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Langmuir 2002, 18, 1294-1301
Kinetics of Gas Discharging in a Glass of Champagne: The Role of Nucleation Sites Ge´rard Liger-Belair,*,† Miche`le Vignes-Adler,‡ Ce´dric Voisin,† Bertrand Robillard,§ and Philippe Jeandet† Laboratoire d’Œnologie, UPRES EA 2069, URVVC, Faculte´ des Sciences de Reims, B.P. 1039, 51687 Reims Cedex 2, France, Laboratoire de Physique des Mate´ riaux Divise´ s et des Interfaces, Universite´ de Marne la Valle´ e, cite´ Descartes, 5 boulevard Descartes, Champs sur Marne, 77454 Marne la Valle´ e Cedex 2, France, and Moe¨ t & Chandon, Laboratoire de Recherches, 6 rue croix de Bussy, F-51504 Epernay, France Received October 25, 2001. In Final Form: November 26, 2001 In this study, bubble production in a glass of champagne was used as a common tool to illustrate and better understand the nonclassical heterogeneous bubble nucleation from pre-existing gas cavities, referred as type IV nucleation in a recent review article (Jones et al. Adv. Colloid Interface Sci. 1999, 80, 27-50). Close-ups of nucleation sites were done during the repetitive and clockwork CO2 bubble production process. Then, by using the mass transfer equations suited to the case of rising champagne bubbles, the growth rate (dR/dt) of expanding bubbles during ascent was modeled and connected with the physicochemical parameters of the liquid medium and especially with the supersaturating ratio S of the solution. This theoretical growth rate was found to be in very good accordance with our experimental results. Several bubble trains of the glass wall were “followed”, during the gas discharging process, until bubble production stops through lack of dissolved gas. Evidence for a critical rising bubble growth rate below which bubble production stops at a given nucleation site enabled us to deduce indirectly the radius of curvature of the meniscus entrapped into the particle acting as a nucleation site.
Introduction Bubble nucleation in liquids occurs in many natural and industrial processes. In a pure homogeneous liquid, bubbles appear when the liquid undergoes a phase change. Another mechanism for the production of bubbles from a liquid is gas desorption, the subject of this paper. Bubble production is a way for the supersaturated liquid medium to recover a stable thermodynamic state, by transferring the excess of dissolved gas molecules into the vapor phase. Lubetkin and Blackwell1 defined the supersaturating ratio used to quantify dissolved gas molecules in excess in a liquid medium as
S)
cL PL -1) -1 c0 P0
(1)
cL and c0 are respectively the concentration of dissolved gas in the supersaturated liquid medium and the equilibrium concentration over a flat surface. PL and P0 are the corresponding equilibrium pressures into the vapor phase, determined by using Henry’s law. As soon as a liquid medium is supersaturated (S > 0), the bulk free energy per unit of volume, ∆gv, associated with the transfer of dissolved gas molecules into the vapor is negative and, therefore, thermodynamically favorable. But the bubble production process also results in the production of interfacial free energy. Below a critical radius, rc ) -2γ/ ∆gv, where γ is the surface tension of the liquid medium, * Corresponding author: tel and fax, 00 (33)3 26 91 33 40; e-mail,
[email protected]. † Laboratoire d’Œnologie, UPRES EA 2069, URVVC. ‡ Laboratoire de Physique des Mate ´ riaux Divise´s et des Interfaces, Universite´ de Marne la Valle´e. § Moe ¨ t & Chandon, Laboratoire de Recherches. (1) Lubetkin, S. D.; Blackwell, M. J. Colloid Interface Sci. 1988, 126, 610-615.
the bubble embryo formation results in a net increase of the total free energy of the system. Henceforth, classical nucleation is characterized by an energy barrier to overcome. As a result, homogeneous nucleation within the liquid bulk or heterogeneous nucleation on smooth surfaces requires very high supersaturating ratios, in excess of 100 or even more.2,3 Bubble nucleation may nevertheless also be observed in weakly supersaturated liquids, such as carbonated beverages.4 At low supersaturating ratios, bubbles need pre-existing gas cavities with radii of curvature greater than the critical radius in order to overcome the nucleation energy barrier and grow freely. In a recent review article, Jones et al.5 proposed a classification for the broad range of bubble formation often encountered. Heterogeneous bubble nucleation from pre-existing gas cavities with radii of curvature greater than the critical radius is referred to as type IV nucleation. In the type IV nucleation, dissolved gas spontaneously diffuses through the meniscus of preexisting gas cavities. Contrary to homogeneous classical nucleation, there is no nucleation energy barrier to overcome in the type IV nucleation process. Effervescence in carbonated beverages is a fantastic and daily example for in vivo bubble formation from preexisting gas cavities. In the case of Champagne wines, the main gas responsible for effervescence is carbon dioxide, which is produced during the second fermentation in the closed bottle. According to Henry’s law, an equilibrium progressively establishes between the dissolved gas into the wine and the vapor phase in the headspace under the cork. At the end of fermentation, the CO2 (2) Wilt, P. M. J. Colloid Interface Sci. 1986, 112, 530-538. (3) Ryan, W. L.; Hemmingsen, E. A. J. Colloid Interface Sci. 1993, 157, 312-317. (4) Bisperink, C. G.; Prins, A. Colloids Surf., A 1994, 85, 237-253. (5) Jones, S. F.; Evans, G. M.; Galvin, K. P. Adv. Colloid Interface Sci. 1999, 80, 27-50.
10.1021/la0115987 CCC: $22.00 © 2002 American Chemical Society Published on Web 01/16/2002
Nucleation Sites in a Glass of Champagne
pressure under the cork is around 6 atm, and the wine may contain up to 12 g/L of dissolved CO2, i.e., cL ≈ 1.6 × 1026 molecules m-3. When the bottle is opened, the pressure of CO2 in the vapor phase suddenly falls. The thermodynamic equilibrium of the closed bottle is broken. To recover a new stable thermodynamic state corresponding to the atmospheric pressure, champagne must degas. In the case of champagne at opening, the supersaturating ratio S is around 5, since c0 ≈ 2.7 × 1025 molecules m-3. When champagne is poured into a glass, two mechanisms enable dissolved CO2 molecules to escape from the supersaturated liquid medium: diffusion through the flat free surface of the liquid, and bubble formation (effervescence). Several kinds of particles stuck on the glass wall are able to entrap gas pockets during the filling of a glass. Most of them are cellulose fibers coming from the surrounding air or remaining from the wiping process. Such particles are responsible for the clockwork and repetitive production of bubbles rising in-line into the form of elegant bubble trains.6,7 The purpose of this paper is to use the repetitive bubble production in a glass of champagne as a common tool to illustrate and better understand the type IV nucleation process. Several bubble trains of the glass wall were investigated during the gas discharging process until bubble production stops through lack of dissolved gas. By use of mass transfer equations suited to the case of rising and expanding champagne bubbles, a theoretical rising bubble growth rate was determined. Evidence for a critical bubble growth rate during ascent, below which bubble production stops at a given nucleation site, enabled us to indirectly deduce the radius of curvature of the meniscus in the cavity acting as a nucleation site. 2. Experimental Section 2.1. Materials. For these experiments, a standard commercial champagne presenting a very typical bubbling behavior was chosen. We used a cylindrical classical crystal flute (Mariana Arystal, Lednicke´, Slovakia) whose inner diameter is 4.9 cm and wall thickness is 0.8 mm. Prior to each experiment, the flute was rinsed several times using distilled water and then air-dried. The champagne surface tension was measured with a pendant droplet apparatus (Kru¨ss). A value of γ ) 46.8 ( 0.6 mN m-1 was found. The kinematic viscosity was measured with an Ubbelohde capillary viscosimeter (type 501 10/I). A value of η ) (1.67 ( 0.02) × 10-3 kg m-1 s-1 was found. Surface tension and viscosity were measured at room temperature (20 ( 2 °C), with a sample of champagne first degassed. 2.2. Experimental Setup. A 150 mL portion of champagne was poured into the flute. To avoid the effect of the initial liquid convection on bubble nucleation, bubble growth, and bubble ascent, nucleation sites and bubble trains were observed at least 3 min after pouring the liquid into the flute. Close-ups of particles stuck on the glass wall acting as nucleation sites and giving rise to bubble trains were done with a high-speed video camera (Speedcam+, Vannier Photelec, Antony, France) fitted with a microscope objective (Mitutoyo, M Plan Apo 5, Japan). A back-light was placed behind the flute. To homogenize the light, a white translucent plastic screen was placed between the flute and the back-light. To have access to bubble nucleation sites, the video camera objective was pointed at the base of each investigated bubble train. To capture the motion of rising bubbles, we benefited from the clockwork repetitive bubble production from nucleation sites. We used this particularity to develop a simple but reliable method, which consists of the association of a photo camera and a stroboscope. The photo camera (Olympus OM2) used to photo(6) Liger-Belair, G.; Marchal, R.; Robillard, B.; Vignes-Adler, M.; Maujean, A.; Jeandet, P. Am. J. Enol. Vitic. 1999, 50, 317-323. (7) Liger-Belair, G.; Marchal, R.; Robillard, B.; Dambrouck, T.; Maujean, A.; Vignes-Adler, M.; Jeandet, P. Langmuir 2000, 16, 18891895.
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Figure 1. Determination of the rising velocity of bubbles during ascent from the photograph of a single bubble train and from the period of bubble production, T, as determined with strobe lighting. graph bubble trains was fitted with bellows and with a 50 mm 1:1.8 objective. The focus plane was chosen near the inner glass wall. A flashlight (Sunpak Auto-Zoom 3600), synchronized with the photo camera diaphragm aperture, was put behind the flute. To homogenize the light, a white translucent plastic screen was placed between the flute and the flashlight. To obtain photographs with a sharp resolution, a low-sensitive film (Ilford Pan F 50 ASA) was used. External light disturbances were eliminated by covering the whole system with a black piece of sheet. A graduated paper stuck on the glass wall gave the scale. Photographs were enlarged 10 times. By comparison of the scale given by the graduated paper stuck on the glass wall with another one plunged into the wine, close to the inner wall, we made sure that the thickness and the curvature of the wall could not have any magnifying effect on a bubble train. On a photograph, bubble diameters and distances between successive bubble centers were measured with a ruler assuming an error of ∆ ) (0.2 mm. Therefore, real distances were measured with a standard error of δ ) ∆/10 ) (20 µm. To have access to bubble radii R, we first measured bubble diameters. Bubble radii were thus measured with a less standard deviation of δradius ) (10 µm. To show up the regularity of bubble production, bubble trains were lit with the stroboscope (Digital Instrument DT-2239-2), flash frequencies ranging from 1.5 to 200 Hz. At a certain frequency of strobe lighting, a regular bubble train appears “motionless”, which means that the frequency f at which the stroboscope emits flashes is the same as the frequency at which the nucleation site generates bubbles. Henceforth, as we know the time interval between two successive bubbles, the photograph of each bubble train can be treated now as a succession of pictures of the same bubble, separated from the period of bubble formation T ) 1/f. The standard error on bubble formation frequency is evaluated by observing the range of frequencies on which the bubble train appears “motionless”. A value of δf ≈ (0.1 Hz was found. Therefore, using the value of the frequency found with strobe lighting, it becomes possible from a single photograph to deduce the velocity of rise for a single bubble. The photograph of a typical bubble train is displayed in Figure 1. The velocity Un of the nth bubble in the train may be deduced from the position of the one that just precedes, indexed n + 1,
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Figure 2. Six particles acting as nucleation sites in a glass poured with champagne. Gas pockets entrapped inside the particles clearly appear. and the one that just follows, indexed n - 1. Thus, Un ) (hn+1 - hn-1)/2T, where h is the traveled distance for a bubble from its nucleation site.
3. Results and Discussion 3.1. Close-Ups of Nucleation Sites. Pictures of six particles acting as nucleation sites are presented in Figure 2. Gas pockets entrapped inside the particles clearly appear. Contrary to a generally accepted idea, nucleation sites are not located on irregularities of the glass itself. The length-scale of such irregularities is far below the critical radius of curvature required for type IV nucleation
at low supersaturating ratios.8 In most cases in our experiments, nucleation sites were located on tiny roughly cylindrical hollow elongated fibers (of the order of 100 µm), with a cavity mouth of a few micrometers. Such characteristics suggest cellulose fibers.9 3.2. Cycle of Bubble Production from Nucleation Sites. A schematic representation of the pre-existing gas cavity attached to a tiny particle plunged into a super(8) Carr, M. W.; Hillman, A. R.; Lubetkin, S. D. J. Colloid Interface Sci. 1995, 169, 135-142. (9) Lehue´de´, P.; Robillard, B. Proceedings of the first in Vino Analytica Scientia Symposium; Bordeaux, France, 1997.
Nucleation Sites in a Glass of Champagne
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Figure 3. Concentrations of dissolved CO2 in the close vicinity of a tiny gas pocket entrapped inside a particle stuck on the glass wall. φ is the angle of the contact line between the solid substrate and the vapor phase.
saturated carbon dioxide solution is presented in Figure 3. CO2 concentrations in the close vicinity of the gas/liquid meniscus are also detailed. The difference in dissolved CO2 concentrations ∆c between the liquid bulk and the bubble surface is the driving force responsible for the diffusion of CO2 molecules through the meniscus of the gas pocket entrapped into the particle. Such a transfer still operates if ∆c > 0, i.e., cL > cB, which means by using Henry’s law, and the excess of pressure into the gas pocket due to its radius of curvature r, cL > H(P0 + 2γ/r), i.e., cL > c0 + c0/P0(2γ/r). Such a condition exists, as long as the radius of curvature r of the meniscus exceeds the critical radius of curvature defined as follows:
r > 2γ/P0S ) r*
(2)
Assuming the vapor pressure of the liquid medium as negligible, it can be noted that this critical radius is identical to that used in the classical theory of homogeneous nucleation. By use of known values of the champagne surface tension, and of its supersaturating ratio, a critical radius of 0.2 µm was found at the opening of a bottle. Assuming a bubble embryo with a radius of curvature greater than the critical radius defined above, CO2 molecules diffuse through the meniscus. A bubble forms and grows anchored to its nucleation site. When buoyancy overcomes the capillary force anchoring the bubble to the cavity, it detaches and rises toward the liquid surface, thereby providing an opportunity for a new bubble to form, grow, and detach at the same size, and so on. Bubble trains in carbonated beverages in general and in champagne in particular are therefore the result of this cycle of bubble production, as schematized in Figure 4. With the help of the stroboscope, this clockwork repetitive bubble production has already been established.6,7
Figure 4. Schematic representation of the cycle of bubble production. FB is the buoyancy and FC is the capillary force anchoring the bubble to the cavity.
Due to bubbling and diffusion through the flat free surface of champagne, CO2 molecules progressively escape from the liquid medium. As a result, the supersaturating ratio S(t) decreases while degassing. Consequently, the critical radius defined in eq 2 increases. Bubble production continues, at a given nucleation site, as long as the supersaturating ratio S exceeds a critical ratio defined from eq 2 as follows:
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S > 2γ/P0r ) S*
Liger-Belair et al.
(3)
Therefore, a given nucleation site with its own geometry (r is fixed) is characterized by its own critical supersaturating ratio S* below which it becomes inactive. In a glass of champagne, a range of particle shapes and sizes probably exists, resulting in a range of meniscus radii entrapped inside these particles. While the champagne degases, the critical supersaturated ratio S* associated with each nucleation site becomes progressively higher than the supersaturating ratio, S(t), which continuously decreases with time. Nucleation sites become in turn inactive, and corresponding bubble trains inexorably extinguish in increasing order of the radius of curvature of the bubble embryo entrapped inside particles acting as nucleation sites. 3.3. Rising Velocity of Champagne Bubbles. After their detachment from nucleation sites, bubbles rise toward the liquid surface. When rising through the supersaturated liquid, bubbles continue to expand by diffusion of CO2 molecules into the vapor phase through the bubble interface. It obviously still concerns a type IV nucleation process. As illustrated in Figure 1, bubbles expand and accelerate during ascent. In Figure 5, the bubble rise velocity U was plotted as a function of its radius R. Close examination of 30 bubble trains enabled us to cover a range of bubble radius between approximately 40 and 480 µm, resulting in intermediate Reynolds numbers (Re ) 2FRU/η) varying from 0.2 to 90. Furthermore, since champagne bubbles progressively expand and accelerate during ascent, the Peclet number (Pe ) 2RU/D), which compares convection and diffusion associated with mass transfer phenomenon, also progressively increases. The diffusion coefficient D of CO2 molecules through the liquid may be approximated by using the Stokes-Einstein equation (D ≈ kBθ/6πηa). kB is the Boltzmann constant, θ is the temperature, and a is the order of magnitude of CO2 molecules hydrodynamic radii (≈10-10 m). Pe varies from approximately 102 immediately after the bubble release to more than 105 near the liquid surface. In terms of power law, Figure 5 suggests a quadratic dependence of the bubble velocity with its radius, as was also found by Maxworthy et al.10 in a liquid of small viscosity and until bubbles are slightly deformed. Therefore, to a quite good approximation, bubble ascent close to the glass wall can be modeled by a modification of the numerical prefactor in Stokes law as follows:
2RFg 2 R U(R) ) 9η
(with R ≈ 0.6-0.8)
(4)
More details about the close examination of the rising velocity of champagne bubbles can be found in Liger-Belair et al.7 3.4. Rising Bubble Growth Rates. Two typical time dependencies of bubble radii during ascent are presented in Figure 6 (at unspecified supersaturating ratios). It clearly appears that the bubble radius increases at a constant rate k ) dR/dt, when rising through the liquid. A constant bubble growth rate was also obtained for the other bubble trains considered. Thus
R(t) ) R0 + kt
(5)
where R0 is the bubble radius as it detaches from the (10) Maxworthy, T.; Gnann, C.; Ku¨rten, M.; Durst, F. J. Fluid Mech. 1996, 321, 421-441.
Figure 5. Velocity of ascending champagne bubbles U(R) (O), compared with the Stokes velocity (s). Re and Pe are respectively the Reynolds and Peclet numbers associated with a rising bubble.
Figure 6. Bubble radius increase vs time for a bubble rising toward the liquid surface. Two typical bubble trains at different steps of gas discharging are considered.
nucleation site. As regards detachment radii R0 in champagne, experimental values straight below 100 µm were found.6 Our experiments on bubble growth rates, k, during ascent led to values ranging from approximately 350400 µm/s 3 min after pouring the champagne into the flute to values around 150 µm/s after 1 h of gas discharging. Shafer and Zare11 demonstrated also the linearity of radius increase with time for bubbles rising in a glass of beer. For bubbles in beer, they nevertheless found a much lower growth rate of 40 ( 10 µm/s (at an unspecified supersaturating ratio). As regards the linearity of bubble radius increase with time, an explanation based upon the transfer of CO2 molecules from the liquid bulk, suited to the case of rising bubbles, is given above. The general equation concerning the mass transfer of molecules from the bulk of a supersaturated liquid to a bubble surface with time is (11) Shafer, W. L.; Zare, E. A. Phys. Today 1991, 44, 48-52.
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Langmuir, Vol. 18, No. 4, 2002 1299
dN/dt ) KA∆c
(6)
where N is the number of transferred CO2 molecules, K the mass transfer coefficient, A the bubble area, and ∆c the difference in CO2 concentration between the bulk of the liquid and the bubble surface in equilibrium with the CO2 gas into the bubble. Assuming the gas into the bubble as ideal (PV ) NkBθ), the number of CO2 molecules transferred into the bubble is connected with the variation of the bubble radius with time as follows
P0 dV P0 dR dN ) ) A dt kBθ dt kBθ dt
(7)
V is the bubble volume, and P0 is the pressure into the bubble assumed to be equal to the atmospheric pressure, since the hydrostatic pressure ∆Ph due to depth h in the liquid and the overpressure inside the rising bubble due to its curvature ∆Pc are negligible (∆Ph ) Fgh < 10-2 atm and ∆Pc ) 2γ/R < 10-2 atm). By combining eq 6 and eq 7, one obtains the rate of expansion of the bubble radius
dR kBθ ) K∆c dt P0
(8)
Generally speaking, heat and mass transfers are functions of two dimensionless numbers, the Sherwood and Peclet numbers: Sh ) 2KR/D and Pe ) 2RU/D. In the case of small and large Pe, asymptotic solutions have been derived in the literature. Most of them are listed in the book by Sherwood et al.12 During ascent, champagne bubbles cover a range of high Pe between approximately 102 and 105. At large Pe, Sh becomes proportional to Pe1/3, with a numerical prefactor very close to unity. Therefore
KR 21/3 RU ≈ D 2 D
1/3
( )
U1/3 R2/3
w K ≈ 0.63D2/3
(9)
Combining eq 8 and eq 9 leads to
kBθ 2/3 U1/3 dR ) 0.63 D ∆c dt P0 R2/3
(10)
By combination of the empirical rising velocity found above (see eq 4) with eq 10, the rate of increase of the bubble radius during ascent becomes
k)
kBθ 2/3 2RFg dR ≈ 0.63 D dt P0 9η
(
1/3
)
∆c
(11)
As a result of the combined effects of power laws in eq 10 and in eq 4, the theoretical growth rate of a champagne bubble rising toward the liquid surface becomes independent of its radius, according to our experimental results. Moreover, since the pressure inside a rising champagne bubble is assumed to be equal to the atmospheric pressure P0, the difference of CO2 concentrations, ∆c ) cL - cB, between the bulk of the liquid and the bubble (12) Sherwood, T. K.; Pigford, R. L.; Wilke, C. R. Mass transfer; Chemical Engineering Series; MacGraw-Hill: Englewood Cliffs, NJ, 1975; Chapter 6.
surface can be written as ∆c ) c0S (by using the definition of the supersaturating ratio S in eq 1). Finally
k)
kBθ 2/3 2RFg dR ≈ 0.63 D dt P0 9η
(
1/3
)
c0S
(12)
Let us test the applicability of relation 12 in the case of rising champagne bubbles. By using known values of F and η in champagne, the intermediate value of R ) 0.7, the equilibrium concentration of CO2 molecules corresponding to a pressure of CO2 in the vapor phase of 1 atm, c0 ≈ 2.7 × 1025 molecules m-3, the value of the diffusion coefficient of CO2 molecules D approximated with the Stokes-Einstein equation, and an intermediate value S ≈ 4 for the supersaturating ratio (due to turbulence and air entrapment when poured into the glass, champagne loses much gas), one finds
k ≈ 0.63
(1.38 × 10-23)293 (1.4 × 10-9)2/3 × 105 2(0.7 × 103)9.8 1/3 (2.7 × 1025)4 ≈ 340 µm/s 9(1.67 × 10-3)
[
]
which is in very good accordance with the order of magnitude of bubble growth rates 3 min after pouring the champagne into the flute. The good agreement between experimental and theoretical bubble growth rates gave us the idea to use the experimental rising bubble growth rate as a probe for measuring the supersaturating ratio S of the liquid medium. From now on, the supersaturating ratio S will be connected to the rising bubble growth rate k as follows, by using relation 12
S)
P0 2RFg D-2/3 0.63kBθ 9η
(
)
-1/3
k c0
(13)
3.5. Kinetics of Bubble Formation Frequencies and Bubble Growth Rates Decrease with Time, for Given Nucleation Sites Followed While Degassing. Since the supersaturating ratio S (the driving force of the mass transfer) decreases with time while degassing, it ensues a progressive decrease of the kinetics of the CO2 molecules transfer from the liquid bulk to the vapor phase. Consequently, the bubble formation frequency f of a given nucleation site and the bubble growth rate k in the corresponding bubble train progressively and inexorably decrease. Such a phenomenon is illustrated in Figure 7, where the kinetics of gas discharging of five different bubble trains were followed while degassing. For each of the five bubble trains, both the bubble formation frequency, f, and the growth rate, k, of bubbles in the corresponding bubble train were followed with time until bubble production stops, i.e., until f tends to zero. Figure 7 indicates that each bubble train is characterized by a critical growth rate k* below which bubble production stops. It means that bubble production stops, although champagne still contains CO2 molecules in excess (i.e., S > 0). Necessarily, S > 0 if k* > 0, as seen in eq 13. Now, bubble production stops for a given nucleation site, if the mass transfer of CO2 molecules from the liquid bulk to the bubble embryo entrapped inside the nucleation site stops. Such a phenomenon occurs when the supersaturating ratio S(t), which continuously decreases with time while degassing, becomes equal to the critical supersaturated ratio S* associated with each given nucleation site, as defined in eq 3. With the help of eq 13, the critical bubble growth rate k*, determined from Figure 7 for each given nucleation site, gives access to this critical supersaturated ratio S*. Then, by using S* into eq 2, it becomes
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length scale of the microscopic nucleation sites. In a preceding research article upon the type IV nucleation process, the authors examined the cycle of bubble production from nucleation sites adsorbed on the surface of a vessel containing carbonated water with low supersaturating ratios.13 Bubble trains were also produced at specific sites on the surface of the vessel. Since the vessel they used was not sealed, nucleation sites in their experiments are certainly comparable with those found in a glass of champagne (in terms of geometrical properties). By modeling the cycle of bubble production, and by measuring the growth time of the last possible bubble, they also indirectly determined the radius of curvature of the bubble embryo entrapped inside a nucleation site. Orders of magnitude of the bubble embryos radii found with their method are compatible with those we found here with another method, which is rather satisfying. Glossary A c0 Figure 7. Correlation between the frequency of bubble production and the bubble growth rate of the corresponding bubble train, for each of the five nucleation sites followed with time while degassing. Table 1. Critical Bubble Growth Rate k*, Critical Supersaturating Ratio S*, Radius of Curvature of the Bubble Embryo Meniscus r, and Dimension of the Nucleation Site Aperture L, for Each of the Five Nucleation Sites Followed with Time While Degassing site no.
k* (µm/s)a
S*b
r (µm)c
L (µm)d
1 2 3 4 5
30 35 50 105 75
0.41 0.49 0.70 1.47 1.05
2.3 1.9 1.3 0.6 0.9
4.6 3.8 2.6 1.2 1.8
a Critical bubble growth rate as obtained in Figure 7. b Critical supersaturating ratio as defined in eq 13: S* ≈ (P0/ 0.63kBT)D-2/3‚ (2RFg/9η)-1/3(k*/c0). c Critical radius of curvature as defined from eq 3: r ) 2γ/P0S*. d Order of magnitude of the nucleation site aperture L ≈ 2r.
possible to deduce the radius of curvature r of the bubble embryo entrapped inside the given nucleation site. Finally, for a given bubble train associated with a given nucleation site, the experimental determination of the critical bubble growth rate during ascent k* gives access to the radius of curvature r of the meniscus inside the particle acting as a nucleation site, which constitutes an order of magnitude of the microcavity aperture L. The following scheme resumes the different steps which enabled us to indirectly deduce the radius of curvature r of the meniscus inside the particle acting as a nucleation site.
f ) 0 w k* w S* wr from Figure 7 from eq 13 from eq 3 Results obtained from the determination of the critical growth rates, k*, to the dimensions, L, of the corresponding nucleation sites are listed in Table 1, for each of the five bubble trains followed with time. To sum up, the close examination of bubble formation frequencies and bubble growth rates in the corresponding bubble train (experimental quantities easily detectable with our simple experimental setup) gives access to the
cL
D f FB FC g H k kB K L n N P0 PB PL Pe r rc r* R R0 Re S Sh t T U V
m2
bubble area, equilibrium concentration of dissolved CO2 molecules in the liquid medium corresponding to a pressure of CO2 molecules in the vapor phase of 1 atm, molecules m-3 concentration of dissolved CO2 molecules in the supersaturated liquid medium corresponding to a pressure of CO2 molecules in the vapor phase of several atmospheres, molecules m-3 diffusion coefficient of dissolved CO2 molecules in the liquid, m2 s-1 frequency of bubble detachment from nucleation sites, s-1 buoyancy, N capillary force anchoring the bubble to its nucleation site, N acceleration due to gravity, 9.8 m s-2 Henry’s law constant bubble growth rate dR/dt, m s-1 Boltzmann constant, 1.38 × 10-23, J K-1 mass transfer coefficient, m s-1 order of magnitude of a nucleation site aperture, m index affected to a bubble of a bubble train number of CO2 molecules atmospheric pressure, ≈105 N m-2 pressure inside a bubble, N m-2 pressure of CO2 molecules over the flat surface of the liquid, N m-2 Peclet number, 2RU/D radius of curvature of the meniscus entrapped inside the particle acting as nucleation site, m critical radius in the classical nucleation theory, m critical radius of curvature below which the nonclassical nucleation referred as type IV becomes impossible (see eq 2), m bubble radius, m bubble radius at detachment, m Reynolds number, 2FRU/η supersaturating ratio as defined in eq 1 Sherwood number, 2KR/D time, s period of bubble formation, s bubble velocity, m s-1 bubble volume, m3
Greek Letters (13) Jones, S. F.; Evans, G. M.; Galvin, K. P. Adv. Colloid Interface Sci. 1999, 80, 51-84.
R
numerical prefactor in eq 4
Nucleation Sites in a Glass of Champagne φ λ η F γ θ
contact angle between the solid substrate and the vapor phase (see Figures 3 and 4) thickness of the boundary layer, m fluid dynamic viscosity, kg m-1 s-1 fluid density, kg m-3 liquid surface tension, N m-1 temperature, K
Langmuir, Vol. 18, No. 4, 2002 1301
Acknowledgment. Thanks are due to the Europoˆl’Agro Institute and to the “Association Recherche Oenologie Champagne Universite´” for financial support and to Champagne Pommery and Verrerie Cristallerie d’Arques for supplying wines and glasses. LA0115987
