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Is the Wall of a Cellulose Fiber Saturated with Liquid Whether or Not Permeable with CO2 Dissolved Molecules? Application to Bubble Nucleation in Champagne Wines Ge´rard Liger-Belair,*,† Daniel Topgaard,‡ Ce´dric Voisin,† and Philippe Jeandet† Laboratoire d’Œnologie et de Chimie Applique´ e, UPRES EA 2069, URVVC, Faculte´ des Sciences de Reims, Moulin de la Housse, B.P. 1039, 51687 Reims, Cedex 2, France, and Department of Chemistry, D62 Hildebrand Hall, University of California at Berkeley, Berkeley, California 94720-1460 Received January 6, 2004. In Final Form: March 11, 2004 In this paper, the transversal diffusion coefficient D⊥ of CO2 dissolved molecules through the wall of a hydrated cellulose fiber was approached, from the liquid bulk diffusion coefficient of CO2 dissolved molecules modified by an obstruction factor. The porous network between the cellulose microfibrils of the fiber wall was assumed being saturated with liquid. We retrieved information from previous NMR experiments on the self-diffusion of water in cellulose fibers to reach an order of magnitude for the transversal diffusion coefficient of CO2 molecules through the fiber wall. A value of about D⊥ ≈ 0.2D0 was proposed, D0 being the diffusion coefficient of CO2 molecules in the liquid bulk. Because most of bubble nucleation sites in a glass poured with carbonated beverage are cellulose fibers cast off from paper or cloth which floated from the surrounding air, or remaining from the wiping process, this result directly applies to the kinetics of carbon dioxide bubble formation from champagne and sparkling wines. If the cellulose fiber wall was impermeable with regard to CO2 dissolved molecules, it was suggested that the kinetics of bubbling would be about three times less than it is.
1. Introduction In glasses poured with champagne and sparkling beverages, most of the bubble nucleation sites were recently found to be located on preexisting gas cavities trapped inside hollow and roughly cylindrical like cellulose fibers,1-4 cast off from paper or cloth which floated from the surrounding air or/and remained from the wiping process. Actually, in weakly carbonated liquids such as carbonated beverages in general, carbon dioxide bubble formation requires preexisting immersed gas cavities with radii of curvature large enough to overcome the nucleation energy barrier and grow freely.5,6 The mechanism of bubble release has already been described in previous articles.1-2 In very short, after opening a bottle or a can of carbonated beverage, the thermodynamic equilibrium of CO2 molecules dissolved in the liquid medium is broken. CO2 dissolved 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 beverage is poured into a glass, the tiny gas pockets trapped inside the 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. When a gas pocket reaches the end of a fiber, a bubble is ejected, * To whom correspondence should be addressed. Phone: (33)3 26 91 86 14. Fax: (33)3 26 91 33 40. E-mail:
[email protected]. † URVVC. ‡ University of California at Berkeley. (1) Liger-Belair, G. Ann. Phys. (Paris) 2002, 27, 1-106. (2) Liger-Belair, G.; Marchal, R.; Jeandet, P. Am. J. Enol. Vitic. 2002, 53, 151-153. (3) Liger-Belair, G.; Vignes-Adler, M.; Voisin, C.; Robillard, B.; Jeandet, P. Langmuir 2002, 18, 1294-1301. (4) Liger-Belair, G. Sci. Am. 2003, 288 (1), 68-73. (5) Jones, S. F.; Evans, G. M.; Galvin, K. P. Adv. Colloid Interface Sci. 1999, 80, 27-50. (6) Lubetkin, S. D. Langmuir 2003, 19, 2575-2587.
but a portion of the gas pocket remains trapped inside the fiber, shrinks back to its initial position, and the cycle starts again until bubble production stops through lack of dissolved gas molecules. Close-ups photographs illustrating the release of bubbles from two cellulose fibers adsorbed on the wall of a glass poured with champagne are displayed in Figure 1. Actually, those elongated gas pockets trapped inside the cavity of cellulose fibers are bounded by two quite hemispherical gas/liquid interfaces directly in touch with the liquid bulk inside the fiber (both ends of the deformed gas pocket) and by a cylindrical border in touch with the fiber inner wall. It is clear from Figure 1 and from our previous observations, that the cylindrical border in touch with the inner fiber wall has a much larger area than the sum of the two hemispherical gas/liquid interfaces.1-3 It is therefore evident that the kinetics of bubble release strongly depends on several factors such as: (i) the thickness of the fiber wall, (ii) the relative size of the cylindrical border and the hemispherical gas/liquid interfaces, and (iii) the ability of CO2 molecules to diffuse through the wall of cellulose fibers. Because the thickness of the fiber wall and the relative size of the cylindrical border and the hemispherical gas/ liquid interfaces are quite easily accessible by the direct observation through a microscope objective, the aim of this study is therefore to reply the following question: Is the wall of a cellulose fiber permeable or not with CO2 dissolved molecules? 2. Discussion 2.1. Structural Levels of a Cellulose Fiber. Cellulose fibers, also commonly referred as wood fibers, are the primary component of plant cell walls, as first recognized by the French chemist Anselm Payen in 1838. Cellulose fibers are in the form of hollow tubes of several hundreds of micrometers long and with a cavity mouth varying from several micrometers to several tens of micrometers wide. The central cavity within the fiber is denoted the lumen.
10.1021/la049960f CCC: $27.50 © 2004 American Chemical Society Published on Web 04/16/2004
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Figure 2. Different structural levels of a cellulose fiber. The fiber wall consists of closely packed cellulose microfibrils oriented mainly in the direction of the fiber.7,9 Figure 1. In vivo close-up on cellulose fibers acting as bubble nucleation sites on the wall of a glass poured with champagne, as seen through the microscope objective of a high-speed video camera. The gas pockets trapped inside the fiber’s lumen and responsible for the repetitive production of bubbles clearly appear in dark.
The fiber wall section consists of densely packed cellulose microfibrils, with a preferential orientation along the fiber axis. Cellulose microfibrils consist of glucose units bounded in a β-conformation favoring straight polymer chains. Microfibrils are in the form of rodlike fibrils with about 10 nm width, and lengths are on the micrometer scale. The microfibrils are partly interwoven and cross-linked by short (≈2 nm) segments.7 Overlapping domains with slightly different microfibril orientation give the fiber its mechanical strength.8 The different structural levels of a cellulose fiber are presented in Figure 2. For a current review on the molecular and supra molecular structure of cellulose, see the article by O’Sullivan9 and references therein. 2.2. Porosity and the Moisture Content of the Fiber Wall. The cellulose fiber wall is a porous material with the microfibrils as the solid porous matrix, the pore space between the microfibrils being filled by liquid or/and air, depending on the moisture content.10,11 Nevertheless, Stone et al. have found that dried pulp fibers are quite compact and essentially nonporous.12 However, as soon (7) Hafre´n, J.; Fujino, T.; Itoh, T. Plant Cell Physiol. 1999, 40, 532541. (8) Reiterer, A.; Lichtenbergger, H.; Tschegg, S.; Fratzl, P. Philos. Magn. A 1999, 79, 2173-2184. (9) O’Sullivan, A. Cellulose 1997, 4, 173-207. (10) Stone, J. E.; Scallan, A. M. Tappi 1967, 50, 496-501. (11) Stone, J. E.; Scallan, A. M. Cellul. Chem. Technol. 1968, 2, 343358. (12) Stone, J. E.; Scallan, A. M.; Aberson, G. M. Pulp Paper Magn. Can. 1966, 67, T263-T268.
as the fiber wall is hydrated, the pore space between these highly hydrophilic microfibrils is progressively filled with liquid. The volume fraction of cellulose is defined as
VC V
ΦC )
(1)
where VC is the volume of the cellulose microfibrils and V is the total volume (cellulose microfibrils + pore space). Actually, in materials with a soft porous matrix as are cellulose microfibrils, ΦC depends on the amount of pore liquid. Addition of liquid results in a widening of the pore space and therefore in lower ΦC. Consequently, molecules in the pore space should have the possibility to move more or less easily throughout the microfibrils, depending on the moisture content of the fiber wall. In such hydrophilic materials, it is also common to specify the cellulose fiber wall porosity in terms of the moisture content defined as follows:
MC )
mL mC
(2)
where mL and mC are the masses of liquid and cellulose microfibrils, respectively. With the assumption of densities of both the liquid and the cellulose phase, eq 1 transforms, and a relationship between ΦC and MC can be obtained
(
ΦC ) 1 +
)
FC MC FL
-1
(3)
where FC and FL are the densities of cellulose and liquid, respectively. Though the kinetics of moisture take-up by the fibers are not so well documented, cellulose fibers adsorbed on
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dHL )
(x ( (x (
dSL )
) )
2π FC MC + 1 - 2 r x3 FL π
) )
FC MC + 1 - 2 r FL
(4)
(5)
where the subscripts HL and SL refer to the hexagonal and square lattice, respectively, and where r is the microfibril radius. At the FSP, by using MC ≈ 0.8 g/g in the two latter equations, one obtains for the pore sizes, dHL ≈ 4.2 nm and dSL ≈ 3.3 nm, respectively. It is worth noting that, at such a high MC, the liquid domains are about the same width as a microfibril. 2.3. Diffusion Through an Anisotropic Porous Medium. 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 one-dimensional rootmean-square distance travelled by a molecule from its initial position and during the time t is ruled by
〈z2〉 ) 2Dt
Figure 3. Schematic transversal cut of the parallel cylindrical microfibrils organized on a 2D hexagonal lattice (A) and on a 2D square lattice (B), respectively; r is the microfibril radius and d is the smallest pore distance between adjacent microfibrils.
(6)
where D is the diffusion coefficient of the molecule in question. For the free diffusion of a molecule far from boundaries, the latter equation holds at all times and D is a constant D0. For molecules diffusing in a porous medium such as the cellulose fiber wall, because of interactions with the pore boundaries, the situation is quite different, and three different time-scales can be distinguished. An apparent diffusion coefficient may be defined as
the inner wall of a glass poured with carbonated beverage are supposed to experience 100% relative humidity (RH). Consequently, the fiber wall pore space is considered as being saturated with liquid. The fiber wall has reached what we call the fiber saturation point10 (FSP). Recent experiments on the changes of the cellulose fiber wall structure during drying were conducted by using NMR self-diffusion of water sorbed in cellulose fibers.13,14 The FSP of the cellulose fiber wall was shown to occur at MC ≈ 0.8 g/g. The authors of the two above-mentioned works used beaten and unbeaten bleached kraft pulps fibers and viscose fibers. By using eq 3, the latter value for the moisture content at the FSP, FC ) 1.55 g cm - 3,15 and assuming FL ≈ 1 g cm - 3, the volume fraction of cellulose in the fiber wall can be deduced at the FSP. A value of ΦFSP ≈ 0.45 is found. The microfibrils network is obviously not perfectly ordered like a crystalline network, but in the following, we will consider for the quite cylindrical microfibrils network, two extreme geometry in terms of compactness: (i) a two-dimensional (2D) hexagonal lattice and (ii) a 2D square lattice. The two models are displayed in Figure 3, together with the geometrically pertinent parameters. By combining the moisture content defined in eq 2 with the volume fraction of cellulose defined in eq 1 and with the geometrical parameters presented in Figure 3, the smallest distance between two adjoining microfibrils may be deduced. In these two models, the smallest distance d between the microfibrils surfaces, defined as the pore size, is given by
where td is the diffusion time. At short times (when x〈z2〉 , d), only very few molecules “feel” the influence of pore boundaries, and Dapp is quite the same as in the liquid bulk, i.e., D0. At longer times (when x〈z2〉 ≈ d), Dapp is decreasing with increasing td due to the increasing number of molecules that reach the pore boundaries. At much longer times (when x〈z2〉 . d), all of the molecules diffusing in the pore space felt the influence of the pore boundaries. Therefore, Dapp reaches a new constant value denoted D∞ and which reflects the long range connectivity and permeability of the porous network. Experimentally, the general trend for the apparent diffusion coefficient versus the diffusion time, as determined by NMR,16-19 is displayed in Figure 4. In the present situation, CO2 molecules must cross the fiber wall by diffusing in the direction perpendicular to the microfibrils. Therefore, the one-dimensional rootmean-square distance traveled by a CO2 molecule is on the order of the fiber wall width (e ≈ 5-10 µm) which is finally much larger than the pores’ size at the FSP (d ≈ 3-4 nm). Consequently, the apparent diffusion coefficient of CO2 molecules diffusing inside the fiber wall, denoted Df, is the same as the long-range diffusion coefficient D∞. However, because the cellulose microfibrils are mainly oriented in the direction of the fiber, this porous medium
(13) Topgaard, D.; So¨derman, O. Cellulose 2002, 9, 139-147. (14) Topgaard, D. Nuclear magnetic resonance studies of water selfdiffusion in porous system. Ph.D. Thesis, Lund University, Sweden, 2003. (15) Fengel, D.; Wegener, G. Wood: Chemistry, Ultrastructure and Reactions; W. de Gruyter: New York, 1984.
(16) Woessner, D. E. J. Phys. Chem. 1963, 67, 1365-1367. (17) Mitra, P. P.; Sen, P. N.; Schwartz, L. M.; Le Doussal, P. Phys. Rev. Lett. 1992, 68, 3555-3558. (18) Mitra, P. P.; Sen, P. N.; Schwartz, L. M. Phys. Rev. B 1993, 47, 8565-8574. (19) Valiullin, R.; Skirda, V. J. Chem. Phys. 2001, 114, 452-458.
Dapp )
〈z2〉 2td
(7)
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Figure 4. In a porous medium, the apparent diffusion coefficient depends on the diffusion time. The long-time value, relevant for the modeling of CO2 diffusion, depends on the porous network tortuosity.
Figure 5. In the fiber wall, the diffusion coefficient of molecules through the anisotropic microfibrils network is different in the direction perpendicular than in the direction parallel to microfibrils; D⊥ and D| are the components perpendicular and parallel to microfibrils, respectively.
is highly anisotropic (cf. Figure 5). Consequently, the diffusion coefficient Df defined above is an average diffusion coefficient which is better defined through
Df )
2D⊥ + D| 3
(8)
where D⊥ and D| are the diffusion coefficients perpendicular and parallel to the director, respectively, the director being oriented in the direction of the fiber. To cross the fiber wall from the liquid bulk to the lumen, CO2 molecules must diffuse in the direction perpendicular to the microfibrils. Consequently, to finally give an answer to our previous question about the permeability of the fiber wall with regard to CO2 molecules, we must focus on the perpendicular component of the diffusion coefficient inside the fiber wall, i.e., D⊥. 2.4. Perpendicular Diffusion of CO2 Molecules Through the Fiber Wall. Direct measurements of D⊥ using NMR are impossible because of unfavorable experimental conditions (concentration, relaxation times, etc.). Instead we will use information from previous NMR experiments on the self-diffusion of water in cellulose fibers and CO2 molecules in champagne wine to arrive at a reasonable estimate of the value needed for the modeling. When modeling the diffusion of a solute through the liquid-filled pore space of a porous material, a number of factors are to be considered:20 (i) the diffusion of the solute (20) Svensson, A.; Topgaard, D.; Piculell, L.; So¨derman, O. J. Phys. Chem. B 2003, 107, 13241-13250.
through the pure pore liquid, (ii) the tortuosity of the porous medium, (iii) the size of the solute molecule in comparison to the size of the pores, and (iv) interactions between the solute molecules and the surfaces of the porous material. We will discuss these factors one by one for the specific case of wine as the pore liquid, a swollen cellulose fiber as the porous material and CO2 as the solute. (i) The bulk value of D0 for CO2 in a variety of liquids, such as champagne and fizzy water, has been very recently determined by use of the NMR pulsed-gradient spinecho technique.21 For the champagne wine, a value of D0 ) 1.41 × 10-9 m2 s-1 was found at 20 °C. At first thought, it seems that the bulk value previously measured for champagne could be used for the pore liquid. However, the composition of the liquid in the pore space will be slightly different from the bulk liquid due to size exclusion effects. Actually, champagne can be viewed as an aqueous carbonated solution of ethanol, glycerol, a variety of sugars, and small amounts of macromolecules such as proteins, glycoproteins, polysaccharides, and polyphenols (see ref 1 and references therein). Macromolecules will simply not fit in the nanometer-scale pores of the fiber wall. It has been shown that sugars even as small as glucose are partially, although not completely, excluded from the pore space.11 The pore liquid will therefore have a composition somewhere between champagne bulk and a carbonated water and ethanol solution. However, because champagne is quite dilute with sugars (several g/L) and macromolecules1,22 (several hundreds of mg/L), we will nevertheless assume in the following that the pore liquid locally diffuses as the champagne bulk, i.e., with D0 ) 1.41 × 10-9 m2 s-1. (ii) The tortuosity of porous materials, or equivalently the obstruction factor for a colloidal dispersion, is the subject of numerous publications.23-30 The tortuosity factor R and the obstruction factor f0 are defined through
D∞ 1 ) f0 ) R D0
(9)
This definition refers to pointlike molecules moving through a porous medium. The tortuosity factor R is a purely geometric property of the pore space. The cellulose fiber wall can be viewed as an irregular stack of rodlike cellulose microfibrils. Approximating the structure as a hexagonal or square array of cylinders, as in section 2.2., we can use the results derived by Jo´hanneson and Halle,28 who performed random-flight simulations to investigate the obstruction factor induced by different shapes and packing of colloidal objects. They derived the two following analytical equations for the obstruction factors for diffusion perpendicular to parallel cylinders organized on hexagonal and square lattice, (21) Liger-Belair, G.; Prost, E.; Parmentier, M.; Jeandet, P.; Nuzillard, J.-M. J. Agric. Food Chem. 2003, 51, 7560-7563. (22) Dussaud, A. Etudes des proprie´te´s de surface statiques et dynamiques de solutions alcooliques de prote´ines: Application a` la stabilite´ des mousses de boissons alcoolise´es. Ph.D. Thesis, ENSIAA, Massy, France, 1993. (23) Jo¨nsson, B.; Wennerstro¨m, H.; Nilsson, P.; Linse, P. Colloid Polym. Sci. 1986, 264, 77-88. (24) Bear, J. Dynamics of fluids in porous media; Dover Publication: New York, 1988. (25) Sahimi, M.; Jue, V. L. Phys. Rev. Lett. 1989, 62, 629-632. (26) Latour, L.; Mitra, P.; Kleinberg, R.; Sotak, C. J. Magn. Reson. A 1993, 101, 342-346. (27) Latour, L.; Kleinberg, R.; Mitra, P.; Sotak, C. J. Magn. Reson. A 1995, 112, 83-91. (28) Jo´hanneson, H.; Halle, B. J. Chem. Phys. 1996, 104, 68076817. (29) Mair, R.; Hu¨rlimann, M.; Sen, P.; Schwartz, L.; Patz, S.; Walsworth, R. Magn. Reson. Imaging 2001, 19, 345-351. (30) Hizi, U.; Bergman, D. J. Appl. Phys. 2000, 87, 1704-1711.
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Figure 6. Obstruction factor f0(Φ) of parallel cylinders organized on a hexagonal lattice (O) and on a square lattice (]), respectively, as determined from the simulations by Jo´hanneson and Halle.28
Figure 7. Illustration of the difficulty of trying to link the tortuosity of a porous structure with its porosity, from theoretical calculations. The three bundles, with porosity increasing from left to right and volume fraction of cellulose decreasing from left to right, consist of 100 cylinders packed in a way that completely blocks diffusion transverse to the cylinder direction.
respectively:
( (
) )
fHL )
2Φ 1 11-Φ 1 + Φ - 0.0754Φ6
(10)
fSL )
1 2Φ 11-Φ 1 + Φ - 0.3058Φ4
(11)
where the subscripts HL and SL refer to the hexagonal and square lattice, respectively. In Figure 6, the obstruction factors of parallel cylinders organized on both hexagonal and square lattice are plotted versus the volume fraction of parallel cylinders. At a volume fraction of cellulose equivalent to that which is found at the FSP (ΦFSP ≈ 0.45), the obstruction factors for both the hexagonal and the square lattice are quite close and about 0.7. However, it should be noted that the way the obstructing objects are packed is an extremely important factor for the tortuosity. In Figure 7, we illustrate the difficulty of trying to link the tortuosity of a structure with its porosity. In this figure, three sets of cylinder packs with different porosity are shown. For all cases, the cylinders are packed in such a way to completely block molecular motion transverse to the cylinders; that is, the tortuosity factor is infinite. Knowing the porosity and
object shape is not sufficient to estimate the tortuosity, the details of how the objects are arranged have paramount influence. Moreover, adding inhomogeneity compared with a periodic structure, as is certainly the case with the real packing of cellulose microfibrils, strongly affects the diffusion through the microfibrils network. As noted by Hizi and Bergman,30 the combination of narrow throats and large liquid compartments gives much higher values for the tortuosity. We logically conclude that it is way too complicated to calculate the tortuosity with any reasonable accuracy, because the exact packing of cellulose microfibrils is still unknown. In contrast, the orientation-averaged tortuosity of the fiber wall can rather easily be measured with NMR using water as a probe molecule.13,14,31 One complication is that the previously published data refer to the diffusion averaged over fiber orientations and not specifically the transverse direction D⊥, which is of interest here. Actually, the anisotropy factor (the ratio D|/D⊥) has been studied at low water contents and the ratio between the diffusion along and across the microfibrils was found to be D|/D⊥ ≈ 6.4 ( 0.9 at a moisture content of 0.15.32 However, because increasing the moisture leads to a widening of the pore space, it is reasonable to assume that the anisotropy factor will decrease with increasing porosity. The value 6.4 can therefore be used as the highest anisotropy factor for the water saturated fiber at the FSP. At the FSP, the inverse tortuosity factor 1/R of the cellulose fiber wall has been measured to Df/D0 ≈ 0.3.13,14 This value nevertheless refers to a two-dimensional average over fiber orientations, i.e., the average between the value along and perpendicular to the microfibrils defined by eq 8. If the average inverse tortuosity factor is 0.3, then the perpendicular component is with certainty comprised between D⊥/D0 ≈ 0.1 (obtained by combining Df/D0 ≈ 0.3 with the highest possible anisotropy factor D|/D⊥ ≈ 6.4 and with eq 8) and D⊥/D0 ≈ 0.3 (if D|/D⊥ ≈ 1), the true value being somewhere between. For modeling purposes, we will assume an intermediate inverse tortuosity factor of D⊥/D0 ≈ 0.2. (iii) The effect of solute size for diffusion in a porous medium has been simulated by Sahimi and Jue.25 According to their results, the diffusion coefficient of a spherical particle in a medium made of a network of cylindrical pores can be written as D ∝ exp(-µa/d), where µ is a constant, a is the radius of the particle, and d is the pore cylinder radius. In the present situation, the pore cylinder radius could correspond to the smallest distance between adjacent microfibrils. They concluded that the particle diffusion was strongly reduced for 10-1 > a/d. In the present system, there are certainly gaps between microfibrils with a size less than 10 times the molecule size. The finite size effect will act to reduce the diffusion more than the tortuosity of the pore space would predict. To what extent the narrow gaps will influence the overall diffusion is difficult to predict, given the lack of knowledge about the detailed porous structure of the fiber wall. It is plausible that the tortuosity estimated with water selfdiffusion also includes some finite size effects. For simplicity, we assume that these effects are the same for water and CO2. (iv) Interactions between the solute and the surfaces of the porous material will inhibit diffusion further.33 As an example of possible interactions, we mention charged (31) Topgaard, D.; So¨derman, O. Biophys. J. 2002, 83, 3596-3606. (32) Topgaard, D.; So¨derman, O. J. Phys. Chem. B 2002, 106, 1188711892. (33) Stallmach, F.; Ka¨rger, J. Adsorption 1999, 5, 117-133.
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evaluated by the following ratio, which is the ratio of the number of CO2 molecules diffusing through the fiber wall to the total number of CO2 molecules that cross the gas pocket interface, per unit of time
JfSf D⊥z ≈ ≈ JfSf + JSCSSC D⊥z + 2D0r 3 × 10-10 × 10-4 ≈ 3 × 10-10 × 10-4 + 2 × 1.4 × 10-9 × 5 × 10-6 2 (14) 3
Figure 8. Real gas pocket trapped inside the lumen of a cellulose fiber acting as a bubble nucleation site in a glass poured with champagne (A), modeled as a slug-bubble trapped inside an ideal cylindrical microchannel and being fed with dissolved CO2 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 ) 100 µm.
solutes that tend to adhere to surfaces with opposite charge.20 Nevertheless, CO2 is a noncharged and nonpolar molecule, and it is very unlikely that it will bind to the cellulose surface. Finally, all of these factors taken together yields to D⊥ ≈ 0.2D0 ≈ 0.2 × 1.4 × 10-9 ≈ 3 × 10-10 m2 s-1 as a good order of magnitude for the diffusion coefficient of CO2 molecules across the fiber wall. 2.5. Application to Bubbling Frequencies from Cellulose Fibers Immersed in a Glass Poured with Champagne. In Figure 8, a gas pocket trapped inside the lumen of a cellulose fiber acting as a nucleation site is modeled as a deformed bubble, being bounded by two spherical caps of radius r ≈ 5 µm and by a cylindrical body of radius r and length z ≈ 100 µm. The number of CO2 molecules that cross the gas pocket interface, per unit of time is ruled by
dN ) dt
∫∫
B J × dS B gaz pocket surface
(12)
where B J is the flux of CO2 molecules defined by the first Fick’s law, B J ) D × ∇c b. In the latter equation, D is the diffusion coefficient of CO2 molecules, and ∇c b is the gradient of CO2 molecules between the liquid bulk and the boundary layer in equilibrium with the CO2 gas molecules in the vapor phase inside the gas pocket. Assuming (i) the flux of CO2 molecules as being constant along each given surface, and (ii) the gradient of CO2 molecules as being constant all around the gas pocket, eq 12 may be rewritten as follows:
dN ) dt
∫∫
∫∫
B+ B) B JSC × dS B Jf × dS spherical caps fiber wall (D0SSC + D⊥Sf) × ∇c (13)
where S corresponds to a given surface, and where the subscripts SC and f correspond to the spherical cap gas/ liquid interface and fiber wall, respectively. Geometrically, SSC ≈ 4πr2 and Sf ≈ 2πrz. The efficiency of the CO2 molecules diffusing through the fiber wall may finally be
Finally, about two-thirds of CO2 molecules that feed the trapped gas pocket diffuse through the fiber wall. Then, if the wall of cellulose fibers was impermeable with regard to CO2 molecules, the flux of CO2 molecules to feed the gas pockets trapped inside the fibers’ lumen would be about three times less. In turn, the kinetics of bubble formation would also be about three times less and the aspect of champagne effervescence would be completely different. Actually, after pouring champagne into a flute, an equilibrium is progressively established between bubbles nucleated on the numerous cellulose fibers adsorbed on the flute wall and bubbles bursting at the liquid surface. The bubble ring which is formed at the champagne surface, the so-called collerette, is of primarily importance during champagne tasting and undoubtedly constitutes one of the hallmark of this traditionally festive wine.4 The number of bubbles which form this bubble ring is approximately equal to nF hT h , where n is the total number of nucleation sites in the glass, F h is the average bubbling frequency of nucleation sites (of about 10-30 Hz after pouring1), and T h is the average bubble lifetime at the liquid surface, which strongly depends on the fine chemical composition of the champagne34-36 and on the age of the liquid surface, i.e., the time elapsed since champagne was poured.1 Finally, if the wall of cellulose fibers were impermeable with CO2 molecules, the average frequencies of bubble formation F h would be about three times less, i.e., only about 3-10 Hz, and champagne would look much less bubbly than it does. 3. Conclusions and Summary The transversal permeability of cellulose fibers saturated with liquid, with regard to dissolved CO2 molecules, was indirectly and quantitatively approached by retrieving information from previous NMR experiments on the selfdiffusion of water in cellulose fibers. The order of magnitude for the transversal diffusion coefficient D⊥ of CO2 molecules through the fiber wall was properly bounded by D⊥/D0 ≈ 0.1 and D⊥/D0 ≈ 0.3, D0 being the diffusion coefficient of CO2 dissolved molecules in the liquid bulk, far from any boundary. For modeling purposes, a value of about D⊥ ≈ 0.2D0 was proposed. This result directly applies to the kinetics of carbon dioxide bubble formation from glasses poured with champagne or sparkling wine, where most of bubble nucleation sites were recently identified as being cellulose fibers. If the cellulose fiber wall was impermeable with regard to CO2 dissolved molecules, it was suggested that the kinetics of bubbling would be about three times less than it does. (34) Sene´e, J.; Robillard, B.; Vignes-Adler, M. Food Hydrocolloids 1999, 13, 15-26. (35) Peron, N.; Cagna, A.; Valade, M.; Bliard, C.; Aguie´-Be´ghin, V.; Douillard, R. Langmuir 2001, 17, 791-797. (36) Peron, N.; Cagna, A.; Valade, M.; Marchal, R.; Maujean, A.; Robillard, B.; Aguie´-Beghin, V.; Douillard, R. Adv. Colloid Interface Sci. 2000, 88, 19-36.
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Because most of the particles adsorbed on the wall of a container or vessel free from any particular treatment are believed to be cellulose fibers coming from the surrounding air, this result about the transversal diffusion coefficient of CO2 dissolved molecules through the wall of hydrated cellulose fibers could be indeed extended to the more general field of heterogeneous bubble nucleation from supersaturated liquids.
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Acknowledgment. Thanks are due to the Europol’Agro institute and to the “Association Recherche Oenologie Champagne Universite´” for financial support, to Etienne Derat for a valuable dicussion, and to Champagne Moe¨t & Chandon and Pommery for their collaborative efforts. LA049960F
