Modeling cement hydration by connecting a nucleation and growth mechanism with a diffusion mechanism. Part II: Portland cement paste hydration

2.1 C₃S hydration in dilute suspensions

The detailed derivation and verification of this model are presented in Part I [8]. Only the main equations will be summarized here. The constants involved in the model include the following: the original cement particle radius R₀, the nucleation density n, the parallel growth rate of the nuclei g₁, the perpendicular growth rate of the nuclei g₂, the volume ratio of the hydration products (formed on the surface of the cement particles, mainly C-S-H) to the cement reacted c, the time (t_d) at which the cement particle enters the diffusion controlled stage, and the diffusion constant D. As discussed in Part I, it is necessary to introduce a parallel growth rate constant S=ng12 and a perpendicular growth rate constant K=g₂/c to couple those parameters that cannot be determined independently by fitting the experimental hydration kinetics curves only. Given that t_d can be determined from S and K for a given R₀, the model only has three independent rate-controlling parameters, i.e., S, K, and D.

The following equations were derived for the NG stage of hydration (i.e., the modified cone model):

where t is the time, α is the degree of hydration at time t, and A_h is the surface area of the particle covered by hydration product at time t. A_h and α can be obtained numerically from the above equations. An implicit variable, the radius of the anhydrous core R, is uniquely related to α by,

When hydration becomes diffusion controlled (t>t_d), the following equation may be obtained:

where

in which t_d can be obtained by numerically solving Equations (1)–(3) for A_h=4πR² corresponding to a condition when the anhydrous cement particle is completely covered by hydration product. From Equations (3) and (4), α for the DC stage of hydration can be obtained by numerical integration. Equations (1)–(5) were developed for the hydration of a single cement particle. The total degree of hydration of a cement sample is the weighted average degree of hydration of all particles in the sample below

where α^T(t) is the total degree of hydration of the sample, α(R₀, t) is the degree of hydration of particles with a mean radius of R₀, f(R₀) is the weight fraction of particles with a mean radius of R₀, and N is the total number of gradations of the particle size distribution (PSD).

2.2 Portland cement paste hydration

The chemical composition of Portland cement undoubtedly has a significant impact on its hydration kinetics. Early hydration of ordinary Portland cement is believed to be dominated by C₃S and C₃A, whose combined hydration mechanism appears to be extremely complex and very difficult to model [11]. The model to be developed in this study is only applied to a special type of Portland cement, namely, the high-sulfate-resistance (HSR) grade Class H oil well cement, which typically contains little to no C₃A. Even with its early hydration dominated by C₃S, the Class H Portland cement paste hydration differs from C₃S hydration in dilute suspensions of constant lime concentration in terms of two major aspects.

First, the hydration product growing on Portland cement particles is likely to be C-S-H mixed with a variety of other hydration products (probably in a small quantity) instead of pure C-S-H. However, no modification to the model is necessary from this aspect as long as the hydration mechanism remains similar. This is especially true as it has already been shown that the shape of the growing nuclei does not significantly affect the fitting results [8]. For application to Portland cement paste hydration, the physical parameters (e.g., nucleation density, growth rate, and density of hydration product) shall be interpreted as the weighted average values of all hydration products growing on the surface of cement particles.

Second, in Portland cement paste, the particles are much more densely packed than in dilute suspensions. The hydration products growing on a cement particle are likely to interfere with those grown on neighboring particles as well as in the pore spaces, especially during later stages of hydration. The extended volume concept (Avrami equation) may be applied to take into account the interactions among different cement particles and CH crystals, i.e.,

where dV_hp is the incremental change of the real volume of hydration products, while dVhpe is the incremental change of the extended volume of hydration products assuming no impingement between different cement particles and CH crystals. In addition, V_T is the total representative volume of a cement paste, while V_S is the total volume of the solid phase in the representative volume. As the total degree of hydration is proportional to the total volume of hydration products, the following equation may be derived:

where α′ is the actual degree of hydration of the cement paste, and α^e is the extended degree of hydration of the cement assuming no impingement among different particles and CH crystals (i.e., the total degree of hydration α^T as presented in Section 2.1). If one assumes particles of all sizes behave like the system average, as suggested by Biernacki and Xie [12], then Equation (8) can be applied to individual cement particles and becomes compatible with the particle-based models. Hence, for Portland cement paste hydration, Equation (2) becomes

Similarly, Equation (4) becomes

It should be mentioned that Equations (9) and (10) do not take into account the shape of the CH crystals, which is known to be different in Portland cement paste than in alite paste [13], [14], [15]. The large CH crystals formed in alite paste may completely stop the hydration of certain C₃S particles enclosed inside. It should also be noted that the method used here to account for inter-particle interactions is different from that introduced in [12], though they are based on similar principles.

Assuming 1 unit mass of cement combines with m_w unit mass of water to form hydration products and results in a volume reduction (i.e., chemical shrinkage) of CS⁰, the Avrami correction factor, as shown in Equations (7)–(10), can be obtained as

where ρ_c and ρ_w are the densities of anhydrous cement and water, respectively. The slight variation of ρ_w with temperature and pressure has been taken into account in the model, although its effect on the fitted results seems to be negligible. R_wc is the w/c ratio of the paste. Note that in this study, the final w/c ratio of the set cement as reported in a previous study [16] was used in the model to account for loss of water due to bleeding. CS⁰ may be determined from a multi-linear empirical model [17] based on cement composition. Finally, m_w can be determined from the simplified cement hydration reaction stoichiometries listed in Equations (12)–(15) below

where p_x is the Bogue weight fraction of the x compound of the cement. The actual reaction stoichiometries of Portland cement hydration is known to be much more complex than the simplified equations listed above. Hence, the accuracy of the estimated value of m_w may be improved in the future with the advancement of our knowledge about cement hydration stoichiometries.

As mentioned earlier, the first two periods of Portland cement hydration are not considered in the proposed model. Given that the length of the induction period may vary depending on several factors, including w/c ratio [5] and mixing method [18], it is necessary to introduce an offset time when fitting the model to experimental data of Portland cement paste hydration. As will be shown later, this offset time is often found to be zero at ambient temperature.

4.1 Test data pre-processing

Given that the chemical shrinkage test is an indirect method of measuring hydration kinetics, the following equation is used to convert experimentally obtained data to the degree of hydration of cement [17]:

where α(t) is the degree of hydration at time t, and CS(t) and CS⁰ are the total chemical shrinkage (ml/g cement) at time t and at the complete hydration of cement, respectively. The variation of CS⁰ with temperature can be estimated by assuming that it decreases linearly at a rate of 0.60% per °C from the reference value obtained at 25°C, according to a recent study that correlates chemical shrinkage and heat evolution test results of different cements at varied temperatures [20]. Note that a slightly different rate was assumed in our previous studies [16], [21], which resulted in different activation energies estimated for the cement.

4.2 Fitting the proposed model and the BNG model to experimental data

As shown in Figure 1, both cements used in this study contain a small quantity of very fine particles (as small as 0.15 μm). Strictly speaking, the model developed in this study may not be applicable to these fine particles because of three reasons. First, the particles may be completely dissolved before the growth stage occurs. Second, the particles may not be perfectly dispersed. Third, the model assumption that the size of a nucleus is much smaller than that of a cement particle may not be valid. Similar to our previous study [16], we find that a slightly better fit to the derivative curve can be obtained by disregarding some of the finest particles (e.g., smallest 2%) of the cement. However, it is difficult to determine the exact particle size that would render the current model inapplicable. Thus, all particle sizes were considered in this study.

Figure 2 shows the hydration kinetics simulation results of the proposed model for individual cement particles (top) as well as for a representative cement sample with an ensemble of particle sizes (bottom). According to the proposed model, the transition of hydration mechanism from NG to DC stage for an individual particle typically occurs shortly after the hydration rate peaked. As in Part I, the time of transition of the hydration mechanism is different for particles of different sizes. Moreover, the different transition time results in the gradual decrease of the total hydration rate of a cement sample.

Figure 2:

Simulated hydration kinetics of individual cement particles (top) and the weighted average result of a sample with an ensemble of particle sizes (bottom) (R₀ is the particle radius in μm, • indicate transition from NG to DC stage).

Fitting to experimental data with the proposed model was obtained by adjusting the three rate-controlling model parameters S, K, and D to provide a good match to both the derivative and the integral curves of hydration kinetics. A representative fit to Test H-1 is shown in Figure 3. For comparison purpose, a fit generated by the widely used BNG model, which has recently been generalized to allow for different nucleation and growth processes [4], is also included in the figure. Apparently, very similar fits are obtained by the two completely different models, suggesting that a good fit with hydration kinetics data is inadequate to prove the validity of a model. A previous study has reported that similar fit to the experimental data can also been obtained by directly applying the model developed in Part I of this study without taking considering inter-particle interactions [16]. The primary difference is that some fitted model parameters (i.e., perpendicular growth rate constant and diffusion constant) in the previous study are relatively lower than those obtained in this study. Scherer has made a similar point by showing that nearly the same fit can be obtained with different confined growth models by slightly changing the model parameters [22]. Therefore, further investigations, possibly by test methods other than hydration kinetics, are needed to determine which of the two hypotheses mentioned in Section 1 is correct.

Figure 3:

Model fits to experimental data of Test H-1.

4.3 Effect of curing temperature

Due to the thermal hysteresis of the heat controllers, test data obtained at elevated temperatures displayed some oscillations, which made it difficult to accurately determine the derivative curves. In these cases, the model is only fitted to the integral curves. Representative fits to experimental results are provided in Figure 4. The fitted model parameters are presented in Table 4. All three rate constants (S, K, and D) increase significantly with increasing curing temperature. The offset time (t₀) also increases with increasing curing temperature, from 0 at ambient temperature to about 1.6 h at 62.9°C. As also discussed in previous studies [21], the effect of curing pressure on cement hydration is almost negligible (<10%) within the pressure range investigated in Test Series I. Therefore, the fitted model parameters vary very slightly with curing pressure.

Figure 4:

Representative model fit to experimental data of Class H cement cured at different temperatures (w/c=0.4).

The temperature dependence of a chemical reaction rate constant, including that for cement hydration, is frequently modeled by the Arrhenius equation [2], [4], [23] given by

where k is the rate constant, A is a proportionality constant that is independent of temperature, E_a is the activation energy of cement (J/mol), R is the gas constant (8.314 J/mol/K), and T is the absolute temperature (K). The Arrhenius equation is derived assuming a constant pressure process. According to chemical kinetics theory, a more generalized description of the dependence of a reaction rate constant on temperature and pressure is given by [24, 25]

where P is the curing pressure (Pa), and ΔV^‡ is the activation volume of cement (m³/mol).

If nucleation occurs in a short burst after mixing, as assumed in the proposed model, then curing temperature should have very little effect on nucleation density, considering that all slurries are mixed at ambient condition for about 15–20 min before being introduced to the pressure cells [16]. Initial mixing temperature has also been shown to have little effect on cement hydration kinetics [18]. Recall that in the current model, the parallel growth rate constant S is the product of the nucleation density (n) and the square of the nuclei parallel growth rate (g₁), while the perpendicular growth rate K is the nuclei perpendicular growth rate (g₂) divided by a more or less constant factor c (i.e., the volume ratio of the hydration product to the reacted cement). Therefore, S^1/2 and K are proportional to the parallel and perpendicular growth rates of the nuclei, respectively. A linear regression plot showing the temperature dependence of the fitted parameters is presented in Figure 5. Assuming that n and c are independent of curing temperature, the activation energies obtained for g₁, g₂, and D are 35.3, 38.3 and 30.7 kJ/mol, respectively. The values determined for the NG stage of cement hydration (i.e., for g₁ and g₂) are similar to those determined by other methods [4], [20], [23]. The activation energy for the DC stage of cement hydration has not been reported previously, but a similar value of 40 kJ/mol has been determined for the diffusion of water into silica glass at temperatures below 550°C [26].

Figure 5:

Temperature dependence of fitted model parameters (Class H cement, w/c=0.4).

4.4 Effect of curing pressure

Curing pressure has a relatively small effect on the hydration rate of cement; thus, ambient temperature fluctuations have to be accounted for to accurately model test results at different pressures. As reported in previous studies [21], the effect of a curing pressure increase of (up to) 17 MPa on cement hydration may be completely canceled by a slightly lower ambient temperature. For the standard Class H cement, only the temperatures of tests H-1, H-2, and H-4 were recorded. The model parameters obtained for these tests may be “corrected” to a uniform temperature using Equation (19) and the previously derived activation energies (Section 4.3). The effect of curing pressure on the growth rate constants (S and K) is found to be similar to that of curing temperature, only at a much smaller magnitude. The diffusion constant (D) also appears to increase with increasing curing pressure; however, the correlation is difficult to quantify with the current test data probably due to temperature fluctuations.

As discussed in Section 4.3, for a constant temperature process, the pressure dependence of a reaction rate constant is related to the activation volume of the reaction by Equation (20). A linear regression plot showing the pressure dependence of the fitted parameters at 26.1°C is presented in Figure 6. Assuming that n and c are independent of curing pressure, the activation volumes obtained for g₁ and g₂ are -28.1 and -33.8 cm³/mol, respectively. Similar values have also been obtained by other studies [21], [27], [28].

Figure 6:

Pressure dependence of fitted model parameters (Class H cement, w/c=0.38).

4.5 Effect of w/c ratio

Figure 7 shows representative results of fitting the proposed model to hydration kinetics data of Class H cement prepared with varied w/c ratios. As can be seen, the induction period (and hence the offset time t₀ of the model) increases slightly with increasing w/c ratio. The fitted model parameters for three different w/c ratios are listed in Table 5. All three rate constants appear to be largely independent of w/c ratio with some small random variations, probably associated with lab temperature fluctuations (not recorded for most tests). The perpendicular growth rate constant appears to decrease slightly with increasing w/c ratio, but the variation is too small to be conclusive. The most interesting part of this set of test results is that the diffusion constant is now found to be largely independent of w/c ratio, which is in line with our understanding of the diffusion mechanism. In previous studies where inter-particle interactions have been ignored, the diffusion constant has to be forced to vary with w/c ratio to obtain a good fit with experimental data [16]. These results suggest that the Avrami correction factor introduced in this study can adequately model inter-particle interactions for different w/c ratios.

Figure 7:

Representative model fit to experimental data of Class H cement prepared with different w/c ratios (ambient curing temperature).

The small lab temperature fluctuations that affect the test results in this study make it difficult to accurately quantify the effect of w/c ratio on cement hydration kinetics. Several studies have shown that the differential equation curve of cement hydration kinetics (defined as rate of hydration vs. degree of hydration) is largely independent of curing temperature and pressure when normalized by the peak rate [17], [20], [21], [29], especially for Class H cement containing no C₃A [20]. The normalized differential equation curves obtained at different pressures were averaged for each w/c ratio and presented in Figure 8 (top). It can be observed very clearly that the lower w/c ratio mainly causes a faster decrease in hydration rate just after the peak. The results are consistent with isothermal calorimetry tests [30]. To further demonstrate that the proposed model can accurately simulate such behavior, the model was first fitted with test data for w/c=0.5. By only changing R_wc in Equation (11) to 0.3, the model accurately mimics the experimental data for w/c=0.3 (Figure 8, bottom). Note that the induction period was removed from the experimental data for the fitting as it was not considered in the proposed model.

Figure 8:

Experimental and modeled effect of w/c ratio on the hydration kinetics of cement.

4.6 Effect of cement composition

Although the early hydration of Class H cement is primarily dominated by C₃S, the relative quantity of the various phases could also have some impact on test results. The proposed model was also fitted to the hydration kinetics data of the Class H-P cement, whose C₃S content is much lower than that of the Class H cement (Table 1). Representative fits to both the derivative curves and the integral curves are presented in Figure 9. The model generated excellent fits to experimental data obtained at different temperatures and pressures; however, the model slightly overestimates the degree of hydration at higher curing temperatures during later ages (t>40 h). While it is not yet clear what caused such discrepancies, the varied hydration mechanisms of different phases could be an important contributing factor as C₃S content is much less dominant in the Class H-P cement.

Figure 9:

Model fit to experimental data of Class H-P cement cured at temperatures and pressures (w/c=0.38).

The parallel and perpendicular growth rate constants are very similar to those obtained for Class H cement cured under similar curing conditions; however, the diffusion constant is more than twice as high as that for Class H cement. The diffusion constant obtained in this study is also about two orders of magnitude higher than that obtained in Part I [8], where pure C-S-H is believed to be the diffusion barrier. These results suggest that the permeability of the inner hydration product (i.e., the diffusion barrier) may be increased with an increasing amount of phases other than C₃S. The dependence of the fitted model parameters on curing temperature and pressure is shown in Figure 10. Assuming that n and c are independent of curing temperature, the activation energies determined for g₁, g₂, and D are 35.7, 36.9 and 24.8 kJ/mol, respectively. In addition, the activation volumes determined for g₁ and g₂ are -28.2 l and -28.1 cm³/mol, respectively. These values are also very similar to those determined for Class H cement in Sections 4.3 and 4.4.

Figure 10:

Temperature and pressure dependence of the fitted model parameters (Class H-P cement, w/c=0.38).