Experimental Results

Figure 8 provides a plot of the normalized experimental results for Cements 115 and 116 for w/c ratios of 0.3, 0.4, and 0.45. In this figure, the heat of hydration values have been normalized by the values calculated based on the Bogue potential phase compositions of the cements and the tabulated heats of hydration of the major phases provided in Table 7. The non-evaporable water contents have been normalized by the values measured experimentally on the high w/c mixes as described in the experimental section. Finally, the chemical shrinkage values have been normalized by the value (within + 0.01 ml/g cement) which gives the best fit to the non-evaporable water content data for the w/c=0.45 systems. The w/c=0.45 data were chosen because these results are not affected by the depercolation of the capillary porosity, as will be discussed below. This value was then held constant at the lower w/c ratios. As can be seen in Figure 8, excellent agreement is observed between the three measured properties (non-evaporable water content, heat release, and chemical shrinkage). A previous study by Parrott et al. [32] has produced similar agreement, finding "a directly proportional relationship between the heat of hydration and chemical shrinkage." Geiker [11] has noted a linear relationship between chemical shrinkage and non-evaporable water content for an ordinary portland cement with w/c=0.5, cured at 20^oC. In addition, in 1935, Powers [33] reported a linear relationship between heat of hydration and water absorbed during hydration for four different cements, with a constant of proportionality of 19.3 (cal/g)/(g water/100 g cement). For the results in Fig. 8, we find values of 16.9 and 20.6 for Cements 115 and 116 respectively, in good agreement with Powers' results.

  

Figure 8: Experimental results for CCRL Cements and 115 and 116 vs. time.

One interesting observation can be made concerning the chemical shrinkage data for the lower w/c ratios in Fig. 8. For both the 0.3 and 0.4 w/c ratios, one can observe that at longer times, the chemical shrinkage curves diverge away from the non-evaporable water content data. In every case, the chemical shrinkage is seen to lie below the non-evaporable water data at these long times. As has been suggested by Geiker [11], this is due to the depercolation of the capillary porosity in the hydrating cement paste. As hydration occurs, depending on the initial w/c ratio, a point will be reached where the capillary porosity is no longer connected, and transport must then occur through the much smaller gel pores in the C-S-H gel [4]. Since this transport will occur at a much slower rate, the rate at which water is absorbed into the specimen will fall below the rate at which empty voids are being generated, leading to the observed divergence in the experimental curves. The horizontal lines provided in Fig. 8 indicate the degree of hydration needed to achieve this capillary pore discontinuity, based on the results of the original C₃S hydration model [3,4]. The agreement between the experimental observations and the predicted point of discontinuity is quite good, particularly for the w/c=0.3 data sets. This pore discontinuity has also recently been observed using impedance spectroscopy measurements on partially frozen cement paste specimens [34].

The heats of hydration measured after seven days using the NIST microcalorimeter can be compared to the values determined using the heat of solution technique in the CCRL proficiency sample program [5]. Table 8 summarizes the results for Cements 115 and 116. The values measured using the microcalorimeter are below, but within 1.3 standard deviations, of the CCRL reported values. These lower values would be expected since the results here do not include the contribution of the initial exothermic mixing peak, as discussed in the experimental section. Additionally, the mixing of the samples for the standard heat of solution method [6] may be more complete than that achieved in the small microcalorimeter cells, also contributing to a greater heat release in the former case. However, it is encouraging to note that the differences in heat release between Cements 115 and 116 are basically identical for the two techniques, being 49 and 50 kJ/kg for the mean heat of solution and microcalorimeter measurements, respectively.

Table 8: Measured Seven Day Heats of Hydration for CCRL Cements 115 and 116

Cement	Heat of Solution Method ( k J / kg ) [5]	Heat of Solution Standard Deviation ( k J / kg ) [5]	NIST microcalorimeter Method ( k J / kg )
115	311	28	277
116	360	26	327

To calibrate the model to the experimental results, both are fitted to the same functional form. In the literature, a variety of models have been used to fit either degree of hydration or strength development vs. time [35], mainly in connection with the application of the maturity method to concrete strength development. Two commonly used models are the linear and parabolic dispersion models originally developed by Knudsen [36]. The linear model for a property of interest, A, is as follows:

$$A=A_u\frac{k(t-t_0)}{1+k(t-t_0)}$$ (6)

where A_u is the ultimate achievable value of the property, t₀ is an induction time, and k is a rate constant.

The equation can be plotted using the interactive graph shown below (simply move one of the scrollbars to activate the plotting) java applet courtesy of Prof. Henry Bungay of Rensselaer Polytechnic Institute:

The parabolic model is similar, taking the form:

$$A=A_u\frac{k\sqrt{(t-t_0)}}{1+k\sqrt{(t-t_0)}}.$$ (7)

Depending on the particular cement being studied, Geiker has noted that one of the two above equations will generally provide the better fit to experimental data [11]. Thus, both equations were fitted to the experimental data for non-evaporable water content vs. time. The non-evaporable water content data was selected (as opposed to heat of hydration or chemical shrinkage), because experimental values were collected for periods of up to 90 days.

Equations 6 and 7 were fitted to the experimental data using non-linear regression analysis available in DATAPLOT [37], a graphical analysis software package developed at NIST. Figure 9 provides a representative example of the fit of the two equations to the experimental data; all of the graphs for the two cements and three w/c ratios are provided in Appendix A. Tables 9 and 10 summarize the results of the regression analysis, including the residual standard deviations of the fits. In general, for Cements 115 and 116, the better fit to the experimental non-evaporable water content data was provided by Equation 7, as indicated by the lower residual standard deviation. In addition, using this model, the values of t₀ are relatively constant for the three w/c ratios for each of the two cements and the values of A_u for the w/c=0.45 data sets are fairly close to the values of 0.226 and 0.235 measured on the high w/c ratio pastes of 115 and 116 respectively, as would be expected for w/c values greater than about 0.42 [8,25]. Based on these considerations, this equation will also be used to fit the model results, for calibration against experiment.

  

Figure 9: Fits of Knudsen's dispersion models to experimental non-evaporable water content (g H₂O/g cement) vs. time.

Table 9: Parameters for Knudsen's Linear Dispersion Model for Cements 115 and 116

Cement	w/c	Au (g H2O/g cement)	k (h-1 )	t0 (h)	Res. Std. Dev.
115	0.3	0.152	0.0382	-1.52	0.0044
115	0.4	0.165	0.0254	-3.55	0.0064
115	0.45	0.176	0.0219	-8.02	0.0086
116	0.3	0.164	0.0607	3.93	0.0026
116	0.4	0.193	0.0385	3.39	0.0043
116	0.45	0.201	0.0338	0.77	0.0055

 

Table 10:Parameters for Knudsen's Parabolic Dispersion Model for Cements 115 and 116

Cement	w/c	Au (g H 2O/g cement)	k (h-1/2 )	t0 (h)	Res. Std. Dev.
115	0.3	0.171	0.218	6.23	0.0033
115	0.4	0.193	0.154	6.22	0.0015
115	0.45	0.207	0.145	5.42	0.0029
116	0.3	0.181	0.299	7.49	0.0048
116	0.4	0.221	0.197	7.54	0.0048
116	0.45	0.231	0.187	7.04	0.0052

Model Results