Carbon dioxide in bottled carbonated waters and subsequent bubble

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Carbon dioxide in bottled carbonated waters and subsequent bubble nucleation under standard tasting condition Gérard Liger-Belair J. Agric. Food Chem., Just Accepted Manuscript • DOI: 10.1021/acs.jafc.9b00155 • Publication Date (Web): 29 Mar 2019 Downloaded from http://pubs.acs.org on March 30, 2019

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Journal of Agricultural and Food Chemistry

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Carbon dioxide in bottled carbonated waters and

2

subsequent bubble nucleation under standard tasting

3

condition

4

Gérard LIGER-BELAIR

5 6 7 8

Equipe Effervescence, Champagne et Applications (GSMA), UMR CNRS 7331, Université

9

de Reims Champagne-Ardenne, BP 1039, 51687 Reims Cedex 2, France.

10 11

Pr. Gérard Liger-Belair

12

telephone: + 333 26 91 33 93

13

e-mail : [email protected]

14 15

Keywords : carbonated waters; carbon dioxide; bubble nucleation; diffusion

16 17 18

ACS Paragon Plus Environment

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Abstract

21 22

Experimental and theoretical developments, including gas-liquid thermodynamics and bubble

23

nucleation, were made relevant to the conditioning and service of three various commercial

24

carbonated bottled waters holding different levels of dissolved carbon dioxide comprised

25

between about 3 g L-1 and 7 g L-1. The strong dependence in temperature of the partial

26

pressure of gas-phase CO2 found within the three batches of bottled carbonated waters was

27

determined. Moreover, in a glass of carbonated water, the process by which the diffusion of

28

dissolved CO2 in tiny immersed gas pockets enables heterogeneous bubble nucleation was

29

formalized, including every pertinent parameter at play. From this assessment, the minimum

30

level of dissolved CO2 below which bubble nucleation becomes thermodynamically

31

impossible was determined, and found to strongly decrease by increasing the water

32

temperature and size of the gas pockets acting as a bubble nucleation sites. Accordingly, the

33

total number of bubbles likely to form in a single glass of sparkling water was theoretically

34

derived in order to decipher the role played by various key parameters. Most interestingly, for

35

a given level of dissolved CO2, the theoretical number of bubbles likely to form in a glass was

36

found to increase by increasing the water temperature.

37


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39

1. Introduction

40

In most western countries, bottled water has become an important concern in people’s daily

41

lives.

42

value, the global bottled water market is expected to reach approximately USD 280 billion by

43

2020. 4 Nowadays, the sparkling water segment has reached about 10 % of the whole bottled

44

water segment, but this average percentage may vary a lot from one country to another. 1 In

45

the context where obesity in the population is increasing, sparkling waters, and especially

46

flavored sparkling waters, are seen as a substitute for sweet beverages. The bottled sparkling

47

water market is therefore booming, involving numerous companies throughout the world,

48

with Europe being the largest producer (75 %) followed by the USA (20 %). 4

49

From the physicochemical point of view, bottled carbonated waters are under a pressure of

50

gas-phase carbon dioxide (CO2), and therefore hold a level of dissolved CO2 within the liquid

51

phase, in accordance with Henry’s law which states that gas-phase and dissolved gas species

52

undergo thermodynamic equilibrium. As detailed in a directive from the European

53

Parliament, commercial bottled carbonated natural mineral waters fall into three categories:

54

(1) ‘‘naturally carbonated natural mineral water’’, when the water content of CO2 comes from

55

the spring and the bottle content is the same as at source; (2) ‘‘natural mineral water fortified

56

with gas from the spring’’ if the content of CO2 comes from the same resource but the bottle

57

content is greater than that established at source; and (3) ‘‘carbonated natural mineral water’’

58

if CO2 comes from an origin other than the groundwater resource is added. A method using

59

gas chromatography-isotope ratio mass spectrometry has been proposed to determine whether

60

or not gas-phase CO2 under pressure in the headspace of a sealed bottle of carbonated water

61

originates from the source spring or is of industrial origin. 6

62

In bottled carbonated waters, and sparkling beverages in general, the level of CO2 is a

63

parameter of paramount importance since it is responsible for the very much sought-after

1-3

In 2014, the global bottled water market stood at around 290 billion liters, and by


5

3


7-13

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64

bubbling process.

65

various sensory properties such as (i) the frequency of bubble formation in a glass, 14,15 (ii) the

66

growth rate of ascending bubbles,

67

CO2 acting on both trigeminal receptors,

68

recently that a minimum concentration of 1.2 g L-1 of dissolved CO2 is required to detect

69

mouthfeel attributes of carbonation and bite by consumers of sparkling wines, as reported by

70

McMahon et al.

71

dissolved CO2 in a carbonated water, subjects rated the carbonation bite equally strong under

72

normal atmospheric pressure (with bubbles), or under a pressure of two atmospheres (at

73

which bubbles could not form).

74

competitive market, it is therefore still looking for new insights and further developments

75

regarding gas-phase and dissolved CO2 equilibrium, as well as bubble dynamics. In the past

76

20 years, the chemical physics behind the production of yeast-fermented CO2 and bubbles has

77

been thoroughly investigated in champagne, sparkling wines, and beers (for recent and global

78

overviews see, for example, refs 25 and 26). Nevertheless, and to the best of my knowledge,

79

the thermodynamic equilibrium of carbon dioxide in carbonated bottled waters, and the

80

subsequent conditions for bubble nucleation under standard tasting conditions remained

81

poorly explored. In ref (12), the kinetics of bubbles’ growth and the progressive losses of

82

dissolved CO2 were closely examined in glasses poured with various batches of commercial

83

carbonated waters holding different levels of dissolved CO2 comprised between 3.2 g L-1 and

84

6.9 g L-1. The present article complements the results evidenced by this previous set of data.

85

Here, experimental and theoretical developments about the thermodynamic equilibrium of

86

dissolved and gas-phase CO2 relevant to the conditioning and service of the same batches of

87

commercial carbonated bottled waters are conducted. The strong dependence in temperature

88

of the partial pressure of gas-phase CO2 found within three batches of bottled carbonated

23

The presence of dissolved CO2 in sparkling water directly impacts

7,8,16

and (iii) the perception of dissolved and gas-phase 17-20

and gustatory receptors. 21,22 It was highlighted

However, and most interestingly, it was reported that for a given level of

24

The bottled sparkling water segment being a very


4

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89

waters is explored. The bubble nucleation theory is used in order to decipher the minimum

90

level of dissolved CO2 below which bubble nucleation becomes impossible in a glass poured

91

with carbonated water. Moreover, and accordingly, the issue of the number of bubbles likely

92

to form in a single glass of sparkling water was theoretically derived in order to evidence the

93

influence of various parameters at play.

94 95

2. Materials and Methods

96

2.1. Water samples

97

Three batches of commercial carbonated natural mineral bottled waters from Poland, holding

98

different levels of CO2, and provided by Danone Research, were investigated in this study.

99

They are the same as the ones used in ref (12). They are described and referenced as follows:

100

1- A batch of low carbonated water (labeled W1);

101

2- A batch of medium carbonated water (labeled W2 ); and

102

3- A batch of highly carbonated water (labeled W3).

103

Each batch contains four identical bottles. Bottles from W1 are conditioned in 0.7 liter

104

polyethylene terephthalate (PET) bottles, whereas bottles from W2 and W3 are conditioned in

105

1.5 liter PET bottles.

106 107

2.2. The temperature-dependent solubility of carbon dioxide in water

108

The dissolution of gas-phase CO2 in bottled carbonated waters is ruled by Henry’s law

109

equilibrium, which states that the concentration c L of dissolved CO2 in the liquid phase is

110

proportional to the partial pressure of gas-phase CO2, according to the following relationship:

111 112

c L  k H PCO 2

(1)

113


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114

with k H being the temperature-dependent Henry’s law constant of gas-phase CO2 in water,

115

27,28

116

The temperature-dependence of the Henry’s law constant of CO2 in water can be conveniently

117

expressed with a Van’t Hoff like equation as follows:

and PCO 2 being the pressure of gas-phase CO2 in the sealed bottle.

118

119

 H diss k H T   k 298 K exp  R 

1  1     T 298 

(2)

120 121

with k 298 K  1.49 g L-1 bar -1 being the Henry’s law constant of CO2 in water at 298 K (i.e., at

122

25 °C), H diss  -19.9 kJ mol-1 being the subsequent dissolution enthalpy of CO2 in water, 29

123

and R being the ideal gas constant (8.31 J K-1 mol-1).

124 125

2.3. Data analysis

126

Dissolved CO2 concentrations in the three various commercial carbonated water samples were

127

determined, at a temperature near 0 °C, by using carbonic anhydrase (labeled C2522 Carbonic

128

Anhydrase Isozyme II from bovine erythrocytes, and provided from Sigma-Aldrich - US).

129

The concentration of dissolved CO2 in each type of carbonated water was conducted

130

immediately after opening the bottle. This titrimetric determination of dissolved carbon

131

dioxide is described in minute details in a previous article. 31

132

The density of each water sample was measured, at 20 °C, with a digital density meter

133

(Mettler Toledo 30PX) based on the oscillating U-tube technique. Moreover, the dynamic

134

viscosity of each water sample was also measured, at 20 °C, with an Ubbelhode capillary

135

viscometer (Schott Gerate). Densities and viscosities were measured with water samples first

136

degassed under vacuum.


30

6

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137

To enable a statistical treatment, measurements of dissolved CO2, density, and viscosity were

138

done on the four different bottles per batch. Table 1 compiles the respective concentrations of

139

dissolved CO2 found in the three various commercial carbonated waters used in connection

140

with this study, as well as their respective density and viscosity. For comparison purposes, the

141

characteristic concentration of dissolved CO2, density, and viscosity of a standard commercial

142

Champagne wine are also reported in Table 1.25

143 144

3. Results and discussion

145

3.1. Deciphering the thermodynamic equilibrium in the sealed bottles

146

In a bottle of carbonated water hermetically sealed with a crown or a screw cap, a volume VG

147

of gas phase in the headspace cohabits with a volume VL of water (i.e., the liquid phase). For

148

the sake of simplicity, we suppose that both volumes remain constant (i.e., we neglect the

149

minute changes of the liquid volume caused by dilation or retraction caused by modifications

150

of the bottle temperature, or due to the progressive dissolution or exsolution of CO2).

151

Moreover, in the bottle hermetically sealed, the total number of moles of CO2, denoted nT , is

152

a conserved quantity that decomposes into nG moles in the gas phase and n L moles in the

153

liquid phase. Therefore,

154 155

nT  nG  n L

(3)

156 157

Furthermore, in the realistic pressure range found in a bottle of carbonated water (a few bars),

158

we may safely suppose that the gas phase is ruled by the ideal gas law. Thus,

159 160

PCO 2 VG  nG RT

(4)


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161 162

with nG being the number of moles of gas-phase CO2 found in the bottle headspace.

163

Finally, in the bottle hermetically sealed, dissolved and gas-phase CO2 always verify the

164

following system of equations, as exemplified in Figure 1:

165

166

nL  c L  V  k H PCO 2 L  nT  n L  nG  P V  n RT G  CO 2 G 

(5)

167 168

With the knowledge of the level of dissolved CO2 initially found in the three various

169

carbonated waters used in connection with this study (see Table 1), the pressure of gas-phase

170

CO2 in the sealed bottles can simply be deduced through Henry’s law as PCO 2  c L k H , with

171

k H determined at 0 °C through equation (2). As a result, for the sealed bottles near 0 °C, the

172

pressures of gas-phase CO2 found in the three carbonated waters showing increasing levels of

173

dissolved CO2 are expected to be 1.04, 1.46, and 2.21 bar, respectively. Actually, with the

174

knowledge of the respective partial pressures of gas-phase CO2 found in the headspace of the

175

various bottles, the mole number of gas-phase CO2 in the headspace of the various bottled

176

carbonated waters through equation (4) as nG  PCO 2 VG RT . The total mole number of CO2 (

177

nT ) found in each sealed bottle of carbonated water can therefore easily be deduced through

178

equation (3). Table 2 compiles the volume of water, the headspace volume, and the total mole

179

number of CO2 trapped in the three various bottled carbonated waters. Finally, by using the

180

system of equations (5) combined with equation (2), the theoretical dependence in

181

temperature of both the partial pressure of gas-phase CO2 and the concentration of dissolved


8

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182

CO2 found in the sealed bottles of carbonated water can be determined according to the

183

following equation (expressed in the MKS system of units):

184

185

nT RT   PCO 2  V  k RTV  G H L  c  nT k H RT  L VG  k H RTV L

(6)

186 187

The dependence in temperature of the pressure of gas-phase CO2 found within the sealed

188

bottles of carbonated waters in connection with the present study is plotted in Figure 2, in the

189

temperature range between 0 and 30 °C. Inevitably, the dependence in temperature of carbon

190

dioxide solubility in water, combined with the ideal gas law, leads to a dependence in

191

temperature of the pressure of gas-phase CO2 in a sealed bottle of carbonated water. As a

192

comparison, the temperature-dependent pressure of gas-phase CO2 found in the headspace of

193

a standard 75 cL bottle of commercial Champagne wine appears in Figure 2. 32 Very clearly,

194

the pressure of gas-phase CO2 found in a bottle of champagne is much higher than that found

195

in a bottle of sparkling water, mainly because the total mole number of yeast-fermented CO2

196

is much higher in a 75 cL bottle of champagne than in a 75 cL bottle of sparkling water (in the

197

usual range of CO2 levels of commercial sparkling waters covered by the present study).

198

Otherwise, and strictly speaking, in the sealed medium of bottled carbonated waters, water

199

vapor in the headspace under the crown or screw cap is indeed under equilibrium with the

200

liquid-phase water. Above the melting point of water, the strongly temperature-dependent

201

saturated water vapor pressure PsatH 2O found in the bottle headspace is correctly approached

202

through the Clausius-Clapeyron equation defined hereafter, provided that water vapor

203

behaves as an ideal gas, and that the specific latent heat of water evaporation Lvap remains

204

reasonably constant in this range of temperatures: 33


9


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205

206

 MLvap  1 1  PsatH 2O  P0 exp      R  373 T 

(7)

207 208

with P0 being the ambient pressure (close to 1 bar), M being the molar mass of water (0.018

209

kg mol-1), and Lvap being the specific latent heat of water evaporation at 20 °C (

210

 2.47  10 6 J kg -1 ).34

211

Nevertheless, by using the latter equation, it appears that the saturated water vapor pressure

212

found in the headspace of the sealed bottles of carbonated waters remains at least two orders

213

of magnitude below the partial pressure of gas-phase CO2, as already noticed in the headspace

214

of corked champagne bottles. 32 Finally, the total pressure of the CO2/H2O gas mixture found

215

in a sealed bottle of carbonated water may safely be considered as being equivalent to the

216

partial pressure of gas-phase CO2, whatever the bottle temperature in a reasonable range of

217

tasting temperatures.

218 219

3.2. Is there a critical dissolved CO2 concentration required for bubbling?

220

In the thermodynamically stable context of a bottle hermetically closed, the capacity of a

221

significant quantity of CO2 to remain dissolved in the liquid phase is achieved by the

222

relatively high pressure of gas-phase CO2 found in the bottle headspace (as shown in Figure

223

2). But, as soon as the bottle is opened, the pressure of gas-phase CO2 falls. The

224

thermodynamic equilibrium of dissolved and gas-phase CO2 is broken. The new stable

225

atm  0.7 mg L-1 (following Henry’s concentration of dissolved CO2 should be only ceq  k H PCO 2

226

atm law at 20 °C, with the partial pressure PCO of gas-phase CO2 in ambient air  0.4 mbar only). 2

227

Therefore, almost all the dissolved CO2 held in the sealed bottled carbonated waters must


10

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228

desorb from the liquid phase to reach ambient air (through bubbling and by diffusion through

229

the free air/water interface). 12

230

Bubbling being the hallmark of sparkling waters, many consumers attach premium

231

importance to both the number and size of bubbles likely to form in a glass. 35,36 But is there a

232

critical concentration of dissolved CO2 below which bubble production could become

233

thermodynamically impossible? In other words, is there a minimum level of dissolved CO2 to

234

overcome in your glass of water to enable the very much sought-after bubbling process?

235

Carbonated waters, and sparkling beverages in general, are weakly supersaturated with

236

dissolved CO2. Bubble formation is therefore limited by an energy barrier to overcome.

237

In order to nucleate, CO2 bubbles need preexisting gas cavities immersed in the liquid phase,

238

with radii of curvature large enough to overcome the nucleation energy barrier and grow

239

freely. 38-41 This process is referred as non-classical heterogeneous bubble nucleation.

240

critical radius of curvature r* required for bubble nucleation can be determined according to

241

the following relationship, with parameters being expressed in the MKS system of units: 40

38

37,38

The

242

243

r 

2k H c L  k H P0 

(8)

244 245

with  being the surface tension of the liquid/gas interface ( 73 mN m-1 in sparkling waters at

246

15 °C), k H being the strongly temperature-dependent Henry’s law constant of CO2 in water

247

(expressed in mol m -3 Pa -1 ), P0 being the ambient pressure (  105 Pa ), and c L being the

248

dissolved CO2 concentration found in the liquid phase (expressed in mol m-3).

249

Strictly speaking, the surface tension of water is also temperature-dependent,

250

temperature-dependent correlation obeys the relationship presented hereafter: 43

42

and its latest

251


11


252

 T   B1   TC

  



  T 1  b1   TC 

Page 12 of 34

  

(9)

253 254

with TC  647.096 K , B  235.8 N m -1 , b  0.625 , and   1.256 .

255

This equation is valid from the supercooled region (to temperatures as cool as 248 K) up to

256

the reference temperature TC .43 Nevertheless, in the range of temperatures between 0 and 30

257

°C,  varies only from about 75 to 71 mN m-1. For the sake of simplicity  can be considered

258

constant in the following, with an average value of 73 mN m-1 (corresponding to the surface

259

tension of pure water at 15 °C). By combining equations (2) and (8), the critical radii r*

260

required to enable non-classical heterogeneous bubble nucleation (at 15 °C) in the three

261

carbonated waters were found to be 2.2 µm for W1, 1.1 µm for W2, and 0.6 µm for W3,

262

respectively. By contrast, r* is rather in the order of only 0.25 µm for a typical Champagne

263

wine at 15 °C. 25 Actually, r* is much smaller in champagne than in carbonated waters (for a

264

given temperature), simply because the typical surface tension of champagne is close to 50

265

mN m-1 (lower than that of pure water), and the level of yeast-fermented dissolved CO2 in

266

champagne is close to 11 g L-1 (higher than that of carbonated waters).

267

inspection of glasses poured with sparkling beverages revealed that most of the bubble

268

nucleation sites were located on pre-existing gas cavities trapped inside tiny crevices from the

269

glass wall, or inside hollow cellulose-fibers of the order of 100 µm long with a cavity mouth

270

in the range of 1-20 µm.

271

gas cavity trapped inside a fiber or a micro-crevice turned out to be much higher than the

272

critical radius r* required for non-classical heterogeneous bubble nucleation. 25

273

Nevertheless, in a glass poured with champagne or sparkling water, the level of dissolved CO2

274

c L was found to inexorably decrease as time proceeds. 12,31,44 In turn, and following equation

275

(8), r* is found to progressively increase with time. Therefore, under standard tasting

8-10,36,40

25

Actually, close

Therefore, and in most cases, the radius of the pre-existing


12

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276

conditions, the radius of curvature of a pre-existing gas cavity inexorably reaches the critical

277

radius r*, thus precluding bubble formation. The minimum level of dissolved CO2 needed to

278

enable non-classical heterogeneous bubble nucleation from a pre-existing gas cavity

279

immersed in a glass of water can be determined by using simple arguments based on

280

molecular diffusion, according to the scheme displayed in Figure 3. The mechanism behind

281

the nucleation of a CO2 bubble is molecular diffusion. The molar flux of gas-phase CO2

282

(denoted J) which crosses the water/CO2 interface to feed the CO2 bubble therefore obeys the

283

first Fick’s law expressed hereafter: 36

284

285

J   Dc  D

c L  c B 

(10)



286 287

with D being the diffusion coefficient of CO2 in water (in the order of 1.8  10 9 m2 s-1, at 20

288

°C, as determined through 13C Nuclear Magnetic Resonance 46), λ being the thickness of the

289

diffuse boundary layer where a gradient of dissolved CO2 exists, c L being the dissolved CO2

290

molar concentration found in the water bulk (far from the bubble), and c B being the dissolved

291

CO2 molar concentration close to the bubble surface in equilibrium with gas-phase CO2

292

within the bubble (as exemplified in Figure 3).

293

Actually, following both Laplace’s and Henry’s law, the concentration c B of dissolved CO2 is

294

in the order of c B  k H P0  2 r  . Indeed, the contribution of hydrostatic pressure is simply

295

negligible in a standard glass filled with a reasonable water depth h in the order of 10 cm

296

gh  10

297

found in the water bulk reaches the dissolved CO2 concentration c B found close to the CO2

298

bubble, bubble nucleation and growth stops because molecular diffusion vanishes (i.e., J  0

299

in equation (10)). Therefore, as the concentration of dissolved CO2 found in the liquid bulk

3



 10  10 1  10 3 Pa  P0 . Therefore, as the dissolved CO2 concentration c L


13


Page 14 of 34

300

reaches the critical value cL expressed hereafter, the transfer of CO2 from the water bulk

301

toward the CO2 bubble ceases, and the nucleation site simply stops releasing bubbles through

302

lack of dissolved CO2:

303

304

2   c L  c B  k H  P0   r  

(11)

305 306

It is also noteworthy to mention that c L is strongly temperature-dependent, because k H is

307

indeed highly temperature-dependent, as seen in equation (2). In the range of temperature

308

between 0 and 30 °C, the temperature dependence of the critical concentration c L below

309

which bubbling would become thermodynamically impossible is plotted in Figure 4 for pre-

310

existing gas cavities showing radii of curvature of 1, 2, 5, and 10 µm, respectively. It can thus

311

be concluded that the colder the sparkling water is in your glass, the higher the critical

312

concentration of dissolved CO2 needed to produce bubbles is. Moreover, the smaller the

313

radius of curvature of the pre-existing gas cavity (and therefore the smaller the particle or the

314

glass anfractuosity responsible for the bubble nucleation process), the higher the subsequent

315

critical concentration of dissolved CO2 c L . From the point of view of the consumer, the

316

bubbling process will stop automatically from a given bubble nucleation site once the level of

317

dissolved CO2 - which continuously decreases after pouring the sparkling water in a glass -

318

reaches the critical concentration of dissolved CO2 defined earlier.

319

Basically, equation (11) applies for the critical concentration required for bubbling in

320

champagne glasses.

321

which bubbling becomes thermodynamically impossible in champagne was also plotted in

322

Figure 4, for a gas cavity with a radius of curvature of 2 µm - a characteristic size for the

47

Likewise, the temperature-dependent critical concentration below


14

Page 15 of 34


323

micro-crevices found in laser-etched glasses often used for champagne tasting. 36 The typical

324

surface tension of champagne being close to 50 mN m-1, and the temperature-dependent

325

Henry’s law constant of CO2 in champagne slightly differing from that in water (mainly due

326

to ethanol, with k 298 K  1.21 g L-1 bar -1 , and H diss  -24.8 kJ mol-1 in a standard commercial

327

champagne 9), it can be clearly evidenced that c L is higher in sparkling water than in

328

champagne (for an identical gas cavity acting as a bubble nucleation site).

329 330

3.3. Exploring the number of bubbles in a single glass of sparkling water

331

The issue of the number of bubbles likely to form in a single glass of bubbly was discussed

332

recently.

333

of dissolved CO2 found in the glass after pouring, tiny pre-existing gas cavities trapped within

334

immersed particles or glass anfractuosities, and ascending bubble dynamics. In a glass of

335

sparkling water, the physicochemical processes being basically the same, the same reasoning

336

set out for champagne glasses may therefore apply to address this question with confidence.

337

The number N of bubbles likely to form in a glass was found to depend on various parameters

338

of both the liquid phase and the glass itself, according to the following relationship: 35

35

This number was found to be the result of a complex interplay between the level

339

340

2  10 7 V  g  N   h  T 

23

 c  k H P0 ln L  c L  k H P0

  

(12)

341 342

with V being the volume of water served in the glass,  being the champagne density (close to

343

the density of water), g being the gravity acceleration (close to 10 m s-2), and h being the

344

distance travelled by a bubble from its nucleation site to the liquid surface (considered as

345

being the water depth in the glass, if most of bubble nucleation sites are located on the bottom

346

of the glass).


15


Page 16 of 34

347

By replacing in equation (12) cL by its expression given in equation (11), and  and g by

348

their respective numerical value, the number of bubbles likely to form in a glass of sparkling

349

water can be rewritten as follows (with every parameter expressed in the MKS system of

350

units):

351

352

N

1010 V  r c L  k H P0    ln 2k H  hT 2 3  

(13)

353 354

In Figure 5, the total number of bubbles likely to form in a glass poured with 20 centiliters of

355

sparkling water and with a reasonable water depth of 10 centimeters, is plotted versus the

356

dissolved CO2 concentration found in the liquid phase. Four tasting temperatures were

357

investigated (namely 5, 10, 15, and 20 °C). An identical average radius of curvature r  5 µm

358

was considered for the pre-existing gas cavities immersed in the liquid phase and acting as

359

bubble nucleation sites. I am aware that each gas cavity immersed in the liquid phase has its

360

own radius of curvature r, but according to my background with the bubbling analysis in

361

sparkling wines, 5 µm is a reasonable mean value for this radius of curvature.25 It is worth

362

noting from Figure 5 that the total number of bubbles likely to nucleate in a glass of sparkling

363

water logically increases with the level of dissolved CO2 found in the liquid phase. Moreover,

364

and most interestingly, the total number of bubbles likely to nucleate in a glass increases with

365

the tasting temperature of water (for a given level of dissolved CO2). Actually, and as shown

366

in Figure 4, c L decreases with increasing the liquid-phase temperature. Therefore, by

367

increasing the tasting temperature of water, the number of bubbles likely to form per glass

368

will increase because bubbles will be able to nucleate at increasingly low dissolved CO2

369

concentrations.

370


16

Page 17 of 34


371

4. Conclusions and Prospects

372

Experimental and theoretical developments including gas-solution thermodynamics, bubble

373

nucleation, and bubble dynamics, were made relevant to explore the conditioning and service

374

of three commercial carbonated bottled waters holding different levels of carbon dioxide

375

comprised between about 3 g L-1 and 7 g L-1. In the range between 0 and 30 °C, the

376

dependence in temperature of the CO2 pressure found in the various carbonated bottled waters

377

sealed with a screw cap was presented. Once the sparkling water is poured in a glass, the

378

diffusion process by which gas-phase CO2 invades the tiny gas pockets immersed in the liquid

379

phase was presented. From this assessment, the minimum dissolved CO2 concentration

380

required to enable heterogeneous bubble nucleation was theoretically derived. This critical

381

level of dissolved CO2 was found to strongly depend on both the water temperature and the

382

size of the tiny gas pockets acting as bubble nucleation sites. Accordingly, and based on a

383

previous study conducted about heterogeneous bubble nucleation in a glass of champagne, the

384

question of how many bubbles are likely to form in a single glass was theoretically discussed

385

in order to decipher the role played by various key parameters. Most interestingly, for a given

386

level of dissolved CO2, the theoretical number of bubbles likely to form in a glass was found

387

to increase by increasing the water temperature.

388

The approach described in the study could also be extended to the very competitive market of

389

non-alcoholic sparkling beverages, still looking for new insights regarding carbon dioxide and

390

bubble dynamics.

391 392

Acknowledgments: Gérard Liger-Belair is indebted to Danone Research for providing the

393

various carbonated water samples, and to the CNRS and the Association Recherche

394

Oenologie Champagne Université (AROCU) for supporting his team and research.


17


Nomenclature

396 397

Page 18 of 34

cB

concentration of dissolved CO2 in Henry’s equilibrium with gas-phase CO2 in the bubble, in g L-1

398 399

ci

initial concentration of dissolved CO2 in the liquid phase, in g L-1

400

cL

concentration of dissolved CO2 in the liquid phase, in g L-1

401

c L

critical concentration of dissolved CO2 in the liquid phase below which bubbling becomes thermodynamically impossible, in g L-1

402 403

D

diffusion coefficient of dissolved CO2 in the liquid phase, in m2 s-1

404

g

gravity acceleration,  10 m s-2

405

h

level of liquid in the glass (water depth), in m

406

J

molar flux of gaseous CO2 which crosses the bubble interface, in mol m-2 s-1

407

kH

Henry’s law constant of dissolved CO2 in water (i.e., its solubility), in mol m-3 Pa-1

408

Lvap

specific latent heat of water evaporation at 20 °C,  2.47  10 6 J kg -1

409

M

molar mass of water, = 18 g mol-1

410

N

total number of bubbles likely to nucleate in a glass

411

nG

mole number of gas-phase CO2 in the headspace of a bottled carbonated water, in mol

412

nL

mole number of dissolved CO2 in a bottled carbonated water, in mol

413

nT

total mole number of CO2 found in a sealed bottle of carbonated water, in mol

414

n

mole number of gaseous CO2 in the bubble, in mol

415

P0

ambient pressure,  105 Pa

416

PCO 2

partial pressure of gas-phase CO2 found in the sealed bottle, in Pa

417

atm PCO 2

partial pressure of gas-phase CO2 found in ambient air,  40 Pa  0.4 mbar


18

Page 19 of 34

418


r

acting as a CO2 bubble nucleation site, in m

419 420

radius of curvature of the pre-existing gas cavity immersed in the liquid phase, and

r

critical radius of curvature required to enable bubble nucleation from a pre-existing gas cavity, in m

421 422

R

ideal gas constant, = 8.31 J K-1 mol-1

423

T

water temperature, in K

424

V

volume of water poured in a glass, in m3

425

VG

volume of gas phase in the headspace of the sealed bottle, in m3

426

VL

volume of liquid in the sealed bottle, in m3

427



surface tension of water, in N m-1

428



thickness of the diffusion boundary layer around the bubble, in m

429



dynamic viscosity of water, in Pa s

430



density of water, in kg m-3


19


References

432 433

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1.

Euzen, A. Bottled water, globalization and behaviour of consumers. Eur. J. Water Qual. 2006, 37, 143-155.

434 435

2.

Storey, M. The shifting beverage landscape. Physiol. Behav. 2010, 100, 10-14.

436

3.

Rani, B.; Maheshwari, R.; Garg, A.; Prasad, M. Bottled water – A global market overview. Bull. Environ. Pharmacol. Life Sci. 2012, 1, 1-4.

437 438

4.

water-market-z39681

439 440

5.

DIRECTIVE 2009/54/EC OF THE EUROPEAN PARLIAMENT AND OF THE COUNCIL of 18 June 2009 on the exploitation and marketing of natural mineral waters.

441 442

Zion Research Analysis, 2015. https://www.marketresearchstore.com/report/bottled-

6.

Calderone, G.; Guillou, C.; Reniero, F.; Naulet, N. Helping to authenticate sparkling 13C/12C

443

drinks with

of CO2 by gas chromatography-isotope ratio mass spectrometry.

444

Food Res. Int. 2007, 40, 324-331.

445

7.

Shafer, N.E.; Zare, R.N. Through a beer glass darkly. Phys. Today 1991, 44, 48-52.

446

8.

Liger-Belair, G. The physics and chemistry behind the bubbling properties of champagne

447

and sparkling wines: A state-of-the-art review. J. Agric. Food. Chem. 2005, 53, 2788-

448

2802.

449 450 451 452

9.

Uzel, S.; Chappell, M.A.; Payne, S.J. Modeling the cycles of growth and detachment of bubbles in carbonated beverages. J. Phys. Chem. B 2006, 110, 7579-7586.

10. Lee, W.T.; McKechnie, J.S.; Devereux, M.G. Bubble nucleation in stout beers. Phys. Rev. E 2011, 83, 051609.

453

11. Perret, A.; Bonhommeau, D.; Liger-Belair, G.; Cours, T.; Alijah, A. CO2 diffusion in

454

Champagne wines: A molecular dynamics study. J. Phys. Chem. B 2014, 118, 1839-

455

1847.


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12. Liger-Belair, G.; Sternenberg, F.; Brunner, S.; Robillard, B.; Cilindre, C. Bubble

457

dynamics in various commercial sparkling bottled waters. J. Food Eng. 2015, 163, 60-70.

458

13. Zenit, R., Rodriguez-Rodriguez, J. The fluid mechanics of bubbly drinks. Phys. Today

459

2018, 71, 44-50.

460

14. Liger-Belair, G.; Marchal, R.; Robillard, B.; Vignes-Adler, M.; Maujean, A.; Jeandet, P.

461

Study of effervescence in a glass of champagne: Frequencies of bubble formation, growth

462

rates, and velocities of rising bubbles. Am. J. Enol. Vitic. 1999, 50, 317-323.

463

15. Liger-Belair, G.; Parmentier, M.; Jeandet, P. Modeling the kinetics of bubble nucleation

464

in champagne and carbonated beverages. J. Phys. Chem. B 2006, 110, 21145-21151.

465

16. Zhang, Y.; Xu, Z. “Fizzics” of bubble growth in beer and champagne. Elements 2008, 4,

466 467 468

47-49 17. Lawless, H.T.; Heymann, H. Sensory evaluation of food: principles and practices; Springer: New York, 2010.

469

18. Dessirier, J.M.; Simons, C.; Carstens, M.; O’Mahony, M.; Carstens, E. Psychophysical

470

and neurobiological evidence that the oral sensation elicited by carbonated water is of

471

chemogenic origin. Chem. Senses 2000, 25, 277-284.

472

19. Kleeman, A.; Albrecht, J.; Schöpf, V.; Haegler, K.; Kopietz, R.; Hempel, J.M.; Linn, J.;

473

Flanagin, V.L.; Fesl, G.; Wiesmann, M. Trigeminal perception is necessary to localize

474

odors. Physiol. Behav. 2009, 97, 401-405.

475

20. Meusel, T.; Negoias, S.; Scheibe, M.; Hummel, T. Topographical differences in

476

distribution and responsiveness of trigeminal sensitivity within the human nasal mucosa.

477

Pain 2010, 151, 516-521.

478 479

21. Chandrashekar, J.; Yarmolinsky, D.; von Buchholtz, L.; Oka, Y.; Sly, W.; Ryba, N.J.; Zucker, C.S. The taste of carbonation. Science 2009, 326, 443-445.


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22. Dunkel, A.; Hofmann, T. Carbonic anhydrase IV mediates the fizz of carbonated beverages. Angew. Chem. Int. Ed. 2010, 49, 2975-2977. 23. McMahon, K. M.; Culver, C.; Ross, C. F. The production and consumer perception of sparkling wines of different carbonation levels. J. Wine Res. 2017, 28, 123–134. 24. Wise, P.M.; Wolf, M.; Thom, S.R.; Bryant, B. The influence of bubbles on the perception carbonation bite. PLoS ONE 2013, 8, e71488. 25. Liger-Belair, G. Effervescence in Champagne and sparkling wines: From grape harvest to bubble rise. Eur. Phys. J. Spec. Top. 2017, 226, 3-116. 26. Vega-Martinez, P.; Enriquez, O.; Rodriguez-Rodriguez, J. Some Topics on the physics of bubble dynamics in beer. Beverages 2017, 3, 38. 27. Carroll, J.J.; Mather, A.E. The system carbon dioxide/water and the KrichevskyKasarnovsky equation. J. Solution Chem. 1992, 21, 607-621.

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28. Diamond, L.W.; Akinfief, N.N. Solubility of CO2 in water from 1.5 to 100 °C and from

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0.1 to 100 MPa: Evaluation of literature data and thermodynamic modelling. Fluid Phase

494

Equilib. 2003, 208, 265-290.

495 496 497 498

29. Lide, D. R.; Frederikse, H. P. Handbook of Chemistry and Physics; 76th ed. CRC Press: Boston, 1995. 30. Caputi, A.; Ueda, M.; Walter, P.; Brown, T. Titrimetric determination of carbon dioxide in wine. Am. J. Enol. Vitic. 1970, 21, 140-144.

499

31. Liger-Belair, G.; Villaume, S.; Cilindre, C.; Jeandet, P. Kinetics of CO2 fluxes outgassing

500

from champagne glasses in tasting conditions: The role of temperature. J. Agric. Food

501

Chem. 2009, 57, 1997–2003.

502 503

32. Liger-Belair, G.; Cordier, D.; Honvault, J.; Cilindre, C. Unveiling CO2 heterogeneous freezing plumes during champagne cork popping. Sci. Rep. 2017, 7, 10938.


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504

33. Stowe K. An Introduction to Thermodynamics and Statistical Mechanics; Cambridge

505

University Press: New York, 2007.

506

34. Provided by the NIST database (http://webbook.nist.gov/chemistry)

507

35. Liger-Belair, G. How many bubbles in your glass of bubbly? J. Phys. Chem. B 2014, 118,

508 509 510 511 512 513 514 515 516

3156-3163. 36. Liger-Belair, G. Modeling the losses of dissolved carbon dioxide from laser-etched champagne glasses. J. Phys. Chem. B 2016, 120, 3724-3734. 37. Wilt, P.M. Nucleation rates and bubble stability in water-carbon dioxide solutions. J. Colloid Interface Sci. 1986, 112, 530-538. 38. Jones, S.F.; Evans, G.M.; Galvin, K.P. Bubble nucleation from gas cavities: A review. Adv. Colloid Interface Sci. 1999, 80, 27-50. 39. Lugli, F.; Zerbetto, F. An introduction to bubble dynamics. Phys. Chem. Chem. Phys. 2007, 9, 2447-2456.

517

40. Liger-Belair, G.; Vignes-Adler, M.;Voisin, C.; Robillard, B.; Jeandet, P. Kinetics of gas

518

discharging in a glass of champagne: The role of nucleation sites. Langmuir 2002, 18,

519

1294-1301.

520 521 522 523 524

41. Lubetkin, S.D. Why is it much easier to nucleate gas bubbles than theory predicts? Langmuir 2003, 19, 2575-2587. 42. Vargaftik, N.B.; Volkov, B.N.; Voljak, L.D. International tables of the surface tension of water. J. Phys. Chem. Ref. Data 1983, 12, 817-820. 43. Dooley, R. B. Revised Release on Surface Tension of Ordinary Water Substance.

525

International

Association

for

the

Properties

526

http://www.iapws.org/relguide/Surf-H2O.html (2014)


of

Water

and

Steam

23


Page 24 of 34

527

44. Liger-Belair, G.; Conreux, A.; Villaume, S.; Cilindre, C. Monitoring the losses of

528

dissolved carbon dioxide from laser-etched champagne glasses. Food Res. Int. 2013, 54,

529

516-522.

530

45. Perret, A. Etude des propriétés de transport du CO2 et de l’éthanol en solution

531

hydroalcoolique par dynamique moléculaire classique : Application aux vins de

532

Champagne. PhD Thesis, Reims, France, 2014.

533

46. Liger-Belair, G.; Prost, E.; Parmentier, M.; Jeandet, P.; Nuzillard, J.-M. Diffusion 13C

534

coefficient of CO2 molecules as determined by

535

beverages. J. Agric. Food Chem. 2003, 51, 7560-7563.

NMR in various carbonated

536

47. Liger-Belair, G.; Carvajal-Pérez, D.; Cilindre, C.; Facque, J.; Brevot, M.; Litoux-

537

Desrues, F.; Chaperon, V.; Geoffroy, R. Evidence for moderate losses of dissolved CO2

538

during aging on lees of a champagne prestige cuvee. J. Food Eng. 2018, 233, 40-48.


24

Page 25 of 34


540

Tables

541

Batch

[CO2]

ci (g

L-1)

Viscosity 

Density 

( 10-3 Pa s)

(kg m-3)

W1

3.25 ± 0.08

0.98 ± 0.01

998 ± 1

W2

4.53 ± 0.15

0.99 ± 0.01

998 ± 1

W3

6.87 ± 0.28

0.99 ± 0.01

998 ± 1

Champagne

 11

 1.6

 103

542 543

Table 1: Concentrations of dissolved CO2 found in the three batches of commercial

544

carbonated waters sealed in PET bottles, as well as their respective dynamic viscosity and

545

density (at 20 °C). Standard deviations correspond to the root-mean-square deviations of the

546

values provided by the four distinct bottles per batch. By way of comparison, the

547

characteristic concentration of dissolved CO2, viscosity, and density of a standard commercial

548

Champagne wine are reported. 25

549

Batch

VL (L)

VG (mL)

nT (mmol)

W1

0.7

10

52.2 ± 1.3

W2

1.5

10

155.0 ± 5.1

W3

1.5

10

235.2 ± 9.5

Champagne

0.75

25

 200

550 551

Table 2: Volume of water, headspace volume, and total mole number of CO2 found in the

552

three batches of commercial bottled carbonated waters sealed in PET bottles. By way of

553

comparison, in a standard commercial 75 cL bottle of champagne, the volume of champagne,

554

headspace volume, and total mole number of CO2 are reported. 25


25


Page 26 of 34

555


26

Page 27 of 34


557

Figure Captions

558 559

Figure 1: Scheme of a sealed bottle exemplifying the thermodynamic equilibrium

560

experienced by dissolved and gas-phase CO2 between the liquid phase and the bottle gaseous

561

headspace.

562 563

Figure 2: Partial pressures of gas-phase CO2 within the three various commercial carbonated

564

natural mineral bottled waters in the range of temperature between 0 and 30 °C. As a

565

comparison, the temperature-dependent pressure of gas-phase CO2 found in the headspace of

566

a standard 75 cL bottle of champagne appears as a grey dotted line. 32

567 568

Figure 3: Close up view of a tiny particle immersed in the liquid phase, as captured through

569

the lens of a high-speed video camera. 45 The pressure of gas-phase CO2 in the pre-existing

570

gas cavity trapped within the tiny particle forces a finite and temperature-dependent

571

concentration of dissolved CO2 (denoted c B ) in a boundary layer surrounding the gas cavity

572

(bar = 10 µm).

573 574

Figure 4: Temperature dependence of the critical dissolved CO2 concentration below which

575

bubble nucleation becomes thermodynamically impossible in a glass of sparkling water from

576

a pre-existing gas cavity immersed in the liquid phase (solid lines). Gas cavities with four

577

various radii of curvature have been investigated (namely 1, 2, 5, and 10 µm, respectively).

578

As a comparison, the temperature-dependent critical concentration below which bubbling

579

becomes impossible in a typical Champagne wine appears as a grey dotted line (for a gas

580

cavity with a radius of curvature of 2 µm).

581


27


Page 28 of 34

582

Figure 5: Theoretical total number of CO2 bubbles likely to form in a glass poured with 20

583

centiliters of sparkling water (and with a water depth of 10 centimeters) plotted versus the

584

dissolved CO2 concentration found in the water. Four tasting temperatures were investigated

585

(namely 5, 10, 15, and 20 °C). It is noteworthy to mention that an identical radius of curvature

586

(with r  5 µm) was considered for all the pre-existing gas cavities acting as bubble

587

nucleation sites in the glass of sparkling water.


28

Page 29 of 34


589

Figures

590 591

VG

VL

Gas phase with a volume VG , and with PCO 2 VG  nG RT

Liquid phase with a volume VL , and with n c L  L  k H PCO 2 VL

CARBONATED WATER

592 593 594

Figure 1


29


Page 30 of 34

596 597

10

W1 W2 W3 Champagne

6

2

PCO (bar)

8

4

2

0

598

0

5

10

15

20

25

30

T (°C)

599 600 601 602

Figure 2


30

Page 31 of 34


604 605 606

CO2 bubble under the pressure

cL  cB

PB 

r

P0  2 r

cB

cL

water bulk supersaturated with dissolved CO2

 boundary layer in equilibrium with gas-phase CO2 in the bubble, where

diffusion boundary layer

cB  k H PB 

k H P0  2 r 

607 608 609 610

Figure 3


31


Page 32 of 34

612 613 614

8 r = 1 µm r = 2 µm r = 5 µm r = 10 µm

7

-1

5

*

cL (g L )

6

4 3 2 1

615

0

5

10

15

20

25

30

T (°C)

616 617 618 619

Figure 4


32

Page 33 of 34


621 622 623

N (bubbles)

106

5 °C 10 °C 15 °C 20 °C

105 2 624

3

4

5

6

7

8

cL (g L-1)

625 626 627 628

Figure 5


33


Page 34 of 34

TOC

630

631 632

cL

N (bubbles)

106

5 °C 10 °C 15 °C 20 °C

105 2 633

3

4

5

6

7

8

cL (g L-1)

634 635 636 637


34

Carbon dioxide in bottled carbonated waters and subsequent bubble

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