Modeling the Kinetics of Bubble Nucleation in ... - ACS Publications

Oct 3, 2006 - This tiny gas pocket was modeled as a slug-bubble growing trapped inside an ideal cylindrical microchannel and being fed with CO2-dissol...
1 downloads 3 Views 371KB Size
Download PDF
J. Phys. Chem. B 2006, 110, 21145-21151

21145

Modeling the Kinetics of Bubble Nucleation in Champagne and Carbonated Beverages Ge´ rard Liger-Belair,* Maryline Parmentier, and Philippe Jeandet Laboratoire d’Œnologie et Chimie Applique´ e, UPRES EA 2069, URVVC, Faculte´ des Sciences de Reims, Moulin de la Housse, B.P. 1039, 51687 Reims, Cedex 2, France ReceiVed: June 28, 2006; In Final Form: August 12, 2006

In champagne and carbonated beverages, bubble nucleation was mostly found to take place from tiny Taylorlike bubbles trapped inside immersed cellulose fibers stuck on the glass wall. The present paper complements a previous paper about the thorough examination of the bubble nucleation process in a flute poured with champagne (Liger-Belair et al. J. Phys. Chem. B 2005, 109, 14573). In this previous paper, a model was built that accurately reproduces the dynamics of these tiny Taylor-like bubbles that grow inside the fiber’s lumen by diffusion of CO2-dissolved molecules. In the present paper, by use of the model recently developed, the frequency of bubble formation from cellulose fibers is accessed and linked with various liquid and fiber parameters, namely, the concentration cL of CO2-dissolved molecules, the liquid temperature θ, its viscosity η, the ambient pressure P, the course of the gas pocket growing trapped inside the fiber’s lumen before releasing a bubble, and the radius r of the fiber’s lumen. The relative influence of the latter parameters on the bubbling frequency is discussed and supported with recent experimental observations and data.

1. Introduction In champagne and sparkling wines, carbon dioxide molecules in excess form together with ethanol when yeast ferments sugars. They are responsible for producing gas bubbles as soon as the bottle is uncorked. In soda drinks and most of fizzy waters, industrial carbonation is the source of effervescence.1 Generally speaking, sparkling beverages are weakly supersaturated with CO2-dissolved gas molecules. In weakly supersaturated liquids such as carbonated beverages in general, bubble formation and growth require preexisting gas cavities with radii of curvature large enough to overcome the nucleation energy barrier and grow freely.2,3 Jones et al. made a classification of the broad range of nucleation likely to be encountered in liquids supersaturated with dissolved gas.4 Bubble formation from preexisting gas cavities larger than the critical size is referred to as nonclassical heterogeneous bubble nucleation (type IV bubble nucleation, following their nomenclature). The close-up observation of glasses poured with carbonated beverages recently revealed that most of the bubble nucleation sites were found to be located on preexisting gas cavities trapped inside hollow and roughly cylindrical cellulose-fiber-made structures on the order of 100 µm long with a cavity mouth of several micrometers.5-7 The hollow cavity inside these fibers where a gas pocket is trapped during the pouring process is called the lumen. The mechanism of bubble release from a fiber’s lumen already has been described and partly modeled in recent papers.5-8 In short, after a bottle of champagne or sparkling wine is opened, the thermodynamic equilibrium of CO2 molecules dissolved in the liquid medium is broken. CO2dissolved molecules become in excess in comparison with what the liquid medium can withstand. Therefore, CO2 molecules will escape from the liquid medium through every available gas/ liquid interface to reach a vapor phase. Actually, once the sparkling beverage is poured into a glass, the tiny gas pockets * Corresponding author. E-mail: [email protected]. Tel: (33)3 26 91 86 14. Fax: (33)3 26 91 33 40.

trapped inside the collection of fibers adsorbed on the glass wall offer gas/liquid interfaces to CO2-dissolved molecules, which cross the interface toward the gas pockets. In turn, gas pockets grow inside the fibers’ lumen. When a gas pocket reaches the tip of a fiber, a bubble is ejected, but a portion of the gas pocket remains trapped inside the fiber’s lumen, shrinks back to its initial position, and the cycle starts again until bubble production stops through lack of dissolved gas molecules (see Figure 1). The fiber displayed in Figure 1 is a sort of textbook case, the behavior of which was recently understood and modeled.8 The fine mechanisms behind the regular production of bubbles from those tiny hollow and cylindrical structures are indeed still under progress.9 Champagne and sparkling wine tasters often pay much importance to the allure of effervescence in the glass, and even if there is no evidence yet to believe that bubbles confer any other sensory advantage to the wine, it often is recognized that bubbles play a major role in the assessment of champagne and sparkling wines. For this reason, champagne makers and champagne houses have spent much effort during the past few years to better detect, understand, and control the various parameters involved in the kinetics of effervescence and foaming. Actually, the average bubbling frequency of a single cellulose fiber is a key parameter that determines the overall number of bubbles produced in a glass per second, and, therefore, the overall kinetics of gas discharging from the supersaturated liquid. The model developed in ref 8 accurately reproduces the dynamics of this tiny Taylor-like bubble that grows inside the fiber’s lumen by diffusion of CO2-dissolved molecules. In the present paper, by use of the model recently developed, the frequency of bubble formation from cellulose fibers is accessed and linked with various liquid and fiber parameters, namely, the concentration cL of CO2-dissolved molecules, the liquid temperature θ, its viscosity η, the ambient pressure P, the course of the gas pocket growing trapped inside the fiber’s lumen before releasing a bubble, and the radius r of the fiber’s lumen. The

10.1021/jp0640427 CCC: $33.50 © 2006 American Chemical Society Published on Web 10/03/2006

21146 J. Phys. Chem. B, Vol. 110, No. 42, 2006

Liger-Belair et al.

Figure 2. The champagne’s viscosity as a function of temperature in the range of usual champagne-tasting temperatures (on a semilogarithmic scale). The solid line is the best fit to experimental data drawn with the Arrhenius-like eq 1.

temperatures, the liquid viscosity η is highly temperaturedependent. The temperature dependence of the champagne viscosity was measured with a thermostated Ubbelhode capillary viscosimeter. The result is displayed in Figure 2, in which the champagne dynamic viscosity is plotted vs temperature in the range of wine-tasting temperatures previously defined. As expected, the temperature dependence of the dynamic liquid viscosity (expressed in kg/m/s) accurately follows an Arrheniuslike equation

η(θ) ≈ 1.08 × 10-7 exp(2806/θ)

Figure 1. (Panels 1-5) Time-sequence illustrating one period of the cycle of bubble production from the lumen of a typical hollow cellulose fiber adsorbed on the wall of a glass poured with champagne (reprinted with permission from ref 8; copyright 2005 American Chemical Society); the time interval between successive frames is about 200 ms. Bar ) 50 µm.

relative influence of the latter parameters on the bubbling frequency is discussed and supported with recent experimental observations and data. 2. Experimental Section 2.1. Material. A standard commercial champagne wine was poured in a classical crystal flute that was first rinsed using distilled water and then was air-dried. Some physicochemical parameters of champagne were already determined at 20 °C with a sample of champagne first degassed.5 The static surface tension of champagne γ was found to be on the order of 47 mN/m, its density F was measured and found to be 998 kg/m3, and its dynamic viscosity η was found to be on the order of 1.5 × 10-3 kg m-1 s-1. In the range of temperature in which champagne and sparkling wine tasting is concerned (approximately 5-15 °C), their surface tension and density do not drastically vary. However, in this range of champagne-tasting

(1)

where θ is the absolute temperature (in K). In a previous work, the carbon dioxide dissolved into the liquid matrix was accurately measured with an imaging spectrometer through use of the 13C magnetic resonance spectroscopy (MRS) technique, which was an original unintrusive and nondestructive method.10,11 Typically, champagne holds about 10 g/L of CO2-dissolved molecules (before gas discharging in a flute). Nevertheless, as time proceeds after pouring this supersaturated liquid in a flute, the concentration cL of CO2dissolved molecules in the liquid progressively decreases to finally almost vanish a few hours later to fulfill Henry’s law, which states that the concentration of dissolved gas in a solution is proportional to its partial pressure P in the vapor phase (cL ) kHP, where kH is the Henry’s law constant). Actually, like the dynamic viscosity η, kH also is highly temperaturedependent. Recently, in the case of our standard champagne wine, the dependence of kH with temperature was expressed with the following van’t Hoff-like equation12

[

kH(θ) ) k298K exp -

∆Hdiss 1 1 R θ 298

(

)]

(2)

where k298 K ) 1.21 × 10-5 kg m-3 ps-1 is the Henry’s law constant at 298 K, ∆Hdiss ≈ -24 800 J/mol is the dissolution enthalpy of CO2 molecules in champagne (in J/mol), and R is the ideal gas constant (8.31 J mol-1 K-1). 2.2. Set-up Used to Observe Bubble Nucleation in Closeup. The dynamics of bubble nucleation was captured with a high-speed digital video camera (Speedcam+, Vannier Photelec, Antony, France) that was able to film up to 2000 frames per second and was fitted with a microscope objective (Mitutoyo, M Plan Apo 5, Japan). The champagne was first poured into the flute, which was placed between the objective and a cold source light (Fiber-Lite, PL-900, dc regulated illuminator). A

Kinetics of Bubble Nucleation in Champagne

J. Phys. Chem. B, Vol. 110, No. 42, 2006 21147

Figure 3. Scheme (a) and photographic detail (b) of the workbench used to observe bubble nucleation sites in close-up.

scheme of the whole workbench is shown in Figure 3A together with a photographic detail of the flute emplacement in Figure 3B. The experimental bubbling frequency fexp was directly accessed simply by measuring the time Texp needed by the nucleation site to release a bubble (therefore, fexp ) 1/Texp). It is worth noting that, in the present paper, we focused on nucleation sites releasing bubbles with clockwork regularity, which is not always the case. Actually, for a given bubble nucleation site the time intervals between the release of successive bubbles may indeed vary. In a previous paper, we reported some transitions from multiperiodic to single periodic bubbling regimes during the formation of sparkling bubbles.13 The present paper deals with modeling the bubbling frequency of nucleation sites that exclusively release bubbles with clockwork regularity. 3. Results and Discussion 3.1. Required Background. The model developed in a previous paper that accurately reproduces the dynamics of the tiny gas pocket, which grows by diffusion inside a fiber’s lumen (see Figure 1), is the starting point of our discussion.8 This tiny gas pocket was modeled as a slug-bubble growing trapped inside an ideal cylindrical microchannel and being fed with CO2-dissolved molecules diffusing (i) directly from both ends of the gas pocket and (ii) through the fiber wall, which consists of closely-packed cellulose microfibrils oriented mainly in the direction of the fiber.14 A scheme is displayed in Figure 4 in which the geometrical parameters of the tiny gas pocket growing by diffusion are defined. Taking into account the diffusion of CO2-dissolved molecules from the liquid bulk to the gas pocket via the two ways defined above, the growth of

Figure 4. Real gas pocket trapped inside the lumen of a cellulose fiber acting as a bubble nucleation site in a glass poured with champagne (A), which is modeled as a slug-bubble trapped inside an ideal cylindrical microchannel and being fed with CO2-dissolved molecules diffusing (i) directly from the liquid bulk through both ends of the gas pocket and (ii) through the wall of the microchannel (B). Bar ) 50 µm.

21148 J. Phys. Chem. B, Vol. 110, No. 42, 2006

Liger-Belair et al. possible to access the frequency of bubble formation f from a single fiber as follows:

f)

1 1 ≈ T τ ln[(zf + Aτ)/(z0 + Aτ)]

(4)

By replacing τ and A in eq 4 by their respective expression given in eq 3, the bubbling frequency of the cellulose fiber may be rewritten as follows:

f≈

Figure 5. Between the release of two bubbles from the fiber’s tip, the gas pocket trapped inside the fiber’s lumen oscillates between two situations where its length varies from its initial length, denoted z0, and its final length, denoted zf.

this gas pocket with time t was linked with both liquid and fiber parameters as follows:

{

z(t) ≈ (z0 + Aτ) exp(t/τ) - Aτ 4RθD0∆c (P + 2γ/r)rλ with τ ) , and A ) 2RθD⊥∆c (P + 2γ/r)λ

(3)

where z is the length of the gas pocket, z0 is the initial length of the gas pocket before it starts its growth through the lumen at each cycle of bubble production (see Figure 1, panel 1), P is the ambient pressure, D0 is the diffusion coefficient of CO2dissolved molecules in the liquid bulk, D⊥ is the diffusion coefficient of CO2-dissolved molecules through the fiber wall (and, therefore, perpendicular to the cellulose microfibrils), ∆c ) cL - cB ) cL - kHPB ) cL - kH(P + 2γ/r) is the difference in CO2-dissolved concentrations between the liquid bulk and the close vicinity of the gas pocket surface in equilibrium with the gaseous CO2 in the gas pocket, and λ is the boundary layer thickness where a linear gradient of CO2-dissolved concentration is assumed. In a previous work, the transversal diffusion coefficient D⊥ of CO2 molecules through the fiber wall was approached and was properly bounded by D⊥/D0 ≈ 0.1 and D⊥/D0 ≈ 0.3.7 For modeling purposes, an intermediate value of about D⊥ ≈ 0.2D0 was proposed and will be used hereafter. 3.2. Modeling the Average Bubbling Frequency. As seen in Figure 1, the whole process leading to the production of a bubble from a cellulose fiber’s tip can be coarsely divided in two main steps: (i) the growth of the gas pocket trapped inside the fiber’s lumen, and (ii) the bubble detachment as the gas pocket reaches the fiber’s tip. Actually, it is clear from the numerous close-up time sequences taken with the high-speed video camera that the time scale of the bubble detachment is always very small (≈1 ms) when compared with the relatively slow growth of the gas pocket (several tens to several hundreds of milliseconds). Therefore, the whole cycle of bubble production seems to be largely governed by the growth of the gas pocket trapped inside the fiber’s lumen. The period of bubble formation from a single cellulose fiber is therefore equal to the total time T required by the tiny gas pocket to grow from its initial length, denoted z0, to its final length, denoted zf, as it reaches the fiber’s tip (see Figure 5). By retrieving eq 3, it is

2RθD⊥∆c rλ(P + 2γ/r) ln[(zf + 10r)/(z0 + 10r)]

(5)

To go further with the dependence of the bubbling frequency with both liquid and fiber parameters, we also can replace the diffusion coefficient D0 in eq 5 by its theoretical expression approached through the well-known Stokes-Einstein equation (D0 ≈ kBθ/6πηd) with kB being the Boltzman constant (1.38 × 10-23 J/K) and d being the characteristic size of the CO2 molecule’s hydrodynamic radius (d ≈ 10-10 m). In eq 5, by replacing each parameter by its theoretical expression and each constant by its numerical value, the variation of the bubbling frequency as a function of the various pertinent parameters involved may be rewritten as follows (in the MKSA system):

θ2[cL - kH(P + 2γ/r)] (6) f ≈ 2.4 × 10-14 ηr(P + 2γ/r)λ ln[(zf + 10r)/(z0 + 10r)] The boundary layer thickness λ was indirectly approached in a recent paper and found to be on the order of 20 µm.8 Let us apply the latter equation to the standard textbook case fiber displayed in Figure 1 and modeled in Figure 4 (i.e., r ≈ 5 µm, z0 ≈ 20 µm and zf ≈ 100 µm). Equation 6 may be rewritten as the following by replacing the fiber’s parameters r, z0, zf, and λ by their numerical values:

f ≈ 5.2 × 10

θ2[cL - kH(P + 0.2)]

-8

η(P + 0.2)

(7)

In the latter expression, cL is expressed in g/L, kH is expressed in g/L/atm, P is expressed in atm, and η is expressed in kg/m/s to fit the standards used in enology. 3.3. Relative Influence of Several Parameters on the Average Bubbling Frequency. We will discuss the relative influence of the following parameters on the average bubbling frequency: (i) the concentration cL of CO2-dissolved molecules, (ii) the liquid temperature θ, and (iii) the ambient pressure P. (i) Through the use of eq 6 with every other parameter being constant, the dependence of the theoretical average bubbling frequency f with the CO2-dissolved concentration cL is in the form f ) acL - b. By use of the high-speed video camera, a few cellulose fibers that act as bubble nucleation sites on the wall of a glass poured with champagne were followed with time during the whole gas discharging process, which lasted several hours. For each given nucleation site, the experimental bubbling frequency fexp was directly accessed simply by measuring the time Texp needed by the nucleation site to release a bubble (therefore, fexp ) 1/Texp). The concentration cL of CO2-dissolved molecules was indirectly accessed by using the growth rate of bubbles just released from the nucleation site as a CO2-probe. This method is developed in minute detail in ref 6 (see Appendix). The dependence of the experimental bubbling frequency fexp with cL is displayed in Figure 6 for four different

Kinetics of Bubble Nucleation in Champagne

Figure 6. Dependence of the experimental bubbling frequency fexp with cL for four different bubble nucleation sites (at 20 °C). Experimental data show a linear-like dependence between fexp and cL, as expected from the theoretical model displayed in eq 6.

nucleation sites. It is clear from Figure 6 that the dependence of the experimental bubbling frequency follows a linear-like cL dependence as was expected from the model developed in the latter paragraph. Furthermore, it is worth noting that the bubbling frequency of a given nucleation site vanishes (i.e., the bubble release ceases, f f 0 bubble/s), although the CO2-dissolved concentration remains higher than a critical value as demonstrated in a previous paper.6 Actually, following both Laplace’s law and Henry’s law, the curvature of the CO2 pocket trapped inside the fiber’s lumen induces a concentration cB of CO2-dissolved molecules on the order of kH(P0 + 2γ/r) in close vicinity of the trapped CO2 pocket . Consequently, as soon as the concentration of CO2-dissolved in the liquid bulk reaches a critical value c/L ) cB ≈ kH(P0 + 2γ/r), the diffusion toward the gas pocket ceases and the given nucleation site stops releasing bubbles, which is shown in Figure 6. Let us apply the latter condition to the characteristic radius of a cellulose fiber6 (r ≈ 2 µm). At 20 °C, the critical concentration cL* in which below that bubble release becomes impossible is

c/L ≈ kH(P0 + 2γ/r) ≈ 1.44 × 10-5(105 + 2 × 5 × 10-2/2 × 10-6) ≈ 2.2 g/L (8) In Figure 6, the wide range of different c/L displayed by extrapolating the data to f ) 0 for each nucleation site followed as time progresses is directly linked with the natural variability of cellulose fibers that may be found in a glass of champagne (in terms of lumen’s radius r). Actually, the typical range of lumens radii may vary from about 1 to 5 µm.5-8 By applying eq 8, it is found that the typical range of c/L may vary from about 1.7 g/L to about 2.9 g/L, which is clearly quite consistent with the range of c/L found experimentally by extrapolating the data to f ) 0 as shown in Figure 6. (ii) The dependence of the bubbling frequency with the liquid temperature is much more difficult to test experimentally in real consuming conditions. Actually, we needed time to decrease or increase the liquid temperature, and we found no satisfying possibility of modifying the liquid temperature without significantly losing CO2-dissolved molecules, which continuously desorb from the supersaturated liquid matrix because of diffusion through the liquid surface and because of bubbling from the numerous nucleation sites found in the flute. We will discuss

J. Phys. Chem. B, Vol. 110, No. 42, 2006 21149

Figure 7. Theoretical dependence of the bubbling frequency f with the temperature θ, as expected from the model displayed in eq 7, in the range of usual champagne-tasting temperatures (from 5 to 15 °C) and for the textbook case fiber displayed in Figure 1.

Figure 8. Theoretical dependence of the bubbling frequency f with the ambient pressure P (at 20 °C), as expected from the model displayed in eq 7 for the textbook case fiber displayed in Figure 1.

the theoretical influence of the liquid temperature by using eq 6. In eq 6, the temperature directly appears as θ2, but the Henry’s law constant kH, as well as the dynamic viscosity η, are strongly temperature-dependent and expressed by eqs 1 and 2, respectively. Increasing the liquid temperature by 10 K (let us say from 278 to 288 K, which is approximately the range of champagne-tasting temperatures) increases the theoretical bubbling frequency by about 50%. For the standard textbook case fiber displayed in Figure 1 (r ≈ 5 µm, z0 ≈ 20 µm, and zf ≈ 100 µm) and with cL ≈ 10 g/L, the theoretical temperature dependence of the bubbling frequency is displayed in Figure 7. (iii) Increasing or decreasing the ambient pressure P also significantly modifies the corresponding average bubbling frequency f. For the standard textbook case fiber displayed in Figure 1 (r ≈ 5 µm, z0 ≈ 20 µm, and and zf ≈ 100 µm) and with cL ≈ 10 g/L, the theoretical pressure dependence of the bubbling frequency is displayed in Figure 8. Reducing the ambient pressure to only 0.3 atm (e.g., on the top of Mount Everest) would increase the average bubbling frequency by a factor of almost 3. This is basically the same phenomenon that is responsible for gas embolism in divers who have breathed high-pressure air under water when they resurface too quickly. Inversely, if the ambient pressure is increased to 2 atm, this decreases the average bubbling frequency by a factor of about 2.

21150 J. Phys. Chem. B, Vol. 110, No. 42, 2006 3.4. A Comparison with the Model by Uzel et al.9 The model developed here is not the only one in the literature. The one recently developed by Uzel et al. makes different assumptions about the way in which the bubbles form.9 Contrary to the present model, which assumes that the dominant time scale that controls bubble release is the growth of the gas pocket growing trapped inside the fiber’s lumen, Uzel’s model essentially assumes that the dominant mechanism is seeding at the top of the fiber, the bubble detachment being controlled by a balance between buoyancy and capillarity. Seeding at the fiber’s tip necessarily involves unfavorable wetting conditions of the fiber’s wall. Actually, because cellulose is a highly hydrophilic material, aqueous liquids such as champagne and carbonated beverages forbid the establishment of a triple contact line at the fiber’s tip. Therefore, a gas bubble released after the breaking of the gas pocket growing trapped inside the fiber’s lumen (Figure 1, panel 5) should be unable to seed at the fiber’s tip (as a droplet of water would do). However, some nucleation sites could eventually experience transitions from wetting to nonwetting conditions (due to lipid contamination, for example). In such conditions, a bubble could definitely grow anchored to the fiber’s tip before detaching because of buoyancy. In such conditions, the total time needed to release a bubble from a nucleation site would be the sum of two characteristic times: (1) the time required by the gas pocket to grow from its initial length, z0, to its final length, zf, as it reaches the fiber’s tip (modeled in the present paper), and (2) the time required for the bubble to grow until it detaches because of buoyancy (modeled in the paper by Uzel et al.9). This is the reason why these two extreme models are highly complementary. They focus on both steps of the bubble release, the best model probably being a combination of the two models. Mixing these two models could be the purpose of future work. 4. Conclusions and Prospects The dynamics of bubble nucleation from the lumen of immersed cellulose fibers stuck on the wall of a glass poured with champagne were accurately observed, in situ, from highspeed video recordings. We used a previously developed model that utilized the dynamics of the tiny gas pockets responsible for the repetitive production of bubbles to access the theoretical frequency of bubble release. The theoretical frequency of bubble release was found to depend on various fiber and liquid parameters. The main parameters likely to significantly influence the frequency of bubble formation were found to be (i) the CO2dissolved concentration cL, (ii) the liquid temperature θ, and (iii) the ambient pressure P. In the future, we plan to organize at a large-scale champagne and sparkling wine tasting under a controlled atmosphere by artificially modifying the intensity of effervescence through the ambient pressure parameter. We could eventually better understand the role played by the intensity of effervescence on flavor release without modifying the intrinsic aromatic properties of the liquid (such as changes in the temperature would do). Actually, the continuous flow of ascending bubbles through the liquid drains flavors and aromas with the result of continuously renewing the layer at the liquid surface, thus providing a better release of the numerous volatile organic compounds above the liquid surface. However, bubbles bursting at the liquid surface also release a sudden and abundant quantity of CO2 above the liquid surface with the result of irritating the nose during the evaluation of aromas. From an enological point of view, there should therefore be a sort of ideal intensity of effervescence that enhances flavor release without being too aggressive in

Liger-Belair et al. terms of CO2 release, which we would like to both theoretically and experimentally evaluate in the future. Acknowledgment. The authors thank the Europol’Agro institute and the “Association Recherche Oenologie Champagne Universite´” for financial support and Champagne Pommery and Arc International for their collaborative efforts. Appendix The general equation concerning the mass transfer of molecules from the bulk of a supersaturated liquid to a bubble surface with time is

dN ) KA∆c dt

(A1)

where N is the number of transferred CO2 moles, K is the mass transfer coefficient, A is the bubble area, and ∆c ) cL - c0 is 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 rising bubble as ideal (PBV ) NRθ), the number of CO2 moles transferred into the bubble is connected with the variation of its radius R with time as follows

dN PB dV PB dR ) ) A dt Rθ dt Rθ dt

(A2)

where V is the bubble volume, and PB is the pressure inside the rising bubble assumed to be equal to the atmospheric pressure P0, because both the hydrostatic pressure (FgH < 10-2P0) and the Laplace pressure (2γ/R < 10-1P0) inside the rising bubble are negligible. By combining eqs A1 and A2, one obtains the rate of expansion of the bubble radius

dR Rθ ) K∆c dt P0

(A3)

Generally speaking, heat and mass transfers are functions of two dimensionless numbers, the Sherwood and Peclet numbers, Sh ) 2KR/D0 and Pe ) 2RU/D0, respectively, with U being the velocity of ascending bubbles. 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.15 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 ≈ D0 2 D0

1/3

U1/3 w K ≈ 0.63D02/3 2/3 R

(A4)

Combining eqs A3 and A4 leads to

dR U1/3 Rθ ) 0.63 D02/3 2/3 ∆c dt P0 R

(A5)

By combining the empirical rising velocity U of ascending champagne bubbles5 with eq A5, the rate of increase of the ascending bubble radius becomes

k)

dR 2Fg 1/3 Rθ ∆c ≈ 0.63 D02/3 dt P0 9η

( )

where  is a numerical prefactor close to 0.7.

(A6)

Kinetics of Bubble Nucleation in Champagne

J. Phys. Chem. B, Vol. 110, No. 42, 2006 21151

Equation A6 was found to be in very good agreement with the order of magnitude of experimental bubble growth rates in champagne and beer glasses.5,6 Therefore, 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 ∆c in the liquid medium. Finally, by retrieving eq A6 in reverse order, ∆c may be linked to the rising bubble growth rate k and some liquid parameters as follows:

∆c ) cL - c0 ≈

P0 9η 1/3 k D0-2/3 0.63Rθ 2Fg

( )

(A7)

where c0 ) kHP0 is the molar concentration of CO2-dissolved molecules in equilibrium with the CO2 gas into the bubble. Finally

c L ≈ c0 +

P0 9η 1/3 k D -2/3 0.63Rθ 0 2Fg

( )

(A8)

It is worth noting that in eq A8, each parameter is expressed in the MKSA system. In particular, cL and c0 are expressed in mol/m3. Nomenclature A ) bubble area, m2. A ) variable depending on several parameters and has the dimension of a velocity (see eq 3), m/s. c0 ) molar concentration of CO2-dissolved molecules in equilibrium (following Henry’s law) with a partial pressure of gaseous CO2 corresponding to the atmospheric pressure (i.e., c0 ) kHP0, mol/m3) cB ) concentration of CO2-dissolved molecules around the gas pocket in equilibrium with the pressure of CO2 molecules in the vapor phase inside the gas pocket (following Henry’s law), mol/m3. cL ) concentration of CO2-dissolved molecules in the bulk of the supersaturated liquid medium, mol/m3. ∆c ) difference in CO2-dissolved concentrations between the liquid bulk and the close vicinity of the bubble surface in equilibrium with the gaseous CO2 into the gas pocket ) cL cB, mol/m3. d ) characteristic size of the CO2 molecule’s hydrodynamic radius (≈10-10 m). D ) diffusion coefficient, m2/s. D0 ) diffusion coefficient in the liquid bulk of CO2-dissolved molecules, m2/s. D⊥ ) transversal diffusion coefficient of CO2-dissolved molecules inside the fiber wall, m2/s. f ) frequency of bubble formation, bubbles/s. fexp) frequency of bubble formation experimentally determined, in situ, from high-speed video recordings, bubbles/s. kH ) Henry’s law constant, kg m-3 ps-1. k ) bubble growth rate dR/dt, m/s. kB ) Boltzman constant, 1.38 × 10-23, J/K.

K ) mass transfer coefficient, m/s. N ) number of CO2 moles. P ) ambient pressure, N/m2. P0 ) atmospheric pressure, ≈ 105 N/m2. PB ) pressure inside a bubble, N/m2. Pe ) Peclet number, 2RU/D. r ) radius of aperture of the lumen of a cellulose fiber acting as a bubble nucleation site, ≈1-5 µm. R ) bubble radius, m. R ) ideal gas constant, 8.31 J K-1 mol-1. Sh ) Sherwood number, 2KR/D. t ) time, s. T ) period of bubble production, s. Texp ) experimental period of bubble production, s. U ) bubble velocity, m/s. V ) bubble volume, m3. z ) length of the gas pocket growing trapped inside a fiber’s lumen. z0 ) initial length of the gas pocket growing trapped inside a fiber’s lumen. zf ) final length of the gas pocket growing trapped inside a fiber’s lumen, when it reaches a fiber’s tip. Greek letters  ) numerical prefactor in eq A6. λ ) thickness of the boundary layer, m. η ) fluid dynamic viscosity, kg m-1 s-1. F ) fluid density, kg/m3. γ ) liquid surface tension, N/m. θ ) temperature, K. τ ) characteristic time scale of the exponential-like bubble growth (see eq 3), s. References and Notes (1) Liger-Belair, G. Uncorked: The Science of Champagne; Princeton University Press: Princeton, NJ, 2004. (2) Blander, M.; Katz, J. L. AIChE J. 1975, 21, 836. (3) Lubetkin, S. D. Langmuir 2003, 19, 2575. (4) Jones, S. F.; Evans, G. M.; Galvin, K. P. AdV. Colloid Interface Sci. 1999, 80, 27. (5) Liger-Belair, G. Ann. Phys. (Paris) 2002, 27, 1. (6) Liger-Belair, G.; Vignes-Adler, M.; Voisin, C.; Robillard, B.; Jeandet, P. Langmuir 2002, 18, 1294. (7) Liger-Belair, G.; Topgaard, D.; Voisin, C.; Jeandet, P. Langmuir 2004, 20, 4132. (8) Liger-Belair, G.; Voisin, C.; Jeandet, P. J. Phys. Chem. B 2005, 109, 14573. (9) Uzel, S.; Chappell, M. A.; Payne, S. J. J. Phys. Chem. B 2006, 110, 7579. (10) Liger-Belair, G.; Prost, E.; Parmentier, M.; Jeandet, P.; Nuzillard, J.-M. J. Agric. Food Chem. 2003, 51, 7560. (11) Autret, G.; Liger-Belair, G.; Nuzillard, J.-M.; Parmentier, M.; Dubois de Montreynaud, A.; Jeandet, P.; Doan, B. T.; Beloeil, J.-C. Anal. Chim. Acta 2005, 535, 73. (12) Liger-Belair, G. J. Agric. Food Chem. 2005, 53, 2788. (13) Liger-Belair, G.; Tufaile, A.; Robillard, B.; Jeandet, P.; Sartorelli, J.-C. Phys. ReV. E 2005, 72, 037204. (14) O’Sullivan, A. Cellulose 1997, 4, 173. (15) Sherwood, T. K.; Pigford, R. L.; Wilke, C. R. Mass transfer; Chemical Engineering Series; MacGraw-Hill: New York, 1975.