Figure 1 presents the experimental data along with the model results for the simplest model based on equation (5). The source of the experimental data is indicated by either the last name of the first author on the corresponding reference [4, 15-17] or by the proficiency sample cement number for CCRL proficiency sample cements C116 [2, 18] and C152 [12]. The value of k1 was chosen to provide a reasonable agreement to the experimentally measured degrees of hydration for the w/c=0.35 and w/c=0.45 cement 152 pastes at an age of 7 d. It is observed that this simplest model underpredicts the early hydration rate, overpredicts the later age hydration rate, and only qualitatively captures the relative degree of hydration trends in Figure 1b. The fits are slightly better for the case of sealed curing, as shown in Figure 2, but still far from adequate. Relating the hydration kinetics only to the amount of available porosity, while providing a general indication of the influence of w/c on degree of hydration, does not provide an adequate quantitative description of the available experimental data.
Figure 1. Predicted a) degree of hydration vs. time and b) relative degree of hydration vs. time for kinetics model based on equation (5) for saturated curing conditions with k1=0.011 h−1. Dark lines (solid, dotted, and dashed) represent model results and symbols connected by grey lines represent experimental data. In b), legend indicates the two w/c for which the ratio of their degrees of hydration is being determined.
Figure 2. Predicted a) degree of hydration vs. time and b) relative degree of hydration vs. time for kinetics model based on equation (5) for sealed curing conditions with k1=0.013 h−1.
However, when the kinetics models are extended to include both the amounts of available water-filled porosity and unhydrated cement, a significant improvement between model predictions and experimental data is observed. Figure 3 shows the results for the case of saturated curing conditions utilizing the model based on equations (6) and (7). Now, the agreement between model and experimental degree of hydration data for cement 152 is much more reasonable and the measured relative degrees of hydration for various w/c pairs are captured adequately by the model predictions, as shown in Figure 3b. Basing the hydration kinetics on a bimolecular basis that involves both of the reactants (cement and water) results in the simple derivation of a model for hydration kinetics that reasonably represents the observed data.
Figure 3. Predicted a) degree of hydration vs. time and b) relative degree of hydration vs. time for kinetics model based on equation (7) for saturated curing conditions with k2=0.05 h−1.
Equations (6) and (7) were also applied to the case of sealed curing (results not shown), but a better fit to the available experimental data for this curing condition was obtained by the application of the hydration kinetics model based on equations (8) and (9). The model predictions and experimental results for this case are provided in Figure 4. With the most complicated of the three models, a quite reasonable agreement with experimental data is observed. But, it should be noted that each of the three models described by equations (5) through (9) is based on only one free parameter (the rate constant ki). This can be contrasted against other models currently available in the literature where as many as six free "fitting" parameters may be employed [19]. In the models presented here, the parameter ki will vary with cement composition, PSD, and curing temperature, so that a calibration will be required to obtain the best fit for each particular cement and curing temperature. However, when determining the relative degrees of hydration, such as those shown in Figures 3b, etc., the model results are relatively insensitive to the chosen value of ki. For this reason, the model predictions are observed to adequately fit experimental data obtained for different cements [2, 4, 15-17]. Thus, the model predictions in Figures 3b and 4b could be used to predict the influence of w/c on achieved degree of hydration for portland cements in general, when cured near 20 ºC.
The results in Figures 1 to 4 indicate that hydrating under sealed conditions as opposed to saturated conditions increases the inherent hydration rate (ki) by about 20 %. This could be due to the presence of a pore solution with higher concentrations of alkali ions and a correspondingly higher ionic strength for the case of sealed curing conditions. This effect is beyond the scope of the current spatial-based models, so that ki must be determined for each curing condition, similar to the manner in which it must be determined for each curing temperature. Of course, as more experimental data becomes available, it may well be that ki can be represented as a (Arrhenius) function of temperature and saturation.
An interesting application of the kinetics equations presented above is to consider the influence of the additions of "inert" fillers, such as limestone, on achieved degree of hydration. In the literature, as summarized by Hawkins et al. [20], mixed results are reported; sometimes significant acceleration in the presence of the limestone is observed, while at other times, no significant effect is noted. Often, the experiments are difficult to interpret due to the fact that the limestone may be interground with the cement, changing the cement's PSD and introducing a confounding factor into the interpretation of any measured degree of hydration data.
Figure 4. Predicted a) degree of hydration vs. time and b) relative degree of hydration vs. time for kinetics model based on equation (9) for sealed curing conditions with k3=0.061 h−1.
From the viewpoint of equations (6) and (7), for example, there are at least three different manners for incorporating a limestone substitution into the model. The simplest approach might be to essentially ignore the limestone and consider only that the w/c is changed by the replacement of cement by limestone. With a 20 % mass substitution at a constant w/s ratio, for instance, the true w/c would change from 0.45 to 0.5625. Second, one could consider that in a constant volume system, the local porosity and cement volume fractions both are reduced by the presence of the inert filler. In this case, equations (1) and (3) become:
where ρfil is the specific gravity of the inert filler (2.71 for limestone present as calcite) and (s/c) is the filler-to-cement mass ratio (0.25 for a 20 % limestone substitution). Third, the possibility of the limestone filler providing nucleating surfaces for the precipitation of cement hydration products could be considered by including the filler volume fraction in the "γ(t)" term in equation (6), e.g.:
In equation (12), the simplifying assumption is being made that the fraction of filler surfaces available for precipitation is reduced in direct proportion to the degree of hydration of the cement. This assumption has been observed to provide a better overall fit to the available experimental data than the alternative of considering that all of the limestone surfaces are always available for the precipitation of hydration products. The latter case only provided a better fit to the 1 d relative degree of hydration experimental data, while vastly overestimating the experimentally measured relative degrees of hydration for 3 d and beyond.
As indicated by the comparison to the measured experimental results shown in Figure 5, the first and third methods both seem to provide reasonable fits to the experimental data for hydration times of 3 d and beyond. Since the two provide predictions that are basically indistinguishable for ages of 28 d and beyond, no conclusive preference can be established at this time. As would be expected, experimentally, the higher surface area fine limestone is seen to generally accelerate the hydration slightly more than its coarser counterpart, and to provide a better agreement with the third model where precipitation of hydration products on the limestone surfaces is directly considered. Finally, as indicated by the solid grey line in Figure 5, it is projected that for lower w/c values (such as 0.3), the influence of filler additions on achieved degree of hydration will be much more pronounced [13]. In this case, since there is insufficient space for complete hydration of the cement in the original unfilled paste, the increase in effective w/c due to the replacement of cement by filler provides a substantial increase in the relative volume of water-filled capillary pore space available for the precipitation of hydration products. A similar heightened influence of silica fume additions on chloride ion diffusivities at lower w/c has been previously noted [21].
Figure 5. Predicted and measured influence of 20% by mass (fine and coarse) limestone filler substitution on relative degree of hydration in an original w/c=0.45 cement paste (CCRL cement 152) hydrated under saturated curing conditions at 20 ºC, with k2=0.05 h−1.