Research ArticleCHEMICAL PHYSICS

Under-expanded supersonic CO2 freezing jets during champagne cork popping

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Science Advances  20 Sep 2019:
Vol. 5, no. 9, eaav5528
DOI: 10.1126/sciadv.aav5528

Abstract

During champagne cork popping, the CO2/H2O gas mixture initially under pressure in the bottleneck freely expands into ambient air and experiences adiabatic cooling. A comparison between the condensation phenomena accompanying cork popping from bottles stored at 20° and 30°C was made. The initial headspace-to-ambient-pressure ratio much exceeded the critical ratio needed for the gas mixture to reach Mach 1, thus forming under-expanded supersonic CO2 freezing jets expelled from the throat of the bottlenecks. It was emphasized that, after adiabatic cooling and with a saturation ratio for gas-phase CO2 about twice higher for the bottles stored at 30°C, dry ice CO2 clusters grow bigger and reach the critical size needed to achieve the Mie scattering of light. Moreover, during the very first millisecond following cork popping, evanescent normal shock waves (or Mach disks) were unveiled in the jets, until the reservoir-to-ambient-pressure ratio goes below a critical ratio.

INTRODUCTION

Even if it is far safer and advised to uncork a champagne bottle with a subdued sigh to prevent serious eye injuries (1–4), uncorking a bottle with a bang has become a festive and iconic action preceding champagne tasting. Champagne and sparkling wines elaborated through the same traditional method are indeed under a high pressure of carbon dioxide (CO2), as gas-phase CO2 forms together with ethanol during a second in-bottle fermentation process promoted by adding yeasts and a certain amount of sugar in bottles hermetically sealed with a crown cap or a cork stopper (5, 6). This is a basic application of Henry’s law, which states that the concentration of dissolved CO2 in the liquid phase is proportional to the partial pressure of gas-phase CO2 in the bottleneck. Therefore, during the cork popping process, a plume mainly composed of gas-phase CO2 (with traces of water vapor) freely expands out of the bottleneck through ambient air. The commonly accepted idea behind champagne cork popping was that the gas mixture gushing from the bottleneck experiences adiabatic expansion and therefore cools adjacent air packages beyond the water dew point. Condensation or even freezing of water vapor found in ambient air can therefore be observed in the form of a characteristic gray-white cloud of fog (7–9). The gray-white color is characteristic of Mie scattering (i.e., the scattering of light by particles with typical sizes larger than the wavelength), well known to be at the origin of cloud color in the sky (10).

Recently, cork popping from clear transparent bottles of champagne stored at different temperatures (namely, 6°, 12°, and 20°C) was filmed through high-speed video imaging (11). On the basis of this set of experiments, the situation has appeared really more complex and subtle than described before. The key role of the champagne temperature was pointed out. It was found that, somewhat counterintuitively, the higher the initial temperature of champagne before cork popping, the lower the temperature reached by the gas mixture freely expanding above the bottleneck after adiabatic cooling. For the bottles stored at 20°C (under a pressure close to 7.5 bar), the temperature of the CO2/H2O gas mixture fell close to −90°C after adiabatic cooling and therefore below the freezing temperature of gas-phase CO2 under 1 bar. Under these conditions, the characteristic gray-white cloud of fog classically observed above the bottlenecks of bottles stored at lower temperatures completely disappeared and was replaced by a more evanescent and unexpectedly blue plume. Blue haze is indeed typical of Rayleigh scattering, which describes the elastic scattering of light by particles much smaller than the wavelength of light (10). Depending on the strongly bottle temperature–dependent saturation ratio experienced by gas-phase CO2 after adiabatic expansion, it was emphasized that blue haze is the signature of a partial and transient heterogeneous freezing of gas phase CO2 on ice water clusters homogeneously nucleated in the bottlenecks (11).

Here, the role of temperature was still more deeply investigated as the CO2/H2O gas mixture initially under pressure in the bottlenecks freely expands during champagne cork popping. A comparison between the condensation phenomena accompanying cork popping from bottles stored at 20° and 30°C was made. Unlike the deep blue haze observed for the cork popping bottles stored at 20°C, heterogeneous freezing of gas-phase CO2 turned white for the 30°C bottles. Moreover, closer examination of the freezing plumes expanding from the champagne bottlenecks revealed another notable feature well known in aircraft and aerospace engineering. From a phenomenological perspective, a parallel between the CO2/H2O gas mixture freely expanding from a champagne bottleneck while cork popping and the phenomena arising in a rocket plume exhaust was done.

RESULTS AND DISCUSSION

Temperature dependence of heterogeneous CO2 freezing plumes

Time sequences displayed in Fig. 1 illustrate minute details of the cork popping process as seen through high-speed video imaging for bottles stored at 20° and 30°C, respectively. As already unveiled in a previous study, for the bottle stored at 20°C, a blue plume appears about 250 μs after the cork popping process and develops above the bottleneck for several milliseconds until it progressively vanishes. It was emphasized that blue haze is the signature of a partial and transient heterogeneous freezing of gas-phase CO2 on ice water clusters homogeneously nucleated in the bottlenecks (11). Blue haze is indeed typical of the Rayleigh scattering of light by clusters much smaller than the wavelengths of ambient light ranging from 0.4 to 0.8 μm. For the bottle stored at 30°C, the CO2 freezing plume gushing from the bottleneck appears much more rapidly, in the very first microseconds following the cork popping process, and seems to be optically more opaque than the blue haze freely expanding from the bottle stored at 20°C. Moreover, the color of the plume expelled from the bottle stored at 30°C rather turns gray-white, a tint which is a characteristic of Mie scattering (as the size of the scattering clusters becomes comparable or larger in size than the wavelengths of ambient light). Why such a modification of the global aspect of the CO2 freezing plume expelled from the throat of the bottleneck as the bottle’s temperature increases?

Fig. 1 High-speed video imaging of CO2 freezing jets formed during champagne cork popping.

Time sequences showing the freezing jet expelled from the throat of a bottleneck for a bottle stored at 20°C (A) as compared with that expelled from a bottle stored at 30°C (B). The time elapsed after cork popping appears in each panel (in microseconds).

Adiabatic expansion and its drop of temperature is the mechanism behind the formation of the CO2 freezing plume observed during the cork popping process. The drop of temperature experienced by the CO2/H2O gas mixture, initially under pressure in the bottleneck and freely expanding above the bottleneck, classically obeys the following equationTf=T×(P0PCB)γ−1γ(1)with T and PCB being the initial temperature and pressure of the CO2/H2O gas mixture, respectively, before cork popping (in the sealed bottle); Tf and P0 being the final temperature and pressure, respectively, reached by the CO2/H2O gas mixture after adiabatic expansion; and γ being the ratio of specific heats of the gas mixture experiencing adiabatic expansion (mostly made of gas-phase CO2 and therefore being equal to 1.3) (12).

In Fig. 2, the respective partial pressures of both gas-phase CO2 and water vapor in a champagne corked bottle are plotted as a function of the bottle’s temperature T, as determined in accordance with the thermodynamic equilibrium (see Materials and Methods). The temperature-dependent partial pressure of gas-phase CO2 being more than two orders of magnitude higher than that of water vapor pressure, the total pressure PCB of the CO2/H2O gas mixture found in the sealed bottles is therefore considered as being equivalent to the partial pressure of gas-phase CO2. In the range of temperatures between 20° and 30°C, the pressure of the gas mixture in the headspace of the sealed bottles ranges between approximately 7.5 and 10.2 bar, as shown in Fig. 2. After adiabatic expansion, the pressure within the bottleneck falls to atmospheric pressure, close to 1 bar. CO2 being the major component of the CO2/H2O freely expanding gas mixture, its partial pressure therefore falls close to 1 bar, whatever the bottle temperature. With regard to the partial pressure of water vapor after adiabatic expansion, it drops to respectively about 2.7 × 10-3 and 3.6 × 10-3 bar, whether the bottle temperature is 20° or 30°C (see Table 1). By combining Eqs. 1 and 11, the final temperature Tf reached by the CO2/H2O gas mixture after adiabatic expansion can therefore be determined as a function of all the parameters of the sealed bottle (i.e., before cork popping) according to the following relationshipTf=T2−γγ(P0(VG+kHRTVL)2nTkHR2VL)γ−1γ(2)

Fig. 2 Partial pressures (in bar) of both gas-phase CO2 and water vapor found in the headspace of a corked bottle in equilibrium with the liquid phase, as a function of the bottle’s temperature (in °C).

Table 1 Parameters of the CO2/H2O gas mixture found in the bottlenecks of the bottles stored at 20° and 30°C before cork popping and after adiabatic expansion.

View this table:

Moreover, the huge drop of temperature experienced by the CO2/H2O gas mixture after adiabatic expansion has some strong effects on the respective saturated vapor pressures of both CO2 and water. At the temperatures reached by the CO2/H2O gas mixture after adiabatic expansion (−89° and −95°C, respectively, whether the bottle’s temperature is 20° or 30°C), and with the knowledge of the respective partial pressures of both water vapor and gas-phase CO2 combined with their corresponding saturated vapor pressures, as determined through Eqs. 13 and 14, respectively, the saturation ratios of both gas-phase CO2 and water vapor (i.e., SCO2=PvapCO2/PsatCO2 and SH2O=PvapH2O/PsatH2O, respectively) can be determined (see Table 1). After the CO2/H2O gas mixture has achieved adiabatic expansion, the saturation ratios reached by water vapor are huge (about 23,800 and 113,800, whether the bottle was stored at 20° or 30°C). Comparatively, the saturation ratios reached by gas-phase CO2 are indeed much smaller but still higher than unity (about 2.5 for the bottles stored at 20°C and about 4.8 for the bottles stored at 30°C). With saturation ratios higher than unity after adiabatic expansion, phase change becomes thermodynamically favorable. Water vapor could therefore transform into ice water clusters, whereas gas-phase CO2 could transform into dry ice CO2 clusters (for the bottles stored at 20° and 30°C).

To further understand the differences observed between the freezing plumes freely expanding from the cork popping bottles stored at 20° or 30°C, the nucleation rates of both ice water and dry ice CO2 clusters must be examined. According to the classical nucleation theory, the nucleation energy barrier ΔG* to overcome the corresponding critical radius r* needed for a cluster to spontaneously grow through condensation of water vapor or gas-phase CO2 and the nucleation rate for homogeneous nucleation Jhom (defined as the number of clusters that grow past the critical radius r∗ per unit volume and per unit time) express as follows (13, 14){ΔG∗=16πσ3νS23(kBT ln S)2r∗=2σνSkBT ln SJhom=NGρVρS(2σπm)1/2exp(−ΔG∗kBT)(3)with σ being the corresponding surface energy of ice water or dry ice CO2, νS being the corresponding volume of a single molecule in the solid phase, kB being the Boltzmann constant, S being the saturation ratio of the corresponding species in the gas mixture (water vapor or gas-phase CO2), m being the mass of a single molecule, ρV being the density of the corresponding specie in the gas mixture (water vapor or gas-phase CO2), ρS being the density of the solid phase (ice water or dry ice CO2) in the clusters, and NG being the molecular concentration of the corresponding specie in the gas mixture (i.e., PvapH2O/CO2/kBT, in m−3).

All the strongly bottle temperature–dependent parameters of the CO2/H2O gas mixture (immediately after adiabatic expansion) are presented in table S1, as a function of the bottle’s temperature. With nucleation rates of ≈ 6 × 1019 cm−3 s−1 (for bottles stored at 20°C) and ≈ 5 × 1020 cm−3 s−1 (for bottles stored at 30°C), the homogeneous nucleation of ice water clusters is very likely to occur whatever the bottle’s temperature. Nevertheless, with homogeneous nucleation rates close to zero for dry ice CO2 clusters, the freezing of gas-phase CO2 through homogeneous nucleation remains thermodynamically impossible for the bottles stored at 20° and 30°C (despite saturation ratios substantially higher than 1 for gas-phase CO2 after adiabatic expansion). In a previous article, the following scenario was proposed to account for the formation of a blue haze as observed above the bottlenecks of a 20°C cork popping bottle (11). After adiabatic expansion of the CO2/H2O gas mixture, clusters of ice water appear in the bottlenecks through homogeneous nucleation (given their very high rate of homogeneous nucleation). The saturation ratio of gas-phase CO2 being significantly higher than 1, the freezing of gas-phase CO2 through heterogeneous nucleation on ice water clusters nuclei becomes therefore thermodynamically possible. Blue haze is nevertheless evidence that dry ice CO2 clusters remain much smaller in size than the wavelengths of ambient light (centered on 0.6 μm) to allow the Rayleigh scattering of light.

After adiabatic expansion of the CO2/H2O gas mixture, the homogeneous nucleation rate of ice water clusters is about eight times higher for the bottles stored at 30°C than that determined for the bottles stored at 20°C. Thus, after an identical period of time following cork popping, ice water nuclei available to allow the freezing of gas-phase CO2 through heterogeneous nucleation are about eight times more likely above the bottlenecks of bottles stored at 30°C than above those stored at 20°C. Therefore, with a number of dry ice CO2 clusters per unit volume much higher in the CO2 freezing plume expelled from the bottle stored at 30°C, the plume appears earlier and is optically more opaque, as shown in the time sequences displayed in Fig. 1. Moreover, unlike the deep blue haze observed for the cork popping bottles stored at 20°C and, presumably, because of dry ice CO2 nuclei smaller in size than the wavelengths of light, the gray-white freezing plume observed for the bottles stored at 30°C is certainly evidence that the dry ice CO2 clusters that scatter ambient light reach a much bigger size allowing Mie scattering (as clusters become comparable in size than the wavelength of ambient light, i.e., ≈0.6 μm). Supersaturation is classically the driving force for the growth of dry ice CO2 clusters in the heterogeneous freezing plumes (15). Therefore, with a saturation ratio for gas-phase CO2 about twice higher for the bottles stored at 30°C (see Table 1), dry ice CO2 clusters logically grow bigger in size in the plume freely expanding from the bottle stored at 30°C, thus probably reaching the critical size needed to achieve the Mie scattering of light regime, as betrayed by the gray-white plume color.

To properly verify the scenario described above and therefore better interpret the color variation between the freezing plumes expanding from the bottlenecks of bottles stored at 20° and 30°C, a “light scattering ratio”, noted as Ssca and defined hereafter, was introducedSsca=ΦblueΦred(4)with Φblue and Φred being the fluxes of scattered photons corresponding to wavelengths λ = 0.4 μm and λ = 0.8 μm, respectively.

Accordingly, a bluish light is observed if Ssca > 1, while a scattering ratio Ssca ≈ 1 results in an achromatic scattering of light. We have estimated the fluxes of scattered light by using the Mie scattering theory valid for spherical clusters (16). According to our scenario, ice clusters embedded within the CO2/H2O freezing plume are represented as tiny spheres with a water ice core coated by a more or less thick layer of dry ice CO2. Numerical simulations based on a simplified two-stream radiative transfer model were conducted, where the adopted medium thickness is the throat diameter of the bottleneck (≈1.8 cm, corresponding also to the diameter of the CO2/H2O freezing plumes gushing from the champagne bottlenecks). The complex refractive indices corresponding to ice water and dry ice CO2 were found in the literature (17, 18), whereas the volume number density of ice clusters found in the freezing plumes gushing from bottlenecks (for the bottles stored at 20° and 30°C) was estimated by multiplying the homogenous nucleation rate of ice water JhomH2O, for each bottle temperature, by the time elapsed after cork popping (typically 750 μs, for example, corresponding to the tenth frames of the time sequences displayed in Fig. 1). The computation of the light scattering ratio Ssca = Φblue/Φred is initiated by using the critical radius for ice water clusters in the freezing plumes after adiabatic expansion, as given in table S1 (namely, 0.23 and 0.21 nm for the bottles stored at 20° and 30°C, respectively). In the frame of our scenario, the growth of dry ice CO2 microcrystals begins through heterogeneous nucleation on ice water nuclei previously formed through homogeneous nucleation (given the huge saturation ratios reached by water vapor after adiabatic expansion of the CO2/H2O gas mixture). For the bottles stored at 20° and 30°C, we have considered two cases described hereafter: (i) The initial ice water nucleus has the critical radius of ice water (corresponding to the given temperature) before it is coated by a growing layer of dry ice CO2 and (ii) the ice water nucleus grows from its critical radius to the critical radius of dry ice CO2 before starting to coat the ice water core with the growing layer of dry ice CO2.

Results, corresponding to both cases, are displayed in fig. S1. For both temperatures, the light scattered by the population of ice clusters is achromatic for clusters with radii larger than ≈10 nm. Moreover, below a radius close to 1 nm, because of the enhanced role of absorption and the influence of water ice core, the light scattered by the population of ice clusters appears also achromatic or even slightly reddish (because Ssca < 1). However, note that in this range of clusters’ sizes, the scattered fluxes are so small that the scattered light is practically undetectable. Following fig. S1, a bluish scattered light is strongly produced by clusters showing radii ranging between about 2 and 10 nm. We have also displayed panels with the RGB (Red, Green, and Blue) color encoding corresponding to Ssca = 2 and to Ssca = 11 (fig. S1A). As shown from these panels, very large light scattering ratios are not required to obtain a blue tint for the scattered light. In view of these simulations, our observations may be understood as follows. The deep blue haze observed for the bottles stored at 20°C would suggest indirect evidence that the CO2/H2O ice clusters grow up to characteristic sizes in the order of several nanometers only (i.e., small enough to promote a bluish scattered light, with Ssca > 1, as shown in Fig. 1A). Nevertheless, the more marked conditions found in the CO2/H2O gas mixture expelled from the bottles stored at 30°C allow an almost instantaneous crossing of the “10-nm threshold” for the population of dry ice CO2 clusters, beyond which an achromatic scattering is observed (i.e., with Ssca ≈ 1), as shown in Fig. 1B.

The transition between the two scattering regimes from a population of CO2/H2O ice clusters with typical radii close to 10 nm, as provided by our numerical simulations, may appear relatively small. However, some particular clouds known as polar mesospheric clouds (PMCs) appear white when hit with sunlight (fig. S2). PMCs are made of ice water crystals with typical sizes about 40 to 50 nm only (19). Nevertheless, in the PMCs, the scattering objects significantly differ from the sphere and are made of ice water crystals and not dry ice CO2 clusters with a core of ice water, as proposed in our numerical simulations aimed at interpreting the plume color observed above champagne bottlenecks.

Heterogeneous condensation caused by the presence of multiple gaseous species was already described in the literature and, particularly, in operational rocket plume exhausts that typically consist in mixtures of simple gaseous species (20–25). Initial nuclei can be created out of the more easily condensable trace species through homogeneous nucleation, followed by heterogeneous condensation of the less condensable species. At a smaller scale indeed, we believe that champagne bottlenecks could be viewed as small rocket nozzles as the CO2/H2O gas mixture, initially under pressure in the sealed bottleneck, freely expands during the cork popping process. In addition, from a phenomenological perspective, the parallel between the CO2/H2O gas mixture freely expanding from a champagne bottleneck while cork popping and the phenomena arising in a rocket plume exhaust does not stop here. Closer examination of the freezing plumes freely expelled from the throat of champagne bottlenecks revealed another notable feature well known in aircraft and aerospace engineering, as described in the following paragraph.

Evanescent Mach disks in the under-expanded freezing jets

Gaseous jets may occur whenever a gas phase is exhausted from a reservoir (through a nozzle exit) at a pressure greater than the ambient pressure into which the jet is exhausting (26). At the nozzle exit, assuming adiabatic expansion, the Mach number M of the jet (defined as the ratio of the local flow velocity to the local velocity of sound) can be expressed as a function of both the reservoir–to–jet pressure ratio and the ratio of specific heats of the gas mixture experiencing adiabatic expansion according to the following relationship (27)M=(2γ−1)1/2[(PRP0)γ−1γ−1]1/2(5)with PR being the pressure of gas phase in the reservoir and P0 being the ambient pressure into which the jet is exhausting.

The Mach number of the flow velocity at the nozzle exit therefore reaches one when the reservoir–to–ambient pressure ratio overcomes the critical ratio defined hereafterPRP0=(γ+12)γγ−1(6)

For a reservoir under a pressure of gas-phase CO2, for example (with γ = 1.3), this critical pressure ratio is PR/P0 ≈ 1.83. When this situation arises, the jet is said to be under-expanded and the flow exiting the nozzle will tend to expand supersonically, adjusting to the ambient pressure through a system of expansions and shock cells within the jet. The flow geometry associated with under-expanded jets has been investigated for several decades, and their steady-state structure is now well understood (28–32). Moreover, for highly under-expanded jets exhausting from nozzles with a reservoir–to–ambient pressure ratio greater than about 5, visually appealing series of normal shockwaves (called Mach disks) will also be present along the jet (29, 31). Mach disks are named after Ernst Mach, the Austrian physicist and philosopher who first described them (33). Highly under-expanded jets are indeed involved in practical engineering situations, such as exhausts and plumes of aircrafts and aerospace rockets, where the thermal signature, jet noise, and flow behavior were carefully studied in the past decades (34–36). The gaseous exhaust is expelled from the engine supersonically, where it is under-expanded adiabatically to form a well-collimated jet that remains stable over a distance many times as long as its diameter (26). A wonderful illustration of the occurrence of Mach disks in the supersonic exhausts of a fighter aircraft can be seen in Fig. 3. More information about the overall structure of under-expanded jets in a quiescent medium, including the main features of Mach disks (apparition, position, and diameter), can be found in a recent review by Franquet et al. (35).

Fig. 3 Evidence for evenly spaced Mach disks in the supersonic exhausts of a pair of U.S. Air Force F-15E Strike Eagles flying over northern Iraq (credit: U.S. Air Force/Senior Airman Matthew Bruch).

From a phenomenological point of view, the CO2/H2O gas mixture initially under pressure in the headspace of the champagne corked bottle freely expands through the throat of the bottleneck into ambient air while cork popping. In our cork popping experiments, the initial reservoir–to–ambient pressure ratio much exceeds the critical ratio needed to reach Mach 1 (equal to 1.83 for gas-phase CO2), resulting in the formation of under-expanded jets. Moreover, in our experiments, the cork popping situation initially shows a reservoir–to–ambient pressure ratio greater than 5 (7.5 for the bottles stored at 20°C against 10.2 for those stored at 30°C), which seems to be the order of magnitude of the critical pressure ratio needed for the formation of Mach disks (29, 31). A close-up time sequence showing details of the freezing plume accompanying the cork popping process of a bottle stored at 20°C is displayed in Fig. 4 (and in movie S1). A Mach disk shock normal to the flow appears on Fig. 4A about 580 μs after cork popping. The Mach disk then progressively moves away from the bottleneck, as seen from Fig. 4 (A to D). It seems to get back toward the bottleneck in Fig. 4E, and it finally vanishes in Fig. 4F about 1 ms after cork popping.

Fig. 4 Details of the freezing jet expelled from the bottleneck of a bottle stored at 20°C.

A Mach disk shock normal to the flow appears in (A), about 580 μs after cork popping (white arrow). The standoff distance L of the Mach disk to the throat of the bottleneck then progressively increases from (A) to (D). It seems to slightly decrease in (E) before the Mach disk finally vanishes in (F) about 1 ms after cork popping. The time elapsed after cork popping appears in each panel (in microseconds).

Mach disks and their characteristics are certainly the most studied features in the literature devoted to under-expanded jets, be it experimentally, theoretically, or numerically, as presented in the exhaustive and recent overview by Franquet et al. (35). Among the huge amount of studies devoted to this feature, the location of the Mach disk with regard to the nozzle exit or vent was particularly investigated. On the largest range of reservoir–to–ambient pressure ratio, the location of the Mach disk shock was first predicted by Crist et al. (37), according to the following relationshipLD=0.65 (PRP0)1/2(7)with L being the standoff distance of the Mach disk shock from the nozzle exit, D being the throat diameter of the nozzle, and PR/P0 being the reservoir–to–ambient pressure ratio.

In Fig. 5, the standoff distance of the Mach disk shocks from the bottleneck’s throat is plotted as a function of time for the bottles stored at 20° and 30°C, respectively. Applying the latter equation to the case of our cork popping experiments, with a bottleneck throat diameter of D ≈ 1.8 cm and with reservoir–to–ambient pressure ratios close to 7.5 and 10.2 (whether the bottles were stored at 20° or 30°C), leads to standoff distances of the Mach disk shocks lying approximately 3.2 and 3.7 cm, respectively, from the bottleneck’s throat. These distances are indeed much higher than those experimentally observed (Fig. 5). But actually, Eq. 7 was determined for unconfined under-expanded supersonic jets freely exhausting in an unbounded infinite atmosphere. In our cork popping experiments, the CO2/H2O gas mixture is underexpanding in ambient air but in the presence of the cork, which is flying with a velocity (≈15 m s−1) much less than that of the jet (9). The gaseous jet following cork popping finally differs from an unconfined under-expanded supersonic jet and rather corresponds to a supersonic jet impinging on a flat obstacle normal to the flow. It was indeed already reported that a supersonic jet impinging on a plate normal to the flow increases the static pressure downstream of the Mach disk shock (28, 38–40). In normal impingement, an outflow parallel to the plate therefore develops the so-called wall jet (39), which is visible on both sides of the flying cork base in Fig. 4. Because of this effect, the Mach disk shock standoff distance was found to decrease with decreasing separation distance between the plate and the nozzle from which the gaseous jet is exhausting (41). It seems self-consistent with our observations indicating that the standoff distance of the Mach disk shock from the throat of the bottleneck is indeed much less than predicted from Eq. 7 and that it progressively increases as a function of time (Fig. 5). The Mach disk shock standoff distance progressively increasing from the throat of the bottleneck should be a consequence of the increasing separation distance between the flying cork base and the throat of the bottleneck from which the CO2/H2O gas mixture is exhausting.

Fig. 5 Standoff distances of the Mach disk to the throat of the bottleneck as a function of the time elapsed since cork popping for the bottles stored at 20° (blue diamond) and 30°C (red diamond).

The panel close to each experimental dot refers to the corresponding panel in Fig. 1.

Otherwise, because the champagne bottleneck has a very small finite volume close to 25 ml, the reservoir–to–ambient pressure ratio quickly decreases as a function of time and should inevitably go below the critical pressure ratio needed for the formation of Mach disks (which were de facto seen vanishing approximately in the very first millisecond after cork popping for the bottles stored at 20° and 30°C, as shown in Figs. 1 and 4). In case of finite overpressurized reservoirs expelling a jet of gas through a nozzle, and assuming an isentropic process, the decreasing with time t of the reservoir–to–ambient pressure ratio was determined according to the following relationship (42)PR(t)P0=PR0P0 (tcRSBVR[(γ−12)(2γ+1)γ+12(γ−1)]+1)2γ1−γ(8)with PR0 being the initial pressure of gas phase in the reservoir (namely, 7.5 and 10.2 bar in the present case for the bottles stored at 20° and 30°C, respectively), VR being the reservoir volume (≈25 ml in the present case), SB being the nozzle area (≈2.5 cm2 in the present case), and cR being the sound speed in the reservoir (i.e., cR=γRT/M, with M = 44 g mol-1 being the molar mass of gas phase in the reservoir, indeed mainly composed of gas-phase CO2).

By application of the latter equation, the characteristic time needed to decrease the reservoir–to–ambient pressure ratio below the critical ratio needed to reach Mach 1 (equal to 1.83 for gas-phase CO2) is equal to 750 μs for the bottle stored at 20°C and to 920 μs for the bottle stored at 30°C. Therefore, the characteristic times needed for the freezing jet expelled from the throat of the bottlenecks to become subsonic (thus precluding the formation of expansions and compression waves within the jet) are in quite good accordance with our experimental observations showing the vanishing of the Mach disks in the very first millisecond following cork popping. It becomes also possible to compare the standoff distance of the Mach disk shocks from the bottleneck’s throat, with the scaling law predicted by Crist et al. (37), by including in Eq. 7 the dependence with time of the reservoir–to–ambient pressure ratio given by Eq. 8. The location of the Mach disk shock normalized by the diameter of the bottleneck (for the bottles stored at 20° and 30°C) is compared with the scaling law predicted from Eq. 7 (fig. S3). As time proceeds, the flying cork moves away from the throat of the bottlenecks, and our experimental data progressively tend toward the scaling law predicted in (37) (determined for unconfined under-expanded supersonic jets). The decreasing influence of the cork is thus evidenced as it moves away from the throat of the bottleneck.

Mach disks were already observed within a jet exhausting from the bottleneck of a plastic soft drink bottle that was pressurized with air on a toy pneumatic rocket launcher (43). A high-speed schlieren video device was used to visualize refractive index gradients within the under-expanded jet, as captured by P. Petersen in fig. S4. After launch, the air jet exhausting the throat of the bottleneck is underexpanding into ambient air at a speed near Mach 1. Far from the launcher, and therefore under unconfined conditions, a series of evenly spaced Mach disks can be observed. Under these unconfined conditions, the distance from the throat of the bottleneck to the first Mach disk is in the order of 2.5 cm, as predicted by Eq. 7. Nevertheless and contrary to most of the better controlled steady flow experiments reported in the literature, the champagne cork popping process is a highly unsteady flow process, with a very quickly time decreasing reservoir–to–ambient pressure ratio and with the base of the flying cork acting as a small plate moving away from the throat of the bottlenecks from which the under-expanded freezing jet is expelled. Two-dimensional computational fluid dynamics simulations based on the resolution of the Navier-Stokes equations would certainly help to a better understanding of the structure of the flow field in the presence of the cork.

CONCLUSION

During champagne cork popping, the CO2/H2O gas mixture initially under pressure in the headspace of the bottle freely expands into ambient air and experiences adiabatic cooling. A comparison between the condensation phenomena accompanying cork popping from bottles stored at 20° and 30°C was made. The initial headspace–to–ambient-pressure ratio (7.5 for the bottles stored at 20°C against 10.2 for those stored at 30°C) much exceeded the critical ratio needed for the CO2/H2O gas mixture to reach Mach 1, thus forming under-expanded supersonic jets expelled from the throat of the bottlenecks. Unlike the deep blue haze observed for the bottles stored at 20°C and, presumably, because of CO2 heterogeneous freezing with dry ice CO2 nuclei smaller in size than the wavelengths of light, the plume freely expanding from the bottles stored at 30°C turned gray-white. After adiabatic expansion, with a saturation ratio for gas-phase CO2 about twice higher for the bottles stored at 30°C, dry ice CO2 clusters logically grow bigger in size, thus probably reaching the critical size needed to achieve the Mie scattering of light and turn gray-white. Moreover, during the first millisecond following the cork popping process, normal shock waves (or Mach disks) were unveiled in the under-expanded CO2 freezing plumes exhausted from the throat of champagne bottlenecks until the reservoir–to–ambient pressure ratio goes below a critical ratio needed for the formation of Mach disks, which were de facto seen vanishing about 1 ms after cork popping. The application of high-speed schlieren technology, which relies on the refractive index of a transparent medium changing with density to produce an image, could be used in a near future to better understand the cork popping process of champagne and sparkling wines. Visualizing the under-expanded jets expelled from the throat of bottlenecks with such a technology could help us achieve a clearer picture of the subsequent formation and dynamics of shock waves accompanying the cork popping process.

MATERIALS AND METHODS

Thermodynamic equilibrium of gas-phase CO2 and water vapor in the sealed bottles

A single batch of six champagne rosé bottles (Vranken Pommery, Marne, France) with 12.5% (v/v) ethanol, elaborated with a blend of chardonnay and pinot noir base wines in standard 75-cl clear transparent bottles, was used for this set of experiments. It is noteworthy to mention that 72 hours before each set of experiments, bottles were stored at the desired temperature (four bottles at 20°C and the two others at 30°C). To promote the in-bottle second alcoholic fermentation (or prise de mousse), yeasts along with sugar (24 g liter−1) were classically added in the bottles hermetically sealed with a crown cap. The key metabolic process behind the production of gas-phase CO2 in the sealed bottles is alcoholic fermentation (i.e., the conversion of sugars into ethanol and CO2 by yeast). Sugar (24 g liter−1) added in the blend leads to the production of 8.8 g of CO2 (or 0.2 mol) in every bottle of champagne (6). Moreover, because the capacity of CO2 to dissolve in champagne is ruled by the so-called Henry’s law equilibrium, the concentration of dissolved CO2 in the liquid phase is proportional to the partial pressure of gas-phase CO2 in the headspace of the sealed bottle. Therefore, the following relationship appliesCL=nLVL=kHP(9)with nL being the mole number of dissolved CO2 in the liquid phase, VL being the volume of the liquid phase in the sealed bottle, P being the partial pressure of gas-phase CO2 in the sealed bottle, and kH being the strongly temperature-dependent Henry’s law constant of gas-phase CO2 in the liquid phase, conveniently expressed with the van’t Hoff–like equation as follows (44)kH(T)=k298K exp[−ΔHdissR(1T−1298)](10)with k298K ≈ 1.21 g liter−1 bar−1, ΔHdiss ≈ 24.8 kJ mol−1 being the dissolution enthalpy of CO2 molecules in the liquid phase (44), R being the ideal gas constant (8.31 J K−1 mol−1), and T being the absolute temperature (in K).

After the prise de mousse, bottles aged in a cool cellar for 42 months. Bottles were then disgorged, which consists in opening the bottles to remove the dead yeast cells and recorking them with traditional natural cork stopper (provided by Amorim & Irmãos, Portugal). During the disgorging process, a bit of CO2 is inevitably lost at this step as gas-phase CO2 escapes from the bottleneck under the action of pressure when opening the bottle. The thermodynamic equilibrium of CO2 is broken, but dissolved and gas-phase CO2 quickly recover Henry’s equilibrium in the recorked bottle where a volume VG of gas phase (the headspace) still cohabits with the volume VL of champagne (i.e., the liquid phase). Note that into every bottle of this batch, the volume VG of gas phase in the headspace under the cork is close to 25 ml, whereas the volume VL of champagne (i.e., the liquid phase) is 75 cl. Recently, the following relationship was theoretically determined for the newly recovered equilibrium pressure of gas-phase CO2 in the corked bottles after the disgorging step was achieved (denoted as PCBCO2 and expressed in Pa) (11)PCBCO2≈nTkH(RT)2VL(VG+kHRTVL)2(11)with nT being the total number of CO2 moles produced in a sealed bottle after the prise de mousse (i.e., 0.2 mol in the present case).

Otherwise, in the sealed medium of a corked champagne bottle, the water vapor in the headspace under the cork is indeed under equilibrium with the liquid-phase water found in champagne. Above the melting point of water, the strongly temperature-dependent saturated water vapor pressure PsatH2O is accurately determined according to the following Antoine equationLog10(PsatH2O)=A−BT+C(12)with PsatH2O being expressed in mmHg, T being expressed in °C, and the coefficients A, B, and C being provided by the National Institute of Standards and Technology database. In the range of temperatures comprised between 1° and 99°C, Antoine coefficients A, B, and C are 8.07131, 1730.63, and 233.426, respectively (45).

The volume fraction of liquid-phase water, denoted as xLH2O, is close to 0.87 for our champagne samples at 12% (v/v) ethanol. The pressure of water vapor in the sealed bottle is therefore expressed as PCBH2O≈xLH2OPsatH2O. In Fig. 2, the respective partial pressures of gas-phase CO2 and water vapor found in a champagne corked bottle, as determined in accordance with the thermodynamic equilibrium described above, are plotted as a function of the bottle’s temperature T. As shown in Fig. 2, the temperature-dependent partial pressure of gas-phase CO2 is more than two orders of magnitude higher than that of water vapor pressure, and the total pressure of the CO2/H2O gas mixture found in the sealed champagne bottles is therefore considered as being equivalent to the partial pressure of gas-phase CO2. For the sake of simplicity, it will be denoted as PCB in the article.

Temperature dependences of ice water and dry ice CO2–saturated vapor pressures

Below the melting point of water, at temperatures between 170 and 250 K (i.e., between −103° and −23°C), which perfectly covers the range of temperatures reached by the gas mixture freely expanding above the bottleneck during the cork popping process, the vapor pressure of ice water PsatH2O was found to obey the following relationship (46)PsatH2O=exp(28.868−6132.9T)(13)with PsatH2O being expressed in Pa and T being expressed in K.

Similarly, the relationship between the saturated vapor pressure of gas-phase CO2 PsatCO2 and temperature T is provided by Antoine equation as follows (47)Log10(PsatCO2)=A−BT+C(14)with PsatCO2 being expressed in bar, T being expressed in K, and the coefficients A, B, and C being provided by the NIST database. In the range of temperatures between 154 and 196 K (i.e., between −119° and −77°C), Antoine coefficients A, B, and C are 6.81228, 1301.679, and −3.494, respectively (45).

High-speed video device

A digital high-speed cinema camera (Phantom Flex Vision Research, USA) was used for this set of experiments. To get synchronized with the split-second timing of cork popping, the shooting of the camera was posttriggered by using a simple microphone recording the “bang” done by the cork popping from the bottleneck. A 5000-W lamp powered by dc, needed to eliminate the so-called flicker phenomenon, was placed behind the corked bottleneck. A filming rate of 12,000 frames/s at a resolution of 512 × 512 pixels was chosen to examine details of the under-expanded CO2 freezing jet formed, as the CO2/H2O gas mixture (initially under pressure in the sealed champagne bottles) freely expands during the cork popping process.

SUPPLEMENTARY MATERIALS

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/9/eaav5528/DC1

Table S1. Parameters of the CO2/H2O gas mixture after adiabatic expansion.

Fig. S1. Numerical simulations of the light scattering ratio.

Fig. S2. Picture of PMCs.

Fig. S3. Location of the Mach disk shock normalized by the diameter of the bottleneck.

Fig. S4. Mach disks formed in the jet exhausting from the bottleneck of a plastic bottle.

Movie S1. High-speed video showing the cork popping process from a bottle stored at 20°C.

Reference (48)

This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is not for commercial advantage and provided the original work is properly cited.

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Acknowledgments: We acknowledge T. Gasco, chef de cave from Champagne Pommery, for providing us with the batch of champagne rosé used in this set of experiments. We are also indebted to the Association Recherche Œnologique Champagne et Université (AROCU) for moral support. Funding: There are no funding sources for this research. Author contributions: G.L.-B. designed and performed research. G.L.-B., D.C., and R.G. analyzed data. G.L.-B. wrote the paper with inputs from all co-authors. All authors discussed the results and commented on the manuscript. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.
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