A discussion based on adiabatic expansion

Adiabatic expansion and its drop of temperature while cork popping, is undoubtedly the mechanism behind the condensation processes observed above and within the bottlenecks. The drop of temperature experienced by the gas mixture gushing out of the bottleneck while cork popping classically obeys the following equation:

$${T}_{f}=T\times {(\frac{{P}_{0}}{{P}_{CB}})}^{\frac{\gamma -1}{\gamma }}$$ (1)

with T and \({P}_{CB}\) being the temperature and pressure of gas phase before cork popping (in the sealed bottle), \({T}_{f}\) and \({P}_{0}\) being the final temperature and pressure of gas phase after adiabatic expansion, and γ being the ratio of specific heats of the gas phase experiencing adiabatic expansion (mainly composed of gaseous CO 2 and being equal to 1.3)8.

By combining equations (1) and (15) (which provides the initial pressure of gas-phase CO 2 within the sealed bottle, before cork popping), and by replacing each and every parameter by its numerical value, the final temperature \({T}_{f}\) reached by the gas mixture after adiabatic expansion and cooling may therefore be determined as a function of the initial temperature T of champagne in the sealed bottles (i.e., before cork popping). Very clearly, and rather counter-intuitively, the higher the initial temperature of champagne before cork popping, the lower the final temperature \({T}_{f}\) reached by the gas mixture after adiabatic cooling, as seen in Fig. 4. By rapidly mixing with adjacent air packages above the bottleneck, holding water vapour with a partial pressure \({P}_{{\rm{vap}}}\approx 0.02\,{\rm{bar}}\) (see the Methods section), ambient air close to the bottleneck cools down. With such a huge drop of temperature of several tens of °C for gas phase expanding out of the bottlenecks, it is no wonder that the temperature of the resulting gas mixture near the bottlenecks falls well beyond the water dew point. It means that locally, and during a very short period of time, ambient air close to the cork popping holds much more water vapour than it can withstand thermodynamically speaking, resulting in the condensation or even freezing of water vapour, as described in the article by Vollmer and Möllman5.

Figure 4 Final temperature \({T}_{f}\) reached by the gas mixture (mostly composed of gas-phase CO 2 ) gushing out of the bottleneck after adiabatic expansion, as a function of the initial storage temperature T of champagne. Full size image

Homogeneous versus heterogeneous nucleation

In the following, focus is made on the gas mixture trapped in the bottlenecks, before, and after adiabatic expansion. Before cork popping, in the sealed bottle, the gas mixture trapped within the bottleneck is indeed mostly composed of gas-phase CO 2 with only traces of water vapour, as detailed in the Methods section. In the sealed bottle, the strongly temperature-dependent partial pressure of gas-phase CO 2 is accurately determined through equation (15), whereas the partial pressure of water vapour (being considered as the saturated vapour pressure corresponding to the temperature of storage of the bottle) is given in equation (17). In the range of temperatures between 6 and 20 °C, the partial pressure of gas-phase CO 2 in the sealed bottles ranges between approximately 4.5 and 7.5 bar (see Fig. 3), whereas the partial pressure of water vapour ranges between about \(9\times {10}^{-3}\), and \(2.3\times {10}^{-2}\,{\rm{bar}}\). After adiabatic expansion, the pressure within the bottleneck falls to atmospheric pressure (close to 1 bar). CO 2 being the major component of the gas mixture in the bottleneck, its partial pressure therefore falls close to 1 bar, whatever the bottle temperature, whereas the partial pressure of water vapour falls between about \(2\times {10}^{-3}\), and \(3\times {10}^{-3}\) bar, depending on the bottle temperature. Moreover, after adiabatic expansion, the huge drop of temperature experienced by the gas mixture found in the bottleneck has some serious effects on the respective saturated vapour pressures of both CO 2 , and water. In the range of very low temperatures reached by the gas mixture after adiabatic expansion (comprised between −76 and −89 °C, depending on the initial temperature of storage before corking), the saturated vapour pressure of CO 2 \({P}_{{\rm{sat}}}^{{{\rm{CO}}}_{2}}\) is approached through Antoine equation with the appropriate coefficients given in equation (18)9. Likewise, in this range of temperatures, the vapour pressure of ice water \({P}_{{\rm{sat}}}^{{{\rm{H}}}_{2}{\rm{O}}}\) was found to obey the relationship displayed in equation (19)10.

With the knowledge of the respective gas phase partial pressures of both CO 2 and water combined with their corresponding saturated vapour pressures, the saturation ratios of both CO 2 and water (i.e., \({S}_{{{\rm{H}}}_{2}{\rm{O}}}={P}_{{\rm{vap}}}^{{{\rm{H}}}_{2}{\rm{O}}}/{P}_{{\rm{sat}}}^{{{\rm{H}}}_{2}{\rm{O}}}\), and \({S}_{{{\rm{CO}}}_{2}}={P}_{{\rm{vap}}}^{{{\rm{CO}}}_{2}}/{P}_{{\rm{sat}}}^{{{\rm{CO}}}_{2}}\)), can be determined in the bottleneck after adiabatic expansion. Depending on the temperature of bottle storage, the saturation ratios of both water and CO 2 are displayed in Table 1, together with the final temperature reached by the gas mixture in the bottlenecks after adiabatic expansion. Whatever the storage temperature of bottles, the saturation ratio of water vapour reached after adiabatic expansion is huge (\({S}_{{{\rm{H}}}_{2}{\rm{O}}} > > 1\)). Phase change from water vapour to ice water is therefore thermodynamically favourable. The situation is different for gas-phase CO 2 . For bottles stored at 6 °C, after adiabatic expansion, the saturation ratio of gas-phase CO 2 remains lower than unity. CO 2 is therefore simply unable to undergo a phase change, from gas to dry ice. Nevertheless, for bottles stored at 12 and 20 °C, the saturation ratio of gas-phase CO 2 goes beyond 1, thus making the bottlenecks a favourable place to freeze gas-phase CO 2 . Actually, as the gas mixture is locally supersaturated with water vapour or gas-phase CO 2 after adiabatic expansion, the Gibbs free energy term regarding the transfer of molecules from the vapour phase (whether water vapour or gaseous CO 2 ) to the solid phase (whether ice water or dry ice) in the form of a cluster of radius r is negative. Following the classical nucleation theory (CNT), the nucleation energy barrier ΔG* to overcome, and the corresponding critical radius r* needed for a cluster to spontaneously grow through condensation of water vapour or CO 2 both express as follows11, 12:

$$\{\begin{array}{c}{\rm{\Delta }}{G}^{\ast }=\frac{16\pi {\sigma }^{3}{

u }_{S}^{2}}{3{({k}_{B}T\mathrm{ln}S)}^{2}}\\ {r}^{\ast }=\frac{2\sigma {

u }_{S}}{{k}_{B}T\,\mathrm{ln}\,S}\end{array}$$ (2)

with \(\sigma \) being the corresponding surface energy of ice water or dry ice CO 2 , \({

u }_{S}\) being the corresponding volume of a single molecule in the solid phase, and \({k}_{B}\) being the Boltzmann constant.

Table 1 Pertinent parameters of the CO 2 /H 2 O gas mixture found in the bottlenecks, before (in the corked bottles), and after adiabatic expansion. Full size table

According to the CNT, the steady state nucleation rate for homogeneous nucleation \({J}_{\hom }\), defined as the number of clusters that grow past the critical radius \({r}^{\ast }\) per unit volume and per unit time, can be written as13:

$${J}_{\hom }={N}_{G}\frac{{\rho }_{V}}{{\rho }_{S}}{(\frac{2\sigma }{\pi m})}^{1/2}\exp (-\frac{{\rm{\Delta }}{G}^{\ast }}{{k}_{B}T})$$ (3)

with the exponential pre-factor being typically determined from gas-kinetic considerations, m being the mass of a single molecule, \({\rho }_{V}\) being the density of the corresponding specie in the gas mixture (water vapour or gas-phase CO 2 ), \({\rho }_{S}\) being the density of the solid phase (ice water or dry ice) in the clusters, and \({N}_{G}\) being the molecular concentration of the corresponding specie in the gas mixture (i.e., \({P}_{{\rm{vap}}}^{{{\rm{H}}}_{2}{{\rm{O}}/\text{CO}}_{2}}/{k}_{B}T\), in m−3).

As far as homogeneous nucleation is concerned in the bottlenecks after adiabatic expansion, critical radii, nucleation energy barriers, molecular concentrations, and nucleation rates of both water and CO 2 are presented in Table 2, depending on the initial temperature of bottle storage. Whatever the bottle storage temperature, homogeneous nucleation of ice water clusters is very likely to occur in the bottleneck after adiabatic expansion, given their huge nucleation rates ranging from \(\approx {10}^{18}\) cm−3 s−1 (for bottles stored at 6 °C) to \(\approx {10}^{20}\) cm−3 s−1 (for bottles stored at 20 °C). Inversely, and despite the fact that bottles stored at 12 and 20 °C show saturation ratios significantly higher than 1 for gas-phase CO 2 after adiabatic expansion, freezing of CO 2 through homogeneous nucleation remains undoubtedly thermodynamically forbidden, because \({J}_{{\rm{\hom }}}^{{{\rm{CO}}}_{2}}\approx 0\) in both cases. It is indeed well-known that significant amount of homogeneous nucleation requires much higher saturation ratios than those experienced for gas-phase CO 2 after adiabatic expansion11, 12. Compared to homogeneous nucleation, heterogeneous nucleation requires relatively low saturation ratios, but foreign particles or aerosols are needed in the system to initiate the process of phase change by condensing molecules on the pre-existing nuclei. It is nevertheless very unlikely that floating particles, which could promote the freezing of gas-phase CO 2 through heterogeneous nucleation after adiabatic expansion, pre-exist in the sealed champagne bottlenecks. If eventually present in the bottleneck immediately after corking the bottle, such particles would have been progressively immersed or wetted on the glass wall during the period of aging, before cork popping. However, even in the absence of foreign particles or aerosols pre-existing in a supersaturated condensable environment, heterogeneous nucleation remains possible. Heterogeneous condensation caused by the presence of multiple gaseous species was already described in the literature, particularly in operational rocket plume exhausts that typically consist of mixtures of simple gaseous species13, 14. 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. In rocket plume exhausts that typically consist of mixtures of simple gaseous species such as N 2 , O 2 , Ar, and CO 2 , condensation can occur when plume temperatures decrease during the expansion process15,16,17,18. Condensation phenomena with the formation of particles within the plumes can even harm sensitive surfaces of a spacecraft19, 20. In the article by Li et al.13, simulations of homogeneous and heterogeneous condensations were performed to study freely expanding mixtures of CO 2 and N 2 condensation plumes. A pure N 2 expanding flow was found to not produce any clusters, whereas in a mixture consisting of 5% CO 2 and 95% N 2 , under the same expansion conditions, heterogeneous condensation of N 2 molecules on homogeneously condensed CO 2 nuclei was reported.

Table 2 Based on the classical nucleation theory (CNT), critical radii, nucleation energy barriers, and corresponding homogeneous nucleation rates of both water and CO 2 after adiabatic expansion. To evaluate the critical radii, homogeneous nucleation energy barriers, and nucleation rates of both ice water and dry ice CO 2 clusters, their respective surface energy, and density were used (i.e., \({\sigma }_{{{\rm{H}}}_{2}{\rm{O}}}\approx 0.106\,{\rm{J}}\,{{\rm{m}}}^{-2}\) , \({\sigma }_{{{\rm{C}}{\rm{O}}}_{2}}\approx 0.08\,{\rm{J}}\,{{\rm{m}}}^{-2}\) , \({\rho }_{{{\rm{H}}}_{2}{\rm{O}}}\approx 920\,{\rm{kg}}\,{{\rm{m}}}^{-3}\) , and \({\rho }_{{{\rm{CO}}}_{2}}\approx 1600\,{\rm{kg}}\,{{\rm{m}}}^{-3}\) 29). Full size table

At a smaller scale indeed, we believe that champagne bottlenecks could be viewed as small rocket nozzles. By drawing a parallel between the gas mixture freely expanding during the champagne cork popping process, and condensation phenomena observed in freely expanding condensation plumes, we therefore propose the following scenario. After adiabatic expansion of the gas mixture following the cork popping process of champagne bottles, clusters of ice water appear in the bottlenecks through homogeneous nucleation due to the very high saturation ratio experienced by water vapour, whatever the storage temperature of bottles. For bottles stored at 6 °C, the saturation ratio of gas-phase CO 2 nevertheless remains lower than 1, thus simply forbidding the freezing of CO 2 (whether through homo- or heterogeneous nucleation). The amount of water vapour being very low in the bottlenecks, the bottleneck remains optically transparent. For bottles stored at 12 and 20 °C, the saturation ratio of gas-phase CO 2 is significantly higher than 1, thus enabling the freezing of gas-phase CO 2 (through heterogeneous nucleation only) on ice water cluster nuclei. Blue haze is therefore attributed to the freezing of gas-phase CO 2 on ice water nuclei much smaller than the wavelength of light. Moreover, heterogeneous freezing of gas-phase CO 2 on ice water nuclei starts earlier, and with a much stronger effect for the bottles stored at 20 °C showing the highest saturation ratio (as clearly observed in Fig. 1b,c).

Rayleigh scattering

Blue haze is typical of Rayleigh scattering, which describes the elastic scattering of light by spherical particles much smaller than the wavelength of light. At the wavelength \(\lambda \), for spherical particles with radii a, and with a refractive index n, the Rayleigh scattering cross-section \({\sigma }_{R}\) is given by the following relationship7:

$${\sigma }_{R}=\frac{128{\pi }^{5}{a}^{6}}{3{\lambda }^{4}}{(\frac{{n}^{2}-1}{{n}^{2}+2})}^{2}$$ (4)

At the wavelength \(\lambda \approx 0.4\) μm (which corresponds to the blue region of the visible light spectrum), dry ice CO 2 has a refractive index \(n\approx 1.35\) 21. Therefore, considering a in the latter equation as being the critical radius for dry ice CO 2 clusters after adiabatic expansion for bottles stored at 20 °C (see Table 2) yields to a cross-section \({\sigma }_{R}\approx 1.7\times {10}^{-23}{{\rm{m}}}^{2}\). This result has to be compared to the Rayleigh scattering cross-section of ambient air surrounding the bottlenecks during our observations. Indeed, and similarly to the Earth atmosphere, ambient air in the laboratory does also scatter light. By keeping the wavelength \(\lambda \approx 0.4\) μm, and by using an equation provided by the literature for atmospheric scattering7, the cross-section for ambient air was found to be \({\sigma }_{{\rm{air}}}\approx 1.7\times {10}^{-30}\,{{\rm{m}}}^{2}\) (i.e., seven orders of magnitude lower than the scattering cross-section of dry ice CO 2 clusters). Therefore, despite the fact that the number of CO 2 clusters per unit volume is still unknown, this huge ratio of order of 107 between \({\sigma }_{R}\) and \({\sigma }_{{\rm{air}}}\) tells us that even a modest number of dry ice CO 2 condensation nuclei would be enough to produce a much stronger scattering in the blue than ambient air (as observed during the cork popping of bottles stored at 20 °C, where gas-phase CO 2 is strongly suspected to freeze in the bottlenecks). We therefore conclude that this characteristic blue haze is the signature of a partial and transient freezing of gas-phase CO 2 initially present in the bottleneck before cork popping. After adiabatic expansion, the progressive growth in size of dry ice CO 2 clusters can even be evidenced by observing the change in colour of the condensation cloud found in the bottleneck of bottles stored at 20 °C, as shown in the time sequence displayed in Fig. 5. It is worth noting that the cloud colour progressively changes from deep blue to white-grey, which pleads in favour of a transition between Rayleigh scattering by nuclei much smaller than the wavelength of ambient light, and Mie scattering as the size of nuclei becomes comparable and larger than the wavelength of light.

Figure 5 After adiabatic expansion, in the bottleneck of bottles stored at 20 °C, the progressive growth in size of dry ice CO 2 clusters can also be evidenced by observing the change in colour experienced by the condensation cloud, which progressively changes from deep blue to grey-white. The time interval between each frame is 83 µs. Full size image

Inhibition of water vapour condensation above the bottlenecks

As already mentioned earlier, the grey-white plume above the bottlenecks (clearly observed for bottles stored at 6 and 12 °C) is the signature of the condensation of water vapour naturally present in ambient air2, 5, 6. The other striking feature revealed by our experiments is the complete disappearance of this grey-white plume above the bottleneck of champagne stored at 20 °C, while the blue haze starts early within the bottleneck, as exemplified in Fig. 6. It is worth noting that the energy required to condense water vapour is brought by the change of internal energy \({\rm{\Delta }}U\) of the gas mixture initially found in the bottlenecks. The first law of thermodynamics states that \({\rm{\Delta }}U=Q+W\), where \(Q\) denotes the exchange of heat during the process, and \(W\) relates to the work of expansion of the gas mixture gushing out of the bottleneck. For adiabatic processes, Q = 0, so that the gas mixture experiences a drop of its internal energy determined by the following relationship:

$${\rm{\Delta }}U={\int }_{{V}_{G}}^{{V}_{f}}PdV$$ (5)

with P being the pressure of the gas mixture freely expanding during adiabatic expansion, \({V}_{G}\) being the volume of the gas mixture in the sealed bottle before cork popping, and \({V}_{f}\) being the volume of the gas mixture after adiabatic expansion.

Figure 6 Three snapshots, taken 1.2 ms after the cork popping process, showing the condensation of water vapour above the bottlenecks of bottles stored at 6 °C (a), 12 °C (b), and the deep blue CO 2 freezing plume gushing from the bottleneck of the bottle stored at 20 °C (c), respectively. In frame (c), the grey-white cloud of condensation droplets found in air packages adjacent to the gas volume gushing out of the bottleneck disappeared. Full size image

Adiabatic expansion keeping the product \(P{V}^{\gamma }\) as constant, the latter equation therefore transforms as follows:

$${\rm{\Delta }}U={P}_{CB}{V}_{G}^{\gamma }{\int }_{{V}_{G}}^{{V}_{f}}\frac{dV}{{V}^{\gamma }}$$ (6)

with \({P}_{CB}\) being the strongly temperature dependent pressure of gas-phase CO 2 in the corked bottle.

Integrating equation (6) between the initial stage in the corked bottle, and the final stage after adiabatic expansion, and developing, leads to the following relationship, function of both the initial pressure, and the volume of gas phase in the corked bottle:

$${\rm{\Delta }}U=\frac{{P}_{CB}{V}_{G}}{(\gamma -1)}[1-{(\frac{{P}_{CB}}{{P}_{0}})}^{\frac{1-\gamma }{\gamma }}]$$ (7)