Pore-scale modeling of chloride ion diffusion in cement microstructures

Accepted Manuscript Pore-scale modeling of chloride ion diffusion in cement microstructures Yuankai Yang, Moran Wang PII:

S0958-9465(16)30372-9

DOI:

10.1016/j.cemconcomp.2017.09.014

Reference:

CECO 2912

To appear in:

Cement and Concrete Composites

Received Date: 12 July 2016 Revised Date:

9 September 2017

Accepted Date: 27 September 2017

Please cite this article as: Y. Yang, M. Wang, Pore-scale modeling of chloride ion diffusion in cement microstructures, Cement and Concrete Composites (2017), doi: 10.1016/j.cemconcomp.2017.09.014. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

3

Yuankai Yang, Moran Wang1

4 5

Department of Engineering Mechanics and CNMM, Tsinghua University, Beijing 100084, China

6

Abstract

7

Understanding the mechanism of chloride ion diffusion in cement is significant to

8

improve the reliability of offshore reinforced concrete structures. The chloride ionic

9

diffusivity in cement-based microstructures is predicted by pore-scale modeling using a

10

modified lattice Boltzmann method. Both the Nernst-Planck equation for ion diffusion

11

and the Poisson equation for electrodynamic effect are fully solved. The predicted

12

effective diffusivities in cement-based microstructures with different porosities are in

13

good agreements with the experiment data. The results show that the pore size

14

distribution and Zeta potential of cement-based microstructures directly influence the

15

effective diffusivities of chloride ions. The cement-based microstructure with smaller

16

pore size and higher negative Zeta potential hinders chloride ions corrosion more

17

effectively. The electrokinetic effect on the chloride ionic transport is negligible when the

18

ratio of the maximum-probability pore size and the Debye length is higher than 32 in the

19

cement-based microstructure. For engineering applications, we provide a predictive and

20

easy-to-use formula by up-scaling to correlate the effective chloride ion diffusivity with

21

electrokinetic effect in cement paste.

22 23 24

Keywords: chloride ion diffusion, cement microstructure, electrokinetic effect, lattice Boltzmann method

AC C

EP

TE D

M AN U

SC

RI PT

2

Pore-scale modeling of chloride ion diffusion in cement microstructures

1

1

Corresponding author. Tel: 86-10-627-87498; E-mail: [email protected] (M. Wang) 1

ACCEPTED MANUSCRIPT

25

1. Introduction Reinforced concrete structures being in the marine environment for a long time, the

27

chloride ions of the seawater are injected into the concrete, causing the corrosion of steel

28

bars [1-4]. The transformation of rusty mild steel is accompanied with the increase of

29

volume inducing the expansion and cracking of concrete, which shortens the service life

30

of reinforced concrete structures [5]. Research on the mechanism of the diffusion of

31

chloride ions in concrete remains a major challenge, but is very significant to improve the

32

reliability of reinforced concrete structure design [6]. The rate of chloride ions ingress

33

depends on numerous conditions including porosity, compactness, hydration, additives

34

and cover thickness [7, 8]. Experiments [3] displayed that the steel in normal reinforced

35

concrete did not rust due to a passive film, formed on the surface of steel, for the pH

36

value of pore solution in the concrete over 10. The passive film, an oxide layer thickness

37

of about 10 nm, can suppress the corrosion of iron. The break of the passive film would

38

occur with the pH value lower than 10, leading to the iron dissolution and corrosion.

39

Besides, the chloride ion can reduce the ferric corrosion barrier. Hausmann [9] found the

40

corrosion of steel would not occur until the [Cl- ] / [OH- ] > 0.6 . Therefore it is very crucial

41

to understand the mechanism correctly of ions transport and distribution in porous

42

microstructures.

TE D

M AN U

SC

RI PT

26

To experimentally study the ion transport and distribution in concrete at pore scale is

44

expensive and tough, especially when the pore size is very small [10-12]. The

45

measurements may be resulted from multi-factorial coupling and hard to reveal the

46

mechanism. Numerical modeling therefore provides an efficient way to help people know

47

what is happening at pore scale and clarify effects from different factors [13-15]. Liu et

48

al. [16] simulated the steady-state diffusion by the Conjugate Gradient method and solved

49

the distributions of ion concentration in porous cement-based microstructures. Zhang et al.

50

[17] applied the lattice Boltzmann (LB) methods to simulate the ion diffusion in the non-

51

saturated cement paste whose microstructure was generated by HYMOSTRUC3D. The

52

gas-water mixture was established by a modified Shan-Chen multi-phase LB model and

53

the diffusion process in the microstructure is solved by another LB model. These recent

54

work opened the door for pore-scale modeling of ion diffusion in concrete, but the

55

electrokinetic effects were ignored which included the ion-ion interaction and the ion-

56

charged-surfaces interaction. As we know that when an electrolyte (even for pure water)

AC C

EP

43

2

ACCEPTED MANUSCRIPT

meets a solid surface, the surface will be automatically charged because of physical

58

adsorptions and chemical reactions [18, 19]. The surface charge will induce an electrical

59

double layer (EDL) near the surface and therefore influence the ion distribution [20]. If

60

the pore size is much larger than the thickness of EDL, the electrokinetic effects on ion

61

diffusion and distribution are negligible; however when the EDL thickness is comparable

62

with the pore size of the cement paste, the electrokinetic effects, including ion-ion

63

interaction and ion-surface interaction, may have a significant impact on ion transport

64

[21-23]. Recently, Monteilhet [24] and Korb [25] have shown through measurements that

65

the microstructures of cement paste mainly consists of nanoscale pores, ranging from 2 to

66

600 nm. Meanwhile the thickness of EDL is about 200~40 nm when the ionic

67

concentration ranges from 0.05 to 1 mM. Therefore the electrokinetic effects play a key

68

role in ion diffusion and distribution in concrete microstructures and have to be

69

considered in pore-scale modeling.

M AN U

SC

RI PT

57

The aim of this present work is to establish a numerical framework to simulate the

71

diffusion of chloride ions in cement-based microstructures with the electrokinetic effects

72

considered. A modified lattice Poisson Boltzmann model is used to solve the Nernst-

73

Planck equation for ion diffusion and the Poisson equation for electrical potential in the

74

three-dimensional microstructure of cement paste. We are to study the influences from

75

microstructure parameters on macroscopic chloride ion diffusion in cement, to reveal the

76

mechanisms and try to provide useful advices on optimizations of engineering

77

applications by up-scaling analysis.

78

2. Simulation of cement-based microstructure

EP

TE D

70

In this study, a friendly-used model is proposed to generate the three-dimensional

80

porous complex microstructure of hydration cement paste. The cement particles are

81

dissolved from external to internal during the hydration reaction [26]. With neglecting the

82

chemical details the cement particles surrounded by water can react to produce solid

83

reaction products (surface products) or spontaneously nucleate in the capillary water to

84

products crystals (pore products) [27]. The surface product is mainly calcium silicate

85

hydrate (C-S-H), as well the major pore product is calcium hydroxide (CH). In this model,

86

the method developed by Wang et al. [28] is adopted to generate the initial cement

87

particles and the hydration process is simulated using the algorithm similar to, but not

AC C

79

3

ACCEPTED MANUSCRIPT

exactly the same as, the microstructural model proposed by Garboczi and Bentz . In our

89

algorithm, we add an extra step, called the “invasion” step. This step ensures the

90

hydration process going on when all remaining cement particles are covered by surface

91

product. Hence, the cement microstructure generated by our model is more consistent

92

with the real cement than that of Garboczi and Bentz. The simulation algorithm is

93

described below.

RI PT

88

The QSGS model [28] is adopted to reproduce the cement patricles in a 3D domain:

95

(1) We use the water-cement ratio to calculate the seed distribution probability sd

96

and stochastically locate the seeds based on the seed distribution probability. The cement

97

particles volume fraction is φs , hence the water volume fraction is 1 − φs and the water-

98

cement ratio is expressed as [27]:

1 − φs . 3.2φs

M AN U

w/c =

SC

94

99

(1)

100

For 3D simulations, the average cement particles volume is π d 3 / 6 and d is the mean

101

grain size of cement particles. Therefore, the 3D seed distribution probability sd is: φ δ3 6δ 3 , sd = s 3 = π d / 6 π d 3 (3.2 w/ c + 1)

103

(2)

where δ denotes the lattice constant (grid size).

TE D

102

104

(2) The seeds grow toward neighboring cells using cellular automata approach.

105

(3) Repeat step (2) until the fraction of the cement paste approaches φs .

106

After the cement particles generated, the procedure of hydration operates by an iteration process and each cycle performs three steps:

EP

107

(i) The dissolution step: count the number of cells on the surface of cement particles

109

and these cells as the reactive cells extend to the neighboring cells. If growing cells locate

110

in pores, these growing cells are judged to be grown successfully as surface product cells.

111

This step stops until the ratio of the number of growing cells and reactive cement cells

112

reach 1.7. By contrast, pore product cells have 0.61 times the number of reactive cement

113

cells. The pore product cells are randomly distributed in the pore space according to the

114

computing number of pore product cells.

115 116

AC C

108

(ii) The diffusion/reaction step: stochastic distributed pore product cells move arbitrarily and stop moving when they contact with other solid cells.

4

ACCEPTED MANUSCRIPT

(iii) The invasion step: because the surface product (C-S-H) is still porous media [29-

118

31] where the cations and small neutral molecules can diffuse through and be reactive,

119

the hydration process can continue even when all cement particles are covered by surface

120

product. Yet, since the pores in C-S-H are mostly at nanoscale, and generally its surfaces

121

are negatively charged, the diffusion of negative chloride ions is super slow. In this step,

122

the hydration reactant cells randomly move one step in the location of un-reacted cement

123

cells and then mark as new reactive cement cells and count. When the cement paste

124

reaches the degree of hydration α or is completely hydrated, the cycles quit and cement

125

microstructure is fully formed.

SC

RI PT

117

Thus, our model essentially uses three parameters (w/c, α and d ) to control the

127

microstructure of cement. The generated porous cement microstructure is sketched in Fig.

128

1(a). When the cement solid is set as hyaline, the capillary pore distribution is shown in

129

Fig. 1(b) and the corresponding pore size distribution shown in Fig. 1(c).

AC C

130

EP

TE D

(a)

M AN U

126

5

ACCEPTED MANUSCRIPT

SC

RI PT

(b)

131

0.9

M AN U

1

(c)

0.7 0.6 0.5 0.4 0.3

TE D

Fraction smaller than

0.8

0.2 0.1

0 0

138

10

20

30

40

Pore size （ µm）

50

60

70

EP

Fig. 1 The three dimension porous microstructure (140×140×140 µm 3 ) of cement generated by our model with a resolution of 2 µm/pixel (a), where the red is the unhydrated cement phase, green is the product of hydration phase and the blue is the capillary pores phase. The capillary pores distribution is shown in (b). The pore size distribution of regenerated early age cement is shown in (c), calculated by the maximum sphere method [32].

AC C

132 133 134 135 136 137

To validate our model on cement microstructure simulation, we generate

139

microstructures with different water-cement ratios and degrees of hydration comparing

140

with the Powers’ law [33]. According to the Powers’ law, the volume fractions of

141

capillary pores φcap (capillary porosity), and unhydrated cement φuc can be respectively

142

determined as [34]:

143 144

w / c − 0.36α , w / c + 0.32 0.32 (1 − α ) φuc = . w / c + 0.32

φcap =

6

(3) (4)

ACCEPTED MANUSCRIPT

Fig. 2 indicates that the capillary porosity and the volume fraction of unhydrated

146

cement by our model have good agreements with the empirical results by Eqs. (3) and (4).

147

Therefore, our model is a friendly-use and robust method to generate porous cement

148

microstructures.

RI PT

145

M AN U

SC

(a)

149

156

AC C

150 151 152 153 154 155

EP

TE D

(b)

Fig. 2 The capillary porosity and the volume fraction of unhydrated cement varying with the degrees of hydration of present simulations and empirical results are shown in (a) and (b), respectively. Lines represent the empirical results and symbols are from our simulations.

3. Mathematical description of ion diffusion in capillary pores

157

The essence of diffusion is the thermal motion of molecular species in fluids, and the

158

concentration gradient of species is the driving force. In 1970, Collepardi [35] firstly

159

described the chloride ion diffusion in the concrete by the Fick’s law with the 7

ACCEPTED MANUSCRIPT

160

assumptions of isotropic concrete structure and negligible physical or chemical

161

interactions between ions and cement-based materials:

162 dCi = ∇ ⋅ ( Di ∇Ci ) , dt

(5)

RI PT

163 164

where Ci , Di are the concentration and the diffusivity of ions, Na + or Cl - , respectively;

165

and t is the time.

However, the cement-based material is not isotropic and homogenous. Actually, it is

167

a complex heterogeneous composite composed by pores, solid, water, bubbles and so on.

168

Especially, to study the mechanism of chloride ion diffusion in pore scale, the results are

169

not convincing without considering the heterogeneous structures of cement-based

170

materials [36]. The scale we focus on is much larger than the atomistic scale, so that the

171

macroscopic continuous equations are still valid [37, 38]. With the electrokinetic effect

172

considered, the mass flux of ions in the pore solution in cement is calculated by [39]:

M AN U

SC

166

173 174

J i = − Di ∇Ci − ezi bC i i ∇ψ ,

175

where e , zi and ψ denote the absolute charge of electron, the valance number for ith ion

176

and the local electrical potential. We assume the seawater as the NaCl solution and

177

neglect other ions in this work. Therefore J i denotes Na + and Cl - species flux, bi

178

denotes the ion mobility of Na + or Cl - . The ionic flux J i and the concentration Ci

179

follow continuity equation:

TE D

EP

181

∂Ci + ∇ ⋅ Ji = 0 . ∂t

Substituting Eq. (6) into Eq. (7) leads to the Nernst-Planck equation.

AC C

180

(6)

8

(7)

Fig. 3

Sketch of electrical double layer with negative charged surface with the binary monovalent

electrolyte [18, 40]. M denote the monovalent metal cations (e.g. Na + , K ) and A the anions (e.g. Cl− ). The d-plane corresponds here to the shear plane and the Zeta potential, (ζ-potential). +

M AN U

182 183 184 185

SC

RI PT

ACCEPTED MANUSCRIPT

Since the product of cement hydration is the composition of calcium oxide, silicon

187

oxide and aluminum oxide, the metal oxide surface begins to decompose when in contact

188

with the electrolyte solution (e.g. NaCl solution) and the surface generates silanol groups

189

[18, 41]. Therefore, the ions in the electrolyte solution form the EDL near the pore wall

190

because of the electrostatic attraction and ionic thermal motion, as shown in Fig. 3. There

191

is a plane to separate mobile fluid from fluid still adhering to the charged surface, called

192

shear plane. The electric potential on this plane is called Zeta potential (ζ-potential). In

193

the cement-based materials, the surface absorbs positive ions (cations) since the Zeta

194

potential is usually negative. Because of the strong electric attraction near the surface,

195

there is area, called stern layer, where the cations are not movable. The ψ denotes the

196

local electric potential caused by the ionic distribution, which is governed by the Poisson

197

equation [20]:

EP

AC C

198

TE D

186

∇ 2ψ = −

ρe N ezCi = −∑ A , ε rε 0 ε rε 0 i

(8)

199

where ρ e , N A and ε r ε 0 are the net charge density, the Avogadro’s number and the

200

dielectric permittivity of electrolyte solution. For dilute electrolyte solutions, the ionic

201 202

concentration in the diffuse layer matches with the Boltzmann distribution [42]: Ci = Ci ,∞ exp(−eziψ / kT ) ,

9

(9)

ACCEPTED MANUSCRIPT

203

where Ci ,∞ denotes the bulk ionic concentration. k and T denote the Boltzmann constant

204

and absolute temperature.

206

Substituting Eq.(8) into Eq.(9) gives the Poisson-Boltzmann equation [43]: ez 1 ∇2ψ = − N Aezi Ci ,∞ exp(− i ψ ) , ∑ kT ε rε 0 i

(10)

RI PT

205

207

which describes the interaction between the electric potential and ionic distribution.

208

Eqs.(6)-(10) govern the electrokinetic ionic transport processes in cement microstructures

209

[44].

The Debye length that characterizes the EDL thickness is defined as:

λD =

211

ε r ε 0 kT 2 z i 2 e 2 N A Ci , ∞

SC

210

.

(11)

It is arduous to determine a suitable boundary condition to depict the electric

213

potential and surface charge distribution at the liquid-solid interface in the porous

214

microstructures [25]. Therefore a constant Zeta potential is still popularly used as the

215

electrical boundary condition: ψ wall = ζ . For the inlet and outlet, we adopt the constant

216

ionic concentration boundaries. At the liquid-solid interfaces, the zero normal flux

217 218

conditions are applied:

219

$ is the unit vector normal to surface. where the n

J i ⋅ n$ = 0 ,

TE D

220

M AN U

212

(12)

The macroscopic effective diffusivity Deff in porous media that characterizes the

221 222

comprehensive ionic transport ability in cement-based materials is defined by: Deff = JL / ∆C ,

223

where J is species flux per unit cross-section in the steady state, L and ∆C are the length

224

and ion concentration difference between inlet and outlet. This effective diffusivity varies

225 226

with the porosity, geometry, chemical and physical properties of cement-based materials.

227

4. Lattice Boltzmann method

EP

AC C

228

(13)

Lattice Boltzmann method (LBM) is a mesoscopic method based on the molecular

229

motion theory and corresponds to clear physical background. In recent years, LBM has

230

shown amazing performances to analyze fluid flow or heat transfer in porous media with

231

complex geometries [28, 44-48]. Benefiting its advantages for modeling transports in

232

porous media, we use LBM to investigate the chloride ions diffusion in cement-based 10

ACCEPTED MANUSCRIPT

materials in this work. The set of coupled ionic electrodynamic transport governing

234

equations, subjected to the appropriate boundary conditions, are solved by our lattice

235

Boltzmann method codes. The developed codes combine an ion diffusion evolution on

236

discrete lattices to solve the Nernst-Planck equation with an electric potential evolution

237

method on the same set of lattices to solve the nonlinear Poisson equation [49].

238 239 240

4.1. Evolution equation for ion diffusion

242

We use the following evolution equation to solve the governing equation for ion diffusion of Na + or Cl − [50-52]:

gαi (r + eα δ x , t + δ t , Di ) − gαi (r , t ) = −

243

1

τD

i

SC

241

RI PT

233

[ gαi (r , t ) − gαi ,eq (r , t )] ,

(14)

where the r denotes the position vector , eα the discrete velocities, where α = 1, 2,L, N

245

representing the discretized directions as shown in Fig. 4. The equilibrium distribution

246 247

function of ionic transport LBM is written as:

TE D

M AN U

244

251 252 253

Fig. 4 The discretized directions in D3Q7 model

AC C

249 250

EP

248

δ ez D ∂ψ   ∂ψ ∂ψ gαi ,eq (r , t ) = 1 − 4 t , D i i ( ex,α + ey,α + ez,α )  ωα Ci , δ x k BT ∂x ∂y ∂z  

with

1/ 4 α = 0 , 1/ 8 α = 1 − 6

ωα = 

(15)

(16)

254

where τ Di is the dimensionless relation time for the Na + or Cl- ionic transport related to

255

the diffusion coefficient Di , and δ t , Di the corresponding time step. The dimensionless

256

relaxation time is calculated by:

11

ACCEPTED MANUSCRIPT

τD =

257

i

4 Di 1 + , cDi δ x 2

(17)

where cDi is the diffusion lattice speed for Na + or Cl - defined as c Di = δ x / δ t , Di , with δ x

259

representing the lattice constant (grid size). The value of cDi can be assigned any positive

260

value as long as the value of τ Di ∈ (0.5, 2) [53]. In the LBM scheme, the macroscopic

261

ionic concentration is calculated by:

RI PT

258

Ci = ∑ gαi .

262

(18)

α

We choose a consistent D3Q7 scheme for both diffusion and electrical potential

264

evolutions in this work because (i) we tried the D3Q19 scheme for diffusion, but found

265

mass flux oscillations near the boundaries; (ii) the D3Q15 scheme was reported unstable

266

for electrical potential evolution due to the extremely large source term near interfaces

267

[49]. The D3Q7 scheme is stable, accurate enough and highly efficient for diffusion and

268

electrical potential evolutions, compared with the other schemes, even though this

269

scheme is reported not of the same high-order accuracy as D3Q19 for fluid flow. For the

270

D3Q7 lattice system for the evolution equation Eq. (10), the discrete velocities are: (0, 0, 0) α =0   eα =  . (±1, 0, 0), (0, ±1, 0), (0, 0, ±1) α = 1 − 6 

M AN U

272 273 274

4.2. Evolution equation for electrodynamics

277

where τ ψ is the dimensionless relaxation time for electric potential transport, cψ the

278

electrodynamic lattice speed, which similar to cDi , defined as cψ = δ x / δ t ,ψ , and δ t ,ψ the

279

time step.

280 281

282 283

EP

276

The evolution equation for electric potential transport can be therefore written as [50]:  0.5  ρe 1 hα (r + ∆r , t + δ t ,ψ ) − hα (r , t ) = − [hα (r , t ) − hα eq (r , t )] + ωα δ t ,ψ 1 −  τ  ε ε , (20) τψ ψ  r 0 

AC C

275

(19)

TE D

271

SC

263

For a D3Q7 lattice system, the equilibrium distribution of electric potential evolution

function hαeq is

α =0  0 hαeq =  , ψ / 6 α = 1 − 6

and the dimensionless relaxation time τ ψ is calculated by:

12

(21)

ACCEPTED MANUSCRIPT

285 286

1 . cψ δ x 2 The macroscopic electrical potential is then calculated by:  ρ  ψ = ∑  hα + 0.5δ t ,ψ ωα e  . ε rε 0  α 

τψ =

4

+

287 288

4.3. Boundary conditions treatments

(22)

(23)

RI PT

284

289 290

For ionic transport, the zero normal flux boundary condition translates into [51]:

291

where the index α and β are the opposite directions normal to the interface and β is the

292

direction towards wall. For inlet and outlet boundaries, the Dirichlet boundary condition

293

follows [54]:

296

M AN U

1 gαi (r , t + δ t , Di ) = − g βi (r , t ) + C0 , 4

294 295

(24)

SC

gαi (r , t + δt , Di ) = gβi (r , t ) ,

(25)

where C0 is the given ionic concentration of inlet or outlet.

As for electric potential, we use non-equilibrium “bounce-back” rule as the boundary condition because it is easy to handle for complex geometries [49].

298

5. Results and discussion

299 300

5.1. Benchmark for ions diffusion in channel

TE D

297

To validate our numerical models and codes, firstly we consider ion diffusion in a

302

two-dimensional microchannel, as shown in Fig. 5, with a height H at 0.65 µm . The

303

electrolyte solution is NaCl solution and the initial bulk ionic concentration is

304

4 ×10−4 mol/L . The inlet and outlet ionic concentrations are given as 2.0 ×10−5 mol/L

305

and 1.0 ×10−5 mol/L , respectively. Meanwhile the other properties and physical

306

parameters are: the diffusivities for both ions DNa + = DCl − = 1.07 ×10−10 m2 /s , the

307

dielectric constant 6.95 × 10−10 C 2 / J ⋅ m and the temperature 273K. The surfaces of

308

microchannel are homogeneously charged and the potential distribution should be

309

consistent with that from the Poisson-Boltzmann model for thin electrical double layers

310

from Ref [55] Chapter 7. The zeta potential for both walls is -20 mV. These parameter

311

ranges are of interest in many civil engineering applications.

AC C

EP

301

312

13

313 314 315 316

RI PT

ACCEPTED MANUSCRIPT

Fig. 5 Ion diffusion in microchannel. The black is the wall. The ionic concentration at inlet and outlet are given.

317

Fig. 6(a) shows good agreements between present simulations and theoretical results,

319

which validates the present algorithm and codes. With increase of the distance from the

320

walls, the electrical potential exponentially decays to zero. When the distance is about 4

321

times of Debye length, the electric potential is less than 3% of zeta potential. Fig. 6(b)

322

illustrates that the concentration gradient of cations along the direction of diffusion near

323

surfaces is much more than the gradient within pores because the cations are converged

324

near the negatively charged walls, and hence the diffusion near surfaces is faster than that

325 326

in pores.

M AN U

SC

318

AC C

EP

TE D

(a)

327

328

14

ACCEPTED MANUSCRIPT

329

Fig. 6 The electrical potential profile of cross section by the present simulation, compared with the theoretical results, is shown in (a). The theoretical solutions are from Ref. [55]. The Zeta potential is ζ = −20mv . Meanwhile the gradient profile of cationic concentration is shown in (b).

M AN U

330 331 332 333 334

SC

RI PT

(b)

5.2. Effective ion diffusion coefficient in cement

The actual domain in which we use to simulate the ion diffusion is a

336

140 ×140 ×140 µm3 cube with two transition regions shown in Fig. 7. The space

337

discretization depends on the balance between numerical accuracy and computational

338

costs. Since the random characteristics of microstructure leads 3-5% fluctuation in our

339

simulations, a too fine grid is not actually necessary. A 70 × 70 × 70 uniform grid is

340

adopted in our simulations after the grid independence checked. We will also show in the

341

following part of this section that the cube with this set of grid satisfies the REV

342

requirement by our modeling results. We only consider the fully saturated cement in the

343

marine environment in this study. For the degrees of hydration and water to cement ratios

344

studied, the porosities of cement microstructures are all above the percolation threshold,

345

and therefore the chloride ion diffusion process only occurs in the capillary pores while

346

the hydrated phase (e.g. C-S-H) has no contribution to chloride ion diffusion in this work

347

[56, 57]. When the porosity is less than percolation threshold, the diffusion process just

348

stops. The bulk diffusivity of chloride ion DCl in this work is 1.5 ×10−9 m2/s for the room

349

temperature 20℃ from reference [58].

AC C

EP

TE D

335

15

SC

RI PT

ACCEPTED MANUSCRIPT

M AN U

350 351 352

Fig. 7 A sketch about simulation domain

We consider a simple binary monovalent electrolyte solution (e.g. NaCl) in the pores.

354

For the absence of electrokinetic effect, the zeta potential for all solid surfaces is set at 0

355

mV. To investigate the effects of porosity, we compute the effective diffusivities in

356

microstructures with different porosities generated by our model and compare with

357

experimental data. Fig. 8 shows the ionic concentration distributions after the diffusion

358

process is steady. Stochastic characteristics are very clear for the ionic concentration

359

across one section. To calculate the effective ion diffusivity more accurately, we use the

360

arithmetic mean value of one cross-section statistically to eliminate the fluctuations. 361

AC C

(a)

EP

TE D

353

362 16

ACCEPTED MANUSCRIPT

SC

RI PT

(b)

364

TE D

M AN U

363 (c)

Fig. 8 The distribution of chloride ion concentration in three dimension cement of different porosity when the diffusion is steady. The cement solid is hyaline and invisible. The water-cement ratio: w/c=0.4 in (a), w/c=0.5 in (b) and w/c=0.6 in (c).

369

Fig. 9 shows the averaged ionic concentration over cross sections varying with

370

position for uncharged or charged porous media under the same ion concentration

371

difference. The effective concentration gradient, defined as ∆C / L = (Ci − Co ) / L , depends

372

on the microstructure and the charge conditions. We use the average concentration at

373

exactly the inlet and the outlet of porous media as Ci and Co respectively, which have

374

been shown as the intersection points between the red lines (inlet and outlet positions)

375

and the black lines (concentration curve). After the concentration gradient determined,

376

the effective ionic diffusivity is then calculated by Eq.(13) for various microstructures.

AC C

EP

365 366 367 368

17

ACCEPTED MANUSCRIPT

377

M AN U

378

SC

RI PT

(a)

383

379

Fig. 9 The schematic mean concentration of each cross section in our simulated domain of cement with uncharged (a) and charged wall (b). The sketch of domain is shown in Fig. 7. The concentration for inlet is 1.5 ×10−4 mol/L and 0.5 × 10−4 mol/L for outlet.

AC C

380 381 382

EP

TE D

(b)

The random factors in generation of microstructures will bring fluctuations to

384

calculation of the effective ion diffusivity. Therefore we have generally generated 3-5

385

samples for each set of parameters of microstructure and used the mean value as the

386

effective predicted one. For the cube and grid we used mostly, the relative standard

387

deviation of effective diffusivities of the microstructures is 3.6%, less than 5%, which

388 389

proves this simulated domain can be treated as a REV.

18

ACCEPTED MANUSCRIPT

390

5.3. Chloride ion diffusion without electrokinetic effect First, simulations are performed in the ordinary Portland cement paste with different

392

porosities. For these cases, we just assign a zero zeta potential to not involve the

393

electrokientic force from solid surfaces. The other physical parameters are the same as

394

the section 5.2. Fig. 10 compares the simulation results with the experimental data

395

reported by others [60-62]. In this study, we ignore the transport process in the gel pores,

396

thus the Deff / DCl rapidly decreases when the capillary porosity is less than 0.2. From Fig.

397

10, the effective diffusivity obtained by our simulation is overestimated with respect to

398

experiment data. When we use the modified diffusivity as 1.07x10-10 m2/s suggested by

399

Pivonika et al.[43] to involve the double layer effect, the results show good agreements

400

with experiment data. It means that the electrokinetic effect on ion diffusion in cement is

401

very important and not negligible.

403 404 405 406 407 408

M AN U

AC C

EP

TE D

402

SC

RI PT

391

Fig. 10 The comparison of effective chloride diffusivity between simulation and experiment in different porosities. The blue squares are the simulation by using the actual chloride diffusivity and the red circles the simulation with the modified diffusivity as 1.07x10-10 m2/s suggested by Pivonika et al.[58] to consider the double layer effect.

5.4. Chloride ion diffusion with electrokinetic effect

409

The electrokinetic effects play an important role on ion diffusion in cement. In this

410

section, we consider the electrokinetic effect and investigate the pore size effect and

19

ACCEPTED MANUSCRIPT

surface charge statue effect on ion diffusion. To investigate the pore size effect on ionic

412

diffuse ability, here let us characterize the pore size and distinguish some definitions of

413

pore size first: the most probable pore size, the means pore size and the median pore size.

414

The most probable pore size, the maximum probability pore size in pore size distribution,

415

can be recognized as the significant pore size of porous media in pore size distribution

416

[67]. Because the gel pores (smaller than 10 nm) is too weak to affect ionic diffuse ability

417

comparing with mesoscale pores if the capillary porosity is over than 0.2 [56, 68], we

418

neglect the gel pores. To study the influence of pore size on the effective diffusivity, a

419

dimensionless number l * , named the dimensionless pore size, is defined: l * = d / λD ,

420

where d is the characteristic size of pores and λD is the Debye length. Therefore after the

421

ionic concentration field is solved out, the significant pore sizes of microstructures are

422

calculated by the scheme proposed by Supriyo and Keith [32].

M AN U

SC

RI PT

411

In this section, we use a computational domain of a 3 × 3 × 3 µm3 porous

424

microstructure plus two 1.5 µm transition regions with a 200 ×100 ×100 uniform grid, as

425

shown in Fig. 7. For the homogeneous charged surfaces of microstructures, the zeta

426

potential for all walls is -20 mV [69, 70]. The chemical composition of pore solution is

427

the symmetric monovalent NaCl electrolyte. The other parameters are the same as stated

428

in the previous section. The distributions of electrical potential and charge density

429

predicted by our numerical frame are shown in Fig. 11.

AC C

EP

(a)

TE D

423

431

430

20

ACCEPTED MANUSCRIPT

SC

RI PT

(b)

432

Fig. 11 The distributions of electric potential (a) and charge density (b) presented by our numerical frame. Zeta potential is -20mV and the direction of diffusion is alone the x axis. Because the bulk concentration in black box is larger than that in red box, the EDL in black box is thinner than that in red box, shown in (a) and (b).

437

To quantify the electrokinetic effect on ion diffusion, we compare the calculated

438

effective ion diffusivity by considering or not the wall charges. The effective diffusivity

439

deviation

γ is defined as:

γ=

* Deff − Deff

TE D

440

M AN U

433 434 435 436

* Deff

,

(26)

441

* where Deff is the effective diffusivity with electrokinetic effect and Deff is that without

442

electrokinetic effect considered of porous microstructures. Fig. 12 shows

443

of l * for different ions. A negative value of deviation denotes that the charged surface

444

can cause the inhibition of ionic transport, while relatively a positive one means the

445

acceleration of ionic diffusion. The results show that the negative zeta potentials can slow

446

down the anionic diffusion and promote the cationic. When l * increases, the magnitudes

447

of effective diffusivity deviations of both anions and cations decrease. For the anion, the

448

effective diffusivity deviation

449

electrokinetic effect can be ignored for the anion diffusion provided that l * > 32 in the

450

cement-based microstructure for the civil engineering application.

AC C

EP

γ as a function

γ is lower than 10% as l * > 32 , which means the

451

21

452 453 454

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

Fig. 12 The effective ionic diffusivities with different l * (Zeta potential= -20 mV)

Fig. 12 also demonstrates that the same l * exhibits different pore size effects on the

456

anion and cation transport. The pore size effect on the promotion of cations is greater

457

than that on the hindrance of anions. The reason for this is still divergent and may be

458

explained by the details of the surface ionic absorption. The cations are gathered on the

459

solid-liquid interface since the negative charged surface adsorption in the microstructure.

460

Therefore, the concentration gradient of cations along the direction of diffusion near

461

surfaces is much more than the gradient within pores and the cation surface diffusion

462

would be faster than that within pores. As a result, the electrokinetic effect can promote

463

the spread of cations remarkably.

EP

TE D

455

The surface charge status is another important factor for electrokinetic effects on ion

465

diffusion in the porous cement-based microstructure [71]. Fig. 13 shows the calculated

466

effective diffusivities through one single porous microstructure for different Zeta

467

potentials. The results indicate that a higher negative Zeta potential enhances the cationic

468

diffusivity and weakens the anionic. The cation is more sensitive to Zeta potential than

469

the anion. The higher negative Zeta potential forms more intense interaction between ions

470

and the charged surfaces, and leads to higher concentration gradient of cations along the

471

direction of diffusion and therefore a higher effective diffusivity.

AC C

464

472 22

473 474 475

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

Fig. 13 The effective ionic diffusivities with different Zeta potentials ( l * = 27.4 )

6. Predictive formula by up-scaling

So far a powerful numerical framework has been established at pore scale, by which

477

one can reveal the miscroscopic mechanisms of ion transport in cement-based

478

microstructures and even help people correct understanding of the ion diffusion behavior.

479

However this tool is too complicated to use and the process is too time-consuming for

480

engineering application. Therefore we need more friendly-use formulas that can bridge

481

the microstructure details with a macroscopic mathematical expression with parameters.

482

This bridging process is called up-scaling.

EP

TE D

476

Generally the saturated cement paste can be regarded as a two-phase composite, and

484

therefore the effective ionic diffusivity can be expressed as a function of the bulk ionic

485

diffusivity, capillary porosity, tortuosity and constrictivity of pore microstructure, and so

486

on. Based on experimental data, empirical formulas have been proposed for calculating

487

the diffusivity of chloride ions in the cement paste in various situations [64, 72]. The

488

fitting parameters in such empirical formulas have to be adapted case by case, which

489

means that they lack generality and physical significance. What is more important is that

490

the previous empirical formulas with fitting parameters can hardly indicate the influences

491

from the microstructure and therefore they are not available for structure design or

492

optimization. However, the process of those empirical formulas inspired us on up-scaling.

AC C

483

23

ACCEPTED MANUSCRIPT

493

If one can find a good mathematical expression, and determine the parameters by our

494

pore-scale modeling, one will obtain the predictive formula successfully.

496 497

Considering the general effective media theory, the effective species diffusivity in a porous media is following the relationship with pore structure parameters [73]: * Deff σ eff = = φcap β , σ0 D0

RI PT

495

(27)

where D0 is bulk diffusivity, σ eff and σ 0 is the conductivity through the porous media

499

and the bulk water respectively, φcap denotes the capillary porosity of pore structure, and

500

β is defined as β = κ / τ 2 [74] with τ representing the tortuosity and κ the

501

constrictivity of the pore structure. τ and κ reflect the complexity of pore network and

502

they are difficult to directly obtain through experiments. Therefore, people have

503

simplified it by using only the capillary porosity, namely the Archie’s law, based on the

504

general composite theories, as the pore-network parameter, given as [65]:

M AN U

SC

498

n Deff* = a ⋅φcap ,

505

(28)

where a is a variable relating to D0 and n is obtained by the constrictivity and

507

tortuosity of the pore network. Arnold et al. [22] introduced specific corrections into

508

Eq.(28) to modified this formula to evaluate effects of other interactions(eg. chemical

509

reactions, electrical potential). The Bruggeman asymmetric medium theory also provided

510

the similar relation to Eqs. (27) and (28), in addition, considering the percolation

511

threshold. To overcome the drawback of the Bruggeman asymmetric medium theory, a

512

more complicated formula was introduced to reveal the chloride ion diffusion in the solid

513

phase of the cement paste, eg. C-S-H gel, which is also diffusive, because of gel pores

514

existed in the C-S-H gel [65].

EP

TE D

506

Near the negative charged surface, the anionic concentration decreases and the

516

cationic concentration increases accordingly in fact. If we still treat the concentration as

517

constant, we can transfer this effect to variations of cross-sectional area for anions.

518

Furthermore, we can even consider charged surfaces change the effective porosity φeff of

519

the cement microstructure for different ions, given as:

520 521 522

AC C

515

φeff = ωe ⋅φcap ,

(29)

where ωe is the correction factor of the electrokinetic effect, written as:

ωe = exp(−ηs b ⋅ e− β l /2 ) , *

s

24

(30)

ACCEPTED MANUSCRIPT

where η s and β s are the correction factors of pores electrical potential and effective

524

pores radius in the porous media, respectively, b is the dimensionless zeta potential,

525

defined as: b = zi eζ / kBT . For the absence of electrokinetic effect or if l *  ∞ , ωe = 1,

526

which means this model is compatible with the previous theoretical models without

527

electrokinetic effects. A full derivation of Eq.(30) is given in the Appendix A. As a result,

528

the effective diffusivity of chloride ions in the cement paste is:

529

Deff = D0 exp(−ηsb ⋅ e− βsl / 2 ) ⋅φcap .

530 531 532 533 534

Fig. 14 The comparison between the proposed predictive formula and pore-scale simulations for different Zeta potentials and microstructures. Effective diffusivities vary with the Zeta potentials of different microstructures. The different colors mean various microstructures with corresponding structure. The detailed parameters of each microstructures are: 1, φcap = 0.95 , l * = 32.5 ; 2, φcap = 0.9 , l * = 30.8 ; 3,

(

)

n

(31)

537

EP

φcap = 0.8 , l * = 27.4 ; 4, φcap = 0.7 , l* = 23.9 ; 5, φcap = 0.6 , l * = 20.5 ; 6, φcap = 0.5 , l * = 17.1 . The lines are analytic predictive results by Eq.(32) and symbols are results from numerical simulations.

AC C

535 536

TE D

M AN U

SC

*

RI PT

523

Eq. (31) is the proposed predictive mathematical formula that can involve the

538

electrokinetic effects on ion diffusion in porous cement-based materials. There are three

539

parameters that can be determined by fitting the pore-scale modeling results. To observe

540

the electrokinetic effect, we generate six groups of cement-based microstructures, whose

541

mean pore size is comparable with EDL thickness. We use these microstructures as the

542

domains for simulation to calculate the effective diffusivity of chloride ion for different

543

ionic bulk concentrations and zeta potentials. The other parameters are the same as

25

ACCEPTED MANUSCRIPT

544

section 5.4. The simulation results shown in Fig. 14 are employed to determine the values

545

of parameters in Eq. (31) by numerical fitting. The results show n = 1.7 , ηs = 0.45 ,

546

βs = 0.17 and we re-write Eq. (31) as:

(

Deff = D0 exp(−0.45b ⋅ e−0.17 l /2 ) ⋅ φcap *

)

1.7

.

(32)

RI PT

547

As a result, we predict the effective diffusivity of chloride ion by Eq.(32) for different

549

ionic bulk concentrations. Fig. 15 shows the comparisons between the predicted results

550

and the pore-scale modeling ones. The preview studies [75] also showed the trend that a

551

low solution concentration could weaken the chloride ion effective diffusivity, which is

552

consistent with Eq.(32) qualitatively. Fig. 16 validates the applicability of Eq.(32) in

553

cement with experiment data. Based on the salinity of pore solution in cement, the

554

parameters of Eq.(32) we used are: λD = 9.6 nm , l * = 0.83 and b = 0, 0.79, 2.4

555

respectively. When the zeta potential is 0 mV, it means that we do not consider the

556

electrical double layer (EDL) effect and again the diffusivity is overestimated by the

557

predictive results without EDL. However, the prediction by Eq.(32) shows a better

558

agreement with experiment data if the EDL effect considered. The good agreements

559

prove the good performance of the proposed predictive formula again.

560 561 562 563 564

Fig. 15 Relationship between effective diffusivities and bulk concentrations. The lines are analytic predictive results by Eq.(32) and symbols are results from numerical simulations. The Zeta potential is -10 mV and the detailed parameters of cement microstructure is φcap = 0.8 , l* = 27.4 .

AC C

EP

TE D

M AN U

SC

548

26

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

Fig. 16 The comparison of results by Eq.(32) and experiment in different porosities. The lines are analytic predictive results by Eq.(32) and symbols are results from experiment. The Zeta potential for the solid line is zero, which represents the diffusion without electrokinetic effect. The Zeta potential of short dash line is 20 mV and that of long dash line is -60 mV.

570

Even though Eq. (32) is finalized through numerical pore-scale modeling, it may

571

provide predictions of different parameters more quickly, efficiently and deeply. For

572

instance, Fig. 14 tells simply that any negatively charged surface will decrease the

573

diffusivity of negatively charged ion, such as chloride ions; it also tells that the effective

574

ion diffusivity decreases in a super-linear way with the pore size ( l * ). Besides, Fig. 14

575

and Eq.(32) may also deduce strategies to reduce chloride ion corrosion of cement paste:

576

(i) reduce the porosity and enhance the compactness; (ii) decrease the pore size in cement

577

paste as much as possible; (iii) increase the Zeta potential of cement solid surfaces if

578

possible.

579

7. Conclusions

EP

AC C

580

TE D

565 566 567 568 569

In this paper, we have established a pore-scale numerical framework to study the ion

581

diffusion with electrokinetic effects in porous microstructures of cements. We developed

582

an algorithm to reconstruct three-dimensional microstructures of saturated cements,

583

whose results agreed well with the Power's law. The transport governing equations were

584

then solved by a high-efficiency coupled LBM code. The numerical results from our

585

pore-scale framework have been validated by comparisons with the available 27

ACCEPTED MANUSCRIPT

experimental data and theoretical solutions. The present modeling results show that the

587

electrokinetic effect is as a function of the most probable pore size and Zeta potential.

588

The electrokinetic effect on ionic diffusion in the cement declines when the most

589

probable pore size increases and negative Zeta potential decreases. The cationic diffusion

590

is more sensitive to the electrokinetic effect than anionic. If pore size is much larger than

591

the Debye length (i.e. l * > 32 ) in the cement-based materials, the electrokinetic effect on

592

the chloride ion diffusion is negligible. By up-scaling analysis, we proposed a predictive

593

formula for the effective chloride ion diffusivity with electrokinetic effect in porous

594

cement paste. To reduce the chloride ion diffusion and corrosion, engineers can design a

595

cement-based microstructure with the smaller pore size and higher negative Zeta

596

potential as much as possible. The present results may improve the understanding of ionic

597

diffusion in cement microstructure.

598 599 600 601 602

This work is financially supported by the NSF grant of China (No. 51676107, 51176089) and National Science and Technology Major Project on Oil and Gas (No.2017ZX05013001).

M AN U

AC C

EP

TE D

Acknowledgements

SC

RI PT

586

28

ACCEPTED MANUSCRIPT

Appendix A

603 604

The subsequent formula derivation is based on the following conditions or

605

assumptions: (i) the capillary pores are accepted as equivalent cylinders; (ii) the Debye-

606

Hückel approximation is tenable.

608

In accordance with Eq.(28), the ionic effective diffusivities in cement paste is given

RI PT

607 as:

609

Deffi = Diporeφeffn ,

610

i where Dpore is the i ionic effective diffusivity (i is Na + or Cl - ) matching with the

611

sodium chloride diffusivity of a pure solution system. φeff is the effective porosity of the

612

cement microstructure, described as following relationship: φeff Seff = , φcap Scap

SC

(A.2)

M AN U

613

(A.1)

where φcap , Scap is the true capillary porosity and cross-sectional area. Seff is the

615

effective cross-sectional area of capillary pores available for ionic diffusion. For the

616

computing of the effective cross-sectional, we postulate total concentration of the cross-

617

section in the charged cylindrical capillary pores is equal to that in the uncharged

618

capillary pores, whose cross-section is the effective cross-section for ionic diffusion,

619 620

regressed as:

TE D

614

i Ctotal =

∫∫ r ⋅ C ( r )drdθ = ∫∫ C i

S cap

i ,∞

ds ,

(A.3)

S eff

621

i is the total concentration of the cross-section and Ci ( r ) satisfies the where Ctotal

622

Boltzmann distribution:

625 626

ez ψ ( r )   Ci ( r ) = Ci ,∞ exp  −ηs i , k BT  

EP

624

(A.4)

where η s is the pores’ electrical potential correction factor in porous media. Based on

AC C

623

the Debye-Hückel approximation, ψ ( r ) is roughly written as [76]:

ψ ( r ) = ζ e−( β R−r )/ λ , s

D

(A.5)

627

where β s is correction factor of effective pores’ radius in the porous media. Summing

628

Eqs.(A.4) and (A.5) with (A.3) results in: βs R= d / 2   ez ζ 2π ∫ r ⋅ Ci ,∞ exp  −η s i e − ( R − r )/ λD  dr = Seff Ci ,∞ . k BT   0

629

29

(A.6)

ACCEPTED MANUSCRIPT

The left integration term is extremely challenging to get the analytic solution. Hence, the

631

Taylor expansion of the term at zero within the integration on the left-hand side of

632

Eq.(A.6) is performed as follows, taking to the second order   ez ζ ηb r ⋅ exp  −η s i e − ( βs R − r )/ λD   e −ηs b exp( − βs R / λD ) ⋅ r − s e −ηs b exp( − βs R / λD ) − βs R / λD ⋅ r 2 . (A.7) λD k BT  

633 634 635

RI PT

630

Substituting Eq.(A.7) into Eq.(A.6) and integrating gives: − β sl* /2  2 * − β s l * /2  Seff / Scap = Seff / (π ( β s R ) 2 ) = e −ηsb⋅e  1 − η s bβ s l e ,  3 

(A.8)

where l * is the dimensionless pore size. For l * > 10 , l * exp(−l * / 2)  1 , and Eq.(A.8)

637

can be simplified as: S eff / S cap = e −ηs b⋅exp( − β s l

638

640 641

/ 2)

.

(A.9)

Therefore, the effective porosity of the cement microstructure with the charged surface is: S * (A.10) φeff = eff φcap = exp( −η s b ⋅ e − β s l / 2 )φcap , S cap

M AN U

639

*

SC

636

then cast Eq.(28) in term of the correction factor:

642

Deff = D0 (ω e ⋅ φcap ) ,

(A.11)

643

ωe = exp(−ηs b ⋅ e− β l /2 ) ,

(A.12)

*

EP

TE D

s

AC C

644

n

30

ACCEPTED MANUSCRIPT

References

EP

TE D

M AN U

SC

RI PT

[1] Obrecht C, Kuznik F, Tourancheau B, Roux J-J. A new approach to the lattice Boltzmann method for graphics processing units. Computers & Mathematics with Applications. 2011;61(12):3628-38. [2] Manera M, Vennesland O, Bertolini L. Chloride threshold for rebar corrosion in concrete with addition of silica fume. Corros Sci. 2008;50(2):554-60. [3] Huet B, L’Hostis V, Miserque F, Idrissi H. Electrochemical behavior of mild steel in concrete: Influence of pH and carbonate content of concrete pore solution. Electrochim Acta. 2005;51(1):172-80. [4] Thomas M. Chloride thresholds in marine concrete. Cem Concr Res. 1996;26(4):513-9. [5] Shin CB, Kim EK. Modeling of chloride ion ingress in coastal concrete. Cem Concr Res. 2002;32(5):757-62. [6] Perko J, Patel RA. Single-relaxation-time lattice Boltzmann scheme for advection-diffusion problems with large diffusion-coefficient heterogeneities and high-advection transport. Physical Review E. 2014;89(5):053309. [7] Bentz D P GEJ, Lagergren E S. Multi-scale microstructural modelling of concrete diffusivity: Identification of significant variables. Cement Concrete and Aggregates. 1998;20(1):129-39. [8] Boddy A, Bentz E, Thomas MDA, Hooton RD. An overview and sensitivity study of a multimechanistic chloride transport model. Cem Concr Res. 1999;29(6):827-37. [9] Hausmann D. Steel corrosion in concrete--How does it occur? Materials Protection. 1967;6(11):19-23. [10] Bentz DP, Jensen OM, Coats AM, Glasser FP. Influence of silica fume on diffusivity in cement-based materials: I. Experimental and computer modeling studies on cement pastes. Cem Concr Res. 2000;30(6):953-62. [11] Friedmann H, Amiri O, Aı̈ t-Mokhtar A, Dumargue P. A direct method for determining chloride diffusion coefficient by using migration test. Cem Concr Res. 2004;34(11):1967-73. [12] Krabbenhøft K, Krabbenhøft J. Application of the Poisson–Nernst–Planck equations to the migration test. Cem Concr Res. 2008;38(1):77-88. [13] Bentz DP, Garboczi EJ. Modelling the leaching of calcium hydroxide from cement paste: effects on pore space percolation and diffusivity. Mater Struct. 1992;25(9):523-33. [14] Zhang MZ, Ye G, van Breugel K. A numerical-statistical approach to determining the representative elementary volume (REV) of cement paste for measuring diffusivity. Materiales de Construcción. 2010;60(300):14. [15] Ukrainczyk N, Koenders EAB. Representative elementary volumes for 3D modeling of mass transport in cementitious materials. Modell Simul Mater Sci Eng. 2014;22(3):035001. [16] Liu L, Chen H, Sun W, Ye G. Microstructure-based modeling of the diffusivity of cement paste with micro-cracks. Construction and Building Materials. 2013;38(0):1107-16. [17] Zhang M, Ye G, van Breugel K. Modeling of ionic diffusivity in non-saturated cement-based materials using lattice Boltzmann method. Cem Concr Res. 2012;42(11):1524-33. [18] Revil A, Leroy P. Constitutive equations for ionic transport in porous shales. Journal of Geophysical Research: Solid Earth (1978–2012). 2004;109(B3):208. [19] Johannesson B. Development of a Generalized Version of the Poisson– Nernst–Planck Equations Using the Hybrid Mixture Theory: Presentation of 2D Numerical Examples. Transp Porous Media. 2010;85(2):565-92. [20] Li D. Electrokinetics in microfluidics, Academic. Oxford; 2004. [21] Nguyen PT, Amiri O. Study of electrical double layer effect on chloride transport in unsaturated concrete. Construction and Building Materials. 2014;50(0):492-8. [22] Arnold J, Kosson DS, Garrabrants A, Meeussen JCL, van der Sloot HA. Solution of the nonlinear Poisson–Boltzmann equation: Application to ionic diffusion in cementitious materials. Cem Concr Res. 2013;44:8-17. [23] Chatterji S. Colloid Electrochemistry of Saturated Cement Paste and Some Properties of Cement Based Materials. Advanced Cement Based Materials. 1998;7(3–4):102-8. [24] Monteilhet L, Korb JP, Mitchell J, McDonald PJ. Observation of exchange of micropore water in cement pastes by two-dimensional T2 -T2 nuclear magnetic resonance relaxometry. Physical Review E. 2006;74(6):061404. [25] Korb JP, Monteilhet L, McDonald PJ, Mitchell J. Microstructure and texture of hydrated cement-based materials: A proton field cycling relaxometry approach. Cem Concr Res. 2007;37(3):295-302. [26] Huber C, Chopard B, Manga M. A lattice Boltzmann model for coupled diffusion. Journal of

AC C

645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699

31

ACCEPTED MANUSCRIPT

EP

TE D

M AN U

SC

RI PT

Computational Physics. 2010;229(20):7956-76. [27] Garboczi EJ, Bentz DP. Computer simulation of the diffusivity of cement-based materials. Journal of Materials Science. 1992;27(8):2083-92. [28] Wang M, Wang J, Pan N, Chen S. Mesoscopic predictions of the effective thermal conductivity for microscale random porous media. Physical Review E. 2007;75(3):036702. [29] Feldman RF. The flow of helium into the interlayer spaces of hydrated portland cement paste. Cem Concr Res. 1971;1(3):285-300. [30] Yip CK, van Deventer JSJ. Microanalysis of calcium silicate hydrate gel formed within a geopolymeric binder. Journal of Materials Science. 2003;38(18):3851-60. [31] Li H, Fratini E, Chiang W-S, Baglioni P, Mamontov E, Chen S-H. Dynamic behavior of hydration water in calcium-silicate-hydrate gel: A quasielastic neutron scattering spectroscopy investigation. Physical Review E. 2012;86(6):061505. [32] Bhattacharya S, Gubbins KE. Fast Method for Computing Pore Size Distributions of Model Materials. Langmuir. 2006;22(18):7726-31. [33] Powers TC, Brownyard TL. Studies of the Physical Properties of Hardened Portland Cement Paste. Journal Proceedings. 1946;43(9): 249-336. [34] Hansen TC. Physical structure of hardened cement paste. A classical approach. Mater Struct. 1986;19(6):423-36. [35] Collepardi M, Marcialis A, Turriziani R. Penetration of chloride ions into cement pastes and concretes. J Am Ceram Soc. 1972;55(10):534-5. [36] Adler P. Porous media: geometry and transports: Elsevier; 1992. [37] Wang J, Wang M, Li Z. Lattice Poisson–Boltzmann simulations of electro-osmotic flows in microchannels. J Colloid Interface Sci. 2006;296(2):729-36. [38] Wang M, Chen SY. On applicability of Poisson-Boltzmann equation for micro- and nanoscale electroosmotic flows. Communications in Computational Physics. 2008;3(5):1087-99. [39] Levich VGe, Tobias CW. Physicochemical Hydrodynamics. J Electrochem Soc. 1963;110(11):251C2C. [40] Leroy P, Revil A. A triple-layer model of the surface electrochemical properties of clay minerals. J Colloid Interface Sci. 2004;270(2):371-80. [41] Iler RK. The chemistry of silica: solubility, polymerization, colloid and surface pro perties, and biochemistry: Wiley; 1979. [42] Wang M, Chen S. On applicability of Poisson–Boltzmann equation for micro-and nanoscale electroosmotic flows. Commun Comput Phys. 2008;3(5):1087-99. [43] Wang M. Structure Effects on Electro-Osmosis in Microporous Media. J Heat Transfer. 2012;134(5):051020. [44] Wang M, Chen S. Electroosmosis in homogeneously charged micro- and nanoscale random porous media. J Colloid Interface Sci. 2007;314(1):264-73. [45] Wang M, Pan N. Predictions of effective physical properties of complex multiphase materials. Materials Science and Engineering: R: Reports. 2008;63(1):1-30. [46] Liu Q-f, Li L-y, Easterbrook D, Yang J. Multi-phase modelling of ionic transport in concrete when subjected to an externally applied electric field. Eng Struct. 2012;42:201-13. [47] Zalzale M, McDonald PJ. Lattice Boltzmann simulations of the permeability and capillary adsorption of cement model microstructures. Cem Concr Res. 2012;42(12):1601-10. [48] Zalzale M, McDonald PJ, Scrivener KL. A 3D lattice Boltzmann effective media study: understanding the role of C-S-H and water saturation on the permeability of cement paste. Modell Simul Mater Sci Eng. 2013;21(8):085016. [49] Wang M, Kang Q. Electrokinetic transport in microchannels with random roughness. Anal Chem. 2009;81(8):2953-61. [50] Wang M, Kang Q. Modeling electrokinetic flows in microchannels using coupled lattice Boltzmann methods. Journal of Computational Physics. 2010;229(3):728-44. [51] Yoshida H, Kinjo T, Washizu H. Coupled lattice Boltzmann method for simulating electrokinetic flows: A localized scheme for the Nernst–Plank model. Communications in Nonlinear Science and Numerical Simulation. 2014;19(10):3570-90. [52] Flekkøy EG. Lattice Bhatnagar-Gross-Krook models for miscible fluids. Physical Review E. 1993;47(6):4247-57. [53] Wang J, Wang M, Li Z. Lattice evolution solution for the nonlinear Poisson–Boltzmann equation in

AC C

700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755

32

ACCEPTED MANUSCRIPT

EP

TE D

M AN U

SC

RI PT

confined domains. Communications in Nonlinear Science and Numerical Simulation. 2008;13(3):575-83. [54] Yoshida H, Nagaoka M. Multiple-relaxation-time lattice Boltzmann model for the convection and anisotropic diffusion equation. Journal of Computational Physics. 2010;229(20):7774-95. [55] Duncan JS. Introduction to colloid and surface chemistry. Elsevier Science Ltd; 1992. [56] Garboczi EJ, Bentz DP. The effect of statistical fluctuation, finite size error, and digital resolution on the phase percolation and transport properties of the NIST cement hydration model. Cem Concr Res. 2001;31(10):1501-14. [57] Bentz DP, Garboczi EJ. Percolation of phases in a three-dimensional cement paste microstructural model. Cem Concr Res. 1991;21(2):325-44. [58] Pivonka P, Hellmich C, Smith D. Microscopic effects on chloride diffusivity of cement pastes—a scale-transition analysis. Cem Concr Res. 2004;34(12):2251-60. [59] Liu L, Sun W, Ye G, Chen H, Qian Z. Estimation of the ionic diffusivity of virtual cement paste by random walk algorithm. Construction and Building Materials. 2012;28(1):405-13. [60] Ngala VT, Page CL, Parrott LJ, Yu SW. Diffusion in cementitious materials: II, further investigations of chloride and oxygen diffusion in well-cured OPC and OPC/30%PFA pastes. Cem Concr Res. 1995;25(4):819-26. [61] Tumidajski PJ, Schumacher AS, Perron S, Gu P, Beaudoin JJ. On the relationship between porosity and electrical resistivity in cementitious systems. Cem Concr Res. 1996;26(4):539-44. [62] Stanish K, Hooton R, Thomas M. Testing the chloride penetration resistance of concrete: a literature review 1997. [63] Page C, Short N, El Tarras A. Diffusion of chloride ions in hardened cement pastes. Cem Concr Res. 1981;11(3):395-406. [64] Bentz DP. A virtual rapid chloride permeability test. Cem Concr Compos. 2007;29(10):723-31. [65] Oh BH, Jang SY. Prediction of diffusivity of concrete based on simple analytic equations. Cem Concr Res. 2004;34(3):463-80. [66] Sun G-w, Sun W, Zhang Y-s, Liu Z-y. Relationship between chloride diffusivity and pore structure of hardened cement paste. J Zhejiang Univ Sci A. 2011;12(5):360-7. [67] Schmidt M, Fehling E. Ultra-high-performance concrete: Research, development and application in Europe. ACI Special Publication. 2005;228:51-78. [68] Gallucci E, Scrivener K, Groso A, Stampanoni M, Margaritondo G. 3D experimental investigation of the microstructure of cement pastes using synchrotron X-ray microtomography (µCT). Cem Concr Res. 2007;37(3):360-8. [69] Chatterji S, Kawamura M. Electrical double layer, ion transport and reactions in hardened cement paste. Cem Concr Res. 1992;22(5):774-82. [70] Popov K, Glazkova I, Myagkov S, Petrov A, Sedykh E, Bannykh L, et al. Zeta-potential of concrete in presence of chelating agents. Colloids Surf Physicochem Eng Aspects. 2007;299(1-3):198-202. [71] Jensen MM. A Coupled Transport and Chem-ical Model for Durability Pre-dictions of Cement Based Ma-terials: Technical University of Denmark; 2014. [72] Wang X-Y, Lee H-S. Modeling of chloride diffusion in concrete containing low-calcium fly ash. Mater Chem Phys. 2013;138(2–3):917-28. [73] Garboczi EJ. Permeability, diffusivity, and microstructural parameters: A critical review. Cem Concr Res. 1990;20(4):591-601. [74] van Brakel J, Heertjes PM. Analysis of diffusion in macroporous media in terms of a porosity, a tortuosity and a constrictivity factor. Int J Heat Mass Transfer. 1974;17(9):1093-103. [75] Chatterji S. Transportation of ions through cement based materials. Part 3 experimental evidence for the basic equations and some important deductions. Cem Concr Res. 1994;24(7):1229-36. [76] Probstein RF. Physicochemical hydrodynamics: an introduction: John Wiley & Sons; 2005.

AC C

756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803

33