Dielectric properties of the pore solution in cement-based materials

Journal of Molecular Liquids 302 (2020) 112548

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Dielectric properties of the pore solution in cement-based materials Tulio Honorio a, * , Thierry Bore b , Farid Benboudjema a , Eric Vourc’h c , Mehdi Ferhat c Université Paris-Saclay, ENS Paris-Saclay, CNRS, LMT - Laboratoire de Mécanique et Technologie, 94235, Cachan, France b School of Civil Engineering, The University of Queensland, Brisbane, Australia c SATIE, UMR CNRS 8029, ENS Paris Saclay, Université Paris Saclay, Cachan, France

A R T I C L E

I N F O

Article history: Received 1 November 2019 Received in revised form 20 December 2019 Accepted 20 January 2020 Available online 24 January 2020 Keywords: Electromagnetic properties Molecular dynamics Porous materials Specific ion effects

A B S T R A C T Composition changes in the pore solution affect significantly the dielectric response of porous materials? The presence of ions is recognized to affect the dielectric response of electrolytes in a complex way that depends on the ion-water and ion-ion interactions and respective concentrations. Pore solutions in cement-based materials are complex and age-dependent exhibiting different ion-pair states, which make them an interesting candidate for the study of the complex pore solution found in geomaterials in general. In cement-based materials, the measurement of dielectric properties is used for non-destructive monitoring and assessment of concrete structures conditions and to unravel details of the hierarchical pore structure of the material, notably at early-age when the microstructure of the material changes significantly. Here, we study the dielectric properties of bulk aqueous solutions representative of the pore solution found in cement-based materials using molecular dynamics simulations. Broadband dielectric spectra and frequency-dependent conductivities are provided. We show that the changes in the ionic composition of the pore solution due to the aging of the material translate in significant changes in the dielectric response: the static dielectric constant varies from 55.2 to 73.9. A mean-field upscaling strategy is deployed to compute the dielectric response at the cement-paste scale. Our results can be used in the interpretation of high frequency electromagnetic methods, such as ground-penetrating radar, enhancing the quality of interpretation and improving the confidence in the corresponding results. © 2020 Elsevier B.V. All rights reserved.

1. Introduction High Frequency Electromagnetic (HF-EM) method [1-3], as well as man-made geomaterials such as cementitious materials (e.g. [4-8]) have been deployed in the last decades to assess water content of soil and rocks. Among these methods, one can quote L-band remote sensing [9], Ground Penetrating Radar (GPR) [10], capacitive probe [11], etc. The success of such methods is caused by the dipolar character of the water molecules resulting in a high permittivity in comparison to other phases such as solid particles or gas voids. However, evidence suggests that the frequency-dependent dielectric permittivity contains far more information than water content only: structure [12], density [13], mineralogy, cohesion forces on the pore solution [14] and even material strength may leave distinct signatures that could potentially be quantified. Nevertheless, despite its great success, HF-EM methods are still confronted with theoretical challenges. Interaction of the pore solutions and solid phases leads

* Corresponding author. E-mail address: [email protected] (T. Honorio).

https://doi.org/10.1016/j.molliq.2020.112548 0167-7322/© 2020 Elsevier B.V. All rights reserved.

to strong contributions to the EM properties. These interactions are not well understood at the moment: theoretical studies are urgently needed. As for other physical properties [15-18], micromechanics can be applied to upscale the dielectric permittivity of heterogeneous materials [19,20]. However, there is a lack of important information concerning the behavior of the constituent phases, and it is this information that is required to provide reliable predictions of the effective dielectric permittivity of nanoporous materials. Dielectric properties are recognized to be dependent on pore size [21,22] and pore solution composition [23-25]. The relative static dielectric permittivity 4(0) of aqueous solutions are in general one order of magnitude larger (e.g. for bulk water is [26]4(0)=80.103 at 20◦ C ) than the dielectric permittivity of solid phases. Therefore, the precise quantification of the dielectric response of the pore solution is crucial to upscale the dielectric properties and to interpret the experimental spectra of porous materials. In cement-based materials, the composition of the pore solution is crucial to the stability and assemblage of phases in these materials [27]. Therefore, cement hydration processes and chemical alterations of cement-based materials (due, for instance, to durability

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T. Honorio, T. Bore, F. Benboudjema, et al. / Journal of Molecular Liquids 302 (2020) 112548

issues) are closely related to the pore solution composition. The composition of the pore solution depends on the composition of the binder (e.g. ordinary Portland cement, alternative cements, and Supplementary Cementitious Materials (SCM)), age and temperature [27-29]. Atomistic simulations have been successfully used to unveil the details of water interaction in cement-based materials, specially in calcium silicate hydrates (e.g.[30-35]). Recently, the authors have studied for the first time the dynamics and structure of bulk aqueous solutions representative of the pore solution found in cementbased materials through molecular dynamics simulations [36]. The repercussions of the dynamics and structure differences observed in these pore solution on the dielectric response are still to be quantified. Here, the dielectric properties of the pore solution of cementbased materials are computed taking into account the variability of ionic concentrations in aqueous solutions. To do so, we perform MD simulations to compute the dielectric response of the pore solutions. The composition of the pore solutions, atomic configurations and force fields, which are recalled in this article, are taken from Honorio et al. [36]. For the first time, the dielectric spectra of cement-based materials pore solutions are computed accounting for the age-dependent composition variations. We observe that the Cole-Cole equation fits well the dielectric spectra obtained for all pore solutions. Finally, we quantify the effect of the variability of the dielectric response of the pore solution on the effective response of cement-based materials using Monte Carlo Micromechanics [37]. Our results contribute to a better understanding of the electromagnetic behavior of cement-based materials and can be readily used in the interpretation of dielectric probing of cement-based materials, enhancing the performance of HF-EM testing and improving the confidence in the corresponding results.

frequency-dependent ionic conductivity s(f) = s  (f) − is  (f) by  f) JI ( f ) = s( f )40 E(

(4)

The frequency-dependent ionic conductivity can be computed using the cross correlations [24,40]: s( f ) =

1 3 VkB T



∞ 0

  ˙ dt e−2pift JI (0).P(t)

Following Rinne et al. [23,24], we define the following autoand cross-correlations functions of the water polarization and ionic current:  0W (t) =

 0IW (t) =

0I (t) =



PW (0).PW (t)



PW (0).JI (t) − JI (0).PW (t)

3 VkB T40   JI (0).JI (t)

3 VkB T40

Dw( f ) = wW ( f ) + wIW ( f ) + DwI ( f )

 wW ( f ) = 0W (0) − i2pf

For a system with n particles i with charge qi , the total system polarization P is defined as the sum of the polarization (or dipole moment) li (t) of each particle i at time t [38]:

wIW ( f ) = −2



DwI ( f ) = −  = P(t)

i=1

li (t) =

qi ri (t)

(1)

i=1

∞

i 2pf



∞ 0

 f ) = w( f )40 E(  f) P(

sIW ( f ) = −i2pf 40



∞

   .P(t) ˙ dt e−2pift P(0)

(3)

0

e−2pift 0W (t)dt



 e−2pift − 1 0I (t)dt

s( f ) = sIW ( f ) + sI ( f )



where 40 is the vacuum permittivity. The dielectric susceptibility w(f) can be computed from the auto-correlation of equilibrium polarization fluctuations using [24,39]: 1 3 VkB T40

(8)

(9)

(10) (11) (12)

The frequency-dependent ionic conductivity can be decomposed into the two terms:

with

(2)

∞

e−2pift 0IW (t)dt

0

where ri (t) is the position of the (center of) particle i. The total system polarization P is related to the electric field E via the complex frequency-dependent dielectric susceptibility w(f) = w  (f) − iw  (f) by

w( f ) = −

(7)

from which the water, ion and ion-water interaction contributions on dielectric susceptibility and frequency-dependent ionic conductivity can be quantified. Accordingly, the regularized susceptibility Dw(f) can be decomposed into three separate contributions:

2.1. Dielectric response and conductivity from molecular simulations

n 

(6)

3 VkB T40

where

2. Theory and calculation

n 

(5)

 sI ( f ) = 40

∞ 0

∞ 0

e−2pift 0IW (t)dt

e−2pift 0I (t)dt

(13)

(14)

(15)

Therefore, the static conductivity can be computed via s( f = 0) = ∞ sI ( f = 0) = 40 0 0I (t)dt.

0

2.2. Molecular modeling of the pore solutions where V and T are the volume and temperature of the system, respec˙ is the time-derivative of tively; kB is the Boltzmann constant and P(t) the total polarization. For a salt aqueous solution, the total polarization is the sum of the contributions of the water PW and ionic PI polarizations: P = PW + PI . The ionic current JI is related to the ionic polarization by: JI (t) = ˙  P I (t). This ionic current JI is linked to the electric field E via the

2.2.1. Pore solutions in cement-based materials: variability and definition of scenarios Here, we study the bulk aqueous solutions mimicking the pore solutions in cement-based materials as in ref. [36]. One can legitimately raise the question of the relevance of analyzing bulk solutions to get insights on pore solution behavior. As previously

T. Honorio, T. Bore, F. Benboudjema, et al. / Journal of Molecular Liquids 302 (2020) 112548 Table 1 Composition of the simulated solutions (as in Honorio et al. [36] based on the experimental data provided by Lothenbach et al. [47]). Adjustments in the ionic concentration were made to ensure the electroneutrality of the system. Sim.

PC1 PC2 PC3 PC4 PC5 PC6 PC7

t [days]

H2 O

Na+

K+

Ca2+

SO2− 4

OH−

0.04 0.25 1 7 28 197 400

2767 2767 2767 2767 2767 2767 2767

4 4 5 9 9 12 16

20 20 20 24 27 29 30

1 1 0 0 0 0 0

8 9 0 0 1 2 2

10 8 25 33 34 37 42

discussed [36], the most significant part of the dynamics of water and ions in pore solution within cement-based materials can be described by the dynamics of the fluid portions less affected by pore walls. The ionic transport occurring in the pore zones affected by the interface (Stern layer) is reported to be negligible when compared to transport in the bulk pore, especially for aqueous solutions with a concentration similar to pore solutions in cement-based materials [41,42]. Furthermore, atomistic simulations of water confined in mesopores show that for a large portion of these pores the fluid behaves as a bulk fluid (e.g. [43]). The smaller pore size found in cement-based materials is the inter-layer calcium silicate hydrates (C-S-H) pores. The total interlayer length (silicate chains, calcium sheet plus interlayer pore) does not exceed 1.4 nm in general, which indicates an interlayer pore thickness of approximately 0.5 nm. CS-H gels exhibit also a mesoporosity ranging from few up to tens of nanometers. Specific techniques [21,22] are required to study the water confined in pore as small as C-S-H inter-layer pores. Regarding larger pores, though, molecular simulations of C-S-H and other mesoporous materials show that for pores as small as 5 nm, in a major portion of the pore the fluid shows bulk-like behavior [35,43,44]. These arguments show that a large portion of the fluids in cement-based materials porosity is expected to exhibit bulklike behavior. Furthermore, most HF-EM remote sensing method is carried out in a frequency range around 1 GHz. In this frequency range, the effect of free water (or bulk water) is predominant. It was shown by previous studies that adsorbed water (chemically or hydrogen-bond) will manifest at lower frequency [45,46]. Therefore,

3

critical physical insights that can be relevant to the understanding of the dielectric response of cement-based materials might be inferred from the analysis of bulk solutions representative of the pore solutions. The composition of the pore solutions is based on the experimental data provided by Lothenbach et al. [47] for an ordinary Portland cement (PC) systems in a formulation with a water-to-cement w/c ratio of 0.4. Seven scenarios PC1-PC7 corresponding to various ages are chosen to be simulated. The species with a concentration below 5 mM are not taken into account. Table 1 shows the compositions of the pore solutions studied here and the associated age in ordinary Portland cement (PC) systems. The ionic concentrations were adjusted to comply with the electroneutrality of the simulated system. No more than 5 ions (preferably potassium, hydroxides and sodium) were added or deleted per type. The resulting aqueous solution is within the variability in ionic composition experimentally observed in pore solution [27]. Fig. 1 shows the evolution of the concentration of the ionic species plotted as a function of the age. For each age, we present a snapshot of the atomic configurations in a system equilibrated in a canonical simulation at 300 K. 2.2.2. Force fields and simulation details A detailed description of the force field parameters is provided in a previous work [36]. The interactions among all the species in the system are modeled by a sum of non-bonded (van der Waals and electrostatic) and bonded (angle and bonds for OH groups) interactions. The van der Waals interactions are described either by the Lennard-Jones (12-6) potential for interactions between water (SPC/E [48]), hydroxide and monatomic ions or by the Buckingham potential for interactions involving sulfates. The force field parameters are provided as Supporting Information. Longrange van der Waals interactions are treated with tail corrections. Coulomb potential describes the electrostatic contribution. Ewald sum method is used to cope with long-range electrostatic interactions. Water and hydroxide bonds and water angles are constrained by SHAKE algorithm. Sulfate ions are constrained using RIGID algorithm. We use LAMMPS [49] to perform the simulations. Initial atomic configurations are obtained by randomly placing the ions in the simulation box following the composition in Table 1. Next, a microcanonical (NVE) simulation with an imposed maximum displacement of 0.01 Å per atom is performed for 1 ps to cope with possible

Fig. 1. Evolution of the concentrations of the main ionic species (concentrations exceeding 5 mM) in ordinary Portland cement (PC) pore solutions. The scenarios PC1–PC7 are retained (snapshots of atomic configurations at the bottom) to study the age-dependency of the pore solution composition.

4

T. Honorio, T. Bore, F. Benboudjema, et al. / Journal of Molecular Liquids 302 (2020) 112548

overlapping atoms. Then, the system is equilibrated for 0.25 ns at 300 K in a canonical simulation (NVT). Nosé-Hoover thermostat is used for equilibration and production with a damping parameter of 100 timesteps. The dipole moments of water and all ions are computed separately each 25 fs in an NVT simulation with a 1 fs timestep for 1.0 ns. As shown in the Supporting Information, 1 ns seems enough to sample the fluctuations of the water polarization and ionic currents. The effects of finite size and fictitious forces from Nosé-Hoover thermostat is discussed in the Supporting Information.

W2

W1

(a)

Fig. 2 (a)–(e) shows the water solvation shells of the ions present in the pore solutions PC1–PC7. As discussed previously [36], cationwater and hydroxide-water structuration is similar in all pore solutions studied; whereas sulfates-water pair exhibits structure variations according to the pore solution. We compute the ion pair states following contact ion pairs (CIP), single solvent-separated ion pairs (SIP) and doubly solvent-separated ion pairs (DSIP) configurations depicted in Fig. 2 (f). Ions that are separated by more than

PC1 K-Ow K-Ow PC2 PC2 K-Ow PC3 PC3 K-Ow PC4 PC4 K-Ow PC5 PC5 K-Ow PC6 PC6 K-Ow PC7 PC7

W3

4

g K,Ow (r)

3. Results and discussion

K-Ow 2

(f) CIP

SIP

DSIP

FI

0

PC1 PC2 PC3 PC4 PC5 PC6 PC7

W3

Na-Ow

2

6

7

8 Ca-Ow PC1

W2r[Å]

Ca-Ow PC2 PC2

Ca-Ow

0.2 0.1

PC 1 PC 2 PC 3 PC 4 PC 5 PC 6 PC 7

7

PC

5

6

4

(h) 3 Oh-Ow PC1 Oh-Ow PC2 PC2 Oh-Ow PC3 PC3 Oh-Ow PC4 PC4 Oh-Ow PC5 PC5 Oh-Ow PC6 PC6 Oh-Ow PC7 PC7

1

5

2

6 r[Å] W2

7

SO4-Ow

8 9 S-Ow PC1 S-Ow PC2 PC2 S-Ow PC5 PC5 S-Ow PC6 PC6 S-Ow PC7 PC7

0 2

3

4

5

6

7

8

0.3 0.2 0.1

9

(i)

PC 1 PC 2 PC 3 PC 4 PC 5 PC 6 PC 7

7

PC

6

PC

PC

3

0.5 CIP SIP

2

1

0

Ca-SO4 coordintation

4

W1

0.4

0

PC 3

Ca-Oh coordintation

2

(e)

CIP SIP DSIP

0.5

0

1

2

2

5

Oh-Ow

0.6 CIP SIP DSIP

PC

W3

6 4

9

4

W2

W1

7

PC

5 r[Å]

3

(d) 8

3

PC

1

K-SO4 coordintation

0

g Oh,Ow (r)

PC

6 2

g SO4,Ow (r)

0.3

0

1

8

CIP SIP DSIP

0.4

0

PC

10

1

PC

g Ca,Ow (r)

12

DSIP

2

4

W1

9

PC

5

3

4

SIP

3

PC

(c) 14

3

0.5

CIP

2

0 16 2

(g) 4

PC

4

2

g Na,Ow (r)

6

Na-SO4 coordintation

r[Å]

W2

W1

Na-Oh coordintation

8

K-Oh coordintation

(b)

0.4

CIP

0.3 0.2 0.1 0

PC1

PC2

r [Å] Fig. 2. Ion hydration shells: radial distribution functions of (a) K+ , (b) Na+ , (c) Ca2+ , (d) OH− and (e) SO2− 4 -Ow pairs. W1, W2 and W3 denotes 1, 2 and 3 hydration shells, respectively. Ion pair states: (f) schematic representation of contact ion pairs (CIP), single solvent-separated ion pairs (SIP), doubly solvent-separated ion pairs (DSIP), and free ions + + 2+ (FI) [24]. Anions coordinated by cations: OH− and SO2− . The ion-ion radial distribution functions used to compute these histograms 4 coordinated by (g) Na , (h) K , and (i) Ca are detailed in the Supporting Information.

T. Honorio, T. Bore, F. Benboudjema, et al. / Journal of Molecular Liquids 302 (2020) 112548

(a)

(b)

5

SPC/E

Empty symbols: (0) from MD Full symbols: (0) as a free parameter

(c) Full symbols:

Empty symbols:

(0) as a free parameter (0) from MD

SPC/E

PC1 PC2 PC3 PC4 PC5 PC6 PC7

Fig. 3. (a) Auto-correlation function of the water polarization 0W (t) of the pore solutions PC1–PC7 and SPC/E water. The inset shows Debye fits 0W (t) = AD exp[−t/tauD ], with AD = 0W (t = 0) being the value obtained from MD simulation, in a Log-plot. (b) 0W (t = 0) for pore solutions PC1–PC7 and SPC/E water: empty symbols are Debye fits with AD = 0W (t = 0) being the value obtained from MD simulation, full symbols are Debye fits using AD as a free parameter. (c) Characteristic relaxation time obtained from Debye fits: empty symbols are Debye fits with AD = 0W (t = 0) being the value obtained from MD simulation, full symbols are Debye fits using AD as a free parameter. In (b) and (c), dashed lines denote the mean value associated with bulk SPC/E water and the thin lines denote the standard deviation. The R2 were computed from the square of the Pearson’s correlation between the fit model and the simulation data; for all cases R2 ≥ 0.97.

three water molecules are considered free ions (FI). DSIP configurations are prevalent in most of the cation-hydroxide pairs while CIP configurations are prevalent in most of the cation-sulfate pairs. Three main stages can be identified regarding the ionic composition of the pore solution for a given cement system [36]:

• very early age up to setting (first hours): PC1 and PC2; • early-age related to property development (within the first days): PC3 and PC4; • late stage associated and the service life of cement-based materials: PC5 to PC7.

such classification has been useful in linking the dynamics of ions to the electrical properties of cement-based materials [37]. As shown in Fig. 2 (g) and (h), ion pair states seem to obey this classification in most cases. Fig. 3 (a) shows the auto-correlation function 0W (t) of the water polarization for all the pore solutions studied and for (pure) SPC/E water. This auto-correlation function is the most relevant for the computation of the dielectric spectra [24]. In all cases, exponential decay is observed. Due to noise, 0W (t) exhibits negative values for times exceeding 25 to 60 ps for all solutions. We fit 0W (t) with the expression 0W (t = 0) = Exp[−t/tD ], where tD is the characteristic relaxation time. The values of 0W (t = 0) and tD are

Fig. 4. (a) Cross-correlation function 0IW (t) of water polarization and ionic current for pore solutions PC1–PC7 and SPC/E water. The inset zooms in smaller timescales. (b) Auto-correlation function of the ionic current (0I (t)) for pore solutions PC1–PC7 and SPC/E water. The inset shows the the decay at longer timescales.

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T. Honorio, T. Bore, F. Benboudjema, et al. / Journal of Molecular Liquids 302 (2020) 112548

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Fig. 5. Dielectric spectra of the pore solutions: real (left) and imaginary (right) parts of the dielectric susceptibility: Dw(f) and Dw(f) , respectively. (a)–(b) very early-age, (c)–(d) early-age, and (e)–(f) late ages pore solutions. Full lines are Cole-Cole fits of the MD results. For comparison, the experimental spectrum obtained by Buchner et al. [50] is also shown (g)–(h).

T. Honorio, T. Bore, F. Benboudjema, et al. / Journal of Molecular Liquids 302 (2020) 112548

(a)

(b)

(c)

(d)

7

Fig. 6. Dielectric spectra of the pore solutions PC1–PC7: (a) real w IW (f) and (b) imaginary w IW (f) parts of the ion-water dielectric contribution; (c) real Dw I (f) and (d) imaginary Dw I (f) parts of the DC-conductivity dielectric ion contribution.

slower relaxation. Compared to the other simpler aqueous solutions, the pore solution shows persistent oscillations even for times above 1 ps.

plotted for each pore solution and for SPC/E water in Fig. 3 (b) and (c), respectively. No clear tendency is observed in the values for the pore solution concerning the three stages classification or the similarity with respect to SPC/E water values. Ion-water cross-correlation 0IW (t) and ion auto-correlation 0I (t) functions are presented in Fig. 4 (a) and (b), respectively. Both functions present a high-frequency oscillatory behavior with the oscillations being more strongly damped in the case of 0I (t). Strong dampening has been also observed in NaCl, NaI, NaBr and NaF aqueous solutions [24]. The ion auto-correlation functions 0I (t) of the pore solution at the very early-age (PC1 and PC2) show an oscillatory behavior with lower amplitude when compared to the other solutions; the larger amplitudes are observed with the solution at later stages (PC5–PC7). As observed to other aqueous solutions (NaCl, NaI, NaBr and NaF), ion-water cross-correlation function 0IW (t) presents an initial steep increase followed by an initial decay and a subsequent

3.1. Dielectric spectra The dielectric spectra of the pore solutions and SPC/E including all contributions are shown in Fig. 5 together with the respective ColeCole fits. The water contribution is predominant (see the Supporting Information for the results without ion contributions). The ion-water dielectric contributions w IW (f) on the dielectric spectra of the pore solutions are shown in Fig. 6 (a) and (b), respectively. The real and the imaginary parts of the DC-conductivity dielectric ion contribution Dw I (f) for all pore solution are shown in Fig. 6 (c) and (d), respectively.

Table 2 Cole-Cole fit parameters from MD simulations of pore solution and SPC/E water, and data from the literature on water.

PC1 PC2 PC3 PC4 PC5 PC6 PC7 SPC/E (This work) SPC/E (Buchner et al. [50] experiments) SPC/E (Rinne et al. [24] MD simulations)

4CC

tCC

a

73.6 ± 2.9 67.1 ± 2.6 73.9 ± 3.2 60.7 ± 2.7 56.0 ± 2.1 55.2 ± 2.4 60.4 ± 3.0 68.2 ± 2.7 78.4 69.9

9.97 ± 0.97 10.26± 0.97 12.55 ± 1.33 9.33 ±1.14 9.80 ± 0.96 9.33 ± 1.27 10.30 ± 1.52 8.35 ± 0.94 8.27 10.72

0.000± 0.060 0.009 ± 0.060 0.000± 0.061 0.000± 0.050 0.040 ± 0.056 0.098 ± 0.055 0.052 ± 0.063 0.033 ± 0.055 0.000 0.014

8

T. Honorio, T. Bore, F. Benboudjema, et al. / Journal of Molecular Liquids 302 (2020) 112548

To fit the normalized dielectric spectrum we use Cole-Cole model [24,51]:

Dw( f ) =

4CC − 4∞ 1 + (i2pf tCC )1−a

SPC/E + 4∞ − 1

(16)

where 4CC and tCC are, respectively, the Cole-Cole amplitude and characteristic relaxation time; and, a is an exponent parameter ranging from 0 to 1 (a=0 corresponding to Debye model). We adopt 4∞ =1 since the force fields employed in MD simulation do not comprise atomic polarization. To fit simultaneously the real and imaginary parts of the dielectric spectrum, we minimize the function Dwi ( f )Dw¯i ( f ), where Dw i (f) refers to the spectrum data to be fitted and ¯. denotes the complex conjugate. The parameters of Cole-Cole fits are gathered in Table 2. Water simulation and experimental data from the literature are also provided. The static dielectric permittivity of SPC/E water is known to be lower than the experimental value (e.g. [52]). We obtain 69.0 for SPC/E water, which is in agreement with other values from the literature (e.g. [53] 67 ± 10). In spite of its simplicity, the SPC/E model shows a remarkable agreement with experiments. Fig. 7 shows the parameters of Cole-Cole fits. From Fig. 1, clear trends are observed in the evolution of the ionic composition of pore solutions in cement-based materials: K+ , OH− and Na+ concentrations increase; Ca2+ concentration decreases and SO2− 4 concentration decreases at the very early-age then increases at late ages. It must be noted however that since the various species can induce different collective effects according to its concentration, we do not necessarily expect the dielectric response to follow a very clear trend. In Fig. 7 the Cole-Cole static dielectric constant 4CC remains close to that of bulk water at the early-ages (PC1–PC3) and then tends to present values lower than that of SPC/E water. The Cole-Cole relaxation time tCC slightly exceeds that of SPC/E water and tends to remain constant (except for the case PC3). The a parameter of very early-age (PC1–PC2) and early-age (PC3–PC4) pore solutions are closer to 0 (i.e. a Debye-like behavior) than the values of late ages solutions (PC5–PC7). An increase in the a parameter is an indication of a symmetrical broadening of the dielectric losses relaxation peak [54]. We compare the frequency-dependent dielectric permittivity 4 = Dw + 1 of the pore solution obtained from MD simulation with the recent experimental results reported by Guihard et al. [55] on synthesized pore solution based on the composition of a CEM I concrete. Note that the experimental results are obtained in a narrower band (0.1–2 GHz) than MD results in Fig. 5. The authors point out that a Debye fit with a static dielectric constant 4D =80, 4∞ =3.13 and a relaxation time of 10 ps fit well their experimental data. In Fig. 8, we compare their Debye fits with the mean value computed from the dielectric spectra of MD simulation on PC1–PC7. A remarkable agreement is observed with respect to the imaginary part of the dielectric spectra. The difference between the experimental part can be attributed to the well-known deviation from the experimental value of the static dielectric constant SPC/E water (e.g. [52]). In both experimental and MD results, the real part remains almost constant for frequencies up, at least, to 2 GHz [55]. In agreement with our Cole-Cole fits that yield low a, Guihard et al. [55] observe that Debye fits can describe fairly well the dielectric spectra of the pore solution.

SPC/E

SPC/E

PC1 PC2 PC3 PC4 PC5 PC6 PC7 Fig. 7. Parameters of Cole-Cole fits from Fig. 5. The dashed lines denote the mean value associated with bulk SPC/E water and the thin lines denote the standard deviation. The R2 were computed from the square of the Pearson’s correlation between the fit model (normalization functions taken as the Dwi ( f )Dw¯i ( f )) and the simulation data; for all cases R2 ≥ 0.96.

static conductivity reported for ordinary PC systems with a water-tocement ratio of w/c =0.4, which can be described by a single master curved weighted by the w/c as discussed by Honorio et al. [37]. Note that this master curve and the associated variability of ±1 S/m, as depicted in the inset of Fig. 9, were obtained from the experimental data base on pore solutions of Vollprach et al. [27] on various cement systems. The static results in Fig. 9 are therefore consistent with the experimental evidence, being representative of cement systems with a w/c of 0.4. The conductivities of the pore solutions are similar to the static value up to frequencies on the order of 10 GHz. At higher frequencies both ion-ion and ion-water contributions exhibit oscillating responses with respect to the frequency.

3.3. Upscaling the effects of the variability pore solution dielectric response 3.2. Electrical conductivity Fig. 9 shows the real and imaginary parts of the frequencydependent conductivity of the pore solutions. The ion-ion and ionwater contributions are shown. Our results are in agreement with the

The homogenization of the dielectric permittivity and electrical conductivity is analogous to the homogenization of the thermal conductivity and diffusivity [20]. Bruggeman (or the self-consistent) scheme has been shown to provide a good estimate of the effective

T. Honorio, T. Bore, F. Benboudjema, et al. / Journal of Molecular Liquids 302 (2020) 112548

electrical conductivity of cement-pastes by capturing the transition from a liquid to a solid matrix during cement hydration [37]. In a polycrystalline-like morphology, for a N-phase heterogeneous materials with N isotropic equiaxed inclusions randomly distributed in representative elementary volume, the Bruggeman (or Self-Consistent) estimate of the effective electrical permittivity 4SC can be computed from the implicit formula [20]: N  r=1

fr

4r − 4SC =0 4r + 24SC

(17)

for all phases r with volume fraction fr and dielectric permittivity 4r . To upscale the effective permittivity 4 = Dw + 1 of the cement paste we use the Cole-Cole fits with the parameters in Table 2 to describe the dielectric permittivity of each pore solution 4PS . Powers cement hydration model [56] is used to estimate the evolution of

9

the capillary porosity (i.e. the volume fraction associated to the pore solution phase) in the system: fPS (n, w/c) = 00cap (w/c) − 1.32(1 − 00cap (w/c))n

(18)

where the degree of hydration n can be described by the sigmoid n = 0.9 [1 − Exp (−t/7)] for t in days as in ref. [37], and 00cap (w/c) = (w/c)/ (qw /qc + (w/c)) is the initial porosity computed from the w/c and densities of water and cement (qc and qw , respectively). It is worth to note that this approach does not take into account interface processes that will dominate the spectra around kHz and MHz range [45,46]. Nevertheless, as far as the authors know, it is the first attempt to use effective medium theory with realistic data on pore solution dielectric response for the modeling of HF-EM properties of cement paste. The real 4 and imaginary 4 parts of the effective permittivity of the cement paste with a water-to-cement ratio (w/c) of 0.4

(a)

(b)

(c)

(d)

(e)

(f)

PC6 PC7 PC4 PC5 PC3 PC1

PC2

Fig. 8. Frequency-dependent dielectric permittivity 4 = Dw + 1 of the pore solution: comparison of the mean value computed from dielectric spectra of PC1–PC7 with the experimental data from Guihard et al. [55] on pore solutions of concrete. The real 4 and imaginary 4 parts are shown. The standard deviation associated with the MD results are depicted by the light blue zones.

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T. Honorio, T. Bore, F. Benboudjema, et al. / Journal of Molecular Liquids 302 (2020) 112548

(a)

PC1

(b)

PC2

PC1

PC3

PC2 PC3 PC4 PC5

PC4 PC6 PC7

(c)

PC5

PC6 PC7

(d)

Fig. 9. Frequency-dependent conductivity of the pore solutions: real (left) and imaginary (right) parts of the dielectric susceptibility: (a)–(b) ion contribution, (c)–(d) ion-water contribution, and (e)–(f) the total electrical conductivity of the pore solutions. The inset in (e) shows the static part of the electrical conductivity compared with the master curve weighted with respect to the w/c of 0.4 (full line) and the corresponding variability of approximately 1 S/m (bounds defined by the dashed curves) as discussed by Honorio et al. [37].

are shown in Fig. 10. The frequency-dependence of both real and imaginary parts are very well pronounced in the range 0 to 1 GHz, especially at the early-ages (t < 10 days). The static values of the real part decrease with age. The imaginary part reaches its maximum at a frequency in the range 0.01 to 0.2 GHz and decreases with age. These results indicate that experimental measurements of effective permittivity can be used to discriminate against the age of the cement paste or, using cross-property relations, to follow property development at early-age. We compare our results concerning the effective static permittivity (presented in Fig. 10 (a)) with the experimental data of Andrade et al. [57] on water-saturated cement pastes with a w/c 0.4 (at 7 days of age) or 0.5 (for various ages) in Fig. 11. A remarkable good agreement is observed between the static values obtained from MD and the experimental data for a paste with w/c =0.4 at 7 days. Due to a dilution effect, cement pastes with higher w/c are expected to exhibit a lower dielectric response. Note however that the tendency of a decreasing effective static permittivity is observed in both experimental and model results.

4. Conclusions The dielectric spectra and frequency-dependent conductivities of bulk aqueous solutions representative of the pore solution found in cement-based materials were computed for the first time using

classical molecular dynamics simulations. Our main conclusions are as follows:

• Classical molecular dynamics simulations provide a powerful tool to compute the dielectric response of pore solutions with complex composition. Pore solutions in cement-based materials are cement type, age, and temperature-dependent. The comparison with experimental data shows that the technique is reliable and that the dielectric spectra and frequencydependent conductivities can be computed for various scenarios, which provide important input to the interpretation of the dielectric response of porous materials with complex fluid phase composition. • The age-dependent composition of the pore solutions is reflected in the dielectric spectra. In terms of static susceptibility values, the difference between the maximum (PC3) and minimum (PC6) is of approximately 20. The variability of the dielectric response associated with the composition of pore solution is, therefore, more significant that the variability can be expected from the other phases present in the material. This result shows that accounting for the composition variability of the pore is crucial to obtain the precise dielectric response of cement-based materials. An important implication of this result is that experimental campaigns should preferably focus on the characterization of pore solution composition variability than on the repartition of other phases in cement-based

T. Honorio, T. Bore, F. Benboudjema, et al. / Journal of Molecular Liquids 302 (2020) 112548

Fig. 10. Effective permittivity 4 = Dw + 1 = 4 − i4 of the cement paste with w/c=0.4. Time evolution of the (a) real 4 and (b) imaginary 4 parts at constant frequency f. Frequency-dependence of the (c) real 4 and (d) imaginary 4 parts at constant time.

materials. The results provided in this paper can readily be used to upscale the dielectric response of ordinary Portland cementbased materials. Approaches aiming at quantifying the effects of the stochastic nature of heterogeneous materials or uncertainty on the properties across scales can adopt the variability observed in the dielectric response as an input. Further studies are necessary, though, to quantify the dielectric response of highly confined water in the micropores found in C-S-H and AF-phases encountered in these materials. Again, molecular dynamics simulations can be a powerful tool in such an analysis. • Cole-Cole fits provide a reasonable description of the dielectric spectra of the pore solution studied. For pore solutions at the early ages (PC1–PC4), the Debye model seems sufficient to capture the real and imaginary parts of the spectrum (i.e.a parameter in the Cole-Cole model close to 0). As expected, the water contribution to the dielectric spectra is the most significant. The ion-ion and ion-water contributions are one to two orders lower than the water contribution. • The age-dependent dielectric response of the pore solution can be used to upscale the dielectric response of cement pastes using effective medium approaches. Both the static values of the real part and the peak in the imaginary part when plotted against the frequency decreases with the age. These results show that the measurement of the (effective) dielectric response can be deployed to discriminate against the age of cement pastes. Our results can be used in the interpretation of dielectric probing of cement-based materials, enhancing the performance of nondestructive monitoring of concrete structures conditions such as radar testing and improving the confidence in the corresponding results. Improvements in the detection of cracking and the diagnostic of structures pathologies (e.g. alkali-silica reactions, delayed ettringite formation) can be envisioned using HF-EM techniques. The next step will be focused on the modeling of the dielectric profile [22] of confined electrolytes in cement hydrates micro- and meso-pores. This will bring fundamental knowledge about the electromagnetic

11

Fig. 11. Effective static permittivity of a cement paste: comparison against the experimental data of Andrade et al. [57] on water-saturated cement paste with w/c (the authors report results on three samples for each age tested).

properties of confined (bound) water in cement-based materials micro- and meso-pores, which is still not understood. Thus, a more realistic model of HF-EM properties of cement-based material could then be developed. CRediT authorship contribution statement Tulio Honorio: Conceptualization, Methodology, Investigation, Formal analysis, Writing - original draft. Thierry Bore: Conceptualization, Methodology, Writing - original draft, Validation. Farid Benboudjema: Conceptualization, Methodology, Supervision. Eric Vourc’h: Conceptualization, Methodology, Writing - original draft, Supervision. Mehdi Ferhat: Conceptualization, Methodology.

Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.molliq.2020.112548.

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