A general framework for ion equilibrium calculations in compacted bentonite

Accepted Manuscript A general framework for ion equilibrium calculations in compacted bentonite Martin Birgersson PII: DOI: Reference:

S0016-7037(16)30647-0 http://dx.doi.org/10.1016/j.gca.2016.11.010 GCA 10013

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Geochimica et Cosmochimica Acta

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Please cite this article as: Birgersson, M., A general framework for ion equilibrium calculations in compacted bentonite, Geochimica et Cosmochimica Acta (2016), doi: http://dx.doi.org/10.1016/j.gca.2016.11.010

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A general framework for ion equilibrium calculations in compacted bentonite Martin Birgerssona,∗

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4

a

Clay Technology AB, Ideon Science Park, S-223 70, Lund, Sweden

5

Abstract

6

An approach for treating chemical equilibrium between compacted bentonite

7

and aqueous solutions is presented. The treatment is based on conceptualiz-

8

ing bentonite as a homogeneous mixture of water and montmorillonite, and

9

assumes Gibbs-Donnan membrane equilibrium across interfaces to external

10

solutions. An equation for calculating the electrostatic potential difference

11

between bentonite and external solution (Donnan potential) is derived and

12

solved analytically for some simple systems. The solutions are furthermore

13

analyzed in order to illuminate the general mechanisms of ion equilibrium

14

and their relation to measurable quantities. A method is suggested for es-

15

timating interlayer activity coefficients based on the notion of an interlayer

16

ionic strength. Using this method, several applications of the framework are

17

presented, giving a set of quantitative predictions which may be relatively

18

simply tested experimentally, e.g.: (1) The relative amount of anions enter-

19

ing the bentonite depends approximately on the square-root of the external

20

concentration for a 1:2 salt (e.g. CaCl2 ). For a 1:1 salt (e.g. NaCl) the

21

dependence is approximately linear, and for a 1:2 salt (e.g. Na2 SO4 ) the

22

dependence is approximately quadratic. (2) Bentonite contains substantially

23

more nitrate as compared to chloride if equilibrated with the two salt so-

24

lutions at equal external concentration. (3) Potassium bentonite generally

25

contains more anions as compared to sodium bentonite if equilibrated at the ∗ Corresponding author Preprint submitted to Elsevier Email address: [email protected] (Martin Birgersson)

November 15, 2016

26

same external concentration. (4) The anion concentration ratio in two ben-

27

tonite samples of different cations (but with the same density and cation

28

exchange capacity) resembles the ion exchange selectivity coefficient for that

29

specific cation pair.

30

The results show that an adequate treatment of chemical equilibrium be-

31

tween interlayers and bulk solutions are essential when modeling compacted

32

bentonite, and that activity corrections generally are required for relevant

33

ion equilibrium calculations. It is demonstrated that neglecting these as-

34

pects may lead to incorrect inferences regarding bentonite structure, and to

35

incorrect interpretation of diffusion data.

36

Keywords: bentonite, montmorillonite, Gibbs-Donnan equilibrium, Ion

37

equilibrium, diffusion

38

1. Introduction

39

Bentonite clay is a key material in many concepts for waste storage –

40

including spent nuclear fuel – due to its swelling ability, which provides ef-

41

fective sealing under confined and water-saturated conditions (Nagra, 2002;

42

Posiva Oy, 2010; Ye et al., 2010; SKB, 2011). For relevant safety and per-

43

formance assessments, an adequate chemical description of bentonite under

44

these conditions is therefore essential.

45

Here bentonite refers to a clay dominated by the mineral montmorillonite.

46

Individual montmorillonite particles are approximately 1 nm thick, typically

47

extend 100 - 1000 nm in the lateral directions, and carry negative charge

48

as a result of atomic substitutions in the crystal structure (Newman &

49

Brown, 1987). The structural charge is compensated by cations located in

50

spaces between adjacent particles (interlayer spaces). The specific charge

51

configuration leads to a strong water affinity, and substantial amounts of 2

52

water may be incorporated as thin (nanometers) films in the interlayer spaces

53

(in the following referred to simply as interlayers). This water uptake is the

54

mechanism behind bentonite swelling.

55

It should be pointed out that some authors define the “interlayer” as hav-

56

ing a maximum extension, typically three or four monolayers of water (Bourg

57

et al., 2006; Churakov et al., 2014). Here, we instead follow the usage of the

58

term as made by e.g. Norrish (1954), who allows for basically any extension,

59

and reports interlayer distances up to approximately 10 nm.

60

Regardless of definition, the pore volume of water-saturated compacted

61

bentonite is, naturally, dominated by interlayers (Holmboe et al., 2012). De-

62

spite this fact, many bentonite models depend critically on the existence of

63

bulk water within the bentonite, while making several unjustified assump-

64

tions regarding interlayers, e.g. by assuming that they are inaccessible to

65

anions (e.g. Bradbury & Baeyens, 2003; Tournassat & Appelo, 2011), and

66

that ions residing there are immobilized (e.g. Oscarson, 1994; Leroy et al.,

67

2006).

68

That anions have access to interlayers is, however, clear from measured

69

water activity in non-pressurized bentonite samples equilibrated with salt

70

solutions; the vapor pressure of such samples is in many cases lower than the

71

vapor pressure of both the corresponding pure samples, and the correspond-

72

ing salt solutions (Karnland et al., 2005). Recently also clear evidence has

73

been presented, showing that diffusion in interlayers dominates mass transfer

74

in compacted montmorillonite (Glaus et al., 2013). Moreover, by assuming

75

chemical equilibrium between external solutions and interlayers, Birgersson

76

& Karnland (2009) showed that ion diffusion in bentonite can be principally

77

described in a completely homogeneous model. A conclusion from that work

78

is that chemical equilibrium between external solutions and interlayers – re-

3

79

ferred to as ion equilibrium – must be at the core of any type of model for

80

bentonite exerting swelling pressure.

81

Several other recently developed models include various forms of ion equi-

82

librium, but do so in conjunction with assuming a non-homogeneous ben-

83

tonite structure (e.g. Leroy et al., 2006; Alt-Epping et al., 2014). In line

84

with the traditional view, they postulate e.g. the existence of pores different

85

from interlayers, and immobilization of exchangeable cations. Consequently,

86

the result of ion equilibrium calculations in these models directly influences

87

the parameter values adopted for describing the bentonite structure, e.g. the

88

amount of various types of pores, or the amount of immobilized exchangeable

89

ions. To avoid drawing incorrect conclusions, it is thus of vital importance

90

that these calculations are sufficiently accurate, in particular when inferences

91

are made by fitting models to experimental data.

92

Here the homogeneous bentonite model of Birgersson & Karnland (2009)

93

is generalized by considering ion activities rather than concentrations. More-

94

over, the equilibrium for anions and cations is put on an equal footing by

95

adopting the so-called Donnan potential. The general framework developed

96

for ion equilibrium calculations is explored by considering a few particularly

97

simple systems for which analytic solutions are achievable. With the aid of

98

these solutions, a whole set of quantitative predictions are presented which

99

can be used to experimentally test the validity of the framework. Lastly,

100

implications of the presented results for other types of bentonite models are

101

discussed, with focus on ion diffusion and structure.

102

2. Bentonite and external solution in chemical equilibrium

103

We will consider an external solution in equilibrium with a volumetrically

104

confined bentonite component, as schematically illustrated in figure 1. It is 4

105

assumed that the interface between bentonite and external solution consists

106

of a semi-permeable component which allows for passage of water and aque-

107

ous species, but not of clay particles. The bentonite is furthermore conceptu-

108

alized as a homogeneous mixture of water and montmorillonite; in particular

109

it is assumed that all bentonite water is distributed in interlayers of uniform

110

width (Birgersson & Karnland, 2009). Apart from containing cations com-

111

pensating structural charge, these pores are also assumed accessible for any

112

other type of aqueous species (including anions).

Pressure

* Electrostatic potential

Figure 1: Schematic illustration of the system considered in the present work.

113

As the interface is assumed impermeable to a charged component (the

114

clay particles), the system formally fulfills the requirement for Gibbs-Donnan

115

membrane equilibrium (Donnan, 1924; Gregor, 1951; Babcock, 1963). In the

116

following, we apply the theory for this type of equilibrium to the defined

117

system. 5

118

119

The electro-chemical potential for an aqueous species Mi can generally be written µi = µ0i + RT · ln ai + F · zi · φ

(1)

120

where ai and zi respectively are activity and charge number of the species,

121

φ is the electrostatic potential, R the universal gas constant, T absolute

122

temperature, F Faraday’s constant, and µ0i a reference chemical potential.

123

124

Activities are generally related to concentrations (mi ) via activity coefficients, γi

ai = γi · mi 125

126

(2)

Throughout this work, concentration is measured in terms of molality, i.e. amount of substance (in mol) per kilogram water.

127

The confinement of montmorillonite results in a lowered electrostatic po-

128

tential in the bentonite as compared to the external solution (figure 1). Here

129

we assume that the potential in the bentonite is constant (Donnan approx-

130

imation), and refer to the potential difference across the interface as the

131

Donnan potential.

132

133

In chemical equilibrium, the electro-chemical potential is everywhere equal, and applying equation 1 to both compartments of our system gives ln aext = ln aint i i +

F · zi ? ·φ RT

(3)

134

where φ? = φint − φext is the Donnan potential. Note that φ? is a negative

135

quantity. Here, and throughout the paper, we use superscripts “ext” and

136

“int” respectively for quantities in the external solution and in the bentonite.

137

Equation 3 can be rewritten as

6

−zi ext aint i = fD · ai 138

(4)

where a “Donnan factor” has been defined

fD = e

F·φ? RT

(5)

139

Note that fD , which takes values between 0 and 1, is simply a transfor-

140

mation of the Donnan potential; for vanishing φ? , fD = 1, and for infinitely

141

negative φ? , fD = 0.

142

Equation 4 shows that the relation between internal and external activ-

143

ities is fully specified by the Donnan factor (or, equivalently, the Donnan

144

potential). In particular, it shows that activities (as well as concentrations)

145

of positively charged species are higher and activities of negatively charged

146

species are lower in the bentonite, while activities for neutral species are

147

equal in the two compartments. The external and internal chemical envi-

148

ronments consequently differ – often drastically – and activity coefficients

149

are therefore also expected to be different. Thus, although the activity is

150

equal for charge neutral species (with activity defined by equation 1), there

151

is typically a prevailing concentration difference for these species as well.

152

The typically much higher total concentration of aqueous species in the

153

clay also implies an osmotic pressure difference (since the chemical potential

154

for water in the two compartments is equal). Thus, the semi-permeable

155

component, which in practice in a laboratory context is e.g. a metal filter, is

156

required to be able to withstand this pressure difference.

157

For expressing concentration relations in the present context, it is con-

158

venient to define an ion equilibrium coefficient as the ratio between internal

159

and external concentration

7

Ξi = 160

mint i mext i

Combining equations 2, 4, and 6 gives Ξi = Γi · fD−zi

161

(6)

(7)

where

Γi =

γiext γiint

(8)

162

Equation 7 shows that ion equilibrium coefficients depend, in addition

163

to the Donnan potential, explicitly on activity coefficients. Thus, although

164

the general trend is that concentrations of positively charged species are

165

higher, and concentrations of negatively charged species are lower in the clay

166

(expressed by the fD−zi -factor), significant corrections may emerge due to the

167

presence of the activity coefficient ratio (Γi ) in equation 7; the larger the

168

value of Γi , the larger is the preference for a specific ion to reside in the

169

bentonite.

170

We are now in position to derive an equation for fD . Given the internal

171

molalities for all charged species, the requirement of zero net charge in the

172

bentonite can be stated

Qnet =

X

zi · F · mint i · Mw + Qsurf = 0

(9)

i

173

where Mw denotes bentonite water mass, and Qsurf total surface charge

174

of montmorillonite in the bentonite. Qsurf can be calculated from the cation

175

exchange capacity, CEC, which expresses the amount of exchangeable ions

176

in terms of charge per mass unit dry bentonite

8

Qsurf = −CEC · Ms 177

where Ms is bentonite solid mass.

(10)

This estimation neglects possible

178

pH-dependent edge charge, which typically only contributes by a few per-

179

cent (Bradbury & Baeyens, 1997). A convenient CEC unit is charge equiv-

180

alents per kg dry mass. In this paper is therefore chosen the charge unit

181

equivalents (= the charge of one mole elementary charges). In this unit,

182

Faraday’s constant is 1 eq/mol.

183

Combining equations 9 and 10 gives X

zi · mint i − mIL = 0

(11)

i

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where

mIL =

CEC F·w

(12)

185

and w = Mw /Ms is the water-to-solid mass ratio of the bentonite. No-

186

tice that mIL quantifies the equivalent concentration of mono-valent cations

187

required to precisely compensate the montmorillonite surface charge.

188

Combining equations 6, 7, and 11 gives the equation for fD X

−zi zi · Γi · mext = mIL i · fD

(13)

i

189

which is to be solved given a complete specification of the external con-

190

centrations (mext i ) and a value of mIL . Note that mIL , which typically has a

191

value of several mol/kgw, is the only parameter characterizing the bentonite

192

in the present framework.

193

As the electrostatic potential is assumed constant, equation 13 gives an

194

average value of the actual potential in the bentonite. As seen e.g. from solv9

195

ing the Poisson-Boltzmann equation, deviations from the average increases

196

with increasing interlayer distance (Hedstr¨om & Karnland, 2012). A straight-

197

forward applicability of the the present framework is therefore restricted to

198

dense systems, with an average interlayer distance of a few nm. This is also

199

the relevant density region for most systems considered for e.g. radioac-

200

tive waste storage, which have average interlayer distances of about 2 nm

201

less (Holmboe et al., 2012). Moreover, Molecular Dynamics simulations con-

202

firm that the use of an averaged constant potential is justified for interlayers

203

distances as short as ∼ 0.6 nm (Hsiao & Hedstr¨om, 2015). In the same den-

204

sity limit, however, influence of the finite size of water molecules becomes

205

important and only interlayer distances corresponding to a discrete number

206

of molecular layers are actually realized. This is not accounted for in the

207

present description, which consequently represent an average of such discrete

208

interlayer configurations.

209

3. Examples

210

In the general case, equation 13 can only be solved numerically. In this

211

section, however, we investigate some specific systems where analytic solu-

212

tions can be achieved (sometimes only in certain parameter limits). These

213

cases serve as examples of using equation 13 for calculating the Donnan po-

214

tential, but they also, and maybe more importantly, give a mean to discuss

215

the general mechanism of ion equilibrium in bentonite and its relation to

216

measurable quantities.

217

To calculate fD , the concentration dependence of all activity coefficients

218

involved must in principle be known. Working out this dependence is of

219

course a main issue, which will be discussed in section 4. In this section we

220

instead solve equation 13 under the assumption that the quantities Γi are 10

221

known constants. This way of separating the problem will turn out to be

222

fruitful.

223

3.1. Systems containing a single type of cation

224

3.1.1. 1:1 system

225

The simplest system to consider is one which contains a single type of

226

monovalent cation (labeled +) and a single type of monovalent anion (labeled

227

−). For this system, which we will refer to as a 1:1 system, equation 13 reads −1 ext Γ+ · mext + · fD − Γ− · m− · fD = mIL

(14)

228

Because this equation can be handled analytically, and because 1:1 sys-

229

tems are quite commonly explored experimentally (Glaus et al., 2010; Tachi

230

& Yotsuji, 2014), we will here analyze its solution in some detail.

231

Equation 14 can be reexpressed utilizing the fact that the requirement

232

ext of charge neutrality implies mext − = m+ (the external salt concentration is

233

moreover labeled mext ) fD2 +

mIL Γ+ f − =0 D Γ− · mext Γ−

(15)

234

Note that in doing this reformulation we have implicitly assumed that the

235

salt dissociate completely. The physically relevant solution to equation 15 is s " # m2IL 1 mIL + Γ+ · Γ− fD = − ext + Γ− 2m 4(mext )2

236

237

(16)

Combining this expression with equation 7, the ion equilibrium coefficient for the anion is given by mIL Ξ− = − ext + 2m

s

m2IL + Γ+ · Γ − 4(mext )2

11

(17)

238

This expression is equivalent to the one derived in Birgersson & Karn-

239

land (2009) for the chloride equilibrium coefficient in a pure NaCl/bentonite

240

system.

241

242

The ion equilibrium coefficient for the cation is most simply derived by expressing equation 14 in terms of Ξ− and Ξ+ , and rearranging, giving mIL + Ξ− (18) mext Combining this expression with equation 17 gives the explicit expression Ξ+ =

243

244

for Ξ+ mIL + Ξ+ = 2mext

s

m2IL + Γ+ · Γ− 4(mext )2

(19)

245

Equation 18 was also derived in Birgersson & Karnland (2009), for a

246

sodium tracer in a pure NaCl/bentonite system. In that work, however, the

247

derivation made use of an argument regarding the mixing equilibrium for

248

the tracer, rather than directly utilizing the Donnan potential (equation 7).

249

These different ways of arriving at the expression for a cation equilibrium

250

coefficient demonstrates that the process usually singled out as “cation ex-

251

change” actually is contained within the general framework here presented.

252

The process of cation exchange is further investigated in later sections.

253

Turning our attention to the dependence of fD and Ξi on activity co-

254

efficients it may be noted that the two derived ion equilibrium coefficients

255

(equations 17 and 19) only depend on activity coefficients in the combination

256

γ+ext · γ−ext (γ±ext )2 = , (20) γ+int · γ−int (γ±int )2 i.e. only as mean ionic activities, γ± , for the 1:1 salt under considera-

257

tion. The Donnan factor, on the other hand, depends explicitly on individual

258

activity coefficient ratios.

Γ+ · Γ− =

12

259

The activity coefficient dependency can be further explored by considering

260

the limit mext << mIL . Moreover, the condition of small mext is usually met

261

in practice, since mIL typically have values of several mol/kgw in systems of

262

interest. Thus, by Taylor expanding equation 16 in the limit of low mext (the

263

formal limit is 4Γ+ · Γ− (mext /mIL )2 << 1), it is found that fD to first order

264

in (mext /mIL ) equals

fD ≈ Γ+

mext mIL

(21)

265

The Donnan potential thus depends primarily on Γ+ while it to leading

266

order is independent of Γ− . This is a very reasonable result, regarding the

267

fact that interlayers are dominated by cations, especially in the limit of low

268

external concentration. The corresponding expressions for Ξ− and Ξ+ are

Ξ− ≈ Γ+ · Γ− 269

mext mIL

(22)

and

Ξ+ ≈

mext mIL + Γ · Γ + − mext mIL

(23)

270

To leading order, Ξ− depends linearly both on Γ+ ·Γ− and on the external

271

concentration. The leading order term of Ξ+ , on the other hand, depends

272

inversely on the external concentration while it is independent of activity

273

coefficients. This independence reflects the fact that ion equilibrium for the

274

cation primarily is governed by the requirement of neutralizing the ever-

275

present bentonite structural charge.

276

3.1.2. 1:2 system

277

The expressions for fD and Ξ are generally different in different kinds of

278

systems. For a 1:2 system, i.e. a system containing one type of mono-valent 13

279

cation (+) and one type of di-valent anion (−−), equation 13 reads Γ+ · 2mext · fD−1 − 2Γ−− · mext · fD2 = mIL

(24)

280

where again mext is used to label the external concentration. In writing

281

ext equation 24, the relation mext = mext −− = m+ /2 has been utilized, which

282

relies on the assumption that the salt dissociates completely.

283

Since the above equation is of third order it has no simple general analytic

284

solution. However, in the limit of small mext it is expected that fD << 1,

285

and the second term on the left-hand side can then be neglected, giving

fD ≈

Γ+ · 2mext mIL

(25)

286

As the external cation concentration in this case equals 2mext , this expres-

287

sion is identical to the one for fD in a 1:1 system (equation 21). This identity

288

demonstrates that the Donnan potential is dominated by the properties of

289

the cation. It follows that also the ion equilibrium coefficient for the cation

290

is identical with the corresponding quantity in a 1:1 dominated system (in

291

the limit of low mext ). The ion equilibrium coefficient for the anion, on the

292

other hand, becomes in the same limit

Ξ−− ≈ 4Γ−− ·

Γ2+



mext mIL

2

(26)

293

In contrast to the linear dependence in the case of a 1:1 system, the

294

anion equilibrium coefficient for the 1:2 system depends quadratically on the

295

external salt concentration.

296

3.1.3. 2:1 system

297

Next, we consider a 2:1 system, i.e. a system containing one type of di-

298

valent cation (++) and one type of mono-valent anion (−). As for the 1:2 14

299

system, the general equation for fD is of third order (utilizing mext = mext ++ =

300

mext − /2) 2Γ++ · mext · fD−2 − Γ− · 2mext · fD = mIL

301

302

(27)

In the limit of small mext (fD << 1), the anion term can be neglected, giving

fD ≈

r

2Γ++ · mext mIL

(28)

303

Note how fD in this case depends on the square-root of the external

304

solution, in contrast to the linear dependence for a mono-valent cation in the

305

same limit. The cation equilibrium coefficient, however, still depends on the

306

inverse of the external solution (to leading order)

Ξ++ ≈

(mIL /2) mext

(29)

307

This expression, which is independent of activity coefficients, reflects the

308

fact that cation equilibrium is mainly governed by the ever-present ben-

309

tonite structural charge – the same conclusion as was made in the case

310

of mono-valent cations. Note that the internal concentration of di-valent

311

cations required for compensating structural charge equals mIL /2. The ratio

312

(mIL /2)/mext thus quantifies mixing equilibrium, in complete analogy with

313

the 1:1 and 1:2 systems (equations 18 and 25).

314

The anion equilibrium coefficient is

Ξ− ≈ Γ−

r

2Γ++ · mext mIL

(30)

315

In a 2:1 system, the anion equilibrium coefficient thus depends on the

316

square-root of the external salt concentration (in the low concentration limit), 15

317

in contrast to the linear dependence in a 1:1 system, and the quadratic de-

318

pendence in a 1:2 system.

319

3.1.4. Tracer ions

320

We now consider the case of having a trace amount of an additional

321

species in an otherwise “pure” system (treated in the previous sections).

322

Strictly, equation 13 will now have an additional term representing the tracer.

323

However, since this term is proportional to the tracer concentration, it can

324

be neglected by definition.

325

The expressions for fD of the pure system are therefore still valid in this

326

case, i.e. the Donnan potential is fully determined by the “main” electrolyte

327

(for which we still use the nomenclature mext ). The ion equilibrium coefficient

328

for the tracer is consequently given by using equation 7 together with the

329

expression for fD for the corresponding pure system.

330

In e.g. the case of adding a tracer to an otherwise pure 1:1 system, the

331

expression for the tracer ion equilibrium coefficient in the limit mext << mIL

332

reads

333

−ztracer mext Ξtracer ≈ Γtracer Γ+ (31) mIL Note that the tracer ion equilibrium coefficient depends explicitly on indi-

334

vidual activity coefficient ratios, in contrast to the corresponding quantities

335

for the ions of the main electrolyte which depend only on mean activity

336

coefficients (equation 20). A way to directly compare individual activity

337

coefficient ratios is therefore to compare ion equilibrium coefficients of ho-

338

movalent tracer ions measured in a specific bentonite system at constant main

339

electrolyte concentration (i.e. measured at constant Donnan potential).



340

Equilibrating a mono-valent cationic tracer with a homo-ionic bentonite

341

(presumably also mono-valent) is the prototype procedure for quantifying the 16

342

processes known as ion exchange (Helfferich, 1995). As already mentioned,

343

the present ion equilibrium framework accounts for this process, which is

344

easily seen by specializing equation 31 to the case of a mono-valent cationic

345

tracer (labeled “tracer+”) Γtracer+ mIL (32) Γ+ mext This equation should be interpreted as follows. The factor mIL /mext acΞtracer+ ≈

346

347

counts for pure mixing of ions; if the ratio of tracer ions to the total number

348

of ions is the same throughout the system, this factor quantifies the corre-

349

sponding ratio between the tracer concentrations in the two compartments.

350

The factor Γtracer+ /Γ+ modifies the pure mixing ratio due to differences in

351

activity coefficients of the tracer ion and the main electrolyte cation. Since

352

Γi measures the preference for an ion to reside in the clay, the ratio between

353

two such parameters quantifies a selectivity coefficient for the ion pair un-

354

der consideration. Actually, there is a formal relationship between interlayer

355

activity coefficients and Gaines-Thomas selectivity coefficients which applies

356

more generally than only to tracer ions. This relation will be derived in the

357

next section.

358

3.2. Systems containing several types of cations

359

3.2.1. Relation between Gaines-Thomas selectivity coefficients and interlayer

360

activity coefficients

361

We here consider chemical equilibrium in a system containing one mono-

362

valent type of cation (labeled 1) and a second type of cation (2) with charge

363

number z2 . The following analysis can in principle be done for the more

364

general case of allowing both cations to have arbitrary charge number. We

365

choose to constrain one of them to be mono-valent, however, in order to get

366

less symbol-burdened equations. 17

367

Traditionally, the equilibration process of cations in bentonite is concep-

368

tualized as sorption reactions where ions in the external solution are viewed

369

as exchanging places with ions on specific sorption sites in the clay. In equi-

370

librium there is a certain relation between the distribution of ions on the

371

sorption sites and the composition of the external solution, quantified by

372

a selectivity coefficient. The Gaines-Thomas convention for expressing se-

373

lectivity coefficients uses activities in the external solution, and equivalent

374

charge fractions of the sorption sites (X) z

KGT =

2 X2 · (γ1ext · mext 1 ) (X1 )z2 · (γ2ext · mext 2 )

(33)

375

Here the activities have been expressed using equation 2.

376

In the case of bentonite, it is obvious that the “sorption sites” must be

377

identified with montmorillonite interlayer pores. Using the present frame-

378

work, the equivalent charge fractions can consequently be expressed using

379

internal concentrations

380

382

mint 1 int z2 · m2 + mint 1

(34)

X2 =

z2 · mint 2 z2 · mint + mint 2 1

(35)

Combining equations 33 – 35 gives

KGT 381

X1 =


z
γ1int 2 int z2 −1 = z2 z2 · mint 2 + m1 int γ2

(36)

In the case where also the second cation is mono-valent (z2 = 1), this expression reduces to mono−valent = KGT

18

γ1int γ2int

(37)

383

384

385

386

The selectivity coefficient is in this case seen to directly quantify the ratio of internal activity coefficients for the two cations. If instead the second cation is di-valent (z2 = 2), the selectivity coefficient reads di−valent KGT

387

388


2
γ1int int = 2 int 2 · mint 2 + m1 γ2

In the limit of vanishing external concentrations the rightmost factor reduces to mIL . Thus, in the limit of low external concentration di−valent KGT

389

(38)


2 γ1int ≈ 2 int mIL γ2

(39)

3.2.2. Two types of mono-valent cations

390

We now consider a system having a single type of monovalent anion (la-

391

beled −) and two types of mono-valent cations (labeled 1 and 2, respectively).

392

We still use the notation mext for the external anion concentration, while the

393

ext cation concentrations are denoted mext 1 and m2 . Due to the requirement of

394

ext charge neutrality, mext = mext 1 + m2 .

395

396

The equation for fD can in this case be written in the same form as for the pure 1:1 system (equation 14) ˜ + · mext · f −1 − Γ− · mext · fD = mIL Γ D

397

398

(40)

where the “pure” cation activity coefficient ratio has been replaced with a weighted average ext mext (2) 2 ˜ + = m1 Γ(1) + Γ+ Γ + ext ext m m

(41)

399

The results from section 3.1.1 can consequently be carried over to the

400

˜ + for Γ+ . For instance, the cation present case by simply substituting Γ 19

401

equilibrium coefficient in the limit of low external concentration now reads (i)

Γ mIL = + ext (42) ˜+ m Γ which is a generalization of equation 32. Furthermore, the ratio of the (i) Ξ+

402

403

two cation equilibrium coefficients is (2)

Ξ+

404

(2)

=

Γ+

γ2ext γ1ext

(43) (1) (1) Ξ+ Γ+ This relation, which hold for any external concentration, is equivalent to

405

equation 33 with z2 = 1.

406

3.2.3. Mixed 2:1 and 1:1 system

= KGT

407

In a system containing one type of di-valent cation (++), one type of

408

mono-valent cation (+), and one type of mono-valent anion (−) the main

409

equation is −2 −1 ext ext 2Γ++ · mext ++ · fD + Γ+ · m+ · fD − Γ− · m− · fD = mIL

410

411

(44)

The anion term can be neglected in the limit of small fD , resulting in a second order equation with the solution mext +

Γ+ · fD ≈ 2mIL

+

s

2 2Γ++ · mext Γ2+ · (mext + ) ++ + 2 4mIL mIL

(45)

412

In contrast to the case of mixing cations with the same charge number, fD

413

cannot in this case be expressed in a form reminiscent of that for a system

414

containing a single type of cation (note that in e.g. the case of mixing

415

two monovalent cations, fD depends on the individual cation concentrations

416

only to the extent that their activity coefficient ratios differ). Note that

417

equation 45 reduces to equation 28 when mext + = 0, and to equation 21 when

418

mext ++ = 0. 20

419

420

Furthermore, the ratio of the cation equilibrium coefficients will in this case depend explicitly on fD Ξ+ Γ+ · fD−1 Γ+ = fD −2 = Ξ++ Γ++ Γ++ · fD

(46)

421

By combining this equation with equation 45 it can be shown to be equiv-

422

alent to equation 33 with z2 = 2 (in the limit of low external concentration).

423

4. Estimating Γi

424

Although the activity coefficient ratios, Γi , depend in a complex way on

425

the composition of the external solution, this dependence was not considered

426

when solving equation 13. The resulting expressions for Donnan factors and

427

ion equilibrium coefficients could still be analyzed e.g. in terms of how they

428

depend on external conditions, bentonite density, etc. However, to put an

429

actual number on fD , knowledge of the relevant activity coefficient ratios is

430

required. Furthermore, as Γi implicitly depends on fD , equation 13 is gen-

431

erally considerably more complex than hitherto acknowledged. A complete

432

framework for ion equilibrium calculations consequently demands a descrip-

433

tion for how Γi depends on the composition of the external solution. In the

434

following, an embryo for such a description is developed.

435

The activity coefficients entering the expression for Γi (equation 8) gen-

436

erally refer to very different chemical conditions. Regarding the bentonite, it

437

could be questioned whether any of the tools available for calculating activity

438

coefficients in ordinary aqueous solutions can be applied to interlayers: from

439

a microscopic perspective, the interlayer is basically a positively charged so-

440

lution, compensated by structural charges of the adjacent montmorillonite

441

surfaces. Nevertheless, viewed from a macroscopic perspective, it may be

442

argued that the charge configuration in bentonite to some degree resem21

443

bles that in a neutral solution – the maximum separation distance between

444

positive ions and structural charge centers in the montmorillonite is on the

445

order of 1 nm. In an attempt to here quantify interlayer activity coefficients,

446

we therefore conceptualize the bentonite as a charge neutral “solution” by

447

treating the structural charges as ordinary mono-valent anions. Their “con-

448

centration” is then quantified by mIL , and an interlayer ionic strength can

449

be defined as I int =

450

451

 1 X 2 zi · mint + m IL i 2

(47)

where the last term should be understood as (−1)2 mIL , acknowledging the negative mono-valent character of montmorillonite structural charge.

452

We furthermore assume that activity coefficients depend only on ionic

453

strength. As a model for the ionic strength dependence we adopt the mean

454

salt method (see e.g. McSween et al., 2003) with the common assumption

455

of equal activity coefficient for potassium and chloride at all ionic strengths m.s. γKm.s. + = γ − = γKCl Cl

(48)

456

where γKCl denotes the activity coefficient for a pure KCl solution. With

457

this choice, the mean salt activity coefficients for Na+ , Ca2+ , and NO− 3 are

458

given by

459

2 γNaCl γKCl 3 γ CaCl2 m.s. (49) γCa = 2+ 2 γKCl 2 γKNO m.s. 3 γNO − = 3 γKCl where, again, the activity coefficients on the right-hand side of these equa-

460

tions refer to pure salt solutions, for which empirical parameterizations are

m.s. γNa + =

22

1.6 K+/ClNa+ Ca2+ NO3-

1.4 1.2

(kgw/mol)

1.0 0.8 0.6 0.4 0.2 0.0 0

1

2

3

4

5

6

Ionic strength (mol/kgw)

Figure 2: Activity coefficients of individual ions derived using the mean salt method (equations 48 and 49). Parameterization of activity coefficients of pure salt solutions was taken from Hamer & Wu (1972) and Staples & Nuttall (1977).

461

available (Hamer & Wu, 1972; Staples & Nuttall, 1977). The resulting activ-

462

ity coefficients as a function of ionic strength for individual ions are displayed

463

in figure 2. The approach adopted for quantifying activity coefficients is summarized

464

465

as γi = γim.s. (I)

466

467

(50)

where the ionic strength (I) is given either by equation 47, for interlayer activity coefficients, or by I ext =

1X 2 zi · mext i 2

(51)

468

for activity coefficients in the external solution.

469

This approach is in all likelihood too simplistic to give a fully reliable

470

quantitative description, but it will be pursued here as a relatively simple

23

3.30 int ± =1.00 int ± =0.75 int ± =0.35 int ± =0.25

3.25 3.20

Iint (mol/kgw)

3.15 3.10 3.05 3.00 2.95 2.90 0

0.05

0.10

0.15

0.20

0.25

0.30

External Concentration (mol/kgw)

Figure 3: Interlayer ionic strength as a function of external concentration in a 1:1 system for various values of the interlayer mean activity coefficient (equation 52). In the calculation, external mean activity of NaCl has been assumed.

471

mean to evaluate e.g. whether the procedure of defining an interlayer ionic

472

strength is at all applicable.

473

4.1. Systems dominated by a single type of cation

474

The results of section 3 show that the interlayer composition is relatively

475

independent of the composition of the external solution, in systems dom-

476

inated by a single type of cation. This behavior is intuitively understood

477

because cations are required to compensate bentonite structural charge un-

478

der all conditions – the interlayer ionic strength is therefore large also in the

479

limit of zero external concentration.

480

481

In e.g. the case of a pure 1:1 system, the interlayer ionic strength may be calculated using equations 20, 22, 23, and 47 I int ≈ mIL + Γ+ · Γ−

482

2 (γ ext )2 (mext )2 (mext ) = mIL + ±int 2 mIL (γ± ) mIL

(52)

This expression, valid in the limit of low mext , is plotted in figure 3 for 24

483

the specific choice mIL = 3 mol/kgw, and for different constant values of γ±int

484

(as external activity coefficient was chosen that of NaCl). Not surprisingly,

485

the slope of this curve depends strongly on the value of γ±int . However, even

486

for the case γ±int = 0.25 kgw/mol (corresponding roughly to the mean activity

487

coefficient of 3 mol/kgw pure KNO3 solution) the interlayer ionic strength

488

increases by only 7.5% when the external concentration is increased from 0

489

to 0.3 mol/kgw.

490

Motivated by the demonstrated negligible impact of external concentra-

491

tion on interlayer composition in systems dominated by a single type of

492

cation we here approximate the interlayer ionic strength in such systems as

493

constant, corresponding to I0int – the interlayer ionic strength at vanishing

494

external concentration (calculated from equation 47) zcation + 1 (53) 2 denotes the charge number of the cation in question. Below, I int ≈ I0int = mIL ·

495

were zcation

496

we apply the presented framework for specific cases. Throughout, we choose

497

mIL = 3 mol/kgw, which for a bentonite of CEC = 0.8 eq/kg (a typical

498

value for high grade material) corresponds to a water-to-solid mass ratio of

499

w = 0.27.

500

4.1.1. Sodium dominated systems

501

First, we consider equilibrating the bentonite with NaCl and NaNO3 solu-

502

int tions. When sodium is the only cation, I0int = 3 mol/kgw, giving γCl − = 0.57

503

int int int = 0.38 kgw/ kgw/mol , γNO − = 0.13 kgw/mol, γK+ = 0.57 kgw/mol, γ Ca2+

504

int mol, and γNa + = 0.90 kgw/mol (figure 2).

3

505

The resulting ion equilibrium coefficients for chloride and nitrate, calcu-

506

lated from equation 22, are plotted in figure 4. The same figure also shows

507

the ion equilibrium coefficient of a mono-valent anion calculated by setting 25

0.45 Cl-

0.40

NO3-

Ideal

0.35

van Loon 07

0.30

Ξ

0.25 0.20 0.15 0.10 0.05 0.00 0

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

External concentration (mol/kgw)

Figure 4: Anion equilibrium coefficients in sodium bentonite as a function of external concentration. mIL = 3 mol/kgw. Also plotted are coefficients for chloride evaluated from tracer through-diffusion experiments performed on untreated bentonite of dry density 1600 kg/m3 (Van Loon et al., 2007). This data is further discussed in section 5.2.

508

Γ+ · Γ− = 1. This is the expression obtained if activity corrections are ne-

509

glected, and will here be referred to as the “ideal” equilibrium coefficient

510

(although the approximation only requires the assumption γ±ext = γ±int ).

511

There are several features of these ion equilibrium coefficients worth com-

512

menting. Firstly, the chloride equilibrium coefficient is similar to the ideal ion

513

equilibrium coefficient, which, in turn, is in fair agreement with experimental

514

values for chloride in sodium dominated bentonite (Birgersson & Karnland,

515

2009). The chloride equilibrium coefficient here calculated consequently re-

516

sembles experimental values (figure 4), which gives support to the presented

517

approach. The present analysis furthermore supports the seemingly ad hoc

518

approximation of putting Γ+ · Γ− = 1 for a NaCl system, which e.g. was

519

made in Birgersson & Karnland (2009).

520

Secondly, the nitrate equilibrium coefficient deviates drastically from ide-

521

ality, and is in the concentration range considered approximately four times 26

522

larger than the chloride equilibrium coefficient. The analysis performed hence

523

predicts that sodium bentonite would contain about four times more nitrate

524

than chloride, if equilibrated with the two salt solutions at the same concen-

525

tration. More generally, the analysis shows that anions of the same charge

526

number but with significantly different interlayer activity coefficients will

527

have a corresponding difference in ion equilibrium coefficient. This predic-

528

tion provides a simple verification test for the presented approach to ion

529

equilibrium; at present, however, there is a lack of published data on e.g.

530

nitrate equilibration in bentonite.

531

Thirdly, the nitrate result also demonstrates that the validity of the ap-

532

proximation Γ+ · Γ− = 1 for NaCl is incidental and only holds for that partic-

533

ular salt, or other sodium salts with anions having similar interlayer activity

534

coefficients. Furthermore, the large deviation from ideality for NaNO3 shows

535

that activity coefficient ratios can have a dominating influence on calculated

536

ion equilibrium coefficients – note that the effect in this case significantly

537

mitigates the reduction in internal concentration implied by considering only

538

the Donnan potential (see equation 7). Thus, generally valid ion equilibrium

539

calculations cannot be performed without accounting for activity coefficients.

540

Finally, since interlayer activity coefficients are treated as constant in the

541

present approach, the non-linear dependence on external concentration of

542

the ion equilibrium coefficients is due to variation of the external activity

543

coefficients with concentration (note that the ideal coefficient is linear). This

544

behavior illustrates that ion equilibrium coefficients, as well as the Donnan

545

potential, are not strictly material parameters of the bentonite, but rather

546

quantities describing the full system (external solution + bentonite).

547

Figure 5 shows ion equilibrium coefficients for a set of cation tracers in

548

Na-bentonite as calculated from equation 31. In this figure is also plotted

27

2000 Na K Ca Ideal di-valent




1500

1000

500

0 0

0.05

0.10

0.15

0.20

External sodium concentration (mol/kgw)

Figure 5: Cation tracer equilibrium coefficients in sodium bentonite as a function of external concentration. mIL = 3 mol/kgw.

549

the “ideal” behavior of a di-valent tracer, obtained by putting the activity

550

coefficient ratios equal to unity both for sodium and the tracer; the “ideal”

551

mono-valent behavior is identical to that for the sodium equilibrium coeffi-

552

cient, as seen from equation 31 (activity ratios cancel).

553

The most striking feature of cation equilibrium coefficients is that they

554

tend toward infinity for vanishing mext , which is a direct consequence of

555

mixing equilibrium. For the same reason, the tendency towards infinity of a

556

di-valent tracer equilibrium coefficient is considerably more pronounced, as

557

it varies as (mext )−2 rather than (mext )−1 . This behavior is in full agreement

558

of what is expected of an ion exchange process and has been evaluated for

559

Na+ and Sr2+ tracers in Na-montmorillonite (Glaus et al., 2007). Figure 5

560

furthermore shows that activity coefficient corrections to the ideal mixing

561

behavior are significant for potassium and calcium. These corrections are

562

intimately related to selectivity coefficients, as shown in section 3.2.1. Using

563

the adopted interlayer activity coefficients, the selectivity coefficient for Na/

564

K can be calculated from equation 37, 28

Na/K

KGT

=

0.90 = 1.6 0.57

(54)

565

Although somewhat low, this value is comparable to reported measure-

566

ments, which are in the range 2 – 5 (Bradbury & Baeyens, 2002). Also the

567

selectivity coefficient for Na/Ca can be calculated from equation 39 Na/Ca

KGT

=2·

0.902 · 3 = 12.8 0.38

(55)

568

This value could be compared with experimental values in Karnland et al.

569

(2011), which, however, report Na/Ca selectivity coefficients expressed in

570

terms of concentrations rather than activities (here labeled K ∗ ). For a mont-

571

morillonite sample with mIL ∼ 3 mol/kgw containing approximately 25% Ca

572

at an external ionic strength of ∼ 0.02 mol/kgw a value of K ∗ = 6.5 mol/L

573

is reported (sample “WyNa 03”). Converting K ∗ to a coefficient based on

574

ext 2 ext activities gives K ∗ · (γNa ) /γCa = 0.8752 /0.59 · 6.5 = 8.4 (using mean salt

575

activity values). Again, the calculated value (12.8) is comparable with the

576

experimental value. The plausible values of calculated selectivity coefficients

577

give additional support to the approach presented for ion equilibrium calcu-

578

lations.

579

4.1.2. Potassium dominated systems

580

The result of equilibrating the same bentonite as in the previous section

581

(mIL = 3 mol/kgw) with KCl and KNO3 solutions is displayed in figure 6.

582

Because the interlayer activity coefficient for potassium is significantly lower

583

than that for sodium, ΓK+ > ΓNa+ in the limit of low external concentration.

584

As a consequence fD is higher (i.e. the Donnan potential is less negative)

585

in a potassium system as compared to a sodium system at the same exter-

586

nal concentration (equation 21). This, in turn, implies larger corresponding 29

ClNO3Ideal

0.50

0.40




0.30

0.20

0.10

0.00 0

0.05

0.10

0.15

0.20

0.25

0.30

External concentration (mol/kgw)

Figure 6: Anion equilibrium coefficients in potassium bentonite as a function of external concentration. mIL = 3 mol/kgw.

587

values for the anion equilibrium coefficients in the potassium system. Thus,

588

according to the present analysis, potassium bentonite generally contains a

589

larger amount of any given anion as compared to sodium bentonite equili-

590

brated at the same external concentration. Testing this prediction provides

591

a mean to further verify the presented approach to ion equilibrium calcula-

592

tions. Published literature, however, lacks relevant data from equilibration

593

of potassium bentonite.

594

The ratio of the ion equilibrium coefficients for a specific anion in the

595

potassium and in the sodium system can be related to the potassium/sodium

596

selectivity coefficient int γNa ΞK Γ− · ΓK+ · (mext /mIL ) + Na/K − ≈ int = KGT = Na ext + Γ− · ΓNa · (m /mIL ) γK+ Ξ−

597

598

(56)

where the second (approximate) equality assumes equal external activity coefficients for sodium and potassium.

599

Note that the present analysis gives the same anion activity coefficient

600

ratio in K-bentonite and in Na-bentonite, since the interlayer ionic strength 30

0.25 Ca-bentonite Ideal 2:1 Na-bentonite 0.20


Cl

0.15

0.10

0.05

0.00 0

0.01

0.02

0.03

0.04

0.05

0.06

External concentration (mol/kgw)

Figure 7: Chloride equilibrium coefficients in calcium and sodium bentonite as a function of external concentration (NaCl or CaCl2 ). mIL = 3 mol/kgw.

601

is basically the same in these systems. A way to test the validity of the

602

assumption that activity coefficients are functions only of ionic strength is

603

therefore to compare the potassium/sodium selectivity coefficient with the

604

ratio between ion equilibrium coefficients for a specific anion.

605

4.1.3. CaCl2

606

Next, we consider the bentonite equilibrated with CaCl2 solution. In this

607

int int case I0int = 4.5 mol/kgw, giving γCa 2+ = 0.66 kgw/mol and γ − = 0.58 Cl

608

kgw/mol. Figure 7 shows the corresponding chloride equilibrium coefficient

609

as a function of external concentration, calculated from equation 30. In the

610

same figure are also plotted the chloride equilibrium coefficient for a sodium

611

system, as well as the “ideal” coefficient for a 2:1 system, achieved by setting

612

all activity coefficient ratios equal to unity.

613

Comparing the chloride equilibrium coefficients in Na-bentonite and Ca-

614

bentonite reveals very different dependencies on external concentration in

615

these two systems. The main reason for this is the different dependence

31

616

of fD in 1:1 and 2:1 systems – in the low concentration limit, the Donnan

617

factor in a 1:1 system depends linearly on external concentration, while it

618

depends on the square-root of the external concentration in a 2:1 system. As a

619

consequence, Ca-bentonite equilibrated with CaCl2 will contain substantially

620

more chloride as compared to Na-bentonite equilibrated with NaCl at the

621

same concentration. This prediction provides yet another simple verification

622

test for the presented approach to ion equilibrium calculations. In fact, there

623

are indications that bentonite dominated by di-valent ions contains non-

624

negligible amounts of chloride even when equilibrated at ionic strength ∼

625

0.001 mol/kgw (Garc´ıa-Guti´errez et al., 2004).

626

Notice that the non-linear dependence on external concentration of the

627

chloride equilibrium coefficient in Ca-bentonite originates both from the

628

square-root form of fD as well as on the non-linear dependence of the ac-

629

tivity coefficient ratios. Comparison with the ideal coefficient shows that the

630

dependence of ΓCl− and ΓCa2+ on external concentration is such as to pro-

631

mote more anions in the clay in the low concentration limit. This behavior,

632

in turn, is a consequence of the ionic strength dependence of the external

633

activity coefficients.

634

Figure 8 shows the ion equilibrium coefficients for a sodium tracer in

635

Ca-bentonite and Na-bentonite, as well as the “ideal” behavior of a mono-

636

valent tracer in Ca-bentonite. As a consequence of having a mono-valent

637

species in a di-valent system, the sodium tracer equilibrium coefficient is

638

significantly suppressed in Ca-bentonite in comparison to Na-bentonite, and

639

tends towards infinity only as (mext )−1/2 .

640

4.2. Systems with several types of cations

641

If the system contains non-negligible amounts of more than one type of

642

cation, the approximation of equation 53 strictly does no longer hold, because 32

100 Na in Ca-bent. Mono-valent Ideal Ca-bent. Na in Na-bent. 80




60

40

20

0 0

0.05

0.10

0.15

0.20

0.25

0.30

External ionic strength (mol/kgw)

Figure 8: Sodium tracer ion equilibrium coefficients in calcium and sodium bentonite as a function of external ionic strength. mIL = 3 mol/kgw.

643

the composition of the interlayer then no longer can be assumed to only have a

644

minor dependence on the external conditions. However, if all cations involved

645

have the same charge number, they contribute equally to ionic strength, and

646

I int still has only a weak dependence on the composition of the external

647

solution. The assumption that activity coefficients only depend on ionic

648

strength (equation 50) can thereby be tested by measuring anion equilibrium

649

coefficients for different external cation configurations while keeping constant

650

ionic strength. In e.g. the case of two types of mono-valent cations, the

651

relation between the anion equilibrium coefficient and the cation fraction is

652

then predicted to be linear, as seen from substituting equation 41 for the

653

factor Γ+ in equation 22.

654

In the general case of having cations with different charge number, the

655

relation between interlayer ionic strength and external solution configuration

656

becomes complex – it is conceivable, for instance, that an increase of the

657

ionic strength in the external solution, lowers it in the bentonite.

658

Consider a system containing a mixture of NaCl and CaCl2 . For fixed Ca/ 33

0.04 XCa = 0.0 XCa = 0.3 XCa = 0.7 XCa = 1.0 mCaext (0.3) mCaext (0.7)

0.20

0.03


Cl

0.15 0.02 0.10

0.01 0.05

0.00 0

0.05

0.10

0.15

0.20

0.25

External calcium concentration (mol/kgw)

0.25

0 0.30

External Ionic Strength (mol/kgw)

Figure 9: Chloride equilibrium coefficients as a function of external ionic strength in pure calcium, pure sodium, and mixed calcium/sodium systems with mIL = 3 mol/kgw. Also plotted are the corresponding external calcium concentrations in the mixed systems.

659

Na ratio in the bentonite, the interlayer ionic strength is relatively insensitive

660

to the external ionic strength and may be approximated by (at low external

661

concentrations)

I

int

≈

I0int



X 2+ = mIL · 1 + Ca 2



(57)

662

where XCa2+ is the equivalent charge fraction of calcium in the ben-

663

tonite (equation 57 is derived from equation 47 and the relations mint Ca ≈

664

XCa2+ · mIL /2 and mint Na ≈ (1 − XCa2+ ) · mIL ). For this particular condition,

665

the interlayer activity coefficients can then be treated as constants – the re-

666

maining challenge is to calculate the external configuration corresponding

667

to given values of XCa2+ and external ionic strength. This has been done

668

numerically for XCa2+ = 0.3 and XCa2+ = 0.7 using equations 7 and 45, and

669

the resulting chloride equilibrium coefficients as a function of external ionic

670

strength are displayed in figure 9, together with the corresponding external

671

calcium concentrations. For comparison, this figure also shows the chloride 34

672

equilibrium coefficients for the pure calcium and sodium systems. Table 1

673

lists the constant values adopted for the interlayer activity coefficients. Table 1: Values of interlayer ionic strength and interlayer activity coefficients adopted in the calculation of chloride equilibrium coefficients in mixed calcium/sodium bentonite (figure 9).

I0int

XCa2+

int γNa +

int γCa 2+

int γCl −

(mol/kgw) (kgw/mol) (kgw/mol) (kgw/mol) 0.0

3.00

0.90

-

0.57

0.3

3.45

0.97

0.45

0.57

0.7

4.05

1.06

0.54

0.58

1.0

4.50

-

0.66

0.58

674

Figure 9 clearly illustrates the inherent variability of the results of ion

675

equilibrium calculations: although the Ca-Na-Cl system is rather simply

676

specified, the interlayer configuration is a complex function of the external

677

conditions.

678

5. Implications for mass transfer and structure

679

Interlayer properties have to a large extent been ignored in traditional

680

modeling of bentonite. It has been commonly assumed, for instance, that

681

anions (i.e. excess salt) do not have access to interlayers (e.g. Bradbury &

682

Baeyens, 2003), and that ions residing there have negligible mobility (e.g.

683

Oscarson, 1994). Instead, the prevailing conceptual view of compacted ben-

684

tonite is that it is multi-porous, and that its physicochemical behavior is

685

critically dependent on the existence of structures different from interlayers.

686

Several recently developed bentonite models, however, include ion equilib-

687

rium considerations to various extents, but do so in a multi-porous context. 35

688

For instance, Gibbs-Donnan equilibrium calculations have been used to sup-

689

port the estimation of non-interlayer pores by comparing with experimental

690

data on the amount of chloride taken up by bentonite at different chemical

691

conditions (e.g. Tournassat & Appelo, 2011). Also, Gibbs-Donnan equilib-

692

rium calculations have been used in conjunction with postulating a Stern-

693

layer – a layer of immobile charge closest to the montmorillonite surface – in

694

order to estimate the extent of this quantity (e.g. Leroy et al., 2006).

695

The work here presented shows that neglecting the interlayer pores in ben-

696

tonite modeling is clearly unjustified – even if the framework in its present

697

form does not represent a fully correct quantitative description, it demon-

698

strates that chemical equilibrium between interlayers and bulk solutions must

699

be accounted for. Moreover, the results presented also indicate that many

700

ion equilibrium calculations performed in multi-porous contexts may not be

701

adequate, as they usually neglect activity corrections. It should be noted that

702

neglecting such corrections nevertheless corresponds to an activity coefficient

703

model (defined by choosing γiint = γiext for all ions under all conditions). Be-

704

low are discussed the implications of the present work on ion diffusion and

705

structure.

706

5.1. Diffusion

707

The traditional treatment of ion diffusion in bentonite assumes that dif-

708

fusion occur in non-interlayer pores (e.g. Muurinen, 1994; Yu & Neretnieks,

709

1997; Jansson, 2002; Molera, 2002; Bradbury & Baeyens, 2003; Bourg, 2004;

710

Shackelford & Moore, 2013). In particular it is assumed that the conditions in

711

these non-interlayer pores are such that ion concentrations vary continuously

712

across interfaces between external water and bentonite. As a consequence, ef-

713

fective diffusion coefficients (De ) – evaluated by relating steady-state fluxes

714

and externally imposed concentration differences – have been assumed to 36

715

quantify diffusion in general in bentonite. These types of diffusion coeffi-

716

cients, however, depend in a complicated manner on the external conditions;

717

De for anion tracers generally increases with background concentration, while

718

the opposite is true for cation tracers. This dependence is, in the case of an-

719

ions, usually accounted for by the concept of “effective porosity”, in which

720

it is assumed that accessible pore volume depends on background concentra-

721

tion.

722

The variation of De for cation tracers is, to the extent that it is identified,

723

sometimes analyzed in terms of the concept of “surface diffusion” (Muurinen

724

et al., 1985). Cations are typically assumed to exist in two (or more) separate

725

phases – as aqueous species, and as sorbed entities associated with a solid

726

phase (section 3.2.1). In the concept of “surface diffusion” it is assumed that

727

the sorbed cations to a certain extent are mobile.

728

Clearly, the traditional description of bentonite is rather complex, involv-

729

ing multi-porosity structure, “effective porosity”, exchange sorption sites,

730

and (occasionally) “surface diffusion”. In contrast, Birgersson & Karnland

731

(2009) showed that diffusion of both anions and cations in bentonite can be

732

described in principle by only taking into account the interlayer pores, by

733

requiring chemical equilibrium with the external solution. Moreover, they

734

provided a general equation connecting De for tracers in through-diffusion

735

tests with ion equilibrium coefficients

De = φ · Ξ ·

1 · Dc 1+ω

(58)

736

where φ is the porosity of the bentonite sample, Dc the interlayer pore

737

diffusion coefficient, and ω a parameter quantifying the influence of the con-

738

fining filters (the larger the value of ω, the more does De reflect the transport

739

capacity of the filters rather than the clay). ω, in turn, is given by 37

ω =2·Ξ·

φ · Dc L f · , φf · Df L

(59)

740

where φf is filter porosity, Df pore diffusivity in the filter, Lf filter length,

741

and L sample length, and it is assumed that the bentonite is sandwiched

742

between two identical filters.

743

From the direct relationship between ion equilibrium coefficients and De ,

744

provided by equations 58 and 59, it is clear that the results of the ion equi-

745

librium calculations here performed have impact on how to interpret De .

746

Under conditions where filter influence is negligible (ω << 1), which are

747

typical in anion tracer diffusion tests, equation 58 implies a simple relation

748

between the ion equilibrium coefficient and “effective porosity” (eff )

eff = φ · Ξ

(60)

749

In view of ion equilibrium theory, “effective porosity” is thus simply a

750

measure of Ξ in the test under consideration, rather than a quantity relating

751

to the amount of non-interlayer pores. The results derived for anions in the

752

present work can be used to test which of these two interpretations of eff is

753

correct:

754

• The ion equilibrium theory predicts that completely different values of

755

eff would result depending on whether it is evaluated using e.g. nitrate

756

or chloride.

757

• The work presented predicts that eff is generally larger in e.g. K-

758

bentonite as compared to Na-bentonite. Note that this result could

759

be obtained for one and the same bentonite sample, if equilibrated in

760

sequence with a potassium and a sodium salt solution.

38

761

• An ion equilibrium coefficient as large as ∼0.5 was calculated for nitrate

762

in K-bentonite (with mIL = 3 mol/kgw). The multi-porosity interpre-

763

tation of such a result, if verified experimentally, is that non-interlayer

764

pores constitute more than 50% of the total pore volume, which is obvi-

765

ously unreasonable in highly compacted bentonite. The framework pre-

766

sented does, in principle, allow for anion equilibrium coefficients larger

767

than unity, which implies the absurdity of having accessible porosity

768

larger than total porosity (such a result might be expected for a salt

769

with very low mean activity coefficient in the interlayer, e.g. CsNO3 ).

770

• The dependence of eff on external ionic strength, evaluated using a

771

single type of anion, is predicted to be completely different in Ca-

772

bentonite as compared to Na-bentonite. eff is furthermore predicted to

773

be much larger in Ca-bentonite at low external concentration. Also this

774

behavior could be tested in a single bentonite sample, if equilibrated

775

in sequence with e.g. NaCl and CaCl2 .

776

The results here presented have impact also on the interpretation of cation

777

diffusion. For instance, the very different equilibrium behavior of a sodium

778

tracer depending on whether it resides in Ca- or Na-bentonite (figure 8) pre-

779

dicts a similar difference in De for sodium, depending on which type of system

780

it is evaluated in. The analysis also suggests that through-diffusion results

781

of e.g. calcium in Na-bentonite will be obscured by transport limitation in

782

the filters (ω >> 1), also at rather high background concentration (figure 5).

783

5.2. Bentonite structure

784

Many studies have attempted to quantify the amount of non-interlayer

785

pores in compacted bentonite using data from diffusion or equilibration tests.

786

The results of the present work not only shows that ion equilibrium clearly 39

0.06

0.05

Ideal 1:1 100% Na 70% Na van Loon 07


Cl

0.04

0.03

0.02

0.01

0 10

50

100

External Concentration (mM)

Figure 10: Comparison between differently calculated chloride equilibrium coefficients and experimental data derived from chloride tracer through-diffusion experiments performed on untreated bentonite of dry density 1600 kg/m3 in external NaCl solutions of constant concentration (Van Loon et al., 2007). The leftmost bar in each group of external concentration shows the ideal anion equilibrium coefficient for a 1:1 system. The second bar shows the chloride equilibrium coefficient calculated in a pure sodium system. The third bar shows the chloride equilibrium coefficient calculated assuming 70% sodium and 30% calcium in the clay. The calculations assume mIL = 2.88 mol/kgw, corresponding to CEC = 0.75 eq/kg and w = 0.26.

40

787

must be taken into account for such quantifications to be relevant, but also

788

demonstrate the necessity to consider activity coefficient corrections, as well

789

as accounting for all types of cations in a system.

790

As an illustration, figure 10 compares calculated chloride equilibrium coef-

791

ficients with experimental data evaluated from tests performed on untreated

792

sodium dominated bentonite with an initial amount of divalent ions in the

793

range 20% – 45% (Van Loon et al., 2007). This particular set of data has been

794

used in attempts to quantify non-interlayer pores in several studies (Van Loon

795

et al., 2007; Tournassat & Appelo, 2011). The leftmost bar in each group of

796

external concentration in figure 10 shows the ideal ion equilibrium coefficient

797

for a 1:1 system; the second bar shows the corresponding quantity for a pure

798

sodium system, with activity coefficient corrections treated as in section 4;

799

the third bar shows the resulting coefficient under the assumption that the

800

clay contains 30% calcium and 70% sodium (with activity corrections in ac-

801

cordance with section 4).

802

Note the comparable size of the “corrections” to the ideal result, achieved

803

by accounting for, respectively, activity coefficients and presence of calcium.

804

As neither of these “corrections” are negligible, they must both be accounted

805

for if chloride content in bentonite should be used for quantifying possible

806

non-interlayer pore volume.

807

Even if the approach here adopted for estimating interlayer activities is

808

too simplistic, activity corrections are typically neglected altogether in most

809

studies which use chloride inventory to estimate non-interlayer pores (e.g.

810

Tournassat & Appelo, 2011; Muurinen et al., 2013). From the present dis-

811

cussion it follows that such studies most probably makes incorrect estimates,

812

and that the relative size of the error may be large.

813

Note also that the actual amount of ions in the clay may be difficult to

41

814

assess, unless they are all measured in the specific test. In e.g. Van Loon et al.

815

(2007), untreated clay was contacted with (initially) pure NaCl-solutions,

816

thus initiating an equilibration process for calcium, magnesium, and sodium.

817

The calcium/magnesium content in the system at the time of the diffusion

818

test therefore depends on e.g. concentration of the NaCl-solution, size of the

819

clay sample, volume of the external solutions, and the number of solution

820

replacements performed. The present analysis shows that this particular

821

calcium/magnesium content must be known if the measured chloride content

822

should be used to estimate non-interlayer pores.

823

6. Conclusions

824

A general framework for calculating Donnan potentials across a ben-

825

tonite/external solution interface has been presented, based on conceptualiz-

826

ing bentonite as a homogeneous mixture with all water located in montmoril-

827

lonite interlayers of uniform width. The framework, which is a generalization

828

of that presented in Birgersson & Karnland (2009), is based on a straight-

829

forward application of the Gibbs-Donnan membrane equilibrium model, and

830

is intended to be applied to high density systems. The primary quantities

831

calculated, apart from the Donnan potential itself, are ion equilibrium co-

832

efficients, defined as the ratio between internal and external concentration

833

for a specific species at the interface. These quantities are experimentally

834

accessible, e.g. through diffusion or equilibration tests.

835

The central equation for the Donnan potential (equation 13) involves ra-

836

tios between activity coefficients in the external solution and in the bentonite

837

for all charged species. Several other Gibbs-Donnan equilibrium calculations

838

in bentonite assume “ideal” behavior by putting these activity coefficient ra-

839

tios equal to unity. In contrast, here is suggested an approach for estimating 42

840

interlayer activity coefficients, based on the mean salt method. Although

841

this approach may be far from a complete description, the derived results for

842

ion equilibrium coefficients are in fair agreement with observations, in cases

843

where these are available. The resemblance with experimental data indicates

844

that the suggested approach is relevant; in particular it supports the notion

845

of an interlayer ionic strength. The analysis also suggests that activity coeffi-

846

cient corrections to “ideal” formulas are in general crucial for making relevant

847

quantifications – they may be of the same order as the “ideal” formula itself.

848

The solutions to the equation for the Donnan potential for several example

849

systems were analyzed in detail, providing a whole set of predictions which

850

can be tested in experiments. Among the predictions are:

851

• Anion equilibrium coefficients in 1:1 systems (e.g. in equilibrium with

852

a NaCl solution) depends approximately linearly on external concen-

853

tration, in the limit of low concentration, while they have an approxi-

854

mately quadratic dependence in 1:2 systems (e.g. in equilibrium with

855

a Na2 SO4 solution). For 2:1 systems (e.g. in equilibrium with a CaCl2

856

solution), anion equilibrium coefficients depends approximately on the

857

square-root of the external concentration in the limit of low external

858

concentration.

859

860

861

862

• The equilibrium coefficient for nitrate is much larger (approximately four times) than that for chloride in highly compacted Na-bentonite. • K-bentonite generally contain more anions as compared to Na-bentonite when equilibrated at the same external concentration.

863

• The ratio between the concentrations of a certain anion in two ben-

864

tonites with different cations (but otherwise similar) resembles the se-

865

lectivity coefficient for that cation pair. 43

866

The analysis furthermore suggests several means to quantify activity co-

867

efficients experimentally, e.g. by systematic measurement of ion equilibrium

868

coefficients in well-defined systems (in particular it is important to prop-

869

erly account for all cations), or by utilizing a relationship derived between

870

Gaines-Thomas selectivity coefficients and interlayer activity coefficients.

871

The conceptualization here advocated contrasts the traditional approach

872

to model bentonite, which neglects ion equilibrium to various degree, and

873

relies critically on the existence of bulk water in bentonite (non-interlayer

874

pores). It was demonstrated that an inadequate treatment of ion equilibrium

875

in such models may lead to incorrect inferences regarding bentonite structure,

876

and to incorrect interpretation of ion diffusion data.

877

7. Acknowledgments

878

This work was partly financed by the Task Force on Engineered Bar-

879

rier Systems of the Swedish Nuclear Fuel and Waste Management Company

880

(SKB). The author thanks Ola Karnland and Magnus Hedstr¨om for stimu-

881

lating discussions. Eva Hofmanov´a and Martin Glaus are thanked for com-

882

menting an earlier version of the manuscript.

883

Alt-Epping, P., Tournassat, C., Rasouli, P., Steefel, C. I., Mayer, K. U.,

884

Jenni, A., M¨ader, U., Sengor, S. S., & Fern´andez, R. (2014). Benchmark

885

reactive transport simulations of a column experiment in compacted ben-

886

tonite with multispecies diffusion and explicit treatment of electrostatic

887

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Birgersson, M., & Karnland, O. (2009). Ion equilibrium between montmoril-

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lonite interlayer space and an external solution – Consequences for diffu-

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Bourg, I. C. (2004). Tracer diffusion of water and inorganic ions in compacted

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saturated sodium bentonite. Ph.D. thesis University of California, Berkeley.

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Compacted, Water-Saturated Bentonite. Clays and Clay Minerals, 54 ,

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49