# Self-Consistent Channel Approach for Upscaling Chloride Diffusivity in Cement Pastes

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## Abstract

Chloride ingress into concrete is a major cause for material degradation, such as cracking due to corrosion-induced steel reinforcement expansion. Corresponding transport processes encompass diffusion, convection, and migration, and their mathematical quantification as a function of the concrete composition remains an unrevealed enigma. Approaching the problem step by step, we here concentrate on the diffusivity of cement paste, and how it follows from the microstructural features of the material and from the chloride diffusivity in the capillary pore spaces. For this purpose, we employ advanced self-consistent homogenization theory as recently used for permeability upscaling, based on the resolution of the pore space as pore channels being oriented in all space directions, resulting in a quite compact analytical relation between porosity, pore diffusivity, and the overall diffusivity of the cement paste. This relation is supported by experiments and reconfirms the pivotal role that layered water most probably plays for the reduction of the pore diffusivity, with respect to the diffusivity found under the chemical condition of a bulk solution.

## Keywords

Diffusion Homogenization Chlorides Multiscale modeling Layered water## List of Symbols

## Mathematical operators

- div
Divergence operator, applied to a vector field

- \(\mathbf {div}\)
Divergence operator, applied to a second-order tensor field

**grad**Gradient operator on the microscopic observation scale of pore space

**GRAD**Gradient operator on the macroscopic observation scale of cement paste

- log
Natural logarithm

- sin
Sine function

- \(\int f(x)\,\text{ d }x\)
Integration of function

*f*with respect to variable*x*- \(\partial f/\partial x\)
Partial differentiation of function

*f*with respect to variable*x*- \(\cdot \)
First-order tensor contraction

## Latin symbols

- \({\mathbf {A}}_{\text {pore}}\)
Localization tensor for the concentration gradient of chloride ions in the pore space

*c*Microscopic concentration of chloride ions

*C*Macroscopic concentration of chloride ions

- \({\mathscr {D}}\)
Characteristic length of heterogeneities within a representative volume element

- \(d_\text {bulk}\)
Chloride diffusion coefficient in bulk solution

- \(d_{\text {pore}}\)
Chloride diffusion coefficient in the pore fluid, when considering isotropic pore diffusivity

- \(d_\text {pore}^\text {long}\)
Longitudinal chloride diffusion coefficient in the pore fluid, when considering anisotropic pore diffusivity

- \(d_\text {pore}^\text {trans}\)
Transverse chloride diffusion coefficient in the pore fluid, when considering anisotropic pore diffusivity

- \({\mathbf {d}}_{\text {pore}}\)
Isotropic chloride diffusivity tensor in the pore fluid

- \({\mathbf {d}}^\text {aniso}_{\text {pore}}\)
Anisotropic chloride diffusivity tensor in the pore fluid

- \({\mathbf {d}}_{\text {solid}}\)
Chloride diffusivity tensor in the solid phase

- \(D_{\text {paste}}^\text {exp}\)
Experimentally obtained diffusion coefficient of chloride ions in cement paste

- \(D_{\text {paste}}^\text {hom}\)
Homogenized diffusion coefficient of chloride ions in cement paste, when considering cylindrical pores with isotropic diffusivity

- \(D_{\text {paste}}^\text {hom,sph}\)
Homogenized diffusion coefficient of chloride ions in cement paste, when considering spherical pores with isotropic diffusivity

- \(D_{\text {paste}}^\text {hom,aniso}\)
Homogenized diffusion coefficient of chloride ions in cement paste, when considering cylindrical pores with anisotropic diffusivity

- \({\mathbf {D}}_{\text {paste}}^\text {hom}\)
Homogenized diffusivity tensor of chloride ions in cement paste, when considering cylindrical pores with isotropic diffusivity

- \({\mathbf {e}}_x, {\mathbf {e}}_y, {\mathbf {e}}_z\)
Unit base vectors of Cartesian coordinate system

- \({\mathbf {e}}_r, {\mathbf {e}}_\vartheta , {\mathbf {e}}_\varphi \)
Unit base vectors of spherical coordinate system

- \(f_\text {air}\)
Volume fraction of air pores

- \(f_\text {cem}\)
Volume fraction of cement clinker

- \(f_\text {hyd}\)
Volume fraction of hydration products

- \(f_\text {pore}\)
Volume fraction of pore space

- \(f_\text {water}\)
Volume fraction of water

*j*Index representing a specific data point

- \({\mathbf {j}}\)
Microscopic diffusive flux vector of chloride ions

- \({\mathbf {j}}_{\text {pore}}\)
Microscopic diffusive flux vector of chloride ions in the pore space

- \({\mathbf {J}}_\text {paste}\)
Macroscopic diffusive flux vector of chloride ions

- \(l_\text {pore}\)
Length of single pore

- \(\ell \)
Characteristic length of representative volume element

- \({\mathscr {L}}_\text {C}\)
Characteristic length of physical quantities related to a representative volume element

- \({\mathscr {L}}_\text {S}\)
Characteristic length of a solid or structure made up by the material defined by a representative volume element

*n*Number of experimental data points

- \({\mathbf {P}}_{\text {pore}}\)
Hill tensor related to the pore space

- \({\mathbf {P}}_{\text {solid}}\)
Hill tensor related to the solid phase

*r*Radial coordinate of spherical coordinate system

- \(R^2\)
Coefficient of determination

- RVE
Representative volume element

*s*Standard deviation of the \((d_{\text {pore},i}/d_{\text {pore},i}^*)\)-population

- \(t_{n-1,\alpha /2}\)
*t*-value- \(V_\text {RVE}\)
Volume of representative volume element

- \(\partial V_\text {RVE}\)
Surface of representative volume element

*w*/*c*Initial water-to-cement mass ratio

- \((w/c)_\text {cur}\)
Water-to-cement mass ratio after curing

- \({\mathbf {x}}\)
Position vector

- \({\bar{x}}\)
Mean value of the \((d_{\text {pore},i}{/}d_{\text {pore},i}^*)\)-population

- \(z_{\alpha /2}\)
*z*-value- \({\mathbf {1}}\)
Second-order unit tensor

## Greek letters

- \(\alpha \)
Statistical significance level

- \(\vartheta \)
Euler angle

- \(\xi \)
Degree of cement hydration

- \(\mu \)
Expected value

- \(\mu _\text {low}\)
Lower bound of the expected value

- \(\mu _\text {up}\)
Upper bound of the expected value

- \(\sigma _\text {up}\)
Upper bound for the standard deviation

- \(\varphi \)
Euler angle

- \(\chi ^2_{n-1,\alpha }\)
\(\chi ^2\)-value

## 1 Introduction

Ingress of chloride ions into steel-reinforced concrete structures is an important factor for initiating corrosion of the embedded steel bars (Glass and Buenfeld 1997). The presence of chloride ions in concrete structures is thus considered a major threat to their durability (Stewart and Rosowsky 1998). The underlying transport processes comprise diffusion, migration, and convection (Glasser et al. 2008), and their interplay turns out as complex, so that studying these processes individually is generally considered as a suitable scientific approach. In this context, the chloride diffusivity of concrete in particular, and of cementitious materials in general, is a (so far only partially resolved) topic of great scientific interest. Here, the key challenge lies in the fact that this diffusivity is not constant, but depends on the composition of the material (governed by the chosen mixture, standardly expressed in terms of the initial water-to-cement mass ratio and the initial aggregate-to-cement mass ratio), of its maturity (quantified by the so-called degree of hydration), as well as of microcracks occurring at the interfaces between aggregate grains (referred to as “bond cracks”) or connecting different aggregate grains (referred to as “matrix cracks”), as shown in (Wong et al. 2009; Wu et al. 2015).

Striving for understanding this composition dependence, numerous experimental campaigns provided valuable insights, and the latter were often condensed into (more or less appropriate) simplified empirical relations (Page and Ngala 1997; Oh and Jang 2004; Sun et al. 2011b). Importantly, the majority of such diffusion tests have been carried out on cement paste specimens (concrete is regarded as composite consisting of cement paste and aggregates—in the present paper, we focus on the chloride diffusivity of cement paste as well. This focus on cement paste implies that the aforementioned microcracks, potentially exerting a substantial influence on the macroscopic diffusivity of mortar or concrete, can be neglected subsequently.

From a more fundamental viewpoint, it is natural to explore the microstructural sources which drive the overall diffusive properties of cement paste, and to derive corresponding mathematical functions taking into account the relevant information available at the microlevel. In this context, repeated inclusion of infinitely small solid spheres into a repeatedly homogenized diffusive medium, as is customarily done in the so-called differential schemes (Dormieux and Lemarchand 2001), proposes that the isotropic diffusivity of cement paste depends on the (capillary) fluid volume fraction to the power of 1.5, times the chloride diffusivity of the pore solution. This allows for translating experimental data obtained from cement paste specimens with different mixtures into one “universal” chloride diffusivity of the cement paste pore solution, \(d_\text {pore}=1.07\times 10^{-10}\) m\(^2\)/s (Pivonka et al. 2004). Interestingly, this value for \(d_\text {pore}\) is about 15 times smaller than the chloride diffusivity of a bulk solution, \(d_\text {bulk}=1.61\times 10^{-9}\) m\(^2\)/s (Robinson and Stokes 1959). This is probably due to the charged pore surfaces causing water structuring, which, in turn, leads to a reduction of the solution’s diffusivity. The aforementioned diffusivity difference in bulk versus electrically influenced solutions was subsequently confirmed by additional studies (Zheng and Zhou 2008; Zheng et al. 2009a, 2010; Liu et al. 2012), and the aforementioned chloride diffusivity of the cement paste solution turned out to be an appropriate input for various simulations dealing with the durability of concrete, see, e.g., (Liu et al. 2013; Du et al. 2014). Alternatively, still considering spherical morphological features, concrete diffusivity upscaling was achieved by means of the so-called Maxwell effective medium approach, for studying the effects of entrapped air pores (Wong et al. 2011) and interphases (Lutz and Zimmerman 2015) on the macroscopic diffusivity.

However, cement paste actually exhibits clearly non-spherical microstructural features. Explicit consideration of the latter has allowed for substantial improvements of microstructural models for the *mechanics* of cement paste and concrete (Sanahuja et al. 2007; Pichler et al. 2009a; Pichler and Hellmich 2011). The same is true for a micromechanical model of gypsum which considers the physically active (solid) parts of the microstructure as infinitely many non-spherical phases (Sanahuja et al. 2010), while the physically non-active (fluid) parts were considered, for simplicity, as spheres. This gypsum model was shown to predict homogenized elastic properties as precisely as corresponding full 3D Finite Element simulations (Meille and Garboczi 2001). Thus, the question arises whether the upscaling-based estimate of the chloride diffusivity of the cement paste pore solution can be as well improved when considering a more realistic microstructural representation of cement paste. This motivates us to adapt, in the present paper, the aforementioned homogenization approach based on infinitely many non-spherical phases, for diffusivity upscaling. In particular, we address (i) whether such a more sophisticated and physically profound approach would replicate the experimental data in a more suitable way than the classical differential method-based approach, and (ii) whether and how the corresponding estimate of the chloride diffusivity in the cement paste pore solution changes with respect to its differential method-based counterpart.

For this purpose, we first present the model representation of cement paste, considering the concepts of scale separation and random homogenization (see Sect. 2). Then, a new self-consistent homogenization scheme for estimating the chloride diffusivity in cement paste is derived (see Sect. 3), and the chloride diffusivity in the cement paste pore solution is assessed (see Sect. 4). After a comprehensive discussion (see Sect. 5), a brief summary and outlook concludes the paper (see Sect. 6).

## 2 Representative Volume Element for Diffusive Transport in Cement Paste

## 3 Homogenization of Macroscopic Diffusion Behavior: Porosity–Diffusivity Relations

## 4 Upscaling Theory-Guided Re-evaluation of Diffusion Experiments: Access to Chloride Diffusivity at the Capillary Pore Level

*w*/

*c*), at which the cement paste was produced, see Table 1. According to the famous Powers–Acker model (Powers and Brownyard 1948; Acker 2001), the water-to-cement ratio relates to the volume fractions of cement paste constituents “cement clinker” (cem), “water,” “hydrates” (hyd), and “air,” via (Pichler et al. 2009a)

Experimentally determined chloride diffusion coefficients in cement paste (\(D_{\text {paste}}^\text {exp}\)); the corresponding pore space volume fractions (\(f_\text {pore}\)) follow from cement paste mixture rules, see (Pivonka et al. 2004; Acker 2001) for details; pore-scale diffusivity \(d_\text {pore}\) follows from evaluation of Eq. (16), for the respective data pairs (\(f_\text {pore}\); \(D_\text {paste}^\text {exp}\))

References | Short name | | \((w/c)_\text {cur}\) [ – ] | \(f_\text {pore}\) [ – ] | \(D_{\text {paste}}^\text {exp}\) [\(10^{-12}\) m\(^2\)/s] | \(d_\text {pore}\) [\(10^{-10}\) m\(^2\)/s] |
---|---|---|---|---|---|---|

Page et al. (1981) | P81 | 0.40 | 0.42 | 0.071 | 2.600 | 1.451 |

0.50 | 0.50 | 0.162 | 4.470 | 0.922 | ||

0.60 | 0.60 | 0.254 | 12.350 | 1.349 | ||

Yu and Page (1991) | Y91 | 0.35 | 0.42 | 0.071 | 1.200 | 0.670 |

0.50 | 0.50 | 0.162 | 5.430 | 1.120 | ||

0.60 | 0.60 | 0.254 | 7.300 | 0.798 | ||

Tang and Nilson (1992) | T92 | 0.40 | 0.42 | 0.071 | 2.900 | 1.162 |

0.60 | 0.60 | 0.254 | 9.400 | 1.027 | ||

0.80 | 0.80 | 0.387 | 21.000 | 1.130 | ||

Hornain et al. (1995) | H95 | 0.55 | 0.55 | 0.210 | 11.250 | 1.621 |

MacDonald and Northwood (1995) | M95 | 0.40 | 0.42 | 0.071 | 2.353 | 1.313 |

0.40 | 0.42 | 0.071 | 2.549 | 1.422 | ||

0.40 | 0.42 | 0.071 | 2.784 | 1.554 | ||

0.50 | 0.50 | 0.162 | 6.412 | 1.322 | ||

0.50 | 0.50 | 0.162 | 6.745 | 1.391 | ||

0.50 | 0.50 | 0.162 | 7.275 | 1.500 | ||

0.60 | 0.60 | 0.254 | 12.290 | 1.343 | ||

0.60 | 0.60 | 0.254 | 12.570 | 1.373 | ||

0.60 | 0.60 | 0.254 | 13.840 | 1.512 | ||

0.70 | 0.70 | 0.327 | 18.730 | 1.356 | ||

0.70 | 0.70 | 0.327 | 21.570 | 1.562 | ||

0.70 | 0.70 | 0.327 | 21.860 | 1.583 | ||

Ngala et al. (1995) | N95 | 0.40 | 0.42 | 0.071 | 3.950 | 2.204 |

0.50 | 0.50 | 0.162 | 7.800 | 1.609 | ||

0.60 | 0.60 | 0.254 | 12.600 | 1.377 | ||

0.70 | 0.70 | 0.327 | 21.460 | 1.554 | ||

Ngala and Page (1997) | N97 | 0.40 | 0.42 | 0.071 | 4.280 | 2.388 |

0.50 | 0.50 | 0.162 | 8.430 | 1.739 | ||

0.60 | 0.60 | 0.254 | 12.300 | 1.344 | ||

0.70 | 0.70 | 0.327 | 21.380 | 1.548 | ||

Castellote et al. (2001) | C01 | 0.40 | 0.42 | 0.071 | 3.646 | 2.035 |

Caré (2003) | C03 | 0.45 | 0.45 | 0.108 | 5.650 | 1.953 |

Huang et al. (2010) | H10 | 0.40 | 0.42 | 0.071 | 5.420 | 3.025 |

0.50 | 0.50 | 0.162 | 8.240 | 1.700 | ||

0.60 | 0.60 | 0.254 | 12.000 | 1.311 | ||

Sun et al. (2011a) | S11 | 0.23 | 0.42 | 0.071 | 1.030 | 0.575 |

0.35 | 0.42 | 0.071 | 4.120 | 2.299 | ||

0.53 | 0.53 | 0.192 | 10.600 | 1.741 |

*i*is an index, \(d_{\text {pore},i}\) is the

*i*th back-analyzed pore-scale diffusion coefficient, and \(d_{\text {pore}}^*\) is a normalization quantity, \(d_{\text {pore}}^*=1\,\text {m}^2/\text {s}\), included to yield dimensionless arguments for the logarithmic function. The sample is characterized by a mean value \({\bar{x}}\) and standard deviation

*s*, according to

*t*-distribution of degree \((n-1)\), see, e.g., (Bortz 1999). This allows one to estimate upper and lower bounds of the statistical populations from the one known mean value of one sample taken from the population, according to

*t*-value, cutting an area equal to \(1-\alpha /2\) of the Student’s

*t*-distribution with \(n-1\) degrees of freedom, and \(\alpha \) is the significance level, i.e., corresponding to \((1-\alpha )\)-confidence interval. Considering the sample size \(n=38\) and the significance level \(\alpha =5\)%, one obtains \(t_{n-1,1-\alpha /2}=2.026\). Then, Eq. (24) allows us to calculate \(\mu _\text {low}=-22.767\) and \(\mu _\text {low}=-22.556\). The upper bound of the respective standard deviation \(\sigma , \sigma _\text {up}\), is based on a one-sided confidence interval, following a standard engineering statistics approach (NIST 2012)

*z*-value that cuts an area equal to \((1-\alpha )\) of the standardized normal distribution. Evaluating Eqs. (26) and (27) for \(\alpha =5\)%, thus for \(z_{0.025}=1.645\), yields \(\text {log}(d_{\text {pore},i}/d_{\text {pore},i}^*)_{5\%}=-23.427\) and \(\text {log}(d_{\text {pore},i}/d_{\text {pore},i}^*)_{95\%}=-21.898\). Eventually, we can calculate the median of the \(d_{\text {pore},i}\)-population, \((d_\text {pore})_{50\%}\), its 5%-quantile, \((d_\text {pore})_{5\%}\), and its 95%-quantile, \((d_\text {pore})_{95\%}\):

Finally, it is further instructive to assess the dependence of the macroscopic diffusivity of cement paste on the underlying cement paste mixture, quantified based on the initial water-to-cement ratio *w* / *c*, and on the hydration degree \(\xi \). For this purpose, we evaluate Eq. (18) for various water-to-cement ratios ranging from 0.2 to 0.7, in order to get access to the mixture- and hydration degree-dependent pore space volume fraction \(f_\text {pore}=f_\text {water}\). Equation (16) then allows us to study how the diffusivity of cement paste changes over its service life, for different cement paste mixtures, see Fig. 5.

## 5 Discussion

The herein presented homogenization approach starts at a local level where the physics is quite well understood, namely the diffusion of chloride ions in the pore space of cement paste, following Fick’s law; we upscale this behavior to the level of cement paste where we again retrieve a Fick-type diffusion behavior. Such a mode of (physically reasonable) upscaling has been repeatedly described in the open literature, see, e.g., (Hashin 1968; Dormieux and Lemarchand 2001; Boutin and Geindreau 2010 and Patel et al. 2016). Thus, we would qualify our approach as a typical “micromechanics strategy.” Concerning the “structural,” “layered,” or “glassy” water phase, it has been shown by the landmark molecular dynamics studies of Ichikawa et al. (2000, 2002) that the changes in physics due to the “structuring” or “layering” of water molecules express themselves simply in a different, namely reduced, pore diffusivity, which, however, may still enter the upscaling strategy related to further above. From a more general viewpoint, we wish to remark that it is well known that the structure of water changes in the immediate vicinity of charged surfaces (such as the surfaces of hydrates), resulting in some kind of layering effect (Pollack 2001, 2013). Such kind of water domains are imagined to be glassy, or of ice-type (“liquid crystalline”) nature, and are alternatively referred to as “surface zone” (Henniker 1949) or “exclusion zone” (Pollack 2013). Layered water exhibits physical properties that vary significantly from the properties of bulk water. For example, as shown by molecular dynamics studies, water layering leads to increased viscosity and reduced diffusivity (Ichikawa et al. 2000, 2002). The thickness of layered water can amount up to a few millimeters (Zheng et al. 2006, 2009b; Pollack 2013; Florea et al. 2014). This suggests that the pores pervading cement paste are entirely and homogeneously filled with layered water (instead of bulk water). In our model, this implies that the (chloride) diffusion coefficient of the pore fluid is, throughout the entire pore space, reduced with respect to the (chloride) diffusion coefficient in bulk water. At the same time, we are aware that there exist other propositions (Bertolini et al. 2004; Yang 2013) which limit the characteristic length scale of water molecule mobility restriction to several tens of nanometers; rather than to micrometers or even to millimeters. However, latest experiments and simulations on pore size distributions in concrete (Huang et al. 2015) reveal that typically 80% of the pore space found in very mature cement paste is made up by pores with characteristic lineal dimensions of less than 20 nanometers. This renders the aforementioned propositions (Bertolini et al. 2004; Yang 2013) as not necessarily at odds with our proposition of approximately homogeneous properties throughout the channel-like pore phases introduced in the herein-presented modeling approach. Furthermore, while we presently analyze, based on our “micromechanical” transport model, only *one* layered water diffusivity value allowing for the prediction of various diffusivity values at the cement paste level, we understand that this approach may be further refined in the future, by performing molecular dynamics studies targeting explicitly at the diffusivity determination of cement paste pore fluid. This is, however, beyond the scope of the present manuscript. We also wish to remark that our upscaling situation is different from that encountered when upscaling, by means of the asymptotic expansion technique, the Navier–Stokes equation to a permeability equation—then, the structure of the governing equations indeed changes upon scale transition, see, e.g., (Auriault and Lewandowska 1997; Auriault 2002; Boutin and Geindreau 2010).

The aforementioned molecular dynamics studies (Ichikawa et al. 2000, 2002) are particularly insightful, as they reveal that for a geomaterial which is somewhat similar to cement paste, i.e., clay, the diffusivity decreases from bulk to pore solution by a factor of 7; from \(d_\text {bulk}=1.61\times 10^{-9}\) m\(^2\)/s to \(d_\text {pore}^\text {Ichikawa}=2.3\times 10^{-10}\) m\(^2\)/s. Our new prediction for the pore solution diffusivity, \({\bar{d}}_\text {pore}=1.476\times 10^{-10}\) m\(^2\)/s (being separated from the bulk solution diffusivity by factor 10.91) is considerably closer to molecular dynamics-derived pore solution diffusivity than the one reported in Sect. 1 of this paper, \(d_\text {pore}^\text {Pivonka}=1.07\times 10^{-10}\) m\(^2\)/s (being separated from the bulk solution diffusivity by factor 15). In other words, the difference between micromechanics- and molecular dynamics-based pore solution diffusivity could be reduced by \(\approx \)32%. This confirms that the more realistic representation of cement paste according to Fig. 1 indeed leads to a more precise prediction of the cement paste diffusivity.

We may also mention that our approach is fully consistent with the current state of the art in the mathematical modeling of concrete: In fact, our microheterogeneous formulation rests on the famous hydration model of Powers and Brownyard (1948) and Acker (2001), which has not only provided the basis for numerous, experimentally validated, micromechanical descriptions (Hellmich and Mang 2005; Sanahuja et al. 2007; Pichler et al. 2009b; Scheiner and Hellmich 2009), but has also been kind of corroborated by very recent statistical physics approaches (Ioannidou et al. 2016). In the aforementioned micromechanics approaches, an RVE of cement paste is either composed of water pores, air pores, hydrates, and unhydrated cement (clinker) grains (Hellmich and Mang 2005; Pichler et al. 2009b; Scheiner and Hellmich 2009), or of clinker grains embedded into a hydrate foam matrix, whereby the latter is, at a smaller scale, resolved into hydrates, water pores, and air pores; i.e., a hierarchical system of two RVEs is used to represent cement paste (Pichler and Hellmich 2011; Pichler et al. 2013). Presently, our transport modeling approach somewhat follows the material phase description of Hellmich and Mang (2005), Pichler et al. (2009b) and Scheiner and Hellmich (2009), involving only one RVE representing the composite material cement paste. Actually, we additionally merge, on the one hand, the water and air pore phases into one phase called “capillary porosity” (this merging results from the experimental conditions realized in standard diffusion tests), and on the other hand, the hydrate and clinker phases are merged as well, into one “solid phase” (given their non-diffusible nature as compared to that of the pores). This concept, involving only one RVE, seems to be particularly well suited for the present situation where we restrict our investigations to fully hydrated cement pastes. In this case, no unhydrated cement clinker is left, and even the aforementioned two-scale formulations degenerate into a one-scale formulation. Still, it is interesting to find out up to which extent our homogenization scheme improves similar earlier developments, such as the differential homogenization scheme of Dormieux and Lemarchand (2001), which has been applied to cement paste diffusivity by Pivonka et al. (2004), and the “self-consistent bispherical model” of Hashin (1968). An analysis as described in Sect. 3 was performed for both alternative homogenization schemes. While the homogenization approach presented in Sect. 4 yields a pore-scale chloride diffusivity separated from the chloride diffusivity in bulk solution by factor 10.91, the homogenization schemes of Dormieux and Lemarchand (2001) and Hashin (1968), respectively, yield pore-scale chloride diffusivities separated from the chloride diffusivity in bulk solution by factors 13.16 and 24.16, respectively. Considering that the molecular dynamics studies of Ichikawa et al. (2000, 2002) suggest that this factor amounts to 7, we conclude that our model yields significantly more accurate results than previous models. All these homogenization approaches may be qualified as “Fickian diffusion,” delivering diffusivity quantities which relate concentration gradients to molar fluxes. Limitations of such approaches have been extensively discussed, and it is of interest to review these limitations in the context of our work: Non-Fickian diffusion induces significant, non-negligible deformations in the respective porous material, because of which the mathematical description of the diffusion process needs to be extended (as compared to the classical Fick’s law), in order to take into account the interaction between deformation-induced stresses and Brownian motion (Ferreira et al. 2015). This is, however, not the case in the problem studied here. Another limitation of the Fickian description related to heterogeneity is: The homogenized diffusivity needs to be defined on an RVE already fulfilling the separation of scales requirement between micro-heterogeneity and RVE size, on the one hand, and between RVE size and size of the structural geometry or loading, on the other hand. In cases where this requirement is violated (Neuman and Tartakovsky 2009; Fourar and Radilla 2009), non-Fickian descriptions are needed.

*always*imply a connected network of capillary pores enabling diffusive transport of species dissolved in the pore solution. Thus, for the consideration of cement paste diffusivities, any percolation thresholds

*not*being close to \(f_\text {pore}=0\) make very limited sense.

It is also interesting to discriminate our approach from the popular “effective medium approaches.” The latter go back to the seminal work of Maxwell Garnett (Garnett 1904) and have been adapted to diffusivity upscaling by Burganos and Sotirchos (1987). However, quoting from (Levy and Stroud 1997), effective medium approaches are *“useful when one of the components can be considered as a host in which inclusions of the other components are embedded.”* This implies that the effective medium theory is probably not useful for estimating the diffusivity of cement paste, at least not when considering the morphology according to Fig. 1. In contrast, the effective medium theory is indeed useful for estimation of concrete, see, e.g., (Oh and Jang 2004), which is, however, beyond the scope of this paper.

## 6 Summary and Outlook

Recent developments in continuum homogenization theories extended to infinitely many material phases related to all orientations in Euclidean space (Fritsch et al. 2009, 2013; Abdalrahman et al. 2015) could be successfully applied to diffusivity upscaling. The corresponding results allow for improved representation of water-to-cement ratio-dependent and hydration-dependent diffusivity of cement paste. This provides a novel, both experimentally and theoretically improved foundation which may support large-scale simulations concerning durability of concrete structures, see, e.g., the approaches presented by Schrefler and Pesavento (2004), Gawin et al. (2006).

Future model developments may include extension of the homogenization scheme from the cement paste to the concrete level. For this purpose, additional interface phenomena, possibly occurring between the cement matrix and the aggregate grains (van Breugel et al. 2004) may deserve particular attention. Another interesting extension of the present model could relate to the case of unsaturated pores. This could be done based on the very interesting recent suggestion (Yang et al. 2016) of covering an Eshelby inclusion representing the solid phase by a liquid layer. This extension should be readily applicable to the Eshelby problem of Fig. 1b; together with the introduction of a new Eshelby problem related to the empty pore spaces, i.e., to a gaseous phase.

## Notes

### Acknowledgements

Open access funding provided by TU Wien (TUW). Financial support by the Austrian Federal Ministry of Science and Research (BMWF), in the frame of the ASEA UNINET-Program, and by the Thailand Research Fund (TRF), via Grant MRG5580222, is gratefully acknowledged.

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