Introduction

The structural analysis of the early-age mechanical behavior of concrete structures heavily relies on a proper description of the hydration reaction, with degree of hydration \(0\le \xi \le 1\) strongly influenced by the temperature history T(t) and vice versa via the second law of thermodynamics [36]. Within the framework introduced by Ulm and Coussy in the early 1990s [36, 37] and successfully employed for the thermochemomechanical analysis of concrete structures at an early age, see, e.g., [8, 9, 17,18,19, 28], the history of the spatial distribution of the degree of hydration \(\xi ({\mathbf{x}},t)\) and the temperature \(T({\mathbf{x}},t)\) is obtained from a thermochemical analysis of the concrete structure. In the subsequent chemomechanical analysis, giving access to the stress and deformation state of the structure, the evolution of material behavior is described by material functions linking parameters (elastic parameters, strength parameters, etc.) to the degree of hydration \(\xi\). On the other hand, the temperature history from thermochemical analysis gives access to thermal eigenstrains \({{\varvec{\varepsilon }}}^{{{\text{th}}}}({\mathbf{x}},t)={\alpha}\,{ \Delta }T({\mathbf{x}},t)\), with \({\alpha }\,[{\text{K}}^{-1}\)] as the coefficient of thermal dilation. In this paper, we revisit the rate law commonly employed in thermochemical analysis of concrete structures.

Employing a stringent thermodynamics framework, [36, 37] introduced a rate law for the degree of hydration \(\xi\) [–], where the latter scales linearly to the mass of water bound in hydration products formed (or cement consumed [3]), i.e., initially amounting to zero and reaching one in case all cement in the considered material volume has reacted with water (given the water/cement ratio w/c is large enough). The time derivative of \(\xi\) or rate of the degree of hydration is given as

$$\begin{aligned} \dot{\xi }(t)=A\left( \xi (t)\right) \exp \left( -\displaystyle \frac{E_{\mathrm{a}}}{{\mathcal{R}}T(t)}\right) . \end{aligned}$$
(1)

In Eq. (1), \(A(\xi )\) [1/s] denotes the so-called chemical affinity of the employed cement, a material function depicting the complex (parallel and/or sequential) hydration reactions in Portland-cement-based material systems, see e.g., [3, 33], which primarily depends on cement composition, grinding fineness, and water/cement ratio w / c; \(E_{\mathrm{ a}}\) [J mol−1] denotes the activation energy, \({\mathcal{R}}=8.314\,{\text{ J mol}}^{-1} {\text{ K}}^{-1}\) is the gas constant.

In the rate law (1), an increase in temperature T [K] results in an increase in the Arrhenius term \(\exp \left[ -E_{\mathrm{ a}}/({\mathcal{R}} T)\right]\) and hence in a higher reaction rate. Note further the decoupling hypothesis in Eq. (1), temperature variations do not influence the chemical affinity, i.e., affinity is a function of \(\xi\) only, \(A=A(\xi )\). In [37], (1) adiabatic calorimetry or (2) isothermal strength gain [assuming a linear relation \(f_{\mathrm{ c}}/f_{{\mathrm{ c}},\infty }=(\xi -\xi _0)/(1-\xi _0)\)] was employed for the determination of \(A(\xi )\). In "Investigated material and calorimetric data" section of this paper, we will employ isothermal calorimetry for the determination of the chemical affinity.

More refined, multiphase, kinetics models, considering the binder composition, e.g., the four clinker phases and gypsum in Portland cement, see, e.g., [3, 24, 25, 27], have been proposed. These models, however, rely heavily on verified material parameters (e.g., rate constants in the scopes of modeling with the Avrami law, which may change with water/cement ratio and grinding fineness) for all clinker phases (and reprocessed product phases, e.g., ettringite and portlandite) and/or separable reactions (e.g., tricalcium aluminate reaction is a multistage reaction in the presence of gypsum). Whereas some of these parameters have been available for some time (e.g., for tricalcium silicate [2, 6, 35]), others are still part of the current research. Furthermore, the bulk of these multiscale models employs kinetic models (Avrami model, i.e., bulk nucleation and growth, diffusion-based Jander model [6, 10]) that have been dismissed in the cement research community in recent years, see, e.g., review papers [4, 32]. On the other hand, substantial effort has been undertaken in recent years as regards hydration kinetics of tricalcium silicate, the abundant phase in all Portland-cement-based material systems, see, e.g., [4, 32]. These models for tricalcium silicate hydration are complex, there is an ongoing debate on the underlying rate-controlling mechanisms (and competing modeling frameworks), models for the interaction with the other clinker phases in Portland cement are not available yet or still at very early stages of development.

The stated reasons make the macroscopic approach (Ulm and Coussy’s thermodynamic approach as described above), relying on, for example, simple calorimetric tests of a certain binder material still attractive as regards an engineering description of hydration in the scopes of thermochemomechanical analysis of early-age concrete structures.

Table 1 Mass fractions obtained by XRD–Rietveld analysis of investigated Portland cement CEM I 425 R; note that standard cement chemistry abbreviations are used throughout this paper: C = CaO, \({\text{S}}={\text{SiO}}_2\), \({\text{A}}={\text{Al}}_2{\text{O}}_3\), \({\text{F}}={\text{Fe}}_2{\text{O}}_3\), \(\bar{\text{S}}={\text{SO}}_3\), \({\text{H}}={\text{H}}_20\)
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Investigated material and calorimetric data

We investigated a commercial, properly sulfated Portland cement type CEM I 425 R with mass fractions of the main clinker phases as obtained by XRD–Rietveld analysis summarized in Table 1. In the scopes of the calorimetric experiments (isothermal heat flow calorimeter TONI Cal 6), the heat release rate of a hydrating cement sample (related to the mass of cement in the sample) \(\dot{Q}\) [J g−1 h−1] was monitored at isothermal conditions, with the experiments conducted at \(T={\text{const.}}=273+22.6=295.6\,{\text{K}}\), \(=273+31.6=304.6\) K, and \(=273+40.6=313.6\) K, respectively (see Fig. 1). The heat release history of the sample, \(Q(t)=\displaystyle \int _0^t \dot{Q}({\tau }){\mathrm{d}}{\tau }\), was determined from time integration of the monitored heat release rate \(\dot{Q}\) till \(\dot{Q}\) drops below 0.5 J \({\mathrm{g}}^{-1}\,{\mathrm{h}}^{-1}\), see Fig.  1c.

Fig. 1
figure 1

a and b heat release rate \(\dot{Q}(t)\), c heat release history \({Q}(t)\) of ordinary Portland cement samples for different isothermal curing temperatures (multiple re-run of experiments)

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The current understanding as regards the parallel hydration/interaction of tricalcium silicate, tricalcium aluminate and sulfate (from gypsum, bassanite or anhydrite) in real-life properly sulfated Portland cement systems can be found in review papers, e.g., [4, 32], in [1, 13] or [7, 20, 21], with the latter authors considering different curing temperatures. Several reaction peaks can be observed from calorimetric data:

  • The initial reaction (first hour) and associated “dissolution peak” are only marginally influenced by the curing temperature (see Fig. 1a). Upon contact with water, besides the dissolution of \({\text{C}}_3{\text{S}}\) releasing a significant amount of heat, tricalcium aluminate and very soluble sulfate carriers (bassanite) dissolve significantly [12] and ettringite precipitates rapidly. Heat flow then declines significantly as tricalcium aluminate dissolution stops or is extremely slow. According to [23], the latter is caused by adsorption of calcium and/or sulfate ions on the tricalcium aluminate surface. [13] recently concluded that the rapid adsorption of sulfur and aluminum leads to the passivation of tricalcium aluminate.

  • A second reaction peak is initiated at \(t\approx\) 2 h, with consensus in the literature that this “silicate peak” is caused by alite dissolution and precipitation of C–S–H and portlandite (silicate reaction, see Fig. 1b) and precipitation of ettringite [1, 11, 13]; there is no consensus on the rate-controlling mechanism. According to [13], the passivation of tricalcium aluminate remains until (dissolved) sulfate is completely consumed by ettringite precipitation occurring continuously until sulfate is depleted, i.e., the amount of \({\text{C}}_3{\text{A}}\) dissolved upon contact with water is sufficient to allow for precipitation of ettringite until further dissolution of \({\text{C}}_3{\text{A}}\) is observed [11].

  • In properly sulfated systems, a third reaction (the so-called shoulder peak) superimposes the second reaction peak (silicate peak). This shoulder peak (as compared to the second reaction peak) is more pronounced for higher curing temperatures (see Fig. 1b). For the mechanisms associated with this “sulfate depletion peak” see [11, 13]. The sulfate necessary for further formation of ettringite comes from C–S–H, where it has previously been absorbed (when sulfate concentration in solution was high) [31]. This sulfate is released back into the solution from C–S–H surfaces. Furthermore, [13] found no relation between adsorbed calcium ions and sulfate depletion (tricalcium aluminate passivation by adsorption of calcium as proposed by [23]). Monosulfate (reaction of tricalcium aluminate and ettringite) only forms later, often not well visible in calorimetric measurement, what seems to be the case in our experiments.

  • What follows is a reaction tail characterized by an ever-decreasing heat release rate.

In the heat flow calorimeter employed, the heat release rate \(\dot{Q}\,[{\text{J/h}}]\) of the hydrating cement paste sample is monitored for isothermal conditions, \(T(t)\,=\,{\text{const}}\). The underlying field equation for the thermochemically coupled problem, the first law of thermodynamics, reads [36]

$$\begin{aligned} ({\rho }c)\dot{T}-{\ell }_\xi \dot{\xi }=-\nabla \cdot {\mathbf{q}}, \end{aligned}$$
(2)

with \(({\rho }c)\,[{\text{J m}}^{-3}{\text{ K}}^{-1}]\) as the volume heat capacity, \({\ell }_\xi \,[{\text{J m}}^{-3}]\) as the specific latent heat of the hydration process and \({\mathbf{q}}\,[{\text{J m}}^{-2}{\text{ h}}^{-1}]\) as the heat flow vector. Specializing Eq. (2) for isothermal conditions, \(T={\text{const.}}\rightarrow \dot{T}=\partial T/\partial t=0\), and integration over the sample volume V (underlying a spatially uniform degree of hydration in the sample) gives

$$\begin{aligned} \dot{\xi } {\ell }_\xi V = \displaystyle \int _V \nabla \cdot {\mathbf{q}}\, {\mathrm{d}}V \end{aligned}$$
(3)

Denoting \(Q_\infty = {\ell }_\xi V\) as the latent heat of the sample [J] and applying the divergence theorem to the right-hand side of Eq. (3), with

$$\begin{aligned} \displaystyle \int _V \nabla \cdot {\mathbf{q}}\, {\mathrm{d}}V = \displaystyle \oint _S {\mathbf{q}}\cdot {\mathbf{n}}\, {\mathrm{d}}S, \end{aligned}$$
(4)

gives

$$\begin{aligned} \dot{\xi } Q_\infty = \displaystyle \oint _S {\mathbf{q}}\cdot {\mathbf{n}}\, {\mathrm{d}}S. \end{aligned}$$
(5)

With the heat flow through the sample surface S being measured, \(\dot{Q}=\displaystyle \oint _S {\mathbf{q}}\cdot {\mathbf{n}}{\mathrm{d}}S\), Eq. (5) gives

$$\begin{aligned} \dot{\xi }=\displaystyle \frac{\dot{Q}}{Q_\infty }. \end{aligned}$$
(6)

Note that in this paper \(\dot{Q}\) and \(Q_\infty\) are scaled by the mass of cement in the paste sample.

Classical way for determination of chemical affinity

Specializing Eq. (1) for isothermal conditions, i.e., T = const., and using Eq. (6) give access to \(A(\xi )\) of the investigated cement sample as (see Fig. 2)

$$\begin{aligned} \begin{array}{l} \dot{\xi }=A[\xi (t)]\exp \left( -\displaystyle \frac{E_{\mathrm{ a}}}{{\mathcal{R}} T}\right) =\displaystyle \frac{\dot{Q}(t)}{Q_\infty } \quad \rightarrow \quad \\ \rightarrow \quad A[\xi (t)]=\displaystyle \frac{\dot{Q}(t)}{Q_\infty } \displaystyle \frac{1}{ \exp \left( -\displaystyle \frac{E_{\mathrm{ a}}}{{\mathcal{R}} T}\right) } \\ {\text{with}}\quad \xi (t)=\displaystyle \frac{\displaystyle \int _0^t \dot{Q}({\tau })\,{\mathrm{d}}{\tau }}{Q_\infty }=\displaystyle \frac{Q}{Q_\infty }\,. \end{array} \end{aligned}$$
(7)
Fig. 2
figure 2

Back-calculated affinity \(A(\xi )\) for different prescribed values of \(E_{\mathrm{ a}}/{\mathcal{R}}\) where \(E_{\mathrm{a}}/{\mathcal{R}}=4330\,{\text{K}}\) corresponds to \(E_{\mathrm{ a}}=36\,{\text{kJ mol}}^{-1}\), the value determined for the apparent activation energy of Portland cement in [34]

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The latent heat (or overall heat release) \(Q_\infty\) [J g−1] was determined by approximating the tail of the heat release rate history with a reaction-order model ansatz, as detailed and justified in [26]: Denoting the concentration of the transformed substance as c, the rate law for an nth order reaction is given as \(-{\mathrm{d}}c/{\mathrm{d}}t=k c^{\text{n}}\) with rate constant k [\({\text{h}}^{-1}\)] and reaction order n [–]. Replacing c (what in our case is the concentration of unreacted Portland cement in the material system) by \((1-\xi )\) gives \(-\displaystyle \frac{{\text{d}}}{{\text{d}}t}(1-\xi )=\dot{\xi }\), hence

$$\begin{aligned} \dot{\xi }=k (1-\xi )^{\text{n}}. \end{aligned}$$
(8)

Using the change of variables, \(\dot{\xi }(t)\rightarrow \dot{Q}(t)\) and \(\xi (t)\rightarrow Q(t)\), as previously described, in Eq. (8), the fitting function reads

$$\begin{aligned} \dot{Q}= k\,Q_\infty \left( 1-\displaystyle \frac{Q}{Q_\infty } \right) ^{\text{n}} \end{aligned}$$
(9)

with parameters k, n and \(Q_\infty\) determined by the Levenberg–Marquardt algorithm [22, 30] and depicted in an Arrhenius diagram in Fig. 3, where the reduction of the overall heat release \(Q_\infty\) with increasing curing temperature is consistent with the previous results in the literature [14, 16].

Fig. 3
figure 3

a Good fit of reaction-order modeling illustrated by linearization of tail data in double logarithmic diagram \(\dot{\xi }\) versus \((1-\xi )\); Arrhenius plot of parameters, bk, n, and c overall heat release \(Q_\infty\) of reaction-order model

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Let us try to assess whether the separation in Eq. (1) into a temperature-dependent part (Arrhenius term) and reaction-extent-dependent part [affinity function \(A(\xi )\)] is justified. Figure 2 shows the obtained affinity functions for various prescribed activation energies. Note that a collapse into a single master curve is not possible. Only (different) parts of the affinity function collapse for certain prescribed activation energies, with \(E_{\mathrm{a}}/{\mathcal{R}} = 4330\,{\text{K}}\) (related to the apparent activation energy of 36 kJ mol\(^{-1}\) as determined by [34]) achieving the collapse for the increasing slope of the silicate reaction (see Fig. 2). The sulfate depletion peak superimposes the silicate reaction. Much higher values for \(E_{\mathrm{a}}/{\mathcal{R}}\) are necessary to collapse the increasing/decreasing slope of the sulfate depletion peak (r.h.s. of Fig. 2).

Alternative formulation and determination of chemical affinity

Based on the result shown in Fig. 2, one may conclude that a single apparent activation energy is not capable for depicting the parallel/subsequent reactions in real-life Portland-cement-based material systems, as all these reactions may be attributed by substantially different activation energies. A viable approach may consist of making the activation energy dependent on the reaction extent \(E_{\mathrm{a}}=E_{\mathrm{a}}(\xi )\) (compare to the previous work, e.g., [5, 15, 29]). Hence, we modify Ulm and Coussy’s rate law toward

$$\begin{aligned} {\dot{\xi }}(\xi ,T)= & {} A(\xi ,T_{\mathrm{ref}})\exp \left( -\displaystyle \frac{E_{\mathrm{a}}(\xi )}{\mathcal{R}} \left( \displaystyle \frac{1}{T} - \displaystyle \frac{1}{T_{\mathrm{ref}}} \right) \right) \nonumber \\&\quad {\text{with}} \quad A(\xi ,T_{\mathrm{ref}})={\dot{\xi }}(\xi ,T=T_{\mathrm{ref}}) \end{aligned}$$
(10)

and \(T_{\mathrm{ref}}=273+30=303\,{\text{K}}\) denoting an arbitrarily chosen reference temperature. Rewriting Eq. (10) as

$$\begin{aligned} -\ln {\dot{\xi }}(\xi ,T) = -\ln \dot{\xi }(\xi ,T=T_{\mathrm{ref}}) + \displaystyle \frac{E_{\mathrm{ a}}(\xi )}{\mathcal{R}} \left( \displaystyle \frac{1}{T} - \displaystyle \frac{1}{T_{\mathrm{ref}}} \right) \end{aligned}$$
(11)

and performing a linear fit of \(-\ln \dot{\xi }(\xi ,T)\) versus \((1/T-1/T_{\mathrm{ref}})/{\mathcal{R}}\) data for a certain value of \(\xi\) gives access to \(A(\xi )=\dot{\xi }(\xi ,T=T_{\mathrm{ref}})\) and \(E_{\mathrm{ a}}(\xi )\), see Fig. 4.

Fig. 4
figure 4

a Data, \(\dot{\xi }=\dot{Q}/Q_\infty\) versus \(\xi =Q/Q_\infty\), employed for determination of material functions, b affinity \(A(\xi ,T_{\mathrm{ref}}=303\,{\text{K}})\) and c activation energy \(E_{\mathrm{ a}}\) in reformulated rate law

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Concluding remarks

In this paper, we revisited Ulm and Coussy’s classical rate law for the description of hydration kinetics of Portland-cement-based material systems. For this purpose, data obtained from heat flow calorimetry conducted at different (constant) temperatures cannot be merged into a unique affinity function. This is due to the fact that several sub-reactions take place in parallel and/or subsequently in real-life material systems, with these different sub-reactions characterized by various activation energies. In order to reflect the latter, we suggested a modification of the classical rate law. This approach seems viable for determination of (1) temperature history in the scopes of thermochemical analyses and (2) thermal eigenstrains in the scopes of chemomechanical analyses of early-age concrete structures, respectively.

We are currently working on material functions linking the evolution of mechanical material parameters to the degree of hydration/degree of reaction. However, linking the strength/stiffness evolution to \(\xi\) as defined in this paper is not satisfactorily. Rather than considering the overall reaction, one has to filter out the strength/stiffness determining sub-reaction in calorimetric data (i.e., silicate reaction in Fig. 1b) in order to determine a reaction degree X governing the evolution of mechanical material properties.