An extended numerical model of the first exothermic peak for three dimensional printed cement-based materials

Abstract

The first exothermic peak of cement-based material occurs a few minutes after mixing, and the properties of three dimensional (3D) printed concrete, such as setting time, are very sensitive to this. Against this background, based on the classical Park cement exothermic model of hydration, we propose and construct a numerical model of the first exothermic peak, taking into account the proportions of C₃S, C₃A and quicklime in particular. The calculated parameters are calibrated by means of relevant published exothermic test data. It is found that this developed model offers a good simulation of the first exothermic peak of hydration for C₃S and C₃A proportions from 0 to 100% of cement clinker and reflects the effect of quicklime content at 8%–10%. The unique value of this research is provision of an important computational tool for applications that are sensitive to the first exothermic peak of hydration, such as 3D printing.

Graphical abstract

Keywords

3D printed cement-based materials / cement hydration / the first exothermic peak / liquid quick-setting agent / numerical model

1 Introduction

The study of hydration properties of cement is an enduring hot topic in concrete science. Hydration involves a series of very complex physical and chemical changes that play an important role in the microstructural evolution, strength development, and volumetric stability of cement-based products [1].

When cement meets water, it immediately undergoes a series of chemical reactions and thermal energy is released. The rate of this release is generally monitored by conduction calorimetry [2]. Based on the measured rate of release, the hydration process is generally divided into five phases, namely the initial mixing reaction, the dormancy, the strength acceleration, the strength deceleration, and the steady development. The exothermic diagram of the five phases is shown in Fig.1.

Scholars have done a great deal of work on cement hydration modeling, building models that have evolved from the initial simulation of individual cement particles [3], to microstructural models [4] that take into account interactions between hydrated particles and microstructural development, and introducing the concept of hydration degree [5,6], which interprets the reaction rate as a function of the degree of hydration.

Most studies have not focused on the first stage of heat generation. Although this part of the exothermic reaction is fast and has the highest reaction rate, the total heat generated during this stage is only a very small proportion of the total exothermic heat of hydration [7,8]; and this heat is generally generated before the concrete is placed and is not carried into the structure, so studies of the early performance of concrete structures generally ignore this phase.

Three dimensional (3D) printed concrete technology has come a long way over the years and features of the hydration process, at the very early age, such as initial setting, final setting and shrinkage [9–13], have become more significant. Performance of 3D printed concrete is closely related to the thixotropic properties of the cement paste [14–16], which is partly dependent on the heat released during the early stage of hydration [17]. On the other hand, quick-setting agents are widely used in 3D printed concrete [18,19], but their influence is currently less considered in models of the first stage of exothermic hydration.

As a result, there is a lack of a model, for the behavior of 3D printed cementitious materials, that can adequately simulate the first hydration exothermic peak and can adequately account for the effects of quick setting agents.

In fact, early cement hydration can be simplified to the reactions of the silica and aluminum phases. The reaction of the silica phase can be reduced to the hydration of tricalcium silicate (C₃S), including the dissolution of C₃S, the precipitation of hydrated calcium silicate and Calcium hydroxide (CH); the reaction of the aluminum phase can be reduced to the dissolution of sulfate, the dissolution of tricalcium aluminate (C₃A) and the precipitation of calcium alumina [20]. In the first stage, the exothermic heat of hydration is mainly derived from the rapid dissolution of the individual cement mineral phases, and the contribution of the rapid reaction of the initial aluminum phase [20].

Specifically, the contribution of C₃S to the first exothermic peak of hydration comes from the heat of dissolution, and studies have shown that only 1% of C₃S reacts in seconds [21,22], followed by a dormant period of about 5 h.

The contribution of C₃A to the first hydration exothermic peak is more complex. The dissolution of C₃A and the quasi-instantaneous precipitation of monosulphur type hydrated calcium sulfate aluminate (AFm) occur within the first 30 s. After 3 min, ettringite is present on the surface of C₃A grains and AFm also appears in the form of “sheets”. In the presence of calcium sulfate, the slowing down of the C₃A reaction rate in the first stage of exothermic hydration may not be due to the formation of calcium alumina. The most plausible explanation of the slowing down of the rate of reaction of C₃A in the presence of calcium sulfate is the specific adsorption of the calcium and/or sulfate ions on the surface of the grains of C₃A. These ions block dissolution sites of C₃A [23].

Aluminum sulfate based alkali-free quick setting agent is currently the most commonly used quick setting agent for 3D printed concrete technology [24]. On the one hand, it promotes the rapid reaction of C₃A, resulting in a denser structure of the hydration products, which facilitates the rapid hardening of the cement. On the other hand, it promotes the hydration of C₃S, producing richer hydration products such as CH crystals and interspersed Aft (Tri-sulfur type hydrated calcium sulfoaluminat), resulting in a denser structure of the hardened C₃S slurry and rapid early strength development.

This paper is based on the hydration model developed by Park et al. [25], a mathematical model, based on a single kinetic equation to predict the degree of hydration and microstructural evolution of cement particles, describing the microstructural evolution as a function of changes in the composition of the hydration products. The model developed in this paper introduces four groups of 16 parameters to characterize the effect of C₃S and C₃A on the exothermic heat of hydration at very early stages of hydration. Subsequently, we introduced one group of four parameters using the Least Squares Regression method to characterize the effect of aluminum sulfate-based alkali-free quicklime, which is done through mathematics software ‘Maple’.

Based on the description and literature cited earlier, it has been demonstrated that 3D printed concrete materials need a high level of thixotropic behavior, and there is a close relationship between thixotropy and heat release. The heat release hydration of cement-based materials is closely related to the setting time, final setting time, and shrinkage performance. Therefore, simulating the first heat release peak of cement is very important, especially for 3D printed concrete. Our model can help quantify the heat release data of the first heat release peak, and can assist in adjusting the relevant parameters of 3D printed concrete materials. The parameters in the model are matched with published hydration exothermic tests, as well as to their own exothermic tests in the test. The model is then compared with other published hydration test data and the results show that the model developed in this paper is able to simulate the first peak of hydration exothermal behavior, of cement-based material containing quicklime, very well.

2 Basic exothermic model of hydration developed by Park

The hydration model developed by Park et al. [25] is adopted as the foundation of this study, and is expressed as follow:

(1)\begin{array}{rl}\frac{\mathrm{d}\alpha }{\mathrm{d}t}=& \frac{3\left(S_{\mathrm{w}}/S_0\right){\rho }_{\mathrm{w}}{C}_{\mathrm{w}-\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}}{\left(v+w_{\text{g}}\right)r_0{\rho }_{\mathrm{c}}}\\ & \cdot \frac{1}{\left({\displaystyle \frac{1}{k_{\mathrm{d}}}}-{\displaystyle \frac{r_0}{D_{\mathrm{e}}}}\right)+{\displaystyle \frac{r_0}{D_{\mathrm{e}}}}{\left(1-\alpha \right)}^{-\frac{1}{3}}+{\displaystyle \frac{1}{k_{\mathrm{r}}}}{\left(1-\alpha \right)}^{-\frac{2}{3}}},\end{array}

in which α is the degree of hydration, S_w is the effective contacting surface area between the cement particles and capillary water, S₀ is the total surface area if hydration products develop unconstrained, ρ_w and ρ_c denote the density of water and cement, respectively, C_w−free is the amount of capillary water at the exterior of hydration products, v is the stoichiometric ratio by mass of water to mass of cement ( = 0.25), w_g is the physically bound water in hydration products ( = 0.15), r₀ is the radius of unhydrated cement particles, k_d is the reaction coefficient in the initial dormant period, D_e is the reaction coefficient in the diffusion-controlled stage, k_r is the reaction coefficient of the boundary reaction process.

In Eq. (1), C_w−free can be expressed as:

(2)C_{\mathrm{w}-\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}={\left(1-0.4\alpha w/c\right)}^r,

where r = 2.6 4w/c for water cement ratio w/c < 0.4 and r = 1 for w/c ≥ 0.4.

The reaction coefficient k_d can be expressed as:

(3)k_{\mathrm{d}}=\frac{B}{{\alpha }^{1.5}}+C{\alpha }^3,

in which B and C are the rate of the initial impermeable layer formation and decay, respectively.

B and C are given by Eqs. (4) and (5), as follows:

(4)B=B_{20}\exp \left(-{\beta }_1\left(\frac{1}{T}-\frac{1}{293}\right)\right),

(5)C=C_{20}\exp \left(-{\beta }_2\left(\frac{1}{T}-\frac{1}{293}\right)\right),

where B₂₀ = 6 × 10⁻¹² × (C₃S% + C₃A%) + 4 × 10⁻¹⁰, C₂₀ = 0.0003 × C₃S% + 0.0186, β₁ = 1000, β₂ = 1000. C₃S% and C₃A% are the proportions of C₃S and C₃A in cement pastes.

The reaction coefficient D_e can be expressed as:

(6)D_{\mathrm{e}}=D_{\mathrm{e}0}\mathrm{l}\mathrm{n}\left(\frac{1}{\alpha }\right),

where D_e0 is:

(7)D_{\mathrm{e}0}=D_{\mathrm{e}20}\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\beta }_3\left(\frac{1}{T}-\frac{1}{293}\right)\right),

where D_e20 = − 8 × 10⁻¹² × C₂S% + 7 × 10⁻¹⁰, β₃ = 7500, and C₂S% is the content of dicalcium silicate.

The reaction coefficient k_r is:

(8)k_{\mathrm{r}}=k_{\mathrm{r}20}\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{E}{R}\left(\frac{1}{T}-\frac{1}{293}\right)\right),

where k_r20 = 8 × 10⁻⁸ × C₃S% + 1 × 10⁻⁶, E/R = 5400.

Considering the influence of water reduction, C_w−free can be expressed as:

(9)C_{\mathrm{w}-\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}={\left(1-0.4\alpha w/c\right)}^r{\beta }_{\mathrm{R}\mathrm{H}},

where {\beta }_{\mathrm{R}\mathrm{H}} is:

(10){\beta }_{\mathrm{R}\mathrm{H}}=\{\begin{array}{ll}{\left[{\displaystyle \frac{RH-0.55}{0.45}}\right]}^4,& RH>0.55,\\ 0,& RH⩽0.55,\end{array}

where RH is the relative humidity, and can be written as:

(11)RH=\mathrm{e}\mathrm{x}\mathrm{p}\left[{\left(\frac{22}{W_{\mathrm{f}}}\right)}^{1/0.05}\right],

and W_f is:

(12)W_{\mathrm{f}}=W_0-W_{\mathrm{n}}C,

where W₀ is the initial amount of water per 1 cm³ of the paste and C is the mass of cement per 1 cm³ of the paste, and W_n is:

(13)\begin{array}{rl}{W}_{\mathrm{n}}=& 0.234{\mathrm{C}}_3\mathrm{S}\mathrm{\%}+0.178{\mathrm{C}}_2\mathrm{S}\mathrm{\%}\\ & +0.514{\mathrm{C}}_3\mathrm{A}\mathrm{\%}+0.158{\mathrm{C}}_4\mathrm{A}\mathrm{F}\mathrm{\%}.\end{array}

3 Modeling of the first exothermic peak of hydration

3.1 Hydration degree model of the first exothermic peak

The model proposed in this paper uses the hydration degree as the base parameter, and this can be expressed as:

(14){\alpha }_t=\alpha +{\alpha }_{\mathrm{d}},

where {\alpha }_{\mathrm{d}} is specifically used to indicate the first exothermic peak of hydration, and \alpha is the degree of hydration that indicates the subsequent hydration process, i.e., the one adopted in Eqs. (1)–(12) above. {\alpha }_{\mathrm{t}} denotes the hydration degree of the whole hydration process.

Equation (14) can be differentiated with respect to time t, which gives

(15)\frac{\mathrm{d}{\alpha }_{\mathrm{t}}}{\mathrm{d}t}=\frac{\mathrm{d}\alpha }{\mathrm{d}t}+\frac{\mathrm{d}{\alpha }_{\mathrm{d}}}{\mathrm{d}t},

where {\displaystyle \frac{\mathrm{d}\alpha }{\mathrm{d}t}} satisfies Eq. (1), for a given time t, the following Eqs. (16) and 17 can be obtained:

(16){\int }_0^{\alpha }{\left(\frac{3\left(S_{\mathrm{w}}/S_0\right){\rho }_{\mathrm{w}}{C}_{\mathrm{w}-\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}}{\left(v+w_{\text{g}}\right)r_0{\rho }_{\mathrm{c}}}\frac{1}{\left({\displaystyle \frac{1}{k_{\mathrm{d}}}}-{\displaystyle \frac{r_0}{D_{\mathrm{e}}}}\right)+{\displaystyle \frac{r_0}{D_{\mathrm{e}}}}{\left(1-\alpha \right)}^{-\frac{1}{3}}+{\displaystyle \frac{1}{k_{\mathrm{r}}}}{\left(1-\alpha \right)}^{-\frac{2}{3}}}\right)}^{-1}\mathrm{d}\alpha =t,

(17){\alpha }_{\mathrm{d}}\left(t\right)={\int }_0^t\frac{\mathrm{d}{\alpha }_{\mathrm{d}}}{\mathrm{d}t}\mathrm{d}t,

where \alpha \mathrm{a}\mathrm{n}\mathrm{d}{\alpha }_{\mathrm{d}} can be obtained by Eqs. (16) and (17), respectively. If the expression of {\displaystyle \frac{\mathrm{d}{\alpha }_{\mathrm{d}}}{\mathrm{d}t}} can be gotten, then the total exothermic model of hydration can be obtained.

{\displaystyle \frac{\mathrm{d}{\alpha }_{\mathrm{d}}}{\mathrm{d}t}} is a function of temperature T and time t:

(18)\frac{\mathrm{d}{\alpha }_{\mathrm{d}}}{\mathrm{d}t}=F\left(T,t\right).

At constant temperature,

(19)\frac{\mathrm{d}{\alpha }_{\mathrm{d}}}{\mathrm{d}t}=f\left(t\right),

where

(20)F\left(T_0,t\right)=f\left(t\right).

According to the definition of hydration,

(21){\alpha }_{\mathrm{d}}=\frac{H\left(t\right)-H_{\alpha }}{H_{\mathrm{m}\mathrm{a}\mathrm{x}}},

where H(t) is the total cumulative heat release at time t, H_{\alpha } is the heat release of hydration model (1) at alpha degree of hydration and H_max is the maximum heat release.

When cement particles are in contact with water and dissolution/hydrolysis occurs, then if it is assumed that the density of cement particles and the heat release per unit mass is constant, then:

(22)\frac{\mathrm{d}X}{\mathrm{d}t}-\frac{\mathrm{d}\alpha }{\mathrm{d}t}=\frac{\mathrm{d}{\alpha }_{\mathrm{d}}}{\mathrm{d}t},

where X is the volume fraction consumed by the chemical reaction. In the first exothermic peak, the changes of {\displaystyle \frac{\mathrm{d}\alpha }{\mathrm{d}t}} are much less than those of {\displaystyle \frac{\mathrm{d}{\alpha }_{\mathrm{d}}}{\mathrm{d}t}}. Then Eq. (22) can be approximated as:

(23)\frac{\mathrm{d}X}{\mathrm{d}t}\approx \frac{\mathrm{d}{\alpha }_{\mathrm{d}}}{\mathrm{d}t}.

Referring to Avrami [26] and Cahn [27], the nucleation growth model of the particles is considered as:

(24)\frac{\mathrm{d}X}{\mathrm{d}t}=K_1\mathrm{e}\mathrm{x}\mathrm{p}\left(\eta t^n\right),

where K₁, η and n are the calculated variables.

According to Avrami’s model,

(25)K_1=n{k}_{\mathrm{a}\mathrm{v}\mathrm{r}}^n{t}^{n-1},\eta =-k_{\mathrm{a}\mathrm{v}\mathrm{r}}^n.

According to Cahn’s model,

(26)K_1=n{k}_{\mathrm{a}\mathrm{v}\mathrm{r}}^n{t}^{n-1},\eta =-k_{\mathrm{a}\mathrm{v}\mathrm{r}}^n.

Before the induction period, let K_1 be a time-dependent function, then we have

(27)K_1=F\left(t\right).

For F(t), when t > 0, the constructor H(t) satisfies

(28)F\left(t\right)=\frac{H\left(t^{\lambda }\right)}{t},

where λ is a real number to be determined. Expanding H on t^λ, we have

(29)H\left(t^{\lambda }\right)\approx \frac{{\sum }_{i=0}^{n_1}{\left(V_i{t}^{\lambda }\right)}^i}{t},

where Vi(i = 1, 2, 3, ..., n1) is the calculated parameter.

Simplifying the treatment by making n = 1 [27] in Eq. (24) and substituting Eqs. (27)–(29) into Eq. (24), we have

(30)\frac{\mathrm{d}{\alpha }_{\mathrm{d}}}{\mathrm{d}t}=\frac{{\sum }_{i=0}^{n_1}{\left(V_i{t}^{\lambda }\right)}^i}{t}\mathrm{e}\mathrm{x}\mathrm{p}\left(\eta t\right).

Taking the 1st order approximation, when T is a constant temperature, then the 4-parameter model can be obtained as follows

(31)\frac{\mathrm{d}{\alpha }_{\mathrm{d}}}{\mathrm{d}t}\approx \frac{V_0+V_1{t}^{\lambda }}{t}\mathrm{e}\mathrm{x}\mathrm{p}\left(\eta t\right),

where V_0=V_0\left(n_{3\mathrm{C}\mathrm{a}\mathrm{O}\cdot \mathrm{S}\mathrm{i}{\mathrm{O}}_2},n_{3\mathrm{C}\mathrm{a}\mathrm{O}\cdot {\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3}\right), V_1=V_1(n_{3\mathrm{C}\mathrm{a}\mathrm{O}\cdot \mathrm{S}\mathrm{i}{\mathrm{O}}_2}, n_{3\mathrm{C}\mathrm{a}\mathrm{O}\cdot {\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3}), and \eta =\eta \left(n_{3\mathrm{C}\mathrm{a}\mathrm{O}\cdot \mathrm{S}\mathrm{i}{\mathrm{O}}_2},n_{3\mathrm{C}\mathrm{a}\mathrm{O}\cdot {\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3}\right).

In Eq. (31), V_0, V_1, and \eta are related to the initial proportions of 3\mathrm{C}\mathrm{a}\mathrm{O}\cdot \mathrm{S}\mathrm{i}{\mathrm{O}}_2, 2\mathrm{C}\mathrm{a}\mathrm{O}\cdot \mathrm{S}\mathrm{i}{\mathrm{O}}_2, 3\mathrm{C}\mathrm{a}\mathrm{O}\cdot {\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3, and 4\mathrm{C}\mathrm{a}\mathrm{O}\cdot {\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3\cdot {\mathrm{F}\mathrm{e}}_2{\mathrm{O}}_3 in the cement.

3.2 First exothermic peak model of C₃S and C₃A

In the first stage of exothermic hydration, the main contributors to the heat of dissolution are C₃S and C₃A.

It is thus that the V_0, V_1, \eta can be expressed as follow in Eq. (31). Based on the derivation of the equations in the previous section and combined with Park’s hydration model, this paper proposes the expression formula of {\displaystyle \frac{\mathrm{d}{\alpha }_{\mathrm{d}}}{\mathrm{d}t}} as shown in Eq. (32):

(32)\frac{\mathrm{d}{\alpha }_{\mathrm{d}}}{\mathrm{d}t}=\frac{\left(l_{{\mathrm{C}}_3\mathrm{S}}{a}_1{\left({\mathrm{C}}_3\mathrm{S}\mathrm{\%}\right)}^{d_1}+l_{{\mathrm{C}}_3\mathrm{A}}{a}_2{\mathrm{C}}_3\mathrm{A}\mathrm{\%}\right)t^{-1}+\left(m_{{\mathrm{C}}_3\mathrm{S}}{b}_1{\left({\mathrm{C}}_3\mathrm{S}\mathrm{\%}\right)}^{d_2}+m_{{\mathrm{C}}_3\mathrm{A}}{b}_2{\left({\mathrm{C}}_3\mathrm{A}\mathrm{\%}\right)}^{d_4}\right)t^{0.5}}{\mathrm{e}\mathrm{x}\mathrm{p}\left(\left(n_{{\mathrm{C}}_3\mathrm{S}}{c}_1{\left({\mathrm{C}}_3\mathrm{S}\mathrm{\%}\right)}^{d_3}+n_{{\mathrm{C}}_3\mathrm{A}}{c}_2{\mathrm{C}}_3\mathrm{A}\mathrm{\%}\right)t\right)},

where C₃S% and C₃A% are the contents of C₃S and C₃A, respectively; l_{{\mathrm{C}}_3\mathrm{S}},m_{{\mathrm{C}}_3\mathrm{S}},n_{{\mathrm{C}}_3\mathrm{S}} and l_{{\mathrm{C}}_3\mathrm{A}},m_{{\mathrm{C}}_3\mathrm{A}},n_{{\mathrm{C}}_3\mathrm{A}} are calculated parameters when the contents of C₃S and C₃A are 100%, as indicated by the subscripts, and are calibrated by actual measurement.

a_1, a_2, b_1, b_2, c_1, c_2, d_1, d_2, d_3, d_4 are when the content of either C₃S or C₃A is not 100%, and the calculated parameters satisfy:

(33)

(34)

where x = a, b, c, d.

The parameters when the proportion of either C₃S or C₃A is 100% are l, m, n, and are calibrated according to the test data of Guo et al. [28]. They stirred pure C₃S and C₃A according to the water-cement ratio of 1:1 and measured the exothermic experimental data containing the first exothermic peak of hydration.

Based on the experimental data, the values of these parameters are shown below:

(35)\{\begin{array}{c}{l}_{{\mathrm{C}}_3\mathrm{S}}=-0.1154215733,\\ m_{{\mathrm{C}}_3\mathrm{S}}=124.9535235,\\ n_{{\mathrm{C}}_3\mathrm{S}}=10.20374822,\end{array}

(36)\{\begin{array}{c}{l}_{{\mathrm{C}}_3\mathrm{A}}=-0.036810525,\\ m_{{\mathrm{C}}_3\mathrm{A}}=359.1488651,\\ n_{{\mathrm{C}}_3\mathrm{A}}=\mathrm{7.181926582.}\end{array}

The comparison graphs of the experimental data and the model at 100% of C₃S and C₃A, respectively, are detailed in Fig.2 and Fig.3.

The parameters when the proportion of either C₃S or C₃A is not 100%, which are a, b, c, d, are calibrated according to the test data from public data.

Tydlitát et al. [29] performed an exothermic test of CEM I 42.5 incorporating the first exothermic peak of hydration. Based on the chemical composition of the cement (Tab.1) from Tydlitát et al. [29], we obtain the contents of C₃S and C₃A as 46.5% and 9.57% (Tab.2), respectively, using the Bogue formula. Han et al. [30] conducted a hydration exothermic test of OPC containing cement and slag, and its chemical composition is detailed in Tab.1. Using the same method, we calculate the C₃S and C₃A proportions are 51.05% and 6.617% (Tab.2), respectively.

Based on the above measured data, we calibrated the four groups of parameters a, b, c, d, as shown in Eqs. (37) and (38):

(37)\{\begin{array}{l}{a}_1=0.3099071276,\\ b_1=2.556970013,\\ c_1=6.656026290,\\ d_1=14.77372283,\\ d_2=-1.197634786,\\ d_3=2.324035937,\end{array}

(38)\{\begin{array}{l}{a}_2=86.03589637,\\ b_2=-2.159869954\times {10}^5,\\ c_2=0.3782212542,\\ d_4=\mathrm{5.0012.}\end{array}

The comparison plots of the computational model and the measured data are shown in Fig.4 and Fig.5, respectively.

It can be seen from Fig.4 and Fig.5 that the model data fits well with the experimental data within the allowable range of errors.

3.3 First exothermic peak model of Quicklime

Aluminum sulfate based alkali-free quick-setting agent is the most commonly used quick-setting agent in 3D printing concrete technology. It contains Al₂(SO₄)₃ and organic and inorganic acids as the main raw materials, compounded with appropriate amounts of other coagulation promoters, early strength enhancers and stabilizers. The role of rapid coagulant is to promote the hydration of C₃S and C₃A. We added calculation parameters and concentration parameters to Eq. (32), as shown in the following equation.

(39)\frac{\mathrm{d}{\alpha }_{\mathrm{d}}}{\mathrm{d}t}=\frac{\left(l_{{\mathrm{C}}_3\mathrm{S}}{a}_1{\left({\mathrm{C}}_3\mathrm{S}\mathrm{\%}\right)}^{d_1}+l_{{\mathrm{C}}_3\mathrm{A}}{a}_2{\mathrm{C}}_3\mathrm{A}\mathrm{\%}+{\lambda }_{1}n\right)t^{-1}+\left(m_{{\mathrm{C}}_3\mathrm{S}}{b}_1{\left({\mathrm{C}}_3\mathrm{S}\mathrm{\%}\right)}^{d_2}+m_{{\mathrm{C}}_3\mathrm{A}}{b}_2{\left({\mathrm{C}}_3\mathrm{A}\mathrm{\%}\right)}^{d_4}\right)t^{0.5}}{\mathrm{e}\mathrm{x}\mathrm{p}\left(\left(n_{{\mathrm{C}}_3\mathrm{S}}{c}_1{\left({\mathrm{C}}_3\mathrm{S}\mathrm{\%}\right)}^{d_3}+n_{{\mathrm{C}}_3\mathrm{A}}{c}_2{\mathrm{C}}_3\mathrm{A}\mathrm{\%}+{\lambda }_{2}n\right)t\right)+{\lambda }_3{\left({\mathrm{C}}_3\mathrm{S}\mathrm{\%}\right)}^{{\lambda }_4}nt},

where {\lambda }_i (i = 1,2,3) is the calculated parameter, n is the concentration of the quicklime.

The study uses isothermal calorimetry to measure the exothermic data of the first stage of hydration of cements with aluminum sulfate based alkali-free quick setting agent contents of 6%, 7%, and 8%. These tests are carried out using P.O42.5 cement with a water-cement ratio of 0.35 and a temperature of 20 °C. The chemical composition of the cement used is listed in Tab.1. Based on these three sets of exothermic hydration test data, we use the Least Squares Regression method to obtain the calculated parameters shown below:

(40){\lambda }_1=3.021,{\lambda }_2=-5.005,{\lambda }_3=700.131,{\lambda }_4=1.80197.

It can be seen from Fig.6 that the model data fits well with the experimental data within the allowable range of errors.

Yang et al. [31] also tested data for the first exothermic peaks during hydration of mixes containing aluminum sulfate based alkali-free quick setting agent with 7% and 9% quick setting agent content. His tests were carried out using P.O. 425 cement with a water-to-cement ratio of 0.35 at 25 °C. The cement components used are detailed in Tab.1 and the calculated C₃S and C₃A contents were 42.53% and 9.9% (Tab.2), respectively.

It can be seen from Fig.7 that the model data fits well with the experimental data within the allowable range of errors.

4 Results and discussion

The exothermic hydration of cement-based materials is closely related to the properties of initial setting, final setting and shrinkage. Therefore the simulation of the first exothermic peak of hydration of cement is of great importance, especially for 3D printed concrete.

In this paper, a hydration exotherm model is constructed on the basis of a crystal nucleus growth model, assuming a rate of change function for the hydration of the first hydration exotherm peak, combined with Park’s single kinetic equation. Although the study establishes a model for the first heat release peak of 3D printed concrete based on measured data, the problem currently faced is that the physical meanings of the parameters in the model are not clear enough, and there are no models in existing literature that are similar to that of this study for comparison. Therefore, there are limitations to the outcome of our model.

Main findings.

1) The model proposed in this paper assumes that the first exothermic peak of hydration is mainly derived from the heat of dissolution of C₃S and the exothermic reaction of C₃A. The parameters of the model include the content of C₃S and C₃A, which take values in the range of 0%–100%. Based on the ingestion method, we also introduce a set of four parameters to characterize the effect of aluminum sulfate based alkali-free quicklime based on the developed model;

2) The model is compared with various types of cement hydration test data obtained in the literature based on isothermal calorimetry with materials such as pure C₃A and C₃S, ordinary Portland cement, slag, etc., with aluminum sulfate based alkali-free quick setting agent quick setting agent contents of 6%, 7%, and 8%. The results show that the model describes well the commonly used first exothermic peak of hydration of cements;

3) This model is an extension and complement to the Park hydration model and focuses on the first exothermic peak of hydration, which occurs in approximately the first one hour of hydration. The model describes the rate of change of hydration as a function of the first exothermic peak and is based on the crystal nucleus growth model. The model also verifies the validity of the two classical models in terms of the first exothermic peak of hydration.

5 Conclusions

The aim of this study is to simulate the first exothermic peak of cement hydration. By integrating the theories of crystalline growth and exothermic hydration, a computational model is proposed and validated under the isothermal exothermic test in literatures. A range of parameters are used to characterize the content of C₃S and C₃A, and the influence of quicklime are carried out by ingestion method.

It was found that our proposed model is able to simulate the first exothermic peak of cement hydration well, and that the model can be applied to C₃S and C₃A contents from 0 to 100%, and can also be used for aluminum sulfate based alkali-free quicklime contents from 7%–9%.

In general, the presented model applied to the first exothermic peak of cement hydration results is in excellent agreement with the isothermal calorimetry tests in the literature and suggests that it can be used in predictive manner for studies of the ultra-early age performance of 3D printed concrete.

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