May reversible water uptake/release by hydrates explain the thermal expansion of cement paste? — Arguments from an inverse multiscale analysis

Quasi-instantaneous thermal expansion of cement pastes is governed by the relative humidity ( RH ) within their air- ﬁ lled pores and by the decrease/increase of this internal RH resulting from a temperature decrease/increase. The latter e ﬀ ect is traced back to quasi-instantaneous water uptake/release by cement hydrates, using micro- poromechanics and a three-scale representation of mature cement pastes. Partially saturated gel and capillary pores are considered to be connected and spherical, with radii following exponential distributions. The Mori- Tanaka scheme provides the scale transition from e ﬀ ective pore pressures to eigenstrains at the cement paste level. This modeling approach, together with considering mass conservation of water, allows for downscaling macroscopic thermal expansion coe ﬃ cients, so as to identify the molecular water uptake/release characteristics of the hydrates. The latter characteristics are mixture-independent, as shown by their use for predicting the thermal expansion coe ﬃ cients of di ﬀ erent mature cement pastes, with w / c -ratios ranging from 0.50 to 0.70.


Introduction
External heating/cooling of a millimeter-to-decimeter-sized sample of partially saturated (drained) cement paste actives two transport processes. Heat transport provides the transition between two different isothermal states [1][2][3]. This takes typically from minutes to hours, depending on the size of the specimen and on the magnitude of the temperature change, ΔT. The increase/decrease of temperature goes hand in hand with both thermal expansion/contraction of the cement paste sample and an increase/decrease of its internal relative humidity. The latter change activates a moisture transport process between the open porosity of the specimen and the ambient air. This takes typically from days to weeks. The corresponding deformation partly recovers the thermal expansion/contraction [1,2,4]. In saturated (or close to saturated) conditions, delayed deformations under constant temperature have been attributed to macroscopic water transport caused by overpressure of water expanding more than the solid skeleton, see Ref. [5].
The coefficient of thermal expansion of mature cement paste ranges from 8 ⋅ 10 −6 K −1 to 22 ⋅ 10 −6 K −1 and appears as an asymmetrical bell-shaped function of relative humidity which prevails in the air-filled pores just before the applied temperature change, see Refs. [6][7][8][9][10] and Fig. 1. Knowledge of such thermal expansion coefficients is important for practical applications in concrete construction, see, e.g. Ref. [11]. In this context, it is noteworthy that concrete aggregates have typically smaller thermal expansion coefficients, ranging between 4 ⋅ 10 −6 K −1 and 12 ⋅ 10 −6 K −1 , depending on the mineral composition [9,12,13]. Therefore, subjecting concrete to a temperature change, yields a mismatch of thermal eigenstrains. This results in thermal (micro-)stresses, which may even cause microcracking [14][15][16], reducing the durability of concrete structures.
Quasi-instantaneous expansion due to the temperature increase ΔT goes hand in hand with an increase of the internal relative humidity ΔRH, see, e.g. measurements documented in Refs. [2,10,[17][18][19]. These experimental data suggest that the ratio ΔRH/ΔT, referred to as the hygrothermic coefficient [3], is a nonlinear function of the internal relative humidity which prevails in the air-filled pores just before the temperature change, see Fig. 2. In addition, the hygrothermic coefficient is considered to depend on the initial water-to-cement mass ratio [3,19], but the underlying physical processes are still not fully understood.
The ongoing scientific discussion regarding the origin of hygrothermic coefficients mainly concerns the redistribution of liquid water. In this context, Bažant [2] attributed hygrothermic coefficients to water migration from gel pores to capillary pores, considering that increasing temperature causes a chemical potential difference between the gel and the capillary water. Grasley et al. [10] and Radjy et al. [19] estimated temperature-induced changes of relative humidity RH. For that, they used Kelvin's equation, considered that the surface tension at liquid-gas interfaces decreases with increasing temperature according to measurements [20], and treated the radius of the corresponding meniscus between pore water and air as a constant. The correspondingly estimated increase of relative humidity underestimates the measured hygrothermic coefficients. Therefore, Grasley and Lange proposed water expansion in ink-bottle pores to be responsible for the aforementioned measured values of hygrothermic coefficients [10]. This indicates that the radius of the meniscus between the pore water and the air increases with increasing temperature. Wyrzykowski and Lura [21] obtained hygrothermic coefficients by means of water activity and sorption measurements. The two independent methods associated with the aforementioned measurements delivered consistent results. Using a model linking capillary pressure to elastic deformation [22][23][24], Wyrzykowski and Lura related the measured early-age evolution of the hygrothermic coefficients to the measured coefficients of thermal expansion. They did so for samples aged between 1 and 7 days [21], when the measured relative humidity decreased from full saturation to values, ranging between 85% and 94%. Still, modeling of hygrothermic coefficients of mature cement pastes in the regime of intermediate relative humidities from 25% to 85% remains an open research question.
Existing models for thermal expansion of cement paste are of hygrothermoporoelastic and macroscopic nature. They typically consider two contributions: (i) thermal expansion of the solid skeleton, α cp sol , and (ii) hygrothermic expansion resulting from pore pressure changes [2,10,21,25], α cp hyg : cp mod cp sol cp hyg (1) As regards the latter, the Kelvin-Laplace equation is used to quantify capillary pressure from knowledge on temperature and relative humidity. Increasing temperature and the associated increase of internal relative humidity result in a decrease of capillary pressure Δp c and an increase of the saturation ratio ΔS r . The resulting hygrothermic expansion Δε hygro is typically quantified as [10,21,24] (2) where k hom and k s , respectively, are the bulk moduli of the porous material and of the solid skeleton. Eq. (2) is theoretically rigorous for liquid saturation, i.e. for S r = 100%, see Refs. [21,24], and it is of approximate nature in case of partial saturation [24,26]. While Eq. (2) could be shown to be reliable for saturation degrees S r ≥ 80%, it underestimates the hygrothermic expansion at smaller saturation degrees [21,24].
In the present paper, hygrothermic coefficients and thermal expansion of mature cement pastes are explained by quasi-instantaneous uptake or release of water by the hydrates. This is an effect for which no physical measurements were available at the time of submitting the present manuscript. However, such measurements were reported shortly thereafter [27]. In the present context of modeling, the hydrates are, in the sense of a reductionist approach, not subdivided explicitly into different hydration products, such as calcium-silicate-hydrates (C-S-H), portlandite, and ettringite. Still, it is noteworthy that water (H) in the C-S-H can be considered to be at least partly mobile [28,29], while the water chemically bound in crystalline products, such as portlandite and ettringite, cannot be considered as at least partially mobile.
Methods of microporomechanics [26,60] in combinations with the mass conservation law are used to identify the quasi-instantaneous uptake or release of water by hydrates, with focus on mature cement pastes and on the central regime of relative humidities ranging from 25% to 85%. To this end, unhydrated clinker grains, hydrates, capillary pores, and gel pores are introduced as elementary constituents in a three-scale representation of mature cement pastes. The pores are considered to be well connected and spherical. They are either filled by water or by air, following the Kelvin-Laplace equation. Thin layers of adsorbed water are taken into account at solid surfaces, including the surfaces of "air-filled" pores [30]. The size distributions of gel and capillary pores are modeled by exponential distributions, following Huang et al. [31]. These distributions are identified such that the model-predicted adsorption isotherms agree as well as possible with Fig. 1. Relation between the coefficient of thermal expansion and the internal relative humidity which prevails in the air-filled pores, just before the temperature change: experimental data from Meyers [6], Mitchell [7], and Dettling [8], measured on mature cement pastes with initial water-to-cement mass ratios w/c ∈ [0.12; 0.40]; the curve described by Emanuel et al. [9]; and the "master curve" given in Eq. (3).

Fig. 2.
Hygrothermic coefficient of cement paste as a function of the internal relative humidity, prevailing just before the temperature change: test results from Nilsson [17], Persson [18], Radjy et al. [19], Grasley et al. [10], and Bažant [2]. corresponding Brunauer-Xi-type adsorption isotherms [32,33]. In the context of fluid-solid interactions, the solid skeleton of cement pastes is assumed to be loaded by effective pore pressures. The latter combine the actions of fluid pressure and surface tension prevailing at solid-fluid interfaces [26,34,35]. Effective pore pressure changes are quantified from temperature changes and from the corresponding changes of the internal relative humidity. The required scale transitions to macroscopic thermal eigenstrains of cement pastes are established by the Mori-Tanaka scheme [36,37,59]. The resulting analytical multiscale model provides the desired link between the macroscopic thermal expansion of cement pastes and hygrothermic coefficients. The water uptake/release by hydration products is finally identified, in the context of inverse multiscale analysis, starting from the hygrothermic coefficients and based on the mass conservation of solid and liquid water inside a tested specimen. This is motivated by the fact that temperature changes typically take place in time intervals which are significantly shorter than the ones required for a significant exchange of water between the cement paste samples and their ambient environment. The corresponding mass balance includes water which is bound inside solid hydrates, bulk pore water, and water adsorbed to solid surfaces.
In order to identify water uptake/release coefficients and to show that they are a material property of hydrates, independent of the composition of cement pastes, the paper is structured as follows. Measured thermal expansion coefficients of cement pastes with initial water-to-cement mass ratio w/c = 0.40 are downscaled in order to identify the temperature-induced water uptake/release by hydrates as a function of the relative humidity and the temperature change, see Section 2. Considering the identified water uptake/release characteristics to be a mixture-independent material property of hydrates, the multiscale hygrothermoporomechanics model is used to predict w/cdependent hygrothermic coefficients and macroscopic thermal expansion coefficients of cement pastes with w/c values ranging from 0.50 to 0.70, see Section 3. A discussion, see Section 4, is followed by concluding remarks, see Section 5.
2. Top-down identification of the water uptake/release by hydration products as a function of the internal relative humidity and the temperature change 2.1. Thermal expansion of mature cement paste as a function of the internal relative humidity Meyers [6], Mitchell [7], and Dettling [8] measured the thermal expansion of mature cement pastes (at material ages up to six months) as a function of the relative humidity, see Fig. 1. Independent of the initial water-to-cement mass ratio, w/c ∈ [0.12; 0.40], the smallest thermal expansion develops at full saturation (RH = 100%). The maximum thermal expansion is virtually twice as large and develops at RH ≈ 65%, and the thermal expansion at the fully dried state (RH = 0%) is only slightly larger than the one at full saturation. The following regression function reproduces the experimental data satisfactorily It is very similar to the non-analytic fitting function used by Emanuel and Hulsey [9], see Fig. 1.

2.2.
Hierarchical organization of mature cement pastes, elastic constants, and volume fractions Cement pastes are (micro-)heterogeneous materials, consisting of unhydrated cement clinker grains, hydration products, called hydrates, and water-or air-filled gel and capillary pores, with typical lengths ranging from a few nanometers to several tens of micrometers. The hierarchical organization of cement pastes is modeled by means of three matrix-inclusion composites, introduced at different scales of observation: (i) cement paste consists of a hydrate foam matrix, hosting spherical unhydrated clinker inclusions (Fig. 3a); (ii) the hydrate foam consists of a hydrate gel matrix surrounding the spherical capillary pores (Fig. 3b); and (iii) the hydrate gel consists of a solid hydrate matrix with embedded spherical gel pores, (Fig. 3c). This is a simplified version of the hierarchical representation introduced in [39], which is admissible in the case of mature cement pastes where the specific morphology of the gel-porous hydrates does not play an important role [38]. This allows for consideration of a continuous solid hydrate matrix (Fig. 3c).
Hydrates and clinker are considered to be isotropic. Their elastic stiffness tensors are expressed in terms of their bulk moduli, k, and shear moduli, μ: In Eq. (4),  dev stands for the deviatoric part of the fourth-order unity tensor, defined as    = − dev vol , where  is the symmetric fourth-order unity tensor with components I ijrs = 1/2 (δ ir δ js + δ is δ jr ), and stands for the volumetric part of the fourth-order unity tensor, where 1 denotes the second-order unity tensor with the Kronecker delta δ ij as its components, i.e. δ ij = 1 for i = j, and δ ij = 0 otherwise. Pores, in turn, are characterized by a vanishing solid stiffness: ℂ cpor = ℂ gpor = 0.
The volume fractions of the cement paste constituents are functions of the initial water-to-cement mass ratio, w/c, and of the hydration degree ξ [41][42][43][44]. They are quantified based on Powers' 1946Powers' /1957 hydration model [41,42], see also Refs. [43,44] for detailed descriptions. At the cement paste scale, the volume fractions of the hydrate foam matrix and of the unhydrated clinker grains read as [39]: At the hydrate gel scale, the volume fractions of the solid hydrate matrix and of the gel pores read as [40]: Pores in cement paste are either filled by pore water or occupied by air, with thin layers of adsorbed water covering the pore surfaces, see Fig. 4. The thickness of these layers, t, increases with increasing relative humidity, as quantified by [30]: The thickness of these layers also appears in the Kelvin-Cohan equation for the radius r ℓg [45], discriminating between water-filled and air-filled pores, Pores with radii r ≤ r ℓg are filled with water, while pores with r > r ℓg are filled with air, see Fig. 4. In Eq. (10), R = 8.31446 J mol −1 K −1 , T, and γ ℓg denote the universal gas constant, the absolute temperature, and the temperature-dependent surface tension of interfaces between liquid and gas [20], respectively; ν m is the molar volume of water, which is equal to the molar mass of water M H O 2 = 18.016 g mol −1 , divided by its temperature-dependent mass density [46], see Table 1; θ is the contact angle between the liquid and the solid, which is so small that it is usually set equal to zero for cement-based materials, i.e. pore water is considered to be an almost perfectly wetting liquid [22,47,48]. Consideration of the temperature-dependent mass density of bulk water (Table 1) is the way thermal expansion of bulk pore water is accounted for in the present contribution, see also Section 4.3.
The solid skeleton of cement pastes is subjected to effective pore pressures p, combining the effects of fluid pressure and surface tension [26,34,35]. Because of the latter, the effective pressures are functions of the pore radius r [34] where γ sℓ and γ sg denote the surface tension at the solid-liquid and the solid-gas interfaces, respectively. The difference between the gas pressure p g and the liquid pressure p ℓ is defined as the capillary pressure p c [35], i.e.
which follows from the Kelvin-Laplace equation, reading as Equilibrium at the 2D liquid-gas interfaces, also called "menisci", entails that capillary pressure is proportional to the mean curvature of the menisci [34] where b denotes the curvature tensor of the menisci, and "tr" stands for "trace". In the present context of spherical pores, the menisci are spherical. Accordingly, |trb| is equal to 2/r ℓg in case of a perfectly wetting fluid, and equal to 2cosθ/r ℓg otherwise [49]. Applying the Cohan-extension to the latter expression yields Insertion of Eq. (15) into Eq. (14), and of the resulting expression for p c into Eq. (13), yields, after re-arrangement of terms, the expression for r ℓg according to Eq. (10). Thus, the present developments consider that the size of the menisci is a nonlinear function of the relative humidity and the temperature, as described by the Kelvin-Laplace equation. As the pores of the cement pastes are assumed to be well-connected, the gas pressure p g is set equal to the atmospheric pressure. Moreover, in the sense of a reference pressure, the latter is set equal to zero. Thus, Eq. (12) results in The surface tensions in Eqs. (10) and (11) occur in Young's equation, describing the thermodynamic equilibrium of a solid-liquid-gas system: Out of the four quantities in Eq. (17), only the surface tension between the liquid and the gas γ ℓg and the contact angle θ are directly measurable [20]. As a remedy, Berthelot's state equation [50][51][52], see Appendix A, is used as a supplementary element for quantification of γ sg and γ sℓ : Considering an almost perfectly wetting fluid, θ ≈ 0, resulting in cosθ ≈ 1, and combining Eqs. (17) and (18), gives Consideration of Eqs. (13), (16), and (19) in Eq. (11) simplifies the expression for the effective pressures as follows: Notably, both the molar volume of water, ν m , and the surface tension γ ℓg are temperature-dependent, see Table 1, and the thickness of the adsorbed water layer, t, is a function of the internal relative humidity, see Eq. (9). For upscaling of these effective pressures to the material scale of Fig. 4. Sketch of the pores coated by absorbed water at their surfaces, and filled either with air or pore water; two dimensional sketches of three dimensional configurations. Table 1 Temperature-dependent surface tension [20], mass density [46], and molar volume of water.  [13][14][15][16][17][18][19][20][21][22][23][24][25][26] cement pastes, average effective pore pressure of gel and capillary pores, p gpor and p cpor , are relevant [34,35]. Their quantification requires a mathematical description of the pore size distributions of the gel and the capillary pores. Following Refs. [31,53], exponential distributions are introduced, i.e. the pore size probability distribution functions read as with R gpro and R cpor as the characteristic radii of the gel and the capillary pores, respectively, and with ϕ r d gpor pdf and ϕ r d cpor pdf representing the portions of the gel and capillary porosity which are associated to pores with radii ranging from r to (r + dr). This allows for averaging the effective pressures (Eq. (20)) individually over the respective pore size distributions of the gel and the capillary pores as Notably, R gpor and R cpor represent the characteristic radii of the gel and the capillary pores, respectively.

Identification of characteristic pore radii, R gpor and R cpor , from adsorption porosimetry
Adsorption porosimetry [54][55][56] starts from a dried reference state and quantifies the mass of liquid water m H O 2 entering the specimen during a step-by-step increase of the relative humidity. Because complete drying is difficult to achieve [54], a small value of relative humidity, (herein: RH = 1%), is associated with the dried reference state. Therefore, the modeled saturation ratio ∈ S [0; 1] r mod follows as: In order to link the water mass m H O 2 in Eq. (23) to the pore size distributions (Eq. (21)), the total pore space (i.e. the sum of gel and capillary pores) is subdivided into a water-filled part (bulk water and adsorbed water) and an air-filled part, discriminated based on r ℓg , see Eq. (10). The corresponding volume fractions add up to one: . These volume fractions provide access to the water mass inside the pores as where V por is the total pore volume and ρ H O 2 is the temperature-dependent water mass density [46]. The volume fraction of air is obtained by integration over the "air-filled" pores as where the term in brackets accounts for the adsorbed water layers, see Fig. 4. Notably, both the lower integration limit, r ℓg , and the thickness of the adsorbed water layer, t, are nonlinear functions of the relative humidity, see Eqs. (10) and (9). The symbol ϕ r ( ) por pdf denotes the pore size probability distribution function of all pores. The latter results from superposition of the pore size probability distribution functions of the gel and the capillary pores, see Eq. (21), as where f gpor por and f cpor por denote the volume fractions of the gel and the capillary pores, respectively, with regard to the total pore volume. These two quantities can be directly derived from the hydration model, Eqs. (6)- (8), as  (27), and considering that r ℓg in Eq. (25) is a function of the relative humidity, see Eq. (10), allows for expressing the modelpredicted saturation ratio S r mod as a function of the relative humidity RH and the characteristic pore radii R gpor and R cpor . The latter are identified in the following. Many experimental adsorption isotherms were fitted by Brunauer et al. [32,57], using the following function For cementitious materials, semi-empirical formulae for C and k were established by Xi et al. [33]: where K stands for the SI unit Kelvin, and n is a function of the age t of the material and of its initial water-to-cement mass ratio w/c, which reads as where d stands for the physical unit day. Finally, N ct is equal to 1.1 for type I Portland cement. The characteristic pore radii R gpor and R cpor are identified by minimization of the square root of the sum of the squared errors, ε SRSS , quantifying the difference between saturation degrees derived from the pore size distribution model, see S r mod in Eq. (23), and from adsorption isotherms, see S r exp in Eq. (28), where RH i = i/100. The optimization problem Eq. (31) is solved numerically using continuously refined search intervals, as detailed in Ref. [58]. Optimal characteristic radii of the gel and the capillary pores are identified for mature cement pastes with initial water-to-cement mass ratios w/c ∈ [0.40; 0.70], made of type I Portland cement, cured at room temperature (T = 293.15 K) for 90 days (ξ ≈ 0.8) [43], such that γ ℓg = 0.07275 N m −1 [20] and ρ H O 2 = 998.2067 kg m −3 [46], see Table 2. The identified pore radii result in a very good agreement between the model-predicted adsorption isotherms and the Brunauer-Xitype counterparts, see the error values ε SSRS listed in Table 2 and Fig. 5 (a). In addition, the identified values of R gpor and R cpor are close to 2 nm Table 2 Volume fractions and characteristic radii of gel and capillary pores of mature cement pastes with initial water-to-cement mass ratios w/c ∈ [0. 40 [13][14][15][16][17][18][19][20][21][22][23][24][25][26] and 12 nm, respectively. This is in very good agreement with previous studies [15], which were also based on sorption isotherms. The cumulative pore size distribution function r Φ ( ) por cdf , representing the integral of the pore size probability distribution function Eq. (26) over the pore radius, reads as It is well suited to illustrate the identified pore size distributions, see Fig. 5 (b). Based on the identified pore radii R gpor and R cpor , the average effective pore pressures p gpor and p cpor can be quantified for any value of the internal relative humidity RH and any temperature T, see Eqs. (22), (20), and (9). Returning to the hierarchical organization of cement pastes as illustrated in Fig. 3, the average effective pore pressures are to be upscaled to the macroscopic scale of cement pastes.

Hygrothermoporoelastic homogenization of mature cement pastes
The microstructure of mature cement pastes is subdivided into three matrix-inclusion composites, see Fig. 3. This renders the Mori-Tanaka scheme appropriate for analytical homogenization of poroelastic properties [26,36,37,59,60,81]. It allows for analytical homogenization of poroelastic properties in the framework of microporomechanics [60]. Each representative volume element (RVE) with volume V RVE is subdivided into a matrix phase and an inclusion phase, occupying the domains V m and V i , respectively. Both material phases k ∈ [m;i] exhibit a specific elastic stiffness ℂ k , a specific eigenstress σ k E , and a known volume fraction f k : The generalized Hooke's law of the homogenized composite follows from Levin's theorem [61], containing the macroscopic stress Σ, the macroscopic strain E, the homogenized stiffness  hom , and the homogenized eigenstress Σ E hom [26,61]: where  m and  i are the phase strain concentration tensors of the two material phases, and symbol ":" refers to second-order tensor contraction. Mori-Tanaka estimates of these two tensors read as [37,59] where  is the Eshelby tensor of a spherical inclusion embedded in an infinite matrix of stiffness ℂ m , defined as [62] The homogenized stiffness tensor is obtained from inserting Eq. (38) into Eq. (37), followed by substituting the result together with Eq. (4) into Eq. (35): In order to homogenize the thermal expansion of cement pastes, analytical upscaling of the elastic stiffness, see Eqs.  steps, for the matrix-inclusion composites of the cement pastes shown in Fig. 3. For the inputs and the homogenization results, see Table 3. To this end, free thermal expansion of cement paste is considered, i.e. Σ cp = 0. Inserting the vanishing macrostress into Eq. (34) yields, after solving for the macrostrain E cp , the macroscopic eigenstrain E cp E as The thermal expansion coefficient of cement paste is proportional to the hygrothermic eigenstrain difference, quantified for a temperature difference ΔT, i.e.
Herein, T = 293.15 K and ΔT = 1 K are considered, for corresponding values of the surface tension γ ℓg and the mass density of water ρ H O 2 , see Table 1. In addition, the change of the internal relative humidity resulting from a temperature change is taken into account, see the first term on the right-hand-side of Eq. (49).

Identification of the solid expansion coefficient
In order to identify the unknown thermal expansion coefficients of hydrates and clinker, see Table 3, they are assumed to be equal, i.e. α hyd = α clin . They are identified by means of the thermal expansion coefficients measured on fully dried cement pastes, i.e. RH = 0%. In this context, the temperature variation is considered to result in thermal dilatation of the solid constituents, while the average effective pore pressures remain constant. Consequently, see Fig. 1. The proof that the homogenized thermal expansion coefficient of the matrix-inclusion composites in Fig. 3 is equal to that of the solids, see Eq. (50), is given in Appendix B. For a more refined approach, which abandons the assumption of α hyd = α clin , see Appendix C.

2.7.
Identification of hygrothermic coefficients of cement pastes with w/c = 0.40 and ξ = 0.80 The homogenization approach described in Section 2.5 is used for identification of hygrothermic coefficients. The resulting inverse problem is solved for relative humidities RH ∈ [20%; 85%]. For each value of the relative humidity, the hygrothermic coefficient ΔRH/ΔT is determined such that the homogenized thermal expansion coefficient α cp , see Eq. (49), is equal to the value of the experimental master curve, see Fig. 1 and the regression function in Eq. (3). The identified hygrothermic coefficients form an asymmetrical bell-shaped function of the internal relative humidity which prevails in the air-filled pores just before the temperature change, see Fig. 6. The results from the identification agree well with measurement results by Nilsson [17] and Grasley [10], see Fig. 6.

Identification of the water uptake/release by hydration products as a function of the internal relative humidity and temperature change
Thermal expansion of cement paste occurs on the way to a uniform temperature distribution inside a tested specimen. Such a process takes typically from several minutes to a few hours, depending on the size of the specimen and on the magnitude of the temperature change. It is significantly shorter than the period of time required to balance the internal and the external relative humidities, prevailing inside and around partially saturated and drained cement paste samples. This balance requires a significant water transport through the pore network and frequently takes several weeks. Based on this "separation of time scales"-argument, it is concluded that the water mass exchange between the cement paste specimens and their exterior environment is insignificant as regards characterization of thermal expansion coefficients, so that the cement pastes may be treated as closed systems, referred to in poromechanics as "sealed conditions". Consequently, mass conservation can be considered as the basis for quantifying the water uptake/release by the hydration products.
A temperature increase results in an increase of the internal relative humidity, see Fig. 2, and this yields both an increase of the thickness of the adsorbed water layer, see Eq. (9), and an increase of the radius r ℓg , see Eq. (10). Both effects underline that the volume of liquid water (both adsorbed water and bulk water) increases. Using the presented model, this cannot be explained by regular or anomalous thermal expansion of water alone, see Section 4.3. Given that the overall mass of the cement paste is constant (= mass conservation, see above) and, Table 3 Step-by-step homogenization of mature cement pastes, subdivided into homogenization of the hydrate gel, the hydrate foam, and the cement paste, see also  (39) and (40).  [17] and Grasley [10] for mature cement pastes with initial water-to-cement mass ratio of 0.40.
thus, that the total mass of water is constant, it is concluded that the hydration products act as a water source, i.e. they release water can be calculated using the developed multiscale poromechanics model, see Eqs. (24)- (27), under consideration that r ℓg in Eq. (25) is a nonlinear function of the relative humidity and of the temperature, see Eq. (10). Notably, the change of the mass of water vapor is by orders of magnitude smaller than the change of the mass of liquid water. Therefore, water vapor was disregarded in Eq. (51). Finally, the dimensionless "specific water uptake/release" of solid hydrates, Δμ, is introduced as the mass loss of the solid hydrates, see Eq.  (53) where ρ hyd = 2.85 ⋅ 10 3 kg m −3 [29] denotes the mass density of the solid hydrates. The initial volume of solid hydrates, V hyd , is expressed, by analogy to Eq. (24), through the pore volume V por as (6)- (8).
The water uptake/release coefficient Δμ/ΔT is identified based on the following expression, resulting from the combination of Eqs. (51)-(54) with Eqs. (24) and (25):   Table 1, w/c = 0.40 and ξ = 0.80, as well as the identified hygrothermic coefficients illustrated in Fig. 6, the water uptake/release coefficient Δμ/ΔT is identified. It forms an asymmetrical bell-shaped function of the internal relative humidity which prevails in the air-filled pores just before the temperature change, see Fig. 7. The maximum of Δμ/ΔT amounts to 6.2 ⋅ 10 −4 K −1 and refers to the internal relative humidity RH = 72%. The characteristic order of magnitude of water uptake/release coefficients underlines that the exchange between solid and liquid water is a rather subtle effect, but still large enough as to result in significant hygrothermic coefficients and in the thermal expansion coefficients illustrated in Fig. 1. The water uptake/release coefficients (Fig. 7) represent a material property of solid hydrates. Therefore, they are valid for all Portland cement pastes, irrespective of their composition, and not only for w/c = 0.40, for which the top-down analysis was carried out. In order to prove the mixture-independence of the water uptake/release coefficients exemplarily, a bottom-up approach is applied to cement pastes with w/c = [0.50; 0.60; 0.70]. The aim is to predict hygrothermic coefficients and thermal expansion coefficients of cement pastes, and to compare them with experimental data that were not involved in the identification of the water uptake/release coefficients.

Prediction of hygrothermic coefficients of mature cement pastes with different initial water-to-cement mass ratios
Hygrothermic coefficients are predicted on the basis of mass conservation, as explained in Section 2.8. They follow from the water uptake/release coefficients shown in Fig. 7 and from the developed pore size distribution model, see Eqs. (26)- (27) and Table 2. The predicted hygrothermic coefficients form similarly shaped asymmetrical bellshaped functions which are the smaller, the larger the initial water-tocement mass ratio, see Fig. 8. The quantitative differences can be explained by the fact that the porosity of mature cement pastes increases with increasing initial water-to-cement mass ratio. As for regression functions rendering model predictions accessible, see Appendix D.
As for the exemplary validation, the hygrothermic coefficients predicted for w/c = 0.50, are compared with experimental data from Persson [18], Radjy [19], and Grasley [10], who tested mature cement pastes with w/c ∈ [0.48; 0.50]. Given that measured hygrothermic coefficients exhibit a rather large intrinsic experimental uncertainty, the obtained agreement is satisfactory, see Fig. 9. This corroborates that the identified water uptake/release coefficients (Fig. 7) are indeed a material property of solid hydrates. Fig. 7. Identified specific water uptake/release by hydration products per unit temperature increase, referred to as "water uptake/release coefficients", see Eq. (55).

Prediction of thermal expansion coefficients of mature cement pastes with different initial water-to-cement mass ratios
Thermal expansion coefficients of cement pastes are predicted by using the hygrothermic coefficients shown in Fig. 8 as input for the developed multiscale poroelastic model, see Eqs. (1)- (49) and Table 3. The predicted thermal expansion coefficients form a dense bunch of similarly shaped asymmetrical bell-shaped functions. Quantitatively, they exhibit only very small fluctuations for initial water-to-cement mass ratio in the range from 0.12 to 0.70, see Fig. 10.
In the context of the exemplary validation, it is remarkable that the thermal expansion coefficients predicted for initial water-to-cement mass ratios in the interval w/c ∈ In addition, the model predictions are compared with measured thermal expansion coefficients of Grasley [10], which were not included in Fig. 1 and, therefore, neither contributed to the master curve of Eq. (3), nor to the identification of the water uptake/release coefficients. The agreement between the model predictions and the experimental data is again satisfactory, see Fig. 10. This further corroborates that the identified water uptake/release coefficients (Fig. 7) are indeed a mixture-independent material property of hydration products.

Discussion
Redistribution of liquid water within the microstructure of cement pastes is standardly considered to govern thermohygroporomechanical phenomena with characteristic duration from a few seconds to several hours. This dates back to Powers [41,63], who proposed (i) that shortterm mechanical or thermal loading of partially saturated concrete disturbs the free-energy equilibrium, such that water redistributes within nanoscopic pores of different sizes, and (ii) that this migration of water results in considerable macroscopic deformation of concrete. This idea has motivated many follow-up research activities related to both modeling and experimentation, see Refs. [2,[64][65][66] and [10,[17][18][19]21,67]. However, the built-up of pore pressure resulting from a temperature increase of liquid-saturated cement pastes cannot be explained by redistribution of liquid water. Therefore, it was related to an anomalous thermal expansion of water [68][69][70]. The proposed quasiinstantaneous water uptake/release by the hydration products, in turn, can explain both the redistribution of liquid water under partially saturated conditions and the pressurization of pore water upon increasing the temperature under liquid-saturated conditions. Water uptake/release by hydration products was so far mainly considered in the context of long-term sorption processes with characteristic duration from several days to several weeks. Related research focuses particularly on calcium-silicate-hydrates (C-S-H), i.e. on the most important type of cement hydrates. Despite very intensive research activities, the molecular composition of C-S-H is not yet fully understood. As regards the water content of C-S-H, ongoing discussions refer to typical chemical compositions ranging, e.g. from 1.7 CaO ⋅SiO 2 ⋅ 1.8 H 2 O, as quantified by small angle scattering techniques [28], to 1.7 CaO ⋅SiO 2 ⋅ 1.3 H 2 O, as determined by means of gravimetry for d-dried C-S-H [29]. These two exemplary formulae underline that the water content of C-S-H is variable, at least during long-term sorption experiments.
Herein, it was essentially proposed that the water uptake/release by the hydration products is not necessarily a very slow process. It appears to be actually fast enough to be the origin of quasi-instantaneous water redistributions, associated with short-term thermohygroporomechanical phenomena. This idea is consistent with the polymerization of mature cement paste under room temperature [71][72][73], where chemical bonds between silicon atoms and hydroxyl groups change upon a moderate temperature increase such that the liquid water is set free, i.e.
Also, water uptake/release by hydrates is reminiscent of quasi-instantaneous changes of internal relative humidity resulting from mechanical loading of cement pastes, as described by Wyrzykowski and Lura [67]. In addition, evidence of some fast water redistribution was reported by Gajewicz et al. [74], in the context of 1 H NMR relaxometry investigation of gel-pore drying shrinkage in cement pastes.

4.1.
Temperature-driven water uptake/release by hydrates: a first direct experimental evidence provided by Wyrzykowski et al. [27] After submission of the original version of the present manuscript, our idea of the hydrates taking up/releasing water upon temperature changes, was strongly supported by direct experimental evidence [27]. Proton nuclear magnetic resonance ( 1 H NMR) studies of mature cement pastes [27] showed heating-induced transport of water from the C-S-H interlayers to the gel/capillary porosities, and vice versa. The observed water redistribution process has the following characteristics: 1. quasi-instantaneous nature and full reversibility, 2. independence of the initial water-to-cement mass ratio, at least within the investigated interval of w/c ∈ [0.25; 0.50], and 3. dependence on the internal relative humidity prevailing inside the Fig. 9. Comparison of hygrothermic coefficients, measured by Grasley [10], Persson [18], and Radjy et al. [19], with model predictions for mature cement paste with an initial water-to-cement mass ratio of 0.50 and hydration degree ξ = 0.80. air-filled pores right before the temperature change, with the maximum water exchange taking place around RH = 75%.
In addition, the observed water redistribution process was conceptually coupled to both hygrothermic effects and thermal expansion of mature cement pastes [27].
In the present paper, values of cooling/heating-driven uptake/release of water by hydrates are identified for mature cement pastes, utilizing top-down multiscale analysis. Thereby, uptake/release of water by hydrates was 1. assumed to be a quasi-instantaneous and fully reversible process, 2. found to be universal in the sense of being independent of the initial water-to-cement mass ratio, at least in the studied interval of w/c ∈ [0.40; 0.70], and 3. found to be a nonlinear function of the internal relative humidity prevailing inside the air-filled pores right before the temperature change, with the maximum intensity around RH = 72%.
In addition, the water redistribution process was linked quantitatively both to hygrothermic coefficients and to thermal expansion coefficients of mature cement pastes. The previous two paragraphs underline that two independent methods, a theoretical method based on multiscale microporomechanics and an experimental method based on 1 H NMR [27] have led to virtually the same conclusions. This supports the idea of water uptake/release by hydrates as the mechanism governing hygrothermic coefficients and thermal expansion of cement paste, particularly so within the herein studied interval of RH ∈ [25%; 85%].

Additional mechanism: rearrangements of the nanostructure
Considering the water uptake/release by hydration products as the origin of hygrothermic coefficients, delivers a satisfactory description of the thermal expansion of mature cement paste in the important range of the relative humidity from 25% to 85%. The regimes of very small and very large relative humidities have not been explored herein. In these regimes, rearrangements of the nanostructure of cement pastes are likely to take place: • At very small relative humidities (RH < 20%), nanostructural transformations may not only comprise removal of interlayer water from C-S-H, as concluded from isothermal sorption experiments [29,75], but also rearrangement of the C-S-H sheets themselves, as concluded from 1 H NMR measurements [74].
• At very large relative humidities (RH > 85%), individual calcium silicate sheets, positioned at the surface of layered calcium-silicatehydrates, may detach from the rest and bend into the adjacent gel pore, subdividing the latter into two smaller pores [27].
These microstructural rearrangements may give rise to hygrothermic coefficients which are significantly larger than the ones identified in the present paper, compare the experimental values reported in Figs. 6, 8, and 9, referring to the range from 85% to 90%, with the results of the measurements performed by Wyrzykowski and Lura [21]. In this context, it is also noteworthy that Wyrzykowski and Lura [21] studied young cement pastes, with material ages ranging from 1 day to 7 days, produced with initial water-to-cement mass ratios ranging from 0.22 to 0.35, and tested under relative humidities ranging from 86% to 98%. Herein, the study focused on mature cement pastes, with material ages amounting up to 6 months, produced with initial water-to-cement mass ratios ranging from 0.40 to 0.50, and tested under relative humidities ranging from some 25% to some 85%. This underlines that the two studies do neither overlap in terms of material composition nor in terms of material age, and that the overlap in terms of the test conditions is very small (if at all existing).

Additional mechanism: anomalous expansion of liquid pore water
Consideration of water uptake/release by hydrates allows for explaining hygrothermic coefficients in the interval of RH ∈ [25%; 85%], of several mature cement pastes, differing in the initial water-to-cement mass ratio. Another mechanism that is quite frequently discussed in this context, is anomalous thermal expansion of confined, rather than bulk, pore water [69]. This is the motivation to show how much of the overall hygrothermic coefficient of cement paste is simply due to thermal expansion of the pore water, and how much this share would depend on whether the thermal expansion coefficient of pore water would be that of bulk water or that of confined water.
The thermal expansion coefficient of bulk water, α bulk H O 2 , is associated with the temperature-dependent densities of bulk water, as given in Table 1 This is significantly larger than the thermal expansion coefficient of the solid constituents of cement paste, see Eq. (50) and Table C. 4. In order to demonstrate the contribution of the thermal expansion of bulk pore water to the hygrothermic coefficients, the mathematical term associated with water uptake/release by hydrates is set equal to zero: Thus, the mass of liquid water, both adsorbed water and bulk water, is constant: The mass of liquid pore water is quantified based on Eq. (24), considering the temperature-dependent mass density of water (Table 1) and the volume fraction of air according to Eq. (25). Values of the hygrothermic coefficient are quantified, as the temperature increases from T = 293.15 K to T + ΔT = 294.15 K, see the dashed line in Fig. 11. It is concluded that consideration of the thermal expansion of bulk pore water alone cannot explain the experimentally measured hygrothermic coefficients. Anomalously larger thermal expansion is reported for pore water, confined in pores smaller than 15 nm [76]. This anomalous thermal expansion increases with decreasing size of the pores and ranges, at room temperature, from 1 to 3 times the thermal expansion of unconfined bulk water [ In order to compute an upper bound for the potential effect of anomalous thermal expansion of pore water on the hygrothermic coefficient, the largest value is taken from Eq. (62), α conf H O 2 = 212.10 ⋅ 10 −6 K −1 , and assigned to the entire pore water, irrespective of the pore size. Corresponding hygrothermic coefficients are calculated by analogy to the procedure described above for the regular bulk thermal expansion of pore water. The results underline that consideration of anomalous thermal expansion of confined pore water indeed yields larger hygrothermic coefficients than the consideration of the thermal expansion of regular bulk pore water, as can be seen from the dashed and dotted graphs in Fig. 11. However, consideration of anomalous thermal expansion of confined pore water alone still cannot explain the experimentally measured hygrothermic coefficients. This implies the existence of another phenomenon, governing the hygrothermic deformations in cement paste, such as the water uptake/release of hydrates, investigated in the present contribution.

Future outlook
As regards quantification of phase volume fractions, the present study is based on Powers' hydration model. The latter was developed based on water vapor sorption experiments and on the assumption that the mass density of hydrates is constant. The model suggests that gel/ capillary porosities are decreasing/increasing linearly with increasing hydration degree.
Powers' model was challenged by nuclear magnetic resonance (NMR) measurements documented by Muller et al. [77,78]. They quantified the amounts of water in different confinement states. Weakly and strongly confined liquid water was associated with capillary pores and gel pores, respectively. A corresponding hydration model was developed, considering that densification of hydrates is governed by restrictions regarding the available precipitation space [79]. The NMRbased model suggests that gel/capillary porosities are decreasing/increasing nonlinearly with increasing hydration degree.
It will be interesting to apply the NMR-based hydration model to micromechanics-based modeling of hydrating cement pastes. This, however, is beyond the scope of the present contribution.

Conclusions
Results from the presented inverse multiscale analysis of mature cement pastes suggest that temperature changes are likely to result in a quasi-instantaneous water uptake/release by hydration products. This is an effect for which no physical measurements were available at the time of submitting the present manuscript. However, such measurements were reported shortly thereafter [27]. This leads to the following conclusions: • Hydration products appear to release liquid water upon heating, and to take up liquid water upon cooling, already within the room temperature regime. This water uptake/release is likely to result (i) in redistribution of water within nanoscopic pores of partially saturated cementitious materials, as studied herein, and (ii) in pressure changes of pore water of liquid saturated cementitious materials.
• The water uptake/release by hydration products was identified as a function of internal relative humidity, prevailing in air-filled pores just before the temperature change (Fig. 2). Notably, this function is independent of the initial water-to-cement mass ratio and, therefore, a mixture-independent material property of the hydration products.
• The water uptake/release by hydration products is much faster than the transport-related water exchange between a tested specimen and its ambient environment. Therefore, mass conservation of solid and liquid water inside a tested specimen allows for linking the water uptake/release by hydration products to hygrothermic coefficients. The latter are the larger, the smaller the overall porosity, i.e. the smaller the initial water-to-cement mass ratio of mature cement pastes.
• Combining model-predicted hygrothermic coefficients with a threescale representation of mature cement pastes allows for explaining mixture-independent macroscopic thermal expansion coefficients of mature cement pastes as a function of relative humidity.  Eq. (A.3) was originally proposed by Berthelot for the quantification of particle interactions in mixtures of two gases, based on knowledge regarding particle interactions in the pure gas phases [50]. Inserting Eq.  [51], and, hence, if θ ≈ 0 such that cosθ ≈ 1. This is the case for cementitious materials.
Appendix B. Proof of Eq. (50) The aim of this Appendix is to prove that the homogenized thermal expansion coefficient of cement paste is equal to that of the hydrates and clinker, provided that (i) the latter two are equal, i.e. α hyd = α clin , and (ii) that pore pressures remain unchanged.
Considering a matrix-inclusion composite with two solid phases, the eigenstresses of the two solids are related to equivalent eigenstrains as .

Appendix C. Refined analysis concerning thermal expansion of cement clinker and hydrates
In Section 2.6, the thermal expansion of cement clinker was assumed, in the sense of a reductionist approach, to be equal to that of the hydrates. In the following, this assumption is assessed from the viewpoint of a more refined analysis.
The thermal expansion coefficient of tricalcium silicate (C 3 S) amounts to [80] = ⋅ − − α 13.5 10 K . Given that 45-75% of typical Portland cements consist of C 3 S provides the motivation to set the thermal expansion coefficient of cement clinker, α clin , equal to α C S 3 according to Eq. (C.1). This allows, in combination with the thermal expansion coefficient of fully dried mature cement pastes, α cp (RH = 0%) = 12.5 ⋅ 10 −6 K −1 , see Fig. 1, for identifying the thermal expansion coefficient of hydrates, based on the presented multiscale model. The identification follows the steps listed in Table 3, whereby all the pore pressures are set equal to zero, because RH = 0%. The identification is carried out for values of the initial water-to-cement mass ratio amounting to 0.40, 0.50, 0.60, and 0.70. The identified thermal expansion coefficients of hydrates differ in the fourth significant digit (Table C.4). In addition, the thermal expansion coefficients of the cement clinker, α clin ≈ 13.5 ⋅ 10 −6 K −1 , and of the hydrates, α hyd ≈ 12.4 ⋅ 10 −6 K −1 , do not differ significantly from that of fully dried mature cement pastes, α cp (RH = 0%) = 12.5 ⋅ 10 −6 K −1 . This justifies the simplifying assumption of α cp (RH = 0%) = α hyd = α clin = 12.5 ⋅ 10 −6 K −1 , see Eq. (50). Table C.4 w/c-dependent thermal expansion coefficients of solid hydrates, identified by means of the presented multiscale model (see Table 3) from the thermal expansion coefficients of unhydrated clinker, α clin = 13.5 ⋅ 10 −6 K −1 , and of fully dried mature cement paste, α cp (RH = 0%) = 12.5 ⋅ 10 −6 K −1 . In order to render the model-predicted hygrothermic coefficients shown in Fig. 8 accessible, the following regression polynomial is introduced: with optimal coefficients k j i listed in Table D1.