Temperature dependence of heterogeneous CO 2 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 CO 2 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 CO 2 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 CO 2 freezing plume expelled from the throat of the bottleneck as the bottle’s temperature increases?

Fig. 1 High-speed video imaging of CO 2 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 CO 2 freezing plume observed during the cork popping process. The drop of temperature experienced by the CO 2 /H 2 O gas mixture, initially under pressure in the bottleneck and freely expanding above the bottleneck, classically obeys the following equation T f = T × ( P 0 P CB ) γ − 1 γ (1)with T and P CB being the initial temperature and pressure of the CO 2 /H 2 O gas mixture, respectively, before cork popping (in the sealed bottle); T f and P 0 being the final temperature and pressure, respectively, reached by the CO 2 /H 2 O 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 CO 2 and therefore being equal to 1.3) (12).

In Fig. 2, the respective partial pressures of both gas-phase CO 2 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 CO 2 being more than two orders of magnitude higher than that of water vapor pressure, the total pressure P CB of the CO 2 /H 2 O gas mixture found in the sealed bottles is therefore considered as being equivalent to the partial pressure of gas-phase CO 2 . 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. CO 2 being the major component of the CO 2 /H 2 O 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 T f reached by the CO 2 /H 2 O 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 relationship T f = T 2 − γ γ ( P 0 ( V G + k H RTV L ) 2 n T k H R 2 V L ) γ − 1 γ (2)

Fig. 2 Partial pressures (in bar) of both gas-phase CO 2 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 CO 2 /H 2 O 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 CO 2 /H 2 O gas mixture after adiabatic expansion has some strong effects on the respective saturated vapor pressures of both CO 2 and water. At the temperatures reached by the CO 2 /H 2 O 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 CO 2 combined with their corresponding saturated vapor pressures, as determined through Eqs. 13 and 14, respectively, the saturation ratios of both gas-phase CO 2 and water vapor (i.e., S CO 2 = P vap CO 2 / P sat CO 2 and S H 2 O = P vap H 2 O / P sat H 2 O , respectively) can be determined (see Table 1). After the CO 2 /H 2 O 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 CO 2 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 CO 2 could transform into dry ice CO 2 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 CO 2 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 CO 2 and the nucleation rate for homogeneous nucleation J hom (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) { Δ G ∗ = 16 π σ 3 ν S 2 3 ( k B T ln S ) 2 r ∗ = 2 σ ν S k B T ln S J hom = N G ρ V ρ S ( 2 σ π m ) 1 / 2 exp ( − Δ G ∗ k B T ) (3)with σ being the corresponding surface energy of ice water or dry ice CO 2 , ν S being the corresponding volume of a single molecule in the solid phase, k B being the Boltzmann constant, S being the saturation ratio of the corresponding species in the gas mixture (water vapor or gas-phase CO 2 ), m being the mass of a single molecule, ρ V being the density of the corresponding specie in the gas mixture (water vapor or gas-phase CO 2 ), ρ S being the density of the solid phase (ice water or dry ice CO 2 ) in the clusters, and N G being the molecular concentration of the corresponding specie in the gas mixture (i.e., P vap H 2 O / CO 2 / k B T , in m−3).

All the strongly bottle temperature–dependent parameters of the CO 2 /H 2 O 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 CO 2 clusters, the freezing of gas-phase CO 2 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 CO 2 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 CO 2 /H 2 O 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 CO 2 being significantly higher than 1, the freezing of gas-phase CO 2 through heterogeneous nucleation on ice water clusters nuclei becomes therefore thermodynamically possible. Blue haze is nevertheless evidence that dry ice CO 2 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 CO 2 /H 2 O 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 CO 2 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 CO 2 clusters per unit volume much higher in the CO 2 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 CO 2 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 CO 2 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 CO 2 clusters in the heterogeneous freezing plumes (15). Therefore, with a saturation ratio for gas-phase CO 2 about twice higher for the bottles stored at 30°C (see Table 1), dry ice CO 2 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 S sca and defined hereafter, was introduced S sca = Φ blue Φ red (4)with Φ blue and Φ red 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 S sca > 1, while a scattering ratio S sca ≈ 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 CO 2 /H 2 O freezing plume are represented as tiny spheres with a water ice core coated by a more or less thick layer of dry ice CO 2 . 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 CO 2 /H 2 O freezing plumes gushing from the champagne bottlenecks). The complex refractive indices corresponding to ice water and dry ice CO 2 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 J hom H 2 O , 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 S sca = Φ blue /Φ red 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 CO 2 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 CO 2 /H 2 O 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 CO 2 and (ii) the ice water nucleus grows from its critical radius to the critical radius of dry ice CO 2 before starting to coat the ice water core with the growing layer of dry ice CO 2 .

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 S sca < 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 S sca = 2 and to S sca = 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 CO 2 /H 2 O 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 S sca > 1, as shown in Fig. 1A). Nevertheless, the more marked conditions found in the CO 2 /H 2 O 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 CO 2 clusters, beyond which an achromatic scattering is observed (i.e., with S sca ≈ 1), as shown in Fig. 1B.

The transition between the two scattering regimes from a population of CO 2 /H 2 O 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 CO 2 clusters with a core of ice water, as proposed in our numerical simulations aimed at interpreting the plume color observed above champagne bottlenecks.