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

We here introduce a representative volume element (RVE) fulfilling the standard “separation of scales”-requirement needed for random homogenization methods or continuum micromechanics (Zaoui 2002; Drugan and Willis 1996; Dormieux and Kondo 2005). Inside the RVE, microscopic ionic fluxes \({\mathbf {j}}\) fulfill the mass conservation law (Dormieux and Lemarchand 2001)

$$\begin{aligned} \forall {\mathbf {x}} \in V_\text {RVE}:~ \text {div}\,{\mathbf {j}}({\mathbf {x}})=0, \end{aligned}$$
(1)

with the position vector \({\mathbf {x}}\) labeling points within the RVE and at its boundary. Equation (1) can be transformed into the so-called weak form, analogous to the principle of virtual power in mechanics, see, e.g., (Germain 1973; Kuhl et al. 2004; Zienkiewicz et al. 2005; Maugin 2013), through multiplication with an arbitrary test function \(c({\mathbf {x}})\), and integration over the domain of the RVE, yielding

$$\begin{aligned} \begin{aligned} \int \limits _{V_\text {RVE}} c({\mathbf {x}}) [\text {div}\,{\mathbf {j}}({\mathbf {x}})]\,\text{ d }V&= \int \limits _{V_\text {RVE}} {\mathbf {j}}({\mathbf {x}}) \cdot \mathbf {grad}\,c({\mathbf {x}})\,\text{ d }V\\&=\int \limits _{V_\text {RVE}}\text {div}\,[{\mathbf {j}}({\mathbf {x}}) c({\mathbf {x}})]\,\text{ d }V \\&=\int \limits _{\partial V_\text {RVE}} [{\mathbf {j}}({\mathbf {x}}) c({\mathbf {x}})\cdot {\mathbf {n}}({\mathbf {x}})]\,\text{ d }S, \end{aligned} \end{aligned}$$
(2)

whereby use of Eq. (1) was made, and with \(\partial V_\text {RVE}\) denoting the boundary of the RVE. It is suitable to assign, to the test function, the nature of a “virtual concentration”; namely when defining boundary conditions for the RVE. Following (Dormieux and Lemarchand 2001), we prescribe “microscopic” concentrations at the boundary of the RVE of cement paste, which are compatible with a homogeneous “macroscopic” concentration gradient \(\mathbf {GRAD}\,C\); this reads mathematically as

$$\begin{aligned} \forall {\mathbf {x}}\in \partial V_\text {RVE}: c({\mathbf {x}})=\mathbf {GRAD}\,C\cdot {\mathbf {x}}. \end{aligned}$$
(3)

Using Eq. (3) in Eq. (2) yields

$$\begin{aligned} \int \limits _{V_\text {RVE}} {\mathbf {j}}({\mathbf {x}}) \cdot \mathbf {grad}\,c({\mathbf {x}})\,\text{ d }V= & {} \int \limits _{\partial V_\text {RVE}} [{\mathbf {j}}({\mathbf {x}}) c({\mathbf {x}})\cdot {\mathbf {n}}({\mathbf {x}})]\,\text{ d } S\nonumber \\= & {} \int \limits _{\partial V_\text {RVE}}(\mathbf {GRAD}\,C \cdot {\mathbf {x}}) [{\mathbf {j}}({\mathbf {x}}) \cdot {\mathbf {n}}({\mathbf {x}})] \,\text{ d }S\nonumber \\= & {} \mathbf {GRAD}\,C \cdot \int \limits _{\partial V_\text {RVE}} [{\mathbf {x}} \otimes {\mathbf {j}}({\mathbf {x}})] \cdot {\mathbf {n}}({\mathbf {x}})\,\text{ d }S\\= & {} \mathbf {GRAD}\,C \cdot \int \limits _{V_\text {RVE}} \mathbf {div}\,[{\mathbf {x}}\otimes {\mathbf {j}}({\mathbf {x}})]\, \text{ d }V\nonumber \\= & {} \mathbf {GRAD}\,C \cdot \int \limits _{V_\text {RVE}} {\mathbf {j}}({\mathbf {x}})\,\text{ d }V.\nonumber \end{aligned}$$
(4)

Equation (4) readily suggests to introduce a macroscopic ionic flux \({\mathbf {J}}_\text {paste}\) in the format

$$\begin{aligned} \int \limits _{V_\text {RVE}}{\mathbf {j}}({\mathbf {x}})\cdot \mathbf {grad}\,c({\mathbf {x}})\,\text{ d }V= V_\text {RVE}{\mathbf {J}}_\text {paste}\cdot \mathbf {GRAD}\,C, \end{aligned}$$
(5)

which implies the following average rule for the ionic fluxes:

$$\begin{aligned} {\mathbf {J}}_\text {paste} =\frac{1}{V_\text {RVE}}\int \limits _{V_\text {RVE}} {\mathbf {j}}({\mathbf {x}}) \;\text{ d }V. \end{aligned}$$
(6)

As the microstructure within the RVE is, as a rule, never known in minute detail, it is represented in the simplest possible (while still sufficiently complex) way, by means of homogeneous subdomains within the RVE, called “material phases.” In the present case, these phases are characterized by quantitative properties, namely volume fractions and (chloride) diffusivities: One (spherical) solid phase with volume fraction \((1-f_\text {pore})\) exhibits negligible diffusivity (\({\mathbf {d}}_\text {solid}\approx 0\)), while infinitely many, arbitrarily oriented (elongated) cylindrical pore phases fill the remaining volume (with volume fraction \(f_\text {pore}\)) and exhibit the (chloride) diffusivity of the pore fluid, \({\mathbf {d}}_\text {pore}\), see Fig. 1. Specification of Eq. (6) for this morphology yields

$$\begin{aligned} {\mathbf {J}}_\text {paste}= f_\text {pore}\int \limits _{\varphi =0}^{2\pi } \int \limits _{\vartheta =0}^\pi {\mathbf {j}}_{\text {pore}}(\vartheta ,\varphi ) \frac{\sin \vartheta }{4\pi }\text{ d }\vartheta \,\text{ d }\varphi , \end{aligned}$$
(7)

with \({\mathbf {j}}_{\text {pore}}(\vartheta ,\varphi )\) denoting the ionic flow characterizing the cylindrical pore phase oriented in \((\vartheta ,\varphi )\)-direction according to Fig. 2. Notably, Eq. (7) considers that no fluxes occur in the solid phase, see also (Dormieux and Lemarchand 2001). Transport in the pores is governed by Fick’s first law of diffusion, reading in a phase-specific format as

$$\begin{aligned} {\mathbf {j}}_\text {pore}(\vartheta ,\varphi )= -{\mathbf {d}}_\text {pore}(\vartheta ,\varphi ) \cdot (\mathbf grad \,c)_\text {pore}(\vartheta ,\varphi ), \end{aligned}$$
(8)

whereby the phase-specific diffusivity tensor always exhibits the same eigenvalues, and eigendirections which are aligned with the pore directions. This tensor relates the orientation-specific ionic flux to the concentration gradient by which it is driven, i.e., to the orientation-specific concentration gradient \((\mathbf grad \,c)_\text {pore}(\vartheta ,\varphi )\).

Fig. 1
figure 1

2D illustration of the 3D micromechanical representation of cement paste: a polycrystal-type arrangement of (spherical) solid and (cylindrical) pore phases, with indication of the characteristic lengths of heterogeneities, \({\mathscr {D}}\), and of the RVE, \(\ell , {\mathscr {D}}\ll \ell \), whereas \(\ell \) is significantly smaller than the characteristic lengths of physical quantities related to the RVE, \({{\mathscr {L}}}_\text {C}\), and of a solid or structure made up by the material defined by the RVE, \({{\mathscr {L}}}_\text {S}, \ell \ll \{{\mathscr {L}}_\text {C},{\mathscr {L}}_\text {S}\}\); b and c show the matrix-inclusion problems used for the derivation of localization tensor \({\mathbf {A}}_{\text {pore}}\), see Sect. 3

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3 Homogenization of Macroscopic Diffusion Behavior: Porosity–Diffusivity Relations

Due to the linearities of both the considered transport law and the mass conservation law, a linear macro-to-micro “downscaling” relation can be established for the concentration gradients,

$$\begin{aligned} \left( \mathbf grad \,c\right) _\text {pore}(\vartheta ,\varphi )= {\mathbf {A}}_{\text {pore}}(\vartheta ,\varphi ) \cdot \mathbf GRAD \,C, \end{aligned}$$
(9)

with \({\mathbf {A}}_{\text {pore}}\) as the second-order “downscaling” or localization tensor (alternatively also referred to as concentration tensor) related to the concentration gradient of chloride ions encountered in the pore space; such localization tensors have been originally defined for linear elasticity (Zaoui 1997, 2002), and later also for pressure gradients driving Darcy-type fluid flow (Dormieux and Kondo 2004, 2005). Derivation of \({\mathbf {A}}_{\text {pore}}\) has been dealt with in great detail in (Abdalrahman et al. 2015), based on Eshelby’s famous inhomogeneity problem (Eshelby 1957), see also the work of Dormieux and Kondo (2005), eventually yielding

$$\begin{aligned} {\mathbf {A}}_{\text {pore}}(\vartheta ,\varphi )= & {} \left[ {\mathbf {1}}+{\mathbf {P}}_{\text {pore}} (\vartheta ,\varphi )\cdot \left( {\mathbf {d}}_{\text {pore}}- D_{\text {paste}}^\text {hom}{\mathbf {1}}\right) \right] ^{-1}\cdot \Bigg \{(1-f_\text {pore}) \left[ {\mathbf {1}}-{\mathbf {P}}_{\text {solid}}\cdot D_{\text {paste}}^\text {hom}{\mathbf {1}}\right] ^{-1}\nonumber \\&+ f_\text {pore}\int \limits _{\varphi =0}^{2\pi } \int \limits _{\vartheta =0}^\pi \bigg [{\mathbf {1}}+ {\mathbf {P}}_{\text {pore}}(\vartheta ,\varphi )\cdot \left( {\mathbf {d}}_{\text {pore}}- D_{\text {paste}}^\text {hom}{\mathbf {1}}\right) \bigg ]^{-1} \frac{\sin \vartheta }{4\pi } \text{ d }\vartheta \,\text{ d }\varphi \Bigg \}^{-1},\nonumber \\ \end{aligned}$$
(10)

where \(D_{\text {paste}}^\text {hom}\) is the homogenized macroscopic diffusion coefficient of chloride ions in cement paste. Furthermore, \({\mathbf {P}}_{\text {pore}}\) and \({\mathbf {P}}_{\text {solid}}\), respectively, denote the so-called Hill or morphology tensors accounting for the shape of an ellipsoidal inhomogeneity of diffusivities \({\mathbf {d}}_\text {pore}\) and \({\mathbf {d}}_\text {solid}=d_\text {solid}{\mathbf {1}}\), respectively, embedded in a fictitious matrix with diffusivity \({\mathbf {D}}_\text {paste}^\text {hom}=D_\text {paste}^\text {hom}{\mathbf {1}}\). Given the micromechanical representation chosen for cement paste, see Sect. 2, the Hill tensors follow as (Dormieux et al. 2006; Abdalrahman et al. 2015)

$$\begin{aligned} {\mathbf {P}}_{\text {pore}}(\vartheta ,\varphi )= \frac{1}{2 D_{\text {paste}}^\text {hom}} \left[ \begin{array}{ccc}~0~&{}~0~&{}~0~\\ 0&{}1&{}0\\ 0&{}0&{}1 \end{array}\right] _ {{\mathbf {e}}_r,{\mathbf {e}}_{\vartheta },{\mathbf {e}}_\varphi }, \end{aligned}$$
(11)

representing a cylindrical shape with its orientation defined through angles \(\vartheta \) and \(\varphi \), and

$$\begin{aligned} {\mathbf {P}}_{\text {solid}}= \frac{1}{3 D_{\text {paste}}^\text {hom}} \left[ \begin{array}{ccc}~1~&{}~0~&{}~0~\\ 0&{}1&{}0\\ 0&{}0&{}1 \end{array}\right] {,} \end{aligned}$$
(12)

representing a spherical shape. The tensor components of \({\mathbf {P}}_\text {pore}\) need to be transformed from the \(({\mathbf {e}}_r,\,{\mathbf {e}}_\varphi ,\,{\mathbf {e}}_\varphi )\)-base to the \(({\mathbf {e}}_1,\,{\mathbf {e}}_2,\,{\mathbf {e}}_3)\)-base, through the standard transformation laws for second-order tensors (Salençon 2001). \({\mathbf {P}}_{\text {solid}}\), in turn, is an isotropic tensor, and its components remain unchanged upon any component transformation.

Fig. 2
figure 2

Cylindrical pore space inclusion oriented along vector \({\mathbf {e}}_r\), and inclined by the Euler angles \(\vartheta \) and \(\varphi \), with respect to the reference base frame defined through the unit vectors \({\mathbf {e}}_1, {\mathbf {e}}_2\), and \({\mathbf {e}}_3\); the local base frame, defined by unit vectors \({\mathbf {e}}_r, {\mathbf {e}}_\vartheta \), and \({\mathbf {e}}_\varphi \), is attached to the cylindrical inclusion

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Using Eq. (9) in Eq. (7) yields

$$\begin{aligned} {\mathbf {J}}_\text {paste}= -\Bigg \{f_\text {pore} \int \limits _{\varphi =0}^{2\pi } \int \limits _{\vartheta =0}^\pi {\mathbf {d}}_{\text {pore}}(\vartheta ,\varphi )\cdot {\mathbf {A}}_{\text {pore}}(\vartheta ,\varphi ) \frac{\sin \vartheta }{4\pi }\text{ d }\vartheta \,\text{ d } \varphi \Bigg \}\cdot \mathbf GRAD \,C. \end{aligned}$$
(13)

Inserting the macroscopic ionic flux according to Eq. (13) into Fick’s first law at the macroscopic observation scale,

$$\begin{aligned} {\mathbf {J}}_\text {paste}= -{\mathbf {D}}_{\text {paste}}^\text {hom}\cdot \mathbf GRAD \,C, \end{aligned}$$
(14)

gives, by comparing Eqs. (13) and (14), finally access to the macroscopic diffusivity tensor \(D_{\text {paste}}^\text {hom}\):

$$\begin{aligned} {\mathbf {D}}_{\text {paste}}^\text {hom}= f_\text {pore}\int \limits _{\varphi =0}^{2\pi } \int \limits _{\vartheta =0}^\pi {\mathbf {d}}_{\text {pore}} (\vartheta ,\varphi )\cdot {\mathbf {A}}_{\text {pore}}(\vartheta ,\varphi ) \frac{\sin \vartheta }{4\pi } \text{ d }\vartheta \,\text{ d }\varphi = D_{\text {paste}}^\text {hom}{\mathbf {1}}. \end{aligned}$$
(15)

For simplicity, we now choose to assume isotropic chloride diffusivity in the pore space, thus \({\mathbf {d}}_\text {pore}(\vartheta ,\varphi )=d_\text {pore}{\mathbf {1}}\). Then, insertion of Eqs. (11) and (12) into Eq. (10), and insertion of the corresponding result into Eq. (15), allows us to derive a quadratic equation that can be straightforwardly solved for \(D_{\text {paste}}^\text {hom}\):

$$\begin{aligned} D_{\text {paste}}^\text {hom}= \frac{d_\text {pore}}{2 (f_\text {pore}+9)} \left[ 3\left( 33 f_\text {pore}^2-26 f_\text {pore}+9\right) ^{0.5} +17f_\text {pore}-9\right] . \end{aligned}$$
(16)

The micro–macro relation given by Eq. (16) reveals the homogenized diffusivity of cement paste as a function of pore diffusivity and porosity, \(D_\text {paste}^\text {hom}=F(f_\text {pore},\,d_\text {pore})\), see Fig. 3(a). Dividing both sides of Eq. (16) by \(d_\text {pore}\) readily yields a dimensionless format of this micro–macro relation, reading as \({D_\text {paste}^\text {hom}/d_\text {pore}}={\mathscr {F}}(f_\text {pore})\), see Fig. 3(b). Comparing the latter ratio to the classical results of Dormieux and Lemarchand (2001), who considered a differential homogenization scheme, yielding \(D_\text {paste}^\text {hom}/d_\text {pore}=f_\text {pore}^{1.5}\), and of Hashin (1968), who considered a self-consistent homogenization scheme based on a bispherical morphology, yielding \(D_\text {paste}^\text {hom}/d_\text {pore}=2f_\text {pore}/(3-f_\text {pore})\) shows that due to the revised morphology, our model leads to significantly lower homogenized cement paste diffusivities at low porosities.

Fig. 3
figure 3

Parametric study showing a how the macroscopic chloride diffusion coefficient \(D_{\text {paste}}^\text {hom}\) is governed by the cement paste porosity \(f_\text {pore}\), and the microscopic (pore-scale) chloride diffusion coefficient \(d_{\text {pore}}\), based on evaluation of Eq. (16); and b the porosity-dependent ratio of \(D_{\text {paste}}^\text {hom}\) to \(d_{\text {pore}}\), comparing the self-consistent homogenization scheme resulting in Eq. (16), on the differential homogenization scheme by Dormieux and Lemarchand (2001), and on the self-consistent bispherical model of Hashin (1968)

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

As direct experimental determination of \(d_\text {pore}\) is out of reach (due to the small pore scales, in comparison with standard diffusivity testing devices), we will check whether many different diffusivity tests at the level of cement paste, with different porosities \(f_\text {pore}\), deliver, via Eq. (16), the same (or at least) similar values for the pore diffusivity. Such diffusivity tests are, as a rule, not directly characterized by the porosity \(f_\text {pore}\), but rather given in term of the initial water-to-cement mass ratio (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)

$$\begin{aligned} f_\text {cem}(\xi )&= \frac{20(1-\xi )}{20+63(w/c)}\ge 0, \end{aligned}$$
(17)
$$\begin{aligned} f_\text {water}(\xi )&= \frac{63\left[ (w/c)-0.42\xi \right] }{20+63(w/c)}\ge 0, \end{aligned}$$
(18)
$$\begin{aligned} f_\text {hyd}(\xi )&= \frac{43.15\xi }{20+63(w/c)}, \end{aligned}$$
(19)
$$\begin{aligned} f_\text {air}(\xi )&= \frac{3.31\xi }{20+63(w/c)}, \end{aligned}$$
(20)
Table 1 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 (\(f_\text {pore}\); \(D_\text {paste}^\text {exp}\))
Full size table

with \(f_\text {cem}(\xi )+f_\text {water}(\xi )+f_\text {hyd}(\xi )+f_\text {air}(\xi )=1\) within an RVE of cement paste. In Eqs. (17) – (20), \(\xi \) denotes the degree of hydration, \(f_\text {cem}\) the volume fraction of unhydrated cement, \(f_\text {water}\) the volume fraction of water, \(f_\text {hyd}\) the volume fraction of hydrates, and \(f_\text {air}\) the volume fraction of air pores. Note that in the presently chosen model representation of cement paste, see Fig. 1, no distinction is made between hydrates and unhydrated cement grains; instead, both are considered as non-diffusible, and thus “merged” to one solid phase. Furthermore, since the considered experimental works, see Table 1, exclusively comprise classical diffusion or migration cell tests, in the course of which the cement paste sample is tested after the relevant pore spaces became water-saturated, also no distinction is required between water and air phase; they can be merged to one pore phase. As regards the cement paste mixture, an initial water-to-cement mass ratio smaller than 0.42 usually leads to incomplete hydration of cement paste; thus, a certain portion of unhydrated cement remains. However, all pastes considered here were cured in water for time periods of at least 4 weeks, which implies sufficient additional water supply beyond that accounted for in the initial water-to-cement ratio, so that eventually the entire cement clinker became hydrated (Pivonka et al. 2004). In order to consider this situation when computing the pore space volume fractions, we follow the strategy proposed by Pivonka et al. (2004) and introduce the water-to-cement mass ratio after curing, \((w/c)_\text {cur}\), being identical to the initial water-to-cement mass ratio for \((w/c) \ge 0.42\), and amounting to 0.42 for pastes with \((w/c) < 0.42\). In addition, the aforementioned curing times also propose consideration of completed hydration, \(\xi =1\). Then, the volume fraction of the chloride diffusion-enabling pore space reads as

$$\begin{aligned} f_\text {pore}(\xi =1)= f_\text {water}(\xi =1)+f_\text {air}(\xi =1) =\frac{29.77+63(w/c)_\text {cur}}{20+63(w/c)_\text {cur}}, \end{aligned}$$
(21)

see the fifth column of Table 1.

Equation (16) allows us to find, for each data pair \(f_\text {pore}\) and \(D_\text {paste}^\text {exp}\), one corresponding value for \(d_\text {pore}\), see the last column of Table 1. Such obtained pore diffusivity values are indeed well clustered around their mean value \({\bar{d}}_\text {pore}=1.476\times 10^{-10}\) m\(^2\)/s, with a standard deviation of \(5.049\times 10^{-11}\) m\(^2\)/s.

Fig. 4
figure 4

Comparison of model-predicted and experimentally determined chloride diffusion coefficients in cement paste: experimental data, see Table 1, versus model predictions provided by Eq. (16), considering the pore diffusivities given in Eqs. (28) and (29) as well as the porosities listed in Table 1

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It is also interesting to check the statistical relevance of the identified mean pore diffusivity value in combination with the upscaling model of Sect. 3, in comparison with the porosity–diffusivity relations given through the experimental values documented in Table 1, as illustrated by different (triangle-, square-, and diamond-shaped) symbols in Fig. 4. For this purpose, we derive, from the statistical sample of pore diffusivities of Table 1, a complete statistical population of such values, and then take the 5 and 95% quantiles of this population for estimating upper and lower bounds for paste-related porosity–diffusivity relations. In more detail, we adapt the procedure proposed by König et al. (1998) in the context of the compressive strength of concrete, and refined by Pichler et al. (2005) for the impact of rock boulders onto gravel: Considering that the chloride diffusivities can take only positive values, the log-normal distribution appears as particularly relevant to capture the type of scattering observed with \(d_\text {pore}\). That implies that the logarithms of the macroscopic diffusion coefficients, back-computed by means of Eq. (16) from whatever experimental realization of a diffusion test on cement paste, make up a normally distributed statistical population. Through taking the logarithms of the values given in Table 1, we obtain one statistical sample of the aforementioned statistical population. The elements of this sample are denoted as \(\text {log}(d_{\text {pore},i}/d_{\text {pore},i}^*)\), where i is an index, \(d_{\text {pore},i}\) is the ith 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

$$\begin{aligned} {\bar{x}}= \frac{1}{n}\sum \limits _{i=1}^n\text {log} \left( \frac{d_{\text {pore},i}}{d_{\text {pore},i}^*}\right) = -22.663, \end{aligned}$$
(22)

and

$$\begin{aligned} s= \sqrt{\frac{1}{n-1}\sum \limits _{i=1}^n\left[ \text {log} \left( \frac{d_{\text {pore},i}}{d_{\text {pore},i}^*}\right) -{\bar{x}}\right] ^2}= 0.323, \end{aligned}$$
(23)

where \(n=38\) is the number of back-analyzed pore-scale diffusion coefficients. According to a famous result published by William Sealy Gosset alias “Student” (Student 1908), the normalized differences between the mean values of any sample taken from a statistical population and the mean of the population itself follow a Student 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

$$\begin{aligned} \mu _\text {low}={\bar{x}}-\frac{(t_{n-1,1-\alpha /2})s}{\sqrt{n}} \le \mu \le \mu _\text {up}={\bar{x}}+\frac{(t_{n-1,1-\alpha /2})s}{\sqrt{n}}, \end{aligned}$$
(24)

where \(t_{n-1,1-\alpha /2}\) is the so-called 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)

$$\begin{aligned} \sigma _\text {up}= \sqrt{\frac{(n-1)s^2}{\chi ^2_{n-1,\alpha }}},\quad \sigma \le \sigma _\text {up}, \end{aligned}$$
(25)

where \(\chi ^2_{n-1,\alpha }\) is the \(\chi ^2\)-value that cuts an area equal to the significance level \(\alpha \) of the Chi-squared distribution with \(n-1\) degrees of freedom (Bortz 1999). For \(n=38\) and \(\alpha =5\)%, one obtains \(\chi ^2_{n-1,\alpha }=24.1\); thus, Eq. (25) delivers \(\sigma _\text {up}=0.400\). Finally, we use the estimates for the characteristics of the normally distributed statistical population, namely the upper and lower bounds for the mean value \(\mu _\text {up}\) and \(\mu _\text {low}\) according to Eq. (24) as well as the upper bound for the standard deviation, \(\sigma _\text {up}\) according to Eq. (25), in order to obtain estimates for 5%- and the 95%-quantiles of the statistical population; i.e., we insert the aforementioned estimates in the standard definitions of quantiles, yielding

$$\begin{aligned} \text {log}\left( \frac{d_{\text {pore},i}}{d_{\text {pore},i}^*} \right) _{5\%}&= \mu _\text {low}-z_{0.025}\sigma _\text {up}, \end{aligned}$$
(26)
$$\begin{aligned} \text {log}\left( \frac{d_{\text {pore},i}}{d_{\text {pore},i}^*} \right) _{95\%}&= \mu _\text {up}+z_{0.025}\sigma _\text {up}, \end{aligned}$$
(27)

where \(z_{\alpha /2}\) is the 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, \((d_\text {pore})_{95\%}\):

$$\begin{aligned} \begin{aligned}&(d_\text {pore})_{50\%} = 1.438\times 10^{-10}\, \text {m}^2/\text {s},\\&(d_\text {pore})_{5\%} = 6.698\times 10^{-11}\, \text {m}^2/\text {s},\\&(d_\text {pore})_{95\%} = 3.088\times 10^{-10}\, \text {m}^2/\text {s}. \end{aligned} \end{aligned}$$
(28)
Fig. 5
figure 5

\(D_{\text {paste}}^\text {hom}\), as a function of the the hydration degree \(\xi \), computed for different water-to-cement ratios

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A graphical representation of these results, see Fig. 4, shows that all but one experimental data point are located within the bounds defined by the 5%-quantile and the 95%-quantile of the \(d_{\text {pore},i}\)-population. The respective outlier, however, is located within the bounds defined by the 2%-quantile and the 98%-quantile of the \(d_{\text {pore},i}\)-population, calculated analogously as demonstrated above:

$$\begin{aligned} \begin{aligned}&(d_\text {pore})_{2\%} = 5.391\times 10^{-11}\, \text {m}^2/\text {s},\\&(d_\text {pore})_{98\%} = 3.837\times 10^{-10}\, \text {m}^2/\text {s}. \end{aligned} \end{aligned}$$
(29)

In addition to this satisfactory bounding characteristics, the trend predicted by the homogenization model of Sect. 3 fed with the median of the pore diffusivity, \((d_\text {pore})_{50\%}=1.438\times 10^{-10}\) m\(^2\)/s, describes the actual, experimentally given porosity–diffusivity trend remarkably well, as underlined by a coefficient of determination of \(R^2= 0.913\).

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.

In view of the actual ordering of water molecules at the sub-nanometer scale of the “structured” or “layered” water phase, our concept involving an isotropic pore diffusivity may appear as bold approximation. Strictly speaking, the aforementioned molecular configuration would result in direction-dependent movement possibilities for the ions. In order to study the implications of the isotropic pore diffusivity modeling followed so far, we now extend our analysis to a transversely isotropic pore diffusivity tensor reading as

$$\begin{aligned} {\mathbf {d}}^\text {aniso}_\text {pore}= \left[ \begin{array}{ccc} d_\text {pore}^\text {long} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad d_\text {pore}^\text {trans} &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad d_\text {pore}^\text {trans} \\ \end{array}\right] _ {{\mathbf {e}}_r,{\mathbf {e}}_\vartheta , {\mathbf {e}}_\varphi }, \end{aligned}$$
(30)

where \(d_\text {pore}^\text {long}\) relates to the diffusivity in the pore direction and \(d_\text {pore}^\text {trans}\) denotes the diffusivity orthogonal to it. Using this format for \({\mathbf {d}}_\text {pore}\) in the expression for the homogenized diffusivity tensor, i.e., Eq. (15), yields the following homogenized cement paste diffusivity:

$$\begin{aligned} \begin{aligned} D^\text {hom,aniso}_\text {paste}&= \frac{1}{2(9+f_\text {pore})} \left\{ 2d_\text {pore}^\text {long}f_\text {pore} -9d_\text {pore}^\text {trans}+ 15d_\text {pore}^\text {trans}f_\text {pore}\right. \\&\left. \quad +\left[ 8d_\text {pore}^\text {long} d_\text {pore}^\text {trans} f_\text {pore}(9+f_\text {pore})+ \left( 2d_\text {pore}^\text {long} f_\text {pore}+3d_\text {pore}^\text {trans} (5f_\text {pore}{-}3)\right) ^2\right] ^{0.5} \right\} . \end{aligned}\nonumber \\ \end{aligned}$$
(31)

For the sake of demonstration, we have performed the back-calculation of the pore-scale diffusivity according to Eq. (31), by means of an evolution algorithm (Schwefel 1977), aiming at minimization of \(1-R^2, R^2\) being the coefficient of determination related to the deviation between the model predictions according to Eq. (31) and the experimental data according to Table 1. Such an optimization yields \(d_\text {pore}^\text {long}=1.470\times 10^{-10}\) m\(^2\)/s and \(d_\text {pore}^\text {trans}=0.806\times 10^{-10}\) m\(^2\)/s, with the maximized coefficient of determination amounting to \(R^2=0.9117\). It is now instructive to compare the performance of the model-predicted cement paste diffusivity based on anisotropic pore diffusivity behavior, to the one based on isotropic pore diffusivity behavior; for the latter, we evaluate Eq. (16), with \(d_\text {pore}=1.476\times 10^{-10}\) m\(^2\)/s, see Fig. 6. It turns out that the performances of the two modeling options are very similar. The most apparent difference is that an anisotropic pore-scale diffusivity consistently leads to lower homogenized diffusivities on the cement paste scale. However, this difference becomes relevant only for high-porosity cement pastes, i.e., for \(f_\text {pore}\) significantly larger than 0.3. Since only one experimental data point is available for \(f_\text {pore}=0.387\), and experimental data for cement paste porosities higher than that are not available to us, it appears that the true benefits of using an anisotropic pore-scale diffusivity tensor are yet to be unveiled. For the time being, we consider prescribing in our model an isotropic pore-scale diffusivity as reasonable model assumption because then the pore-scale diffusion coefficient can be simply back-calculated from the experimental data, not requiring the use of a more expensive optimization algorithm.

Fig. 6
figure 6

Comparison of \(D_\text {paste}^\text {exp}\) according to the experimental studies listed in Table 1, \(D^\text {hom,aniso}_\text {paste}\) for anisotropic pore diffusivity, according to Eq. (31), with \(d_\text {pore}^\text {long}=1.470\times 10^{-10}\) m\(^2\)/s and \(d_\text {pore}^\text {trans}=0.806\times 10^{-10}\) m\(^2\)/s, and \(D^\text {hom}_\text {paste}\) for isotropic pore diffusivity, according to Eq. (16), with \(d_\text {pore}=1.476\times 10^{-10}\) m\(^2\)/s

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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.

From the viewpoint of classical self-consistent modeling (Hershey 1954; Hill 1965), it may seem surprising that our approach does not exhibit a percolation threshold, i.e., a minimum porosity needed to allow for ionic transport. This owes to the elongated shape of the cylindrical pores, ensuring a connected, and thus diffusion-enabling network even for arbitrarily small porosities. This changes if the pore space is considered to be of non-cylindrical shape. For example, evaluating our homogenization scheme for spherical pores yields a cement paste diffusivity of

$$\begin{aligned} D_\text {paste}^\text {hom,sph}= {\left\{ \begin{array}{ll} \dfrac{d_\text {pore}}{2}\left( 3f_\text {pore}-1\right) &{}\quad \text {for}~f_\text {pore}\ge \frac{1}{3}\\ 0 &{}\quad \text {for}~f_\text {pore}<\frac{1}{3},\\ \end{array}\right. } \end{aligned}$$
(32)

indicating a percolation threshold of \(f_\text {pore}={1\over 3}\), below which no ionic transport is possible. As discussed in (Pichler et al. 2009a), cylindrical pores imply the lower bound for the diffusivity-related percolation threshold of cement paste, while spherical pores imply the respective upper bound, see also Fig. 7 for a graphical comparison of \(D_\text {paste}^\text {hom}\) according to Eq. (16) and \(D_\text {paste}^\text {hom,sph}\) as defined above; prescribing oblate or prolate spheroids (with finite aspect ratios) as pore shapes would imply percolation thresholds somewhere between \(0\le f_\text {pore}\le {\frac{1}{3}}\). In the present case, we focus on fully hydrated cement paste, with \(f_\text {pore}\ge {0.071}\). As demonstrated in experimental studies, see Table 1, such cement paste porosities 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.

Fig. 7
figure 7

Comparison of \(D_\text {paste}^\text {hom}\) according to Eq. (13) of the revised paper, considering cylindrically shaped pores, to \(D_\text {paste}^\text {hom,sph}\) according to Eq. (32), considering spherically shaped pores; the former does not exhibit any percolation threshold, while the latter features percolation at \(f_\text {pore}={\frac{1}{3}}\)

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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.