# Dielectric properties of the pore solution in cement-based materials

Tulio Honorio  LMT, ENS Paris-Saclay, CNRS, Université Paris-Saclay, 94235
Cachan, France<EMAIL_ADDRESS>Thierry Bore
School of Civil Engineering, The University of Queensland, Brisbane, Australia
Farid Benboudjema  LMT, ENS Paris-Saclay, CNRS, Université Paris-Saclay,
Cachan, France  Eric Vourc’h  Mehdi Ferhat  SATIE, UMR CNRS 8029, ENS Paris
Saclay, Université Paris Saclay, Cachan, France

###### Abstract

Dielectric properties are intimately related to the content and composition of
the liquid phase present in micro- and mesoporous 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. 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. A mean field upscaling
strategy is deployed to compute the dielectric response at cement-paste scale.
In cement-based materials, the measurement of dielectric properties can be
used for non-destructive monitoring 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. 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.

††preprint: APS/123-QED

##  I Introduction

High Frequency Electromagnetic (HF-EM) method have been deployed last decades
to assess water content of soil and rocks [1, 2, 3], as well as man made
geomaterials such as cementitious materials (e.g. [4, 5, 6, 7, 8]). 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 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, 16, 17, 18], micromechanics can be
applied to upscale the dielectric permittivity of heterogeneous materials [19,
20]. However, important information concerning the behavior of the constituent
phases is required to provide reliable predictions of the effective dielectric
permittivity of nanoporous materials, is lacking. Dielectric properties are
recognized to be dependent on pore size [21, 22] and pore solution composition
[23, 24, 25]. The relative static dielectric permittivity $\epsilon(0)$ of
aqueous solutions are in general one order of magnitude larger (e.g. for bulk
water is [26] $\epsilon(0)$=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 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) a), age and temperature [28,
29, 27]. 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, 31, 32, 33, 34, 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 cement-based 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 cement-based 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.

##  II Theory and Calculation

###  II.1 Dielectric response and conductivity from molecular simulations

For a system with $n$ particles $i$ with charge $q_{i}$, the total system
polarization $\vec{P}$ is defined as the sum of the the polarization (or
dipole moment) $\vec{\mu_{i}}(t)$ of the each particle $i$ at time $t$ [38]:

$\vec{P}(t)=\sum_{i=1}^{n}\vec{\mu_{i}}(t)=\sum_{i=1}^{n}q_{i}\vec{r_{i}}(t)$
(1)

where $\vec{r_{i}}(t)$ is the position of the (center of) particle $i$.

The total system polarization $\vec{P}$ is related to the electric field
$\vec{E}$ via the complex frequency-dependent dielectric susceptibility
$\chi(f)=\chi^{\prime}(f)-i\chi^{\prime\prime}(f)$ by

$\vec{P}(f)=\chi(f)\epsilon_{0}\vec{E}(f)$ (2)

where $\epsilon_{0}$ is the vacuum permittivity. The dielectric susceptibility
$\chi(f)$ can be computed from the auto-correlation of equilibrium
polarization fluctuations using [39, 24]:

$\chi(f)=-\frac{1}{3Vk_{B}T\epsilon_{0}}\int^{\infty}_{0}e^{-2\pi
ift}\left\langle\vec{P}(0).\dot{\vec{P}}(t)\right\rangle dt$ (3)

where $V$ and $T$ are the volume and temperature of the system, respectively;
$k_{B}$ is the Boltzmann constant and $\dot{\vec{P}}(t)$ is the time-
derivative of the total polarization.

For a salt aqueous solution, the total polarization is the sum of the
contributions of the water $\vec{P}_{W}$ and ionic $\vec{P}_{I}$
polarizations: $\vec{P}=\vec{P}_{W}+\vec{P}_{I}$.

The ionic current $\vec{J}_{I}$ is related to the ionic polarization by:
$\vec{J}_{I}(t)=\dot{\vec{P}}_{I}(t)$. This ionic current $\vec{J}_{I}$ is
linked to the electric field $\vec{E}$ via the frequency-dependent ionic
conductivity $\sigma(f)=\sigma^{\prime}(f)-i\sigma^{\prime\prime}(f)$ by

$\vec{J}_{I}(f)=\sigma(f)\epsilon_{0}\vec{E}(f)$ (4)

The frequency-dependent ionic conductivity can be computed using the cross
correlations [40, 24]:

$\sigma(f)=\frac{1}{3Vk_{B}T}\int^{\infty}_{0}e^{-2\pi
ift}\left\langle\vec{J}_{I}(0).\dot{\vec{P}}(t)\right\rangle dt$ (5)

Following Rinne et al. [24, 23], we define the following auto- and cross-
correlations functions of the water polarization and ionic current:

$\phi_{W}(t)=\frac{\left\langle\vec{P}_{W}(0).\vec{P}_{W}(t)\right\rangle}{3Vk_{B}T\epsilon_{0}}$
(6)

$\phi_{IW}(t)=\frac{\left\langle\vec{P}_{W}(0).\vec{J}_{I}(t)-\vec{J}_{I}(0).\vec{P}_{W}(t)\right\rangle}{3Vk_{B}T\epsilon_{0}}$
(7)

$\phi_{I}(t)=\frac{\left\langle\vec{J}_{I}(0).\vec{J}_{I}(t)\right\rangle}{3Vk_{B}T\epsilon_{0}}$
(8)

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 $\Delta\chi(f)$ can be decomposed
into three separate contributions:

$\Delta\chi(f)=\chi_{W}(f)+\chi_{IW}(f)+\Delta\chi_{I}(f)$ (9)

where

$\chi_{W}(f)=\phi_{W}(0)-i2\pi f\int^{\infty}_{0}e^{-2\pi ift}\phi_{W}(t)dt$
(10)

$\chi_{IW}(f)=-2\int^{\infty}_{0}e^{-2\pi ift}\phi_{IW}(t)dt$ (11)

$\Delta\chi_{I}(f)=-\frac{i}{2\pi f}\int^{\infty}_{0}\left(e^{-2\pi
ift}-1\right)\phi_{I}(t)dt$ (12)

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

$\sigma(f)=\sigma_{IW}(f)+\sigma_{I}(f)$ (13)

with

$\sigma_{IW}(f)=-i2\pi f\epsilon_{0}\int^{\infty}_{0}e^{-2\pi
ift}\phi_{IW}(t)dt$ (14)

$\sigma_{I}(f)=\epsilon_{0}\int^{\infty}_{0}e^{-2\pi ift}\phi_{I}(t)dt$ (15)

Therefore, the static conductivity can be computed via
$\sigma(f=0)=\sigma_{I}(f=0)=\epsilon_{0}\int^{\infty}_{0}\phi_{I}(t)dt$.

###  II.2 Molecular modeling of the pore solutions

####  II.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 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 of approximately 0.95 nm [44],
but C-S-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, 45]. These arguments show that a large
portion of the fluids in cement-based materials porosity is expected to
exhibit bulk-like 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 for
lower frequency [46, 47]. Therefore, 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. [48] 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. Figure
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 300K.

Table 1:  Composition of the simulated solutions (as in Honorio et al. [36] based on the experimental data provided by Lothenbach et la. [48]). Adjustments in the ionic concentration were made to ensure the electroneutrality of the system.  Sim. | $t$ | H2O | Na+ | K+ | Ca2+ | SO${}_{4}^{2-}$ | OH-
---|---|---|---|---|---|---|---
| [days] |  |  |  |  |  |
PC1 | 0.04 | 2767 | 4 | 20 | 1 | 8 | 10
PC2 | 0.25 | 2767 | 4 | 20 | 1 | 9 | 8
PC3 | 1 | 2767 | 5 | 20 | 0 | 0 | 25
PC4 | 7 | 2767 | 9 | 24 | 0 | 0 | 33
PC5 | 28 | 2767 | 9 | 27 | 0 | 1 | 34
PC6 | 197 | 2767 | 12 | 29 | 0 | 2 | 37
PC7 | 400 | 2767 | 16 | 30 | 0 | 2 | 42
Figure 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.

####  II.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 [49]), hydroxide and monatomic ions or by the Buckingham
potential for interactions involving sulfates. The force field parameters are
provided as Supporting Information. Long-range 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 [50] to perform the simulations. Initial atomic configurations
were obtained by randomly placing the ions in the simulation box following the
composition in Tab.1. Next, a microcanonical (NVE) simulation with an imposed
maximum displacement of 0.01 Å per atom was performed during 1 ps to cope with
possible overlapping atoms. Then, the system was equilibrated during 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 were computed separately each 25 fs in an
NVT simulation with a 1 fs timestep during 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 are discussed in the Supporting
Information.

##  III Results and Discussion

Figure 2 (a)-(e) shows the water solvation shells of the ions present in the
pore solutions PC1-PC7. As discussed previously [36], cation-water and
hydroxide-water structuration are 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). 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 seems to obey this classification in most cases.

Figure 2: Ion hydration shells: radial distribution functions of (a) K+, (b)
Na+, (c) Ca2+, (d) OH- and (e) SO${}_{4}^{2-}$ -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 (FI)
[24]. Anions coordinated by cations: OH- and SO${}_{4}^{2-}$ coordinated by
(g) Na+, (h) K+, and (i) Ca2+. The ion-ion radial distribution functions used
to compute these histograms are detailed in the Supporting Information.

Figure 3 (a) shows the auto-correlation function $\phi_{W}(t)$ 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, $\phi_{W}(t)$ exhibit negative values for times exceeding 25 to 60 ps
for all solutions. We fit $\phi_{W}(t)$ with the expression
$\phi_{W}(t=0)=Exp[-t/\tau_{D}]$, where $\tau_{D}$ is the characteristic
relaxation time. The values of $\phi_{W}(t=0)$ and $\tau_{D}$ are plotted for
each pore solution and for SPC/E water in Figs. 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.

Figure 3: (a) Auto-correlation function of the water polarization
$\phi_{W}(t)$ of the pore solutions PC1-PC7 and SPC/E water. The inset shows
Debye fits in a Log-plot. (b) $\phi_{W}(t=0)$ for pore solutions PC1-PC7 and
SPC/E water. (c) Characteristic relaxation time obtained from Debye fits.

Ion-water cross-correlation $\phi_{IW}(t)$ and ion auto-correlation
$\phi_{I}(t)$ functions are presented in Figs. 4 (a) and (b), respectively.
Both functions present a high-frequency oscillatory behavior with the
oscillations being more strongly damped in the case of $\phi_{I}(t)$. Strong
dampening has been also observed in NaCl, NaI, NaBr and NaF aqueous solutions
[24]. The ion auto-correlation functions $\phi_{I}(t)$ 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 $\phi_{IW}(t)$ present an initial steep increase followed by an
initial decay and a subsequent slower relaxation. Compared to the other
simpler aqueous solutions, the pore solution shows persistent oscillations
even for times above 1 ps.

Figure 4: (a) Cross-correlation function $\phi_{IW}(t)$ 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
($\phi_{I}(t)$) for pore solutions PC1-PC7 and SPC/E water. The inset shows
the the decay at longer timescales.

###  III.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 Cole-Cole fits.
The water contribution is predominant (see the Supporting Information for the
results without ion contributions). The ion-water dielectric contributions
$\chi_{IW}(f)$ 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 $\Delta\chi_{I}(f)$ for all pore
solution are shown in Fig. 6 (c) and (d), respectively.

Figure 5: Dielectric spectra of the pore solutions: real (left) and imaginary
(right) parts of the dielectric susceptibility: $\Delta\chi(f)^{\prime}$ and
$\Delta\chi(f)^{\prime\prime}$, 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. [51] is also shown (g)-(h). Figure 6: Dielectric spectra of the
pore solutions PC1-PC7: (a) real $\chi_{IW}(f)^{\prime}$ and (b) imaginary
$\chi_{IW}(f)^{\prime\prime}$ parts of the ion-water dielectric contribution;
(c) real $\Delta\chi_{I}(f)^{\prime}$ and (d) imaginary
$\Delta\chi_{I}(f)^{\prime\prime}$ parts of the DC-conductivity dielectric ion
contribution.

To fit the normalized dielectric spectrum we use Cole-Cole model [52, 24]:

$\Delta\chi(f)=\frac{\epsilon_{CC}-\epsilon_{\infty}}{1+\left(i2\pi
f\tau_{CC}\right)^{1-\alpha}}+\epsilon_{\infty}-1$ (16)

where $\epsilon_{CC}$ and $\tau_{CC}$ are, respectively, the Cole-Cole
amplitude and characteristic relaxation time; and, $\alpha$ is an exponent
parameter ranging from 0 to 1 ($\alpha$=0 corresponding to Debye model). We
adopt $\epsilon_{\infty}$=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
$\Delta\chi_{i}(f)\bar{\Delta\chi_{i}(f)}$, where $\Delta\chi_{i}(f)$ refers
to the spectrum data to be fitted and $\bar{.}$ denotes the complex conjugate.

The parameters of Cole-Cole fits are gathered in Tab. 2. Water simulation and
experimental data from the literature are also provided. The static dielectric
permittivity of SPC/E water is known to be inferior to the experimental value.
We obtain 69.0 for SPC/E water, which is in agreement with other values from
the literature (e.g.[53] 67 $\pm$ 10). In spite of its simplicity, the SPC/E
model shows a remarkable agreement with experiments.

Figure 7 shows the parameters of Cole-Cole fits. The amplitude of Cole-Cole
fits is inferior to that of SPC/E water for pore solution at late ages
(PC5-PC7). Cole-Cole relaxation times of all pore solutions exceed that of
SPC/E water, with very early-age (PC1-PC2) and late ages (PC5-PC7) pore
solution exhibiting similar $\tau_{CC}$. The $\alpha$ 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).

In agreement with our results, recent experimental measurements of the
dielectric response of a concrete pore solution show that the real part
remains almost constant for frequencies up, at least, to 2 GHz [54]. The same
authors also showed that a Debye model can describe fairly well the dielectric
spectra of the pore solution, which is also consistent with our results that
show values of $\alpha$ close to zero.

Figure 7: Parameters of Cole-Cole fits from Fig. 5. Table 2:  Cole-Cole fit parameters from MD simulations of pore solution and SPC/E water, and data from the literature on water. | $\epsilon_{CC}$ | $\tau_{CC}$ | $\alpha$
---|---|---|---
PC1 | 73.6 | 9.97 | 0.000
PC2 | 67.1 | 10.26 | 0.009
PC3 | 73.9 | 12.55 | 0.000
PC4 | 60.7 | 9.33 | 0.000
PC5 | 56.0 | 9.80 | 0.040
PC6 | 55.2 | 9.33 | 0.098
PC7 | 60.4 | 10.30 | 0.052
SPC/E 111This work. | 68.2 | 8.35 | 0.033
SPC/E 222Buchner et al. [51] experiments. | 78.4 | 8.27 | 0.000
SPC/E 333Rinne et al. [24] MD simulations. | 69.9 | 10.72 | 0.014

###  III.2 Electrical conductivity

Figure 8 shows the real and imaginary parts of the frequency-dependent
conductivity of the pore solutions. The ion-ion and ion-water contributions
are shown. Our results are in agreement with the static conductivity reported
for ordinary PC systems with a water-to-cement 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]. 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.

Figure 8: 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].

###  III.3 Upscaling the effects of the variability pore solution dielectric
response

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 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 $\epsilon^{SC}$ can be computed from the
implicit formula [20]:

$\sum_{r=1}^{N}f_{r}\frac{\epsilon_{r}-\epsilon^{SC}}{\epsilon_{r}+2\epsilon^{SC}}=0$
(17)

for all phases $r$ with volume fraction $f_{r}$ and dielectric permittivity
$\epsilon_{r}$.

To upscale the effective permittivity $\epsilon=\Delta\chi+1$ of the cement
paste we use the Cole-Cole fits with the parameters in Tab. 2 to describe the
dielectric permittivity of each pore solution $\epsilon_{PS}$. Powers cement
hydration model [55] is used to estimate the evolution of the capillary
porosity (i.e. the volume fraction associated to the pore solution phase) in
the system:

$f_{PS}(\xi,w/c)=\phi^{0}_{cap}(w/c)-1.32(1-\phi^{0}_{cap}(w/c))\xi$ (18)

where the degree of hydration $\xi$ can be described by the sigmoid
$\xi=0.9\left[1-\text{Exp}\left(-t/7\right)\right]$ for $t$ in days as in ref.
[37], and $\phi^{0}_{cap}(w/c)=(w/c)/\left(\rho_{w}/\rho_{c}+(w/c)\right)$ is
the initial porosity computed from the $w/c$ and densities of water and cement
($\rho_{c}$ and $\rho_{w}$, 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 [46, 47]. 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 $\epsilon^{\prime}$ and imaginary $\epsilon^{\prime\prime}$ parts of
the effective permittivity of the cement paste with a water-to-cement ratio
($w/c$) of 0.4 are shown in Fig. 9. 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 ($t<10$). 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 the age of
the cement paste or, using cross-property relations, to follow property
development at early-age.

Figure 9: Effective permittivity
$\epsilon=\Delta\chi+1=\epsilon^{\prime}-i\epsilon^{\prime\prime}$ of the
cement paste with $w/c$=0.4. Time evolution of the (a) real
$\epsilon^{\prime}$ and (b) imaginary $\epsilon^{\prime\prime}$ parts at
constant frequency $f$. Frequency-dependence of the (c) real
$\epsilon^{\prime}$ and (d) imaginary $\epsilon^{\prime\prime}$ parts at
constant time.

##  IV 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 dielectric spectra and frequency-dependent 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. This
result shows that accounting for the composition variability of the pore is
crucial to obtain the precise dielectric response of cement-based materials.
The results provided in this paper can readily be used to upscale the
dielectric response of ordinary Portland cement-based materials. Further
studies are, though, necessary 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 (ie. $\alpha$ parameter in 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 inferior to 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 the age of cement pastes.

Our results can be used in the interpretation of dielectric probing of cement-
based materials, enhancing the performance of non-destructive monitoring of
concrete structures conditions such as radar testing and improving the
confidence in the corresponding results. Improvements on the detection of
cracking and the diagnostic of structures pathologies (e.g. alkali-silica
reactions, delayed ettringite formation, etc) 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
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.

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