and $\tau_{i}$ is the relaxation time ($\tau_{\rm rot},\>\tau_{\rm vib}$) Chapman and Cowling (1970). A frequency-dependent version of this formula, depending on the product $f_{\rm s}\tau_{i}$ was given by Meijer et al. (2010). Carbon dioxide is a linear molecule with 2 rotational degrees of freedom. In case of frozen vibrations ($f_{\rm s}\tau_{\rm vib}\gg 1$), Eq. (1) reduces to

[TABLE]

The vibrational relaxation time strongly decreases with temperature, and so does the bulk viscosity. In a simple model Landau and Teller proposed an exponential dependence of the chance of relaxation on the ratio of the collision interaction time and the vibration period Herzfeld and Litoviz (1959). This leads to a scaling prediction for the temperature dependence of the bulk viscosity which agrees with experiment Cramer (2012). On the other hand, a classical analysis of collisions of rigid rotators by Parker (1959) results in a scaling expression for $\tau_{\rm rot}$ which predicts an increase of $\tau_{\rm rot}$ with increasing temperature.

That the bulk viscosity changes with temperature motivated the present study in which laser-based light scattering measurements in CO2 gas are carried out in a pressure regime of 0.5 – 4 bar and in a temperature regime of 257 – 355 K. Accurate, highly spectrally resolved and high signal-to-noise scattering line profiles are measured at a scattering angle of 55.7 degrees and a scattering wavelength of $\lambda_{i}=532$ nm, a wavelength commonly used in Lidar applications. Under these conditions of a longer wavelength and a smaller scattering angle than in a previous study Gu et al. (2014), the Brillouin-side peaks become more pronounced in the scattering spectrum.

Experimental data are analyzed in the context of two models for the spectral lineshape. One model applies at low pressures, the kinetic regime where the mean free path between collisions is comparable to the wavelength of light (Boley et al., 1972; Tenti et al., 1974), while the other one is valid in the hydrodynamic regime (Hammond and Wiggins, 1976). Values of the bulk viscosity are determined in a least squares method by comparing model spectra to measured ones.

The remainder of the article consists of an experimental section, a section discussing the bulk viscosity and model descriptions for RB-scattering, a presentation of results in the context of applicable models to describe the scattering spectrum, followed by a conclusion.

II Experiment

The experimental apparatus for measuring the RB-scattering spectral profiles of CO2 is displayed in Fig.  1. The laser source provides continuous wave radiation at $\lambda_{i}=532.22$ nm at a bandwidth of less than 5 MHz. RB-scattered light is produced from the laser beam of 5 Watt intensity traversing a gas cell equipped with a gas inlet valve and a pressure sensor. Brewster-angled windows are mounted at entrance and exit ports, and black paint covers the inside walls, to reduce unwanted scatter contributions. A temperature control system equipped with Peltier elements is employed for heating, cooling and keeping the cell at a constant temperature with uncertainty less than $0.1\>^{\circ}{\rm C}$. RB-scattered light is captured at a scattering angle $\theta=(55.7\pm 0.3)^{\circ}$ defined by the setting of a slit in the scatter beam path (see Fig. 1), that also limits the opening angle for collecting RB-scattered light to less than 0.5∘. The exact scatter angle and uncertainty are determined with a rotatable stage operated as a goniometer. A reference beam as depicted in Fig. 1 is used for aligning the collection and detection system. The scattering angle determines the scattered light wave vector $k_{\rm sc}$,

[TABLE]

with $n$ the index of refraction.

The scattered light propagates through a bandpass filter (Materion, T $>$ 90$\%$ at $\lambda_{i}$ = 532 nm, bandwidth $\triangle\lambda$ = 2.0 nm) onto a Fabry-Perot interferometer (FPI) via an optical projection system consisting of a number of lenses and pinholes to reduce stray light and contributions from Raman scattering. Finally the scattered photons are detected on a photomultiplier tube (PMT), processed and stored in a data acquisition system. The FPI is half-confocal, the curved mirror having a radius of curvature of $r=-12.5$ mm. Mirror reflectivities are 99%. The FPI has an effective free spectral range (FSR) of $2.9964$ GHz, which is determined through frequency-scanning a laser (a narrowband tunable cw-ring dye laser) over more than 1000 modes of the FPI while measuring the laser wavelength by a wavelength meter (Toptica HighFinesse WSU-30), hence yielding an uncertainty in the FSR below 1 MHz. The instrument width, yielding a value of $\sigma_{\nu_{\rm instr}}$ = 58.0 $\pm$ 3.0 MHz (FWHM), is determined by using the reference beam while scanning the piezo-actuated FPI, following methods discussed by Gu et al. (2012). It includes the bandwidth of the incident laser. The instrument function is verified to exhibit the functional form of an Airy function, which may be well approximated by a Lorentzian function during data analysis.

RB-scattering spectral profiles were recorded by piezo-scanning the FPI at integration times of 1 s for each step, usually over 18 MHz. Typical detection rates were $\sim 2000$ count/s for conditions of 1 bar pressure. A full spectrum covering lots of consecutive RB-peaks and 10,000 data points was obtained in about 3 h. The piezo-voltage scans were linearized and converted to frequency scale by fitting the RB-peak separations to the calibrated FSR-value. The linearization procedure also corrects for frequency drifts of the laser, which were measured to amount to $10-100$ MHz/hour, depending on temperature drifts in the laboratory. Finally, a collocated spectrum was obtained by cutting and adding all individual recordings over $\sim 60$ RB-peaks Gu et al. (2012). In a final step the RB scattering profiles were averaged to improve the signal to noise ratio. This procedure yields a noise level of $\sim 0.4$% (with respect to peak height) for the 1 bar pressure case. A single typical light scattering spectrum recorded at 1 bar and room temperature, measured in a typical recording time of $\sim 3$ h, is displayed in Fig. 2. This figure and its inset demonstrate the signal-to-noise ratio attainable in the present setup.

III RB-scattering and line shape models

In light scattering the key quantity is the uniformity parameter $y$, which -up to a constant- is defined as the ratio of the scattering wavelength over the mean free path between collisions, which can be shown to equal

[TABLE]

with thermal velocity $v_{0}=(2k_{\rm B}T/m)^{1/2}$, where $k_{\rm B}$ is the Boltzmann constant, $m$ the molecular mass and $\eta_{\rm s}$ the shear viscosity.

Values $y={\cal O}(1)$ pertain to the kinetic regime, and models must be based on the Boltzmann equation. There, spectra do not deviate strongly from the Rayleigh (Maxwellian) line shape. At larger values of $y$ many mean free paths fit in a wavelength, and a hydrodynamic, continuum, approach applies. The Brillouin-side features become more and more prominent with increasing $y$, and occur at frequency shifts $f_{\rm s}=\pm\upsilon_{\rm s}k_{\rm{sc}}/2\pi$, with $\upsilon_{\rm s}$ the speed of sound. Our data are in the interval $y=[0.7-9]$, and, therefore range from the kinetic into the hydrodynamic regime.

III.1 The Tenti model

The Tenti model, originally developed for analyzing RB-scattering in molecular hydrogen and diatomic molecules Boley et al. (1972); Tenti et al. (1974), is a widely used model for light scattering spectra in the kinetic regime. It uses the Wang Chang–Uhlenbeck eigentheory which takes known values of the macroscopic transport coefficients as input Wang Chang and Uhlenbeck (1951). This input consist of values for the shear viscosity $\eta_{\rm s}$, thermal conductivity $\lambda_{\rm th}$, the molar heat capacity $C_{\rm int}$ of internal modes of motion (rotations, vibrations), and the bulk viscosity $\eta_{\rm b}$. The model agrees well with experiment Pan et al. (2004); Vieitez et al. (2010); Gu et al. (2014), where it was established that the six-mode version of the Tenti model (hereafter called the Tenti-S6 model) yields a better agreement with experiment than the seven-mode variant.

We use the values for a temperature dependent shear viscosity $\eta_{\rm s}(T)$ for CO2 from Boushehri et al. (1987). Of the transport coefficients needed in the model, $\lambda_{\rm th}$ and $C_{\rm int}$ and $\eta_{\rm b}$ depend on the participation of intra-molecular modes of motion. We assume that at the frequencies associated with light scattering, only rotations participate in the exchance of internal and kinetic energy, so that $C_{\rm int}$, for the linear CO2 molecule becomes $C_{\rm int}=2/2\>R$ with $R$ the gas constant.

For the thermal conductivity $\lambda_{\rm th}$, Uribe et al. (1990) list temperature-dependent values. As in the present study only the thermal conductivity associated with rotational relaxation is considered, these values are not straightforwardly applicable to RB-scattering data. We need a value that reflects rotational internal energy only. For polyatomic gases, a high-frequency value for $\lambda_{\rm th}$ was estimated from the Eucken relation, which expresses $\lambda_{\rm th}$ as a function of the shear viscosity $\eta_{\rm s}$, the diffusivity $D$ and the heat capacity $C_{\rm int}$ of internal motion:

[TABLE]

with $C_{\rm t}=3/2\>R$, the heat capacity of kinetic motion, $C_{\rm int}=2/2\>R$, and the temperature-dependent diffusivity $D$ taken from Boushehri et al. (1987). At temperature $T=296.55\>{\rm K}$, the low-frequency value is $\lambda_{\rm th}=1.651\times 10^{-2}$ W/m$\cdot$K (Ref. Uribe et al., 1990), whereas Eq. (3) predicts a high-frequency value $\lambda_{\rm th}=1.452\times 10^{-2}$ W/m$\cdot$K. As will be demonstrated below, the smaller value indeed produces a better fit of the kinetic model. The bulk viscosity $\eta_{\rm b}$, our prime quantity of interest, is determined in a least squares procedure.

III.2 The Hammond–Wiggins model

At the other end of the uniformity scale we seek confrontation with a hydrodynamic, continuum, model by Hammond and Wiggins (1976). Unlike the Tenti model, which is built on (tensorial) eigenvectors of the linearized collision operator, the hydrodynamic model is built on (tensorial) moments of the space-time distribution function, i.e. hydrodynamic quantities. The hydrodynamic model takes the shear viscosity $\eta_{\rm s}$, the diffusivity $D$ and the heat capacity of internal motion $C_{\rm int}$ as parameters. The rotational relaxation time $\tau_{\rm rot}$ is determined in a least squares procedure, from which the bulk viscosity $\eta_{\rm b}$ is computed using Eq. (2). Allowance for rotational relaxation only is done through the choice for $C_{\rm int}=2/2\>R$.

The evaluation of both kinetic and hydrodynamic models can be done extremely quickly. Where their range of validity overlaps, the derived values of the bulk viscosity should agree.

IV Results and Discussion

In this section we will first present experimental data on light scattering in CO2, followed by an analysis in terms of two complementary spectral line models. We finally summarize the results of the temperature-dependent bulk viscosity.

IV.1 Measurements: Light scattering in CO2

Measurements of the RB-scattering spectral profile of CO2 gas were performed for conditions of 0.5 – 4 bar pressure and temperatures in the range between 258 and 355 K, as listed in Table 1. In this table the accurately measured $p$ and $T$ conditions for 29 ($p,T$) measurement combinations are listed as well as the temperature-dependent transport coefficients: shear viscosity $\eta_{\rm s}$ and thermal conductivity $\lambda_{\rm th}$. For all measurements a value for the internal molecular heat capacity of $C_{\rm int}=2/2\>R$ is adopted.

In Fig. 3 the RB light spectra for the 29 different ($p$,$T$) combinations are graphically displayed. A qualitative inspection shows, when comparing profiles from the top- row down, the pressure $p$ is increased and therewith the $y$-parameter is increased as well, and hence the spectra show more pronounced Brillouin-side peaks. Indeed at higher uniformity parameters $y$ the hydrodynamic regime is approached resulting in well-isolated acoustic side modes. Similarly, while going from left to right along the columns, the temperature $T$ is increased, associated with a lowering of the $y$-parameter, and hence the Brillouin-side peaks become less pronounced. In the following the experimental profiles will be compared to the Tenti-S6 model and the Hammond–Wiggins hydrodynamic model.

IV.2 Comparison with the two models

For a quantitative analysis of the data a comparison is made with the Tenti-S6 model, that was developed into a code Pan (2003); Pan et al. (2004) that was included in fitting routines for analyzing both spontaneous and coherent RB-scattering Meijer et al. (2010); Vieitez et al. (2010). In comparing model and experiments, the bulk viscosity $\eta_{\rm b}$ was determined in a least-squares procedure, minimizing the mean squared deviation

[TABLE]

where $I_{e}(f_{i})$ and $I_{m}(f_{i})$ are the experimental and modeled amplitude of the spectrum at (discrete) frequency $f_{i}$, and $N$ is total number of the experimental data. The error $\delta(f_{i})$ of $I_{e}(f_{i})$ is estimated as the square root of the number of collected photons.

First an analysis is made of the spectrum presented in Fig. 2. Least-squares fits are performed for the value of the bulk viscosity $\eta_{\rm b}$ invoking two values for the thermal conductivity, first $\lambda_{\rm th}=14.3$ mW/m$\cdot$K as resulting from the modified Eucken approach and, second the value obtained from a direct measurement at acoustic frequencies $\lambda_{\rm th}=16.2$ mW/m$\cdot$K Boushehri et al. (1987). The residuals plotted in Fig. 2 show that in the first approach a peak residual of 3% is found, where in the latter approach the peak deviation amounts to 5%. This supports validity of the treatment of thermal conductivity following the modified Eucken relation in this study conducted at hypersound frequencies. In such an approach focusing on hypersound the internal heat capacity is set at $C_{\rm int}=2/2\>R$, signifying that two rotational degrees of freedom are involved and vibrational relaxation is ’frozen’.

Least-squares fitting procedures based on the Tenti-S6 model were applied to the large body of 29 data sets on CO2 for ($p,T$) values as displayed in Fig. 3. With inclusion of values for the transport coefficients as listed in Table 1 optimized values for $\eta^{T}_{\rm b}$ were derived. Resulting values are listed in Table 1. Based on these fits and optimized $\eta^{T}_{\rm b}$ values, residuals between experimental data and the Tenti-S6 model description are calculated and displayed in Fig. 4. These residuals provide insight in the quality of the fit, its accuracy and the applicability of the model.

Similarly, the hydrodynamic model is used to compare the experimental and model spectra for the data of pressures above 1 bar. Here, the rotational relational time $\tau_{\rm rot}$ was adopted as a free parameter, from which the bulk viscosity was calculated using Eq. (2). Fig. 5 displays the residual between the experimental data and this model.

IV.3 Bulk viscosities

Our main result, the ratio $\eta_{\rm b}/\eta_{\rm s}$ as a function of temperature and pressure is summarized in Fig. 6. At $p=0.5\>{\rm bar}$ the spectral lineshape is not very sensitive to variation of $\eta_{\rm b}$, at the highest pressure the models deviate significantly from the experiment. At $p=1\>{\rm bar}$ (corresponding to $y=[1.4-2.3]$), the hydrodynamic model does not yet apply, and the values of $\eta_{\rm b}/\eta_{\rm s}$ of the two models differ significantly.

Fig. 6 also shows the prediction of the Landau–Teller scaling, which captures low-frequency experimental data Cramer (2012). However, with a crucial proviso: it is scaled down by a factor $10^{4}$. This vividly illustrates the dramatic effect of high frequencies on the ratio $\eta_{\rm b}/\eta_{\rm s}$. At these high frequencies, only rotational relaxation remains. Based on the analysis of classical trajectories, Parker (1959) derived a scaling expression for the ratio $\eta_{\rm b}/\eta_{\rm s}$,

[TABLE]

with $T^{*}=82.6\>{\rm K}$ the temperature associated with the well depth of the O-O interaction potential Zhang and Duan (2005). This prediction, scaled on $\eta_{\rm b}/\eta_{\rm s}=0.5$ at $T=250\>{\rm K}$, is also shown in Fig. 6.

We find no significant dependency on pressure or temperature. At pressures where both kinetic and hydrodynamic models apply, we find an average $\eta_{\rm b}/\eta_{\rm s}=0.33\pm 0.06$ at $p=2\>{\rm bar}$ and $\eta_{\rm b}/\eta_{\rm s}=0.48\pm 0.06$ at $p=3\>{\rm bar}$. These averages, together with their uncertainty, are also indicated in Fig. 6. Our present numbers are consistent with the finding obtained from the light scattering experiments on CO2 in the UV-range ($\lambda=366.8$ nm) covering the parameter space $y=[0.9-3.7]$, and yielding $\eta_{\rm b}=(5.7\pm 0.6)\times 10^{-6}$ Pa$\cdot$s Gu et al. (2014), which gives rise to $\eta_{\rm b}/\eta_{\rm s}=0.39\pm 0.04$. These values should be compared to $\eta_{\rm b}=4.6\times 10^{-6}$ Pa$\cdot$s (for $y=[3.3-8.2]$) by Lao et al. (1976), corresponding to $\eta_{\rm b}/\eta_{\rm s}=0.31$, and $\eta_{\rm b}=3.7\times 10^{-6}$ Pa$\cdot$s for $y=[0.44-3.54]$ by Pan et al. (2005), for which the ratio $\eta_{\rm b}/\eta_{\rm s}=0.25$.

V Discussion and conclusion

In this paper we study Rayleigh-Brillouin scattering over a range of pressures with the aim of determining the bulk viscosity using two different types of models for the spectral lineshape. Where the range of applicability of these two models overlaps, we find consistent values of the bulk viscosity.

At low frequencies the bulk viscosity depends strongly on temperature, which is caused by the temperature dependence of the vibrational relaxation rate. We do not find a significant temperature dependence, not even the one predicted for the increase of the bulk viscosity with temperature due to the increase of the rotational relaxation time. We find $\eta_{\rm b}/\eta_{\rm s}=0.41\pm 0.10$ at pressures of 2 and 3 bar.

For N2, a measurement of sound absorption at low pressure yielded a value of the bulk viscosity, expressed relative to the shear viscosity as $\eta_{\rm b}/\eta_{\rm s}=0.73$ (Ref.Prangsma et al., 1972). Cramer (2012) showed that this ratio should increase from 0.4 to 1 as the temperature changes from 100 K to 420 K. Indeed Gu and Ubachs (2013) experimentally determined a ratio of $\eta_{\rm b}/\eta_{\rm s}=[0.46-1.01]$ from RB-scattering for the temperature range $254.7-336.6$ K. Hence, for measurements at lower sound frequencies ($f_{\rm sl}$) and at hypersound frequencies ($f_{\rm sh}$), this ratio yields a similar value. The vibrational relaxation time of N2 is larger than $10^{-4}$ seconds at room temperature Taylor et al. (2013), thus $f_{\rm sl}\tau_{\rm vib}\gg 1$ as well as $f_{\rm sh}\tau_{\rm vib}\gg 1$. In other words, the vibrational degrees remain frozen under both conditions Emanuel (1990).

For CO2 a different situation is encountered. At atmospheric pressure, the vibrational relaxation time is $\tau_{\rm vib}=6\times 10^{-6}$ s, while the rotational relaxation time is $\tau_{\rm rot}=3.8\times 10^{-10}$ s (Ref. Lambert, 1977; Gu et al., 2014). Hence, for sound frequency measurements (at the MHz scale), $f_{\rm sl}\tau_{\rm vib}\thickapprox 1$ and $f_{\rm sl}\tau_{\rm rot}\ll 1$, which means that both rotation and vibration are excited and take effect during energy exchange with translation in collisions. The method of sound absorption delivers an experimental ratio of bulk viscosity to shear viscosity ($\eta_{\rm b}/\eta_{\rm s}$) of ${\cal O}(10^{4})$ (Ref. Tisza, 1942; Prangsma et al., 1972; Emmanuel, 1996). For hypersound frequencies (at the GHz scale), $f_{\rm sh}\tau_{\rm vib}\gg 1$, causing the vibrational modes not to take effect.

In order to describe macroscopic flow phenomena on time scales ranging from microseconds to nanoseconds, the used value of the bulk viscosity will range over four orders of magnitude. An example of such a flow phenomenon is a shock in a high Mach number flow. The comparison of scattered light spectra to kinetic and hydrodynamic models in this paper shows that this dramatic frequency dependence of the bulk viscosity is due to the to the (gradual) cessation of vibrational relaxation.

Acknowledgements

This research was supported by the China Exchange Program jointly run by the Netherlands Royal Academy of Sciences (KNAW) and the Chinese Ministry of Education. YW acknowledges support from the Chinese Scholarship Council (CSC) for his stay at VU Amsterdam. WU acknowledges the European Research Council for an ERC-Advanced grant under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 670168). The core part of the code that computes the Tenti models has been kindly provided to us by Xingguo Pan.