Modeling the Losses of Dissolved CO2 from Laser-Etched

Article pubs.acs.org/JPCB

Modeling the Losses of Dissolved CO2 from Laser-Etched Champagne Glasses Gérard Liger-Belair* Equipe Effervescence, Champagne et Applications - Groupe de Spectrométrie Moléculaire et Atmosphérique (GSMA), UMR CNRS 7331 - Université de Reims Champagne-Ardenne, UFR Sciences Exactes et Naturelles, BP 1039, Reims 51687 Cedex 2, France ABSTRACT: Under standard champagne tasting conditions, the complex interplay between the level of dissolved CO2 found in champagne, its temperature, the glass shape, and the bubbling rate definitely impacts champagne tasting by modifying the neuro-physicochemical mechanisms responsible for aroma release and flavor perception. On the basis of theoretical principles combining heterogeneous bubble nucleation, ascending bubble dynamics, and mass transfer equations, a global model is proposed, depending on various parameters of both the wine and the glass itself, which quantitatively provides the progressive losses of dissolved CO2 from laser-etched champagne glasses. The question of champagne temperature was closely examined, and its role on the modeled losses of dissolved CO2 was corroborated by a set of experimental data.

1. INTRODUCTION Since the end of the 17th century, champagne has been a renowned French sparkling wine, praised worldwide for the fineness of its effervescence (the very much sought-after bubbling process). Despite the huge body of research, initiated by Louis Pasteur in the 19th century, aimed at progressively unlocking wine science in general, only quite recently much interest was devoted to depict each and every parameter involved in the bubbling process characteristic of champagne and sparkling wines.1 For more information about the whole life cycle of champagne bubbles under standard tasting conditions, see the recent overview by Liger-Belair,2 and references therein. From a strictly chemical point of view, Champagne wines are multicomponent hydroalcoholic systems supersaturated with dissolved carbon dioxide (CO2) formed together with ethanol during the second fermentation process, called prise de mousse (promoted by adding yeasts and sugar inside bottles filled with a base wine and sealed with a cap). Champagnes, or sparkling wines elaborated through the same traditional method, therefore hold a concentration of dissolved CO2 proportional to the amount of sugar consumed by yeast to promote this second fermentation. The process of fermentation was first scientifically described by the French chemist Joseph-Louis Gay Lussac in 1810, when he demonstrated that glucose is the basic starting block for producing ethanol and gaseous CO2: C6H12O6 → 2CH3CH 2OH + 2CO2

per liter) is therefore roughly equivalent to half of the concentration of sugar (in grams per liter) added into the base wine in order to promote the prise de mousse.2 After the prise de mousse, champagne ages in a cool cellar for at least 15 months, but sometimes much longer, in order to develop its socalled bouquet.3−5 During this period called aging on lees, dissolved CO2molecules very slowly escape through the cap because of the huge difference in partial pressure of gaseous CO2 between the headspace of the bottle and the atmosphere. Bottles then undergo disgorging. Caps are removed in order to remove the sediment of dead yeast cells. Bottles are then quickly corked with traditional cork stoppers to prevent an excessive loss of dissolved CO2. After corking the bottle, dissolved and gas phase CO2molecules quickly recover equilibrium (i.e., dissolved CO2 molecules progressively desorb from the liquid phase to promote the increase of gaseous pressure under the cork, which finally and quickly recovers a stable value, inevitably slightly lower than that before disgorging). Experiments with early disgorged champagne samples were done recently. The characteristic concentration of dissolved CO2 measured inside freshly opened standard bottles was found to be of the order of 11.5 g L−1, and therefore in very good accordance with what is expected from fermenting 24 g L−1 of sugar to promote the prise de mousse in the sealed bottles.6−9 In sparkling beverages in general, and in champagne in particular, the level of dissolved CO2 in the liquid phase is indeed a parameter of paramount importance, since it is responsible for the visually appealing and very much sought-

(1)

Traditionally, 24 g/L of sugar is added in the base wine to promote the prise de mousse. Following eq 1, 24 g/L of sugar added in closed bottles to promote the second alcoholic fermentation produces approximately 11.8 g of CO2/L of wine. The concentration of dissolved CO2 in champagne (in grams © XXXX American Chemical Society

Received: February 10, 2016 Revised: March 29, 2016

A

DOI: 10.1021/acs.jpcb.6b01421 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B after repetitive bubbling process (the so-called ef fervescence). Moreover, dissolved CO2 is also responsible for the very characteristic tingling sensation in mouth. Indeed, the mouth feel perception of dissolved CO2 involves a truly very complex multimodal stimulus.10 During carbonated beverage tasting, dissolved CO2 acts on both trigeminal receptors11−15 and gustatory receptors, via the conversion of dissolved CO2 to carbonic acid,16,17 in addition to the tactile stimulation of mechanoreceptors in the oral cavity (through bursting bubbles). In recent years, glass-shape was found to play a key role concerning the rate at which dissolved CO2 escapes from champagne, with the narrow flute holding much more efficiently the dissolved CO2 as time proceeds than the wide and rather old-fashioned coupe.7,18,19 Recently also, both losses of dissolved CO2 and fluxes of gaseous CO2 escaping from a Champagne wine served in laser-etched glasses showing an increasing number of bubble nucleation sites were monitored.9 Logically, the more intense the effervescence is into a glass, the more rapid its progressive loss of dissolved CO2. Moreover, the temperature of champagne was also found to be a key parameter as concerns the losses of dissolved CO2 from ̈ champagne glasses.20,21 Even more recently, the falsely naive question of how many bubbles are likely to form per glass was discussed.22 A theoretical relationship was derived, which provides the whole number of bubbles likely to form per glass, depending on various parameters of both the wine and the glass itself.22 For all the aforementioned reasons, no wonder that the complex interplay between the level of dissolved CO2 found in champagne, its temperature, the glass shape, and the bubbling rate finally impacts champagne tasting by modifying the neurophysicochemical mechanisms responsible for aroma release and flavor perception. Following these recent highlights, better understanding the strong interplay between the various parameters at play during champagne and sparkling wine tasting could be of interest for sommeliers, wine journalists, champagne elaborators, glass makers, and any experienced tasters wondering about the complex mechanisms happening in a single glass of bubbly. Other research groups have indeed conducted several studies about the crucial role of glass on the wine expression during tasting, being, for example, the influence of glass shape on perceived aroma,23 on the evaporation rate of a still wine as a function of time,24 or more generally the influence of the container on the perception of the contents.25 In this Article, on the basis of theoretical principles combining the physics of heterogeneous bubble nucleation, ascending bubble dynamics, and mass transfer equations, a global model is proposed which quantitatively provides the progressive losses of dissolved CO2 from laser-etched champagne glasses, under standard tasting conditions. The question of temperature was closely examined, and its role on the modeled losses of dissolved CO2 is compared with a set of experimental data.

Table 1. Average Chemical Composition of a Standard Commercial Brut Labeled Champagne Wine (with pH ≈ 3.2 and ionic strength 0.02 M)26 compound

concentration

ethanol dissolved CO2 glycerol tartaric acid lactic acid sugars proteins polysaccharides polyphenols amino acids volatile organic compounds (VOCs) lipids K+ Ca2+ Mg2+ SO42− Cl−

≈12.5% v/v 11−12 g/L ≈5 g/L ≈2.5−4 g/L ≈4 g/L 6−12 g/L 5−10 mg/L ≈200 mg/L ≈100 mg/L 0.8−2 mg/L ≈700 mg/L ≈10 mg/L 200−450 mg/L 60−120 mg/L 50−90 mg/L ≈200 mg/L ≈10 mg/L

champagne style today (i.e., with relatively low levels of sugar), although throughout the entire 19th century and into the early 20th century, it was generally much sweeter than modern-day Champagne wines.27 In a wide range of temperature varying from approximately 2 to 22 °C, the experimental temperature dependence of the Brut champagne viscosity was found to classically obey an Arrheniuslike equation as follows28 ⎛ 2806 ⎞ ⎟ η(θ ) ≈ 1.08 × 10−7 exp⎜ ⎝ θ ⎠

(2)

with the champagne dynamic viscosity η being classically expressed in kg m−1 s−1 and the absolute temperature θ being expressed in K. 2.2. Temperature Dependence of Dissolved CO2 Diffusivity in Champagne. Diffusion is the process by which a molecule elbows its way through a medium and spreads out. Diffusion is a consequence of the constant stochastic thermal motion of molecules. The diffusion coefficient D of molecular species in a liquid phase, such as dissolved CO2 molecules in champagne, for example, is therefore strongly temperature-dependent. Recently, the diffusion coefficients of dissolved CO2 molecules in carbonated hydroalcoholic solutions, and in a standard commercial Brut Champagne wine, were accurately determined as a function of temperature, by classical molecular dynamics simulations, and by 13C nuclear magnetic resonance (NMR) spectroscopy, respectively.29,30 The temperature dependence of the dissolved CO2 diffusion coefficient in champagne was found to be in very good accordance with the well-known Stokes−Einstein relationship expressed hereafter

2. THEORETICAL METHODS 2.1. Temperature Dependence of the Champagne Viscosity. Champagne wines are multicomponent hydroalcoholic systems, with a density close to unity, a surface tension of γ ≈ 50 mN m−1 (indeed highly ethanol-dependent), and a viscosity about 50% larger than that of pure water, also mainly due to the presence of 12.5% v/v ethanol (see Table 1 which provides the average chemical composition of a typical Brut Champagne wine).26 Brut is indeed the most common

DCO2 ≈

kBθ 6πηl

(3) −23

−1

with kB being the Boltzmann constant (1.38 × 10 J K ) and l being the hydrodynamic radius of the solute molecule (namely, l ≈ 10−10 m for the CO2 molecule). By combining eqs 2 and 3 and by replacing each and every parameter by its numerical value, the temperature dependence B


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The Journal of Physical Chemistry B

nucleation sites).7−9,19,40,41 To promote bubble formation, such glasses were simply etched on their bottom, with a ring-shaped structure done with adjoining laser beam impacts (as seen in Figure 1). A sudden cooling after the laser beam impact causes

of the CO2 diffusion coefficient in a Brut labeled champagne (expressed in m2 s−1) may therefore be approached as follows: ⎛ 2806 ⎞ ⎟ DCO2 ≈ 6.8 × 10−8θ exp⎜ − ⎝ θ ⎠

(4)

To facilitate the reading of the Article, the subscript CO2 will be omitted hereafter, as well as the label Brut, referring to the most common type of champagne considered in this Article. 2.3. Temperature Dependence of CO2 Solubility in Champagne. Generally speaking, the solubility of gas species in a liquid phase is strongly temperature-dependent. Thermodynamically speaking, the solubility of gaseous CO2 in champagne (with kH being expressed in mol m−3 Pa−1) was found to be conveniently expressed with a classical Van’t Hofflike equation as follows31 ⎡ ΔHdiss ⎛ 1 1 ⎞⎤ ⎜ ⎟⎥ kH(θ ) = k 298K exp⎢ − − ⎝ ⎣ R 298 ⎠⎦ θ

(5)

with k298K being the Henry’s law constant of dissolved CO2 (its solubility) at 298 K (≈2.75 × 10−4 mol m−3 Pa−1),31 ΔHdiss being the dissolution enthalpy of CO2 molecules in the liquid phase (≈ −24 800 J mol−1),31 and R being the ideal gas constant (8.31 J K−1 mol−1). 2.4. A Critical Radius Required for Bubble Nucleation. After uncorking a bottle of champagne, sparkling wine, or carbonated beverage in general, the partial pressure of gaseous CO2 falls, and the liquid phase becomes supersaturated with dissolved CO2. To recover thermodynamic equilibrium, gaseous CO2 must escape the liquid phase. In Champagne wines, despite dissolved CO2 concentrations rising up to almost 12 g L−1, homogeneous bubble nucleation (i.e., ex nihilo) is still limited by a huge energy barrier to overcome, and is therefore thermodynamically forbidden.32−36 In order to nucleate and grow freely, bubbles need preexisting gas cavities immersed in the liquid phase, with radii of curvature larger than a critical size. This is the so-called nonclassical heterogeneous bubble nucleation process, also referred to as type IV nucleation, following the nomenclature by Jones et al.35 The critical radius r* of gas pockets required to enable bubble production in a liquid phase supersaturated with dissolved CO2 is expressed as follows37 r* ≈

2γkH (c L − kHP0)

Figure 1. Photographic evidence for bubble production from the ring shaped structure done at the bottom of a laser-etched flute poured with 100 mL of champagne (a and b); detail of the ring-shaped structure (c), done with adjoining laser beam’s points of impact (bar = 100 μm) (d); in frame a, dimensions are indicated in cm.

a network of crevices at the glass surface, as seen in the micrographs of adjoining laser beam impacts displayed in Figure 2. As champagne is served in such laser-etched glasses, this network of tiny crevices at the bottom of the glass surface enables the entrapment of tiny air pockets. The mechanism of gas pockets entrapment by a liquid front advancing over the surface of a solid substrate was probably first examined by Bankoff.42 In addition to some geometrical considerations such as both the depth and open aperture of such crevices, it is clear that unfavorable wetting conditions increase the probability of trapping air filled sites within crevices. For a global overview, see the article by Jones et al.,35 and references therein. As far as radii of curvature of these tiny air pockets are larger than the critical radius r* reported in eq 6, favorable conditions for the so-called nonclassical heterogeneous bubble nucleation process are met, as discussed in previous articles.37,43−45 Glasses showing adjoining laser beam impacts are thus able to promote repetitive bubble nucleation into the form of a very characteristic bubble column rising on their axis of symmetry, as seen in the photograph displayed in Figure 1. 2.6. Proposed Scenario for Heterogeneous Bubble Nucleation from the Bottom of the Glass. For modeling purposes, the network of fractures found at the bottom of a laser-etched glass is considered as a network of linear and thin crevices (i.e., with the fracture open aperture, denoted a, being much smaller than its linear extension, denoted L, as seen in

(6)

with γ being the surface tension of the air/liquid interface (of the order of 50 mN m−1 in Champagne wines2), P0 being the atmospheric pressure (≈105 N m−2 ≈ 1 bar), and cLbeing the concentration of dissolved CO2 found in the liquid bulk (expressed in mol m−3). Let us consider a typical champagne at a tasting temperature of 12 °C, showing a concentration of dissolved CO2 of the order of 8 g L−1 (i.e., cL ≈ 180 mol m−3) just after pouring into a glass, as reported by Liger-Belair et al.38,39 Retrieving eq 6 and replacing each and every parameter by its numerical value yields a critical radius r* required to enable bubble nucleation of the order of 0.3 μm. 2.5. Bubble Nucleation from Laser-Etched Champagne Glasses. Quite recently, glassmakers showed particular interest in proposing to consumers a new generation of laseretched champagne glasses, specially designed, with standardized conditions of effervescence (i.e., a rate of bubbling accurately controlled and promoted through laser-beam-driven bubble C


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Figure 3. Scheme of an air pocket trapped in a crevice modeled as an “ideal” fracture showing an open aperture a much smaller than its linear extension L.

quickly develops in the gas pocket emerging from the fracture, thus causing the breakup of the gas pocket. A tiny bubble is therefore ejected from the fracture, but a portion of the trapped gas pocket would still remain trapped within the linear crevice, shrink back to its initial position, and the cycle would start again until bubble production would stop through lack of dissolved CO2 in the liquid phase. For the sake of clarity, a 2D scheme displayed in Figure 4 shows how the diffusion of dissolved CO2 from the liquid phase to the gas pocket trapped within the crevice provides repetitive bubble production (the so-called and very much sought-after effervescence). In the present scenario, it is worth noting that the dominant time scale

Figure 2. Micrograph of adjoining laser beam impacts found at the bottom of a laser-etched champagne glass (bar = 500 μm) (a); detail of a single impact showing the network of linear crevices responsible for nonclassical heterogeneous bubble nucleation (bar = 100 μm) (b); the microscope software shows a network of fractures with characteristic average open apertures close to 2 μm (bar = 50 μm) (c).

Figure 2). Air pockets trapped in such a network of fractures are therefore considered as slug-bubbles filling the linear crevices, with a radius of curvature being half of the fracture aperture (of the order of 2 μm). A scheme is displayed in Figure 3. As they are immersed in a liquid phase supersaturated with dissolved CO2, every air pocket with a radius of curvature larger than the critical radius r* ≈ 0.3 μm defined above will thus ensure the bubbling process following the basic principles described hereafter. Dissolved CO2 molecules diffuse inside the trapped air pocket through the gas/liquid interface. In turn, the trapped gas pocket grows inside the microchannel until it reaches the tip of the fracture. Actually, with the linear extension of the fracture being much larger than its open aperture (L ≫ a), it causes a very unfavorable bubble seeding condition at the solid substrate by forbidding the establishment of a stable triple contact line between the gas, the liquid, and the solid substrate at the tip of the cavity. Under such conditions, Rayleigh−Plateau instability

Figure 4. Schematic layout of the cycle of bubble production from a single crevice releasing bubbles with clockwork regularity. D


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Figure 5. Time sequence showing a tiny gas pocket trapped within a cellulose fiber, and being fed by diffusion of dissolved CO2 from the champagne bulk (a and b); as the gas pocket emerges from the fiber’s tip, it quickly breaks up under the action of Rayleigh−Plateau instability (c); a bubble is released without seeding on the tip because cellulose is a material showing favorable wetting conditions (d) (bar = 100 μm).

boundary layer where a gradient of dissolved CO2 concentration exists. Therefore, with the total surface offered to mass transfer being of the order of S ≈ aL in the linear crevice, the number of CO2 moles which crosses the gas pocket interface per unit of time transforms as

which controls the periodicity of bubble production from linear crevices is therefore the time needed for the gas pocket growing by diffusion within the laser-etched fracture to emerge from its open aperture. Assuming that only one bubble with a volume v0 is released for each round trip of the gas pocket in the fracture (i.e., for each cycle), and following the characteristic dimensions presented in Figure 4 (i.e., with a fracture open aperture a, its linear extension L, and the vertical path Δz that the front part of the trapped gas pocket has to grow in the crevice from its initial position to the tip of the fracture), it is possible to approach the characteristic diameter d0 of a bubble released from a linear crevice as follows: d0 ≈ (2v0)1/3 ≈ (2aLΔz)1/3

dn Δc = SJ ≈ aLJ ≈ aLD dt λ

Otherwise, gas phase CO2 within the gas pocket trapped within the linear crevices is considered as an ideal gas, which therefore reasonably obeys the following classical relationship

PBv = nRθ

(7)

Δc λ

(10)

with PB being the gas pressure within the gas pocket and with v and n being the volume of gas and the number of gaseous moles in the gas pocket, respectively. Because the gas pocket trapped within the linear crevice is constrained by the vertical fracture’s walls, the variation of the number of moles which crosses the bubble interface per unit of time therefore causes a corresponding variation of the vertical extension of the gas pocket as

With the characteristic open aperture a of fractures being of the order of 2 μm, their linear extension L varying from several tens to several hundreds of μm, and Δz being reasonably assumed of the same order of magnitude than the fracture aperture, the characteristic diameters of bubbles released from linear crevices should reasonably range between several μm and several tens of μm. It is worth noting that such diameters seem compatible with the diameters of bubbles released from the ring-shaped laser-etched structure, as seen in Figure 1, and therefore argue in favor of no bubble seeding at the bottom of the glass. A similar bubbling scenario was indeed already experimentally observed in situ from tiny cellulose fibers blowing bubbles in champagne and beer glasses, under standard tasting conditions.28,44−47 Actually, with cellulose being a highly hydrophilic material, aqueous liquids such as champagne or beer forbid bubble seeding at the fiber’s tip, as shown in Figure 5, so that the dominant time scale needed to blow a bubble was also considered as the time needed for the tiny gas pocket to grow by diffusion from its initial position in the lumen to the fiber’s tip.28,46 2.7. Modeling the Bubble Frequency from a Linear Crevice. With the mechanism behind the growth of a gas pocket trapped within a linear crevice being molecular diffusion, the molar flux J of gaseous CO2 which crosses the gas pocket interface therefore obeys the so-called Fick’s first law, which stipulates that J = −D∇c ≈ D

(9)

P dv P dz P aL dz dn ≈ B ≈ BS ≈ B dt Rθ dt Rθ dt Rθ dt

(11)

Therefore, by combining eqs 9 and 11, the variation per unit of time for the vertical extension of the gas pocket trapped within the fracture may be rewritten as follows: RθD(c L − c B) dz ≈ dt PBλ

(12)

Actually, near the gas pocket surface, the concentration of dissolved CO2 cB is forced by PB (namely, the pressure of gas phase CO2 in the gas pocket trapped within the linear crevice). Therefore, Henry’s law locally applies (i.e., cB = kHPB), with kH being the strongly temperature-dependent Henry’s law constant. Strictly speaking, PB is the sum of three terms: (i) the atmospheric pressure P0, (ii) the hydrostatic pressure ρgh (g being the gravity acceleration and h being the level of liquid as the glass is filled with champagne), and (iii) the Laplace pressure of the order of 2γ/a, originated in the bubble’s curvature. However, with h being of the order of several centimeters, the contribution of hydrostatic pressure is clearly negligible in front of both atmospheric and Laplace pressures. Thus, the concentration of dissolved CO2 close to the gas pocket may be expressed as cB = KHPB ≈ kH(P0 + 2γ/a). Moreover, and strictly speaking, cL in eq 12 (namely, the concentration of dissolved CO2 found in the liquid bulk) is

(8)

with D being the diffusion coefficient of CO2 molecules in champagne, Δc = cL − cB being the dissolved CO2 molar concentration difference between the champagne bulk and the gas pocket interface, and λ being the thickness of the diffuse E


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λ ∝ (aDt )1/3

time dependent simply because dissolved CO2 progressively escapes from the liquid phase during the bubbling process. Nevertheless, during the very short period of time required for a single cycle of bubble production, cL may also be considered as a constant in eq 12. Finally, it is worth noting that every parameter lying in the right side of eq 12 is known, with the exception of the thickness of the diffuse boundary layer λ. The aim of the following section is therefore to approach the thickness of this boundary layer in order to account for the rate of bubble production, and finally propose a reasonable model for the progressive losses of dissolved CO2 from laser-etched champagne glasses under standard tasting conditions.

(14)

By replacing the latter time-dependence relationship for the diffuse boundary layer in eq 12, it becomes evident that the velocity at which the gas pocket would grow inside the crevice should continuously decrease as time proceeds (with every other parameter remaining constant). In turn, the time needed to release a bubble from the crevice should inexorably increase as time proceeds. Nevertheless, each time a bubble is released from the crevice and rises through buoyancy, the diffuse boundary layer should inevitably by perturbed and finally unable to progressively expand, as would be the case in a purely static liquid phase. Moreover, several bubbles nucleate close to each other from the laser-etched zone lying at the bottom of the glass. The liquid environment around the several laseretched crevices is therefore far from being stagnant, thus forbidding the progressive expansion of diffuse boundary layers above crevices. The mass transfer of dissolved CO2 from the liquid bulk to the gas pocket trapped within the crevice is therefore definitely believed to be governed by diffusionconvection rather than by pure diffusion. 3.2. Mass Transfer Controlled under DiffusionConvection. In the case of a liquid medium agitated with flow patterns, convection forbids the growing of the diffusion boundary layer, thus keeping it roughly constant by continuously supplying the liquid phase around the gas pocket with dissolved CO2 freshly renewed from the liquid bulk. Finally, eq 12 may therefore be easily integrated (with λ being constant), and the period of time needed to release a bubble from the crevice (for each round trip of the gas pocket growing trapped within the fracture) should therefore obey the following relationship

3. RESULTS AND DISCUSSION Actually, and generally speaking, diffusion of dissolved gas species may be ruled by pure diffusion or by diffusionconvection, whether the liquid phase is perfectly stagnant, or in motion.48 The two aforementioned situations must therefore, a priori, be taken into account in our forthcoming discussion. 3.1. Mass Transfer Controlled under Pure Diffusion. In a purely diffusive mass transfer situation, each CO2 molecule which crosses the gas pocket interface is removed from the diffuse boundary layer around. In turn, a boundary layer depleted with dissolved CO2 molecules progressively expands around the gas pocket interface during the purely diffusive process, which means that λ progressively increases. The boundary layer depleted with dissolved CO2 therefore progressively expands around the linear crevice, in the form of a hemicylindrical shell showing a characteristic radius λ, as seen in the scheme displayed in Figure 6. During the diffusing

T≈

(P + 2γ /a)λΔz PBλΔz ≈ 0 RθDΔc RθD(c L − c B)

(15)

with Δz being the vertical path that the front part of the trapped gas pocket has to grow in the crevice from its initial position to the tip of the fracture (see Figure 4). Generally speaking, in a liquid environment showing flow patterns, the higher the characteristic velocity of the flow patterns, the thinner the diffuse boundary layers. By observing the repetitive production of CO2 bubbles from cellulose fibers immersed in glasses poured with champagne, it was clearly underscored that the transfer of dissolved CO2 was indeed controlled by a diffusion-convection process (i.e., with a constant diffuse boundary layer showing a thickness of the order of 20 μm).44 In a more recent article, diffusionconvection conditions were also clearly identified for the growth of spherically shaped CO2 bubbles stuck by capillary action on the wall of a plastic goblet poured with various sparkling waters.49 In sparkling water, the characteristic thickness of the diffuse boundary layers was rather on the order of 100 μm. Actually, diffuse boundary layers with thicknesses higher in sparkling water than in champagne are believed to be the result of significantly lower concentrations of dissolved CO2 in sparkling waters than in Champagne wines.49 Higher levels of dissolved CO2 cause indeed higher bubbling activity in champagne glasses than in fizzy water (according to common experience). In champagne glasses, the liquid environment should therefore logically show more bubbledriven convection, thus causing finally thinner diffuse boundary layers around growing bubbles.

Figure 6. Schematic layout showing the diffuse boundary layer (where a gradient of dissolved CO2 exists), which progressively expands in the form of a semicylindrical zone depleted with dissolved CO2 above the gas pocket trapped within the linear crevice, and being fed by diffusion.

process, mass conservation between the diffuse hemicylindrical boundary layer (with a total volume of the order of Vλ ≈ πλ2L), which progressively loses dissolved CO2 in favor of the gas pocket trapped within the linear crevice, can therefore be written as Δc dVλ ≈ πλdλLΔc (13) 2 By combining eqs 9 and 13, and by integrating, the thickness of the diffuse boundary layer, growing with time under pure molecular diffusion, may therefore be approached as follows: dn ≈

F


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The Journal of Physical Chemistry B ⎛ 2γ ⎞ ⎟ c L* = c B ≈ kHPB ≈ kH⎜P0 + ⎝ a⎠

For modeling purposes, a constant diffuse boundary layer of the order of 20 μm will be used hereafter, as that previously identified in the case of growing bubbles heterogeneously nucleated from tiny cellulose fibers immersed in champagne and beer glasses.44,46 Moreover, it is worth noting that, with every other parameter being constant under identical tasting conditions, the time required to release a bubble from a crevice logically varies with the main geometrical characteristics of the crevice (i.e., the fracture open aperture a and the path Δz needed to travel for the front part of the gas pocket before emerging from the fracture aperture). Under a standard champagne tasting temperature of 12 °C, by retrieving eq 15 for a fracture with a reasonable open aperture of a ≈ 2 μm and a path of Δz ≈ 5 μm, the theoretical time needed to periodically release a single bubble from a linear crevice fed with dissolved CO2 under diffusion-convection conditions is therefore of the order of 0.05 s. The corresponding theoretical frequency of bubble release is therefore of the order of 20 Hz, which seems in quite good agreement with the typical frequencies of bubble formation ranging from several Hz to a few tens of Hz, as experimentally measured in glasses just poured with champagne.50 Otherwise, it is clear that a network of fractures with various sizes should coexist in laser-etched champagne glasses (in terms of open aperture, linear extension, and depth). Nevertheless, and for modeling purposes, several fractures with average both monodisperse open aperture a ≈ 2 μm (as seen through microscopy) and vertical path Δz ≈ 5 μm will be considered hereafter in our laser-etched glasses. Finally, by replacing each and every parameter being constant in eq 15 by their numerical values and by replacing the diffusion coefficient of dissolved CO2 in champagne by its temperature-dependent relationship given in eq 4, it provides the following relationship (in the MKS system of units) for the frequency of bubble formation from a linear crevice acting as a bubble nucleation site:

(17)

It is worth noting that c*L is highly temperature dependent, because the dissolved CO2 solubility kH is indeed highly temperature dependent, as seen in eq 5. For example, by applying eq 17, the critical concentration cL* below which bubble release becomes thermodynamically impossible from a linear crevice with an open aperture a ≈ 2 μm is therefore c*L ≈ 3.9 g L−1, at 4 °C, whereas it is only cL* ≈ 2.2 g L−1, at 20 °C. Above this critical concentration cL* needed to feed the gas pocket with dissolved CO2 from the champagne bulk, each nucleation site is therefore responsible for the release of a small fraction of gaseous CO2 from the liquid phase supersaturated with dissolved CO2. Per unit of time (i.e., every second) and assuming a number N of identical nucleation sites lying at the bottom of the glass, the volume of gaseous CO2 escaping from champagne through the bubbling activity therefore obeys the following relationship ⎛ dV ⎞ ⎜ ⎟ = N × f × v B ⎝ dt ⎠ B

(18)

(16)

with vB being the volume of a bubble as it reaches the champagne surface. In order to fully access the losses of dissolved CO2 under standard tasting conditions, a reasonable model describing the dependence of the bubble size as it reaches the champagne surface has therefore to be used. The aim of the following paragraph is to discuss the dependence of the bubble volume with a collection of various parameters as it reaches the champagne surface. 3.4. Ascending Bubble Dynamics Model. A very typical high-speed photograph of a single bubble train showing bubbles released with clockwork regularity from a single nucleation site is displayed in Figure 7. Bubbles are seen growing in size while ascending through the liquid phase supersaturated with dissolved CO2, as indeed first reported in a glass of beer by Shafer and Zare.51 In a previous work, and by using ascending

3.3. The Critical Concentration of Dissolved CO2 as Bubbling Stops. As already shown experimentally in situ, the frequency of bubble formation from various nucleation sites was found to progressively decrease as time goes on,37 simply because the concentration cL of dissolved CO2 progressively decreases as CO2 continuously escapes from the liquid phase. Furthermore, it is worth noting that the bubbling frequency f of a given nucleation site physically vanishes (i.e., the bubble release stops), as the dissolved CO2 concentration cL reaches a finite value.37 Actually, following both Laplace’s and Henry’s law, the diameter of curvature a of the gaseous CO2 bubble being trapped within the crevice forces a concentration of dissolved CO2 on the order of cB ≈ kH(P0 + 2γ/a) near the CO2/liquid interface (because, as seen earlier, the contribution of hydrostatic pressure is negligible in the present situation). Consequently, as the dissolved CO2 concentration cL found in the champagne bulk reaches the dissolved CO2 concentration found in the close vicinity of the gas pocket trapped within the crevice (i.e., cL = cB), bubble nucleation stops because the driving force of molecular diffusion vanishes. Therefore, as the concentration of dissolved CO2 found in the liquid bulk reaches the critical value c*L expressed hereafter, diffusion of dissolved CO2 toward the gas pocket ceases, and the nucleation site stops releasing bubbles:

Figure 7. Macroscopic view of a bubble train promoted by the repetitive nucleation of bubbles from a nucleation site lying at the bottom of a glass poured with champagne (a); detail showing bubbles regularly growing in size by diffusion of dissolved CO2 as they rise toward the free surface (b) (bars = 1 mm).

f≈

⎛ 2806 ⎞ (c L − c B) 1 ⎟ ≈ 5.7 × 103θ 2 exp⎜ − ⎝ T θ ⎠ (P0 + 2γ /a)

G


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of bubbles, whereas the other 80% escapes directly through the free surface of champagne.52 Actually, because only about 20% of dissolved CO2 escapes in the form of bubbles, the total volume of gaseous CO2 escaping from the flute, per unit of time, may therefore be expressed as follows:

bubble dynamics coupled with mass transfer equations, the following relationship was derived (in the MKS system of units), which links the diameter d of a single CO2 bubble rising in a liquid phase supersaturated with dissolved CO2, with several parameters of both the liquid phase and the glass itself31 −3 5/9

dB ≈ 5.4 × 10 θ

⎛ 1 ⎞2/9⎛ c L − kHP0 ⎞1/3 1/3 ⎜ ⎟ ⎜ ⎟ h P0 ⎠ ⎝ ρg ⎠ ⎝

⎛ dV ⎞ ⎛ dV ⎞ ⎜ ⎟ ≈ 5⎜ ⎟ ⎝ dt ⎠ T ⎝ dt ⎠ B

(19)

⎛ 2806 ⎞ ⎟ ≈ 5 × 10−6Nθ11/3 exp⎜ − ⎝ θ ⎠

−3

with ρ being the champagne density (≈10 kg m ), g being the acceleration due to gravity (≈9.8 m s−2), and h being the distance traveled by a bubble from its nucleation site lying at the bottom of the glass to the champagne free surface. It is worth noting that, to propose the previous relationship, ascending bubbles were considered as rigid spheres (i.e., hydrodynamically speaking, the boundary conditions on the bubble surface are reduced from slip to nonslip due to surface active compounds forbidding interfacial mobility). Moreover, in the latter equation, the concentration cL of dissolved CO2 is implicitly considered as being constant all along the vertical distance traveled by the rising bubble. This assumption is nevertheless reasonable, because continuous and vigorous recirculation of champagne most probably forbids the formation of zones depleted with dissolved CO2 around ascending bubble trains, thus keeping the concentration of dissolved CO2 constant in the whole bulk of champagne. Such flow patterns were evidenced in previous works, through laser tomography techniques, and numerical simulations.19,40,41 Therefore, and following eq 19, the volume of a CO2 bubble which reaches the liquid surface before bursting finally is expressed as follows: 3

vB ≈

⎛ 1 ⎞2/3⎛ c − kHP0 ⎞ dB3 ≈ 7.9 × 10−8θ 5/3⎜ ⎟ ⎜ L ⎟h 2 P0 ⎠ ⎝ ρg ⎠ ⎝

(c L − kHP0)(c L − kHPB) h P0(P0 + 2γ /a)

Following eq 23, it is worth noting that the corresponding loss of dissolved CO2 concentration from the champagne glass per unit of time, both through bursting bubbles and invisible diffusion through the free surface (expressed in mol m−3 s−1), may therefore be expressed in its differential form as follows dc L 1 ⎛⎜ dV ⎞⎟ =− dt VMVL ⎝ dt ⎠T

dc L ⎛ 2806 ⎞ ⎟ ≈ −6 × 10−7Nθ 8/3 exp⎜ − ⎝ θ ⎠ dt (c L − kHP0)(c L − kHPB) h VL(P0 + 2γ /a)

(20)

(25)

Finally, and after integrating, the continuously decreasing concentration of dissolved CO2 found in champagne glasses, both through bubbling and surface diffusion, should be modeled as follows ⎧ ⎪ c L(t ) ⎪ −ΓkH(2γ / a)t + kHPB(kHP0 − c i) ⎪ ≈ kHP0(c i − kHPB)e −ΓkH(2γ / a)t ⎪ + kHP0 − c i (c i − kHPB)e ⎪ ⎨ with ⎪ ⎪ Nθ 8/3 exp( −2806/θ )h ⎪Γ = 1.7 × 106VL(P0 + 2γ /a) ⎪ ⎪ ⎩ PB = P0 + 2γ /a

(21)

3.5. Modeling the Time-Decreasing Losses of Dissolved CO2. Finally, by replacing in eq 18 both the average frequency of bubble formation and the bubble volume given in eqs 16 and 21, respectively, the volume of gaseous CO2 escaping from champagne through the bubbling activity per unit of time may be approached as (still in the MKS system of units) ⎛ dV ⎞ ⎛ 2806 ⎞ −6 11/3 ⎟ ⎜ ⎟ ≈ 10 Nθ exp⎜ − ⎝ ⎝ dt ⎠ B θ ⎠ (c L − kHP0)(c L − kHPB) h P0(P0 + 2γ /a)

(24)

with VL being the volume of champagne served into the glass (in m3) and VM being the molar volume of an ideal gas (i.e., Rθ/P0, in m3 mol−1). By replacing VM in eq 24 by Rθ/P0 and the ideal law constant R by its numerical value, the following relationship is obtained which provides the losses of dissolved CO2 concentration from the champagne glass per unit of time as a collection of various parameters (namely, the number of nucleation sites, the temperature, the concentration of dissolved CO2 in the champagne bulk, the solubility of dissolved CO2, the ambient pressure, and the level of champagne poured in the glass):

Actually, from one champagne to another (and even from one sparkling beverage to another), the density ρ does not significantly vary and remains close to ≈103 kg m−3. Moreover the gravity acceleration being considered also as constant all around the Earth, the prefactor (1/ρg)2/3 in eq 20 does not vary. Therefore, the latter relationship providing the bubble volume as it reaches the champagne surface may be rewritten as follows: ⎛ c − kHP0 ⎞ vB ≈ 1.7 × 10−10θ 5/3⎜ L ⎟h P0 ⎝ ⎠

(23)

(26)

with ci being the initial concentration of dissolved CO2 found in champagne immediately after the pouring step. Now that the model has been fully developed, a confrontation with some experimental data would be particularly useful to enrich the discussion. The aim of the following paragraph is to compare the main features of our dissolved CO2 decreasing model with a set of experimental data monitoring the decrease of dissolved CO2 with time after

(22)

Actually, a previous set of experimental data proved that, in a classic crystal flute, and contrary to what could have been expected, only about 20% of carbon dioxide escapes in the form H


Article

The Journal of Physical Chemistry B champagne was poured into a glass (at different tasting temperatures). 3.6. Confrontation between Model and Experiments. In a previously published set of data, losses of dissolved CO2 concentrations were accurately monitored with time, from a 100 mL volume of champagne served in a series of six identical laser-etched flutes.20 The series of six identical flutes were exactly similar to the one displayed in Figure 1. The same ringshaped structure done with 22 adjoining laser beam impacts was etched by the glassmaker at the bottom of each flute. A batch of commercial Brut Champagne wine was used for this set of experiments (see Table 1), with a very typical concentration of dissolved CO2 in the sealed bottles of the order of 11.5 g L−1 (i.e., cL ≈ 260 mol m−3).20 In order to test the influence of temperature on the progressive losses of dissolved CO2 from the laser-etched flutes with time, experiments were performed at three sets of champagne temperature (namely, 4, 12, and 20 °C), respectively, in a temperature-controlled room. Losses of dissolved CO2 concentrations were deduced from the finetuning cumulative mass loss experienced by the glasses progressively discharging their level of CO2 (escaping through the bubbling activity and through the free surface). More details about the methodology used to monitor the losses of dissolved CO2 from champagne poured in laser-etched glasses can be found in Liger-Belair et al., 20 and references therein. Experimental concentrations of dissolved CO2, monitored with time for each champagne temperature, are displayed in Figure 8. It is worth noting that the initial concentration of dissolved CO2 found in champagne immediately after the pouring step (namely, ci at t = 0) is (i) much lower than the level of dissolved CO2 found in the sealed bottle and (ii) significantly depends on the temperature of champagne. Actually, turbulences of the pouring step were found to be far from inconsequential as concerns losses of dissolved CO2, with low temperatures preserving a bit better dissolved CO2 in the champagne bulk during the pouring step.38,39 Moreover, it clearly appears in Figure 8 that low champagne temperatures help to keeping dissolved CO2 in the champagne bulk, all along the first 30 min following the pouring step. The time-decreasing dissolved CO2 concentration model given in eq 26 is to be confronted with this experimental set of data evidencing the important role played by the temperature on the losses of dissolved CO2 from champagne glasses. By replacing the temperature θ in eq 26 and each of the two other temperature-dependent parameters (namely, ci and kH) by its corresponding value in the appropriate unit, the timedecreasing dissolved CO2 model can be compared with the three time-decreasing dissolved CO2 experimentally determined (for each investigated temperature). As shown in Figure 8, the best fit between the time-decreasing dissolved CO2 model and the experimental set of data was obtained with a number of active nucleation sites N = 26 in our model (i.e., with 26 fractures releasing bubbles together). The correspondence between the model showing the time-decreasing dissolved CO2 concentration and the experimental set of data seems indeed very satisfying, especially at champagne temperatures of 12 and 20 °C. Nevertheless, it is worth noting that a single laser beam impact creates several crevices close to each other, as seen on the micrographs displayed in Figure 2. The real number of crevices in the case of our laser-etched structure showing 22 laser beam impacts therefore certainly exceeds the number of

Figure 8. Progressive losses of dissolved CO2 concentrations (in mol m−3), as experimentally measured all along the first 30 min following the pouring process in a laser-etched flute filled with 100 mL of champagne (similar to that displayed in Figure 1), and at three champagne temperatures (dot points);20 each experimental dot of each time series is the arithmetic average of six successive values recorded from six successive pourings, standard deviations corresponding to the root-mean-square deviations of the values provided by the six successive data recordings; experimental data are compared with the time-decreasing dissolved CO2 concentration model given in eq 26 (dashed lines); the best fit between the model and the experimental set of data was obtained with a number of active nucleation sites N = 26 in eq 26.

26 active nucleation sites needed to best fit the set of experimental data shown in Figure 8. Actually, bubbles released so close to each other from adjoining crevices can certainly sometimes coalesce, thus modifying the overall bubbling frequency of nucleation sites, as already observed in situ in the case of tiny cellulose fibers blowing bubbles close to each other on the wall of a glass poured with champagne.53 The real number of crevices releasing bubbles being indeed certainly much higher than 26, the average number N = 26 of active nucleation sites in our model (needed to best fit the experimental set of data in Figure 8), should rather be seen as the average number of nucleation sites releasing bubbles without interacting together, and with a corresponding modeled frequency given by eq 16.

4. CONCLUSIONS In conclusion, on the basis of theoretical principles combining heterogeneous bubble nucleation, ascending bubble dynamics and mass transfer equations, a global model is proposed, depending on various parameters of both the wine and the glass itself, which quantitatively provides the progressive losses of dissolved CO2 from laser-etched champagne glasses, under standard tasting conditions. The theoretical model showing the time-decreasing dissolved CO2 concentration in champagne was found to be in very good agreement with a recent set of experimental data. The question of temperature, which is known to be a parameter of paramount importance in champagne and sparkling wine tasting, was closely examined, and its role on the time-decreasing modeled losses of dissolved I


Article

The Journal of Physical Chemistry B k298K

CO2 was also in good agreement with the set of experimental data. Moreover, because the model depends on a collection of various parameters of both the liquid phase and the glass itself, it should be compared in the near future with experimental data collected with various glass shapes, and different sparkling beverages also, such as beers or soft drinks, still looking for new insights and novelties. In the near future also, the model could be used in our shared analysis with glassmakers and sommeliers to create and imagine future glass shapes especially designed for specific types of sparkling beverages. Otherwise, it is indeed well-known that the progressive release of both gaseous CO2 and volatile organic compounds (VOCs) in the headspace above champagne glasses progressively modifies the chemical headspace perceived by the consumer. Investigating the role of the laser-etched champagne glasses compared with traditional champagne glasses during a real tasting session, in order to verify if the panelists can really discriminate the CO2 evolution as a function of the glass, could therefore be the purpose of a future work.

■

l L n N P0 PB r* R t v vB V

AUTHOR INFORMATION

Vλ

Corresponding Author

*Phone: 00 (33) 3 26 91 33 93. Fax: 00 (33) 3 26 91 31 47. Email: [email protected].

VL VM z

Notes

The authors declare no competing financial interest.

■

γ λ

ACKNOWLEDGMENTS Thanks are due to the Association Recherche Oenologie Champagne Université for financial support and to Champagne Houses Moët & Chandon, Pommery, and Veuve Clicquot for regularly supplying us with champagne samples. G.L.-B. is also indebted to the CNRS, the Région Champagne-Ardenne, the Ville de Reims, and the Conseil Général de la Marne for supporting our research.

ρ η θ ΔHdiss

■

Δz

NOMENCLATURE a open aperture of the linear crevice, ≈2 μm cB concentration of dissolved CO2 in the close vicinity of the gas pocket trapped within the crevice (see Figure 6), in mol m−3 ci initial concentration of dissolved CO2 found in the liquid bulk after pouring champagne in the glass, in mol m−3 cL concentration of dissolved CO2 found in the liquid bulk, in mol m−3 c*L critical concentration of dissolved CO2 found in the champagne bulk as bubbling stops, in mol m−3 D diffusion coefficient of dissolved CO2 in champagne, in m2 s−1 dB ascending bubble diameter, in m f frequency of bubble formation from a linear crevice acting as a bubble nucleation site, in s−1 g acceleration due to gravity, ≈9.8 m s−2 h level of liquid as the glass is filled with champagne (i.e., distance traveled by a bubble from its nucleation site to the champagne free surface), in m J molar flux of gaseous CO2 which crosses the gas pocket interface, in mol m−2 s−1 kB Boltzmann constant, =1.38 × 10−23 J K−1 kH temperature-dependent solubility of dissolved CO2 in champagne, in mol m−3 Pa−1

■

temperature-dependent solubility of dissolved CO2 in champagne, at 298 K, ≈2.75 × 10−4 mol m−3 Pa−1 hydrodynamic radius of the CO2 molecule in the Stokes−Einstein equation, ≈10−10 m linear extension of crevices, ≈10−100 μm number of CO2 moles in the gas pocket, in mol number of active nucleation sites in the glass of champagne atmospheric pressure, ≈105 N m−2 = 1 bar pressure of CO2 in the gas pocket trapped within the linear crevice, in N m−2 critical radius required to enable nonclassical heterogeneous bubble nucleation, in m ideal gas constant, =8.31 J K−1 mol−1 time, in s volume of gas phase CO2, in m3 bubble volume as it reaches the champagne surface, in m3 volume of gaseous CO2 escaping from champagne, in m3 volume of the diffuse hemicylindrical boundary layer expanding around the crevice under pure diffusion, in m3 volume of champagne served into the glass, in m3 molar volume of an ideal gas (i.e., RT/P0), in m3 mol−1 vertical extension of the gas pocket trapped within the crevice, in m air/champagne surface tension, ≈50 mN m−1 thickness of the diffuse boundary layer around the gas pocket trapped within the crevice, and where a gradient of dissolved CO2 exists, ≈20 μm champagne density, ≈103 kg m−3 champagne dynamic viscosity, in kg m−1 s−1 absolute temperature, in K dissolution enthalpy of gas phase CO2 in champagne, ≈ −24 800 J mol−1 vertical path that the front part of the trapped gas pocket has to grow in the crevice from its initial position to the tip of the fracture, ≈5 μm

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K


Modeling the Losses of Dissolved CO2 from Laser-Etched

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