Figure 7 provides a plot of the normalized experimental results for Cements 115 and 116 for w/c ratios of 0.30, 0.40, and 0.45 at T = 25ºC. 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 VI. The nonevaporable water contents have been normalized by the values measured experimentally on high w/c mixes. 14 Finally, the chemical shrinkage values have been normalized by the value (within ±0.01 mL/(g of cement)) that gives the best fit to the nonevaporable water content data for the w/c = 0.45 systems. The w/c = 0.45 data have been chosen because these results should not be affected by the depercolation of the capillary porosity, as discussed below. This value then is held constant at the lower w/c ratios. As shown in Fig. 7, excellent agreement is observed between the three measured properties, except for the lower w/c ratios at longer times, as explained below. A previous study by Parrott et al.49 has produced similar agreement, finding "a directly proportional relationship between the heat of hydration and chemical shrinkage." Geiker29 has noted a linear relationship between chemical shrinkage and nonevaporable water content for an ordinary Portland cement with w/c = 0.5, cured at 20º C. In addition, in 1935, Powers39 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 of water/(100 g of cement)). For the results in Fig. 7, we find values of 16.9 ± 0.9 and 20.6 ± 0.9 for Cements 115 and 116, respectively, in good agreement with Powers' results.
One interesting observation can be made concerning the chemical shrinkage data for the lower w/c ratios in Fig. 7. For both the 0.30 and 0.40 w/c ratios, we can observe that, at longer times, the chemical shrinkage curves diverge away from the nonevaporable water content data. In every case, the chemical shrinkage is below the nonevaporable water content data at long times. As has been suggested by Geiker, 29 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 is reached where the capillary porosity is no longer connected, and transport then must occur through the much smaller gel pores in the C-S-H gel.11 Because this transport occurs at a much slower rate, the rate at which water is absorbed into the specimen falls 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. 7 indicate the degree of hydration needed to achieve this capillary pore discontinuity, based on results from the original C3 S hydration model.11, 13 The agreement between the experimental observations and the predicted point of discontinuity is quite good, particularly for the w/c = 0.30 data sets. This pore discontinuity also has been observed recently using impedance spectroscopy measurements on partially frozen cement paste specimens.50
|Table VI. Enthalpy of Complete
Hydration for Major Phases of Cement
|Phase||Enthalpy (kJ/(kg phase))||Source|
|C3S||517||Ref. 26 1|
|C3A||1144||Ref. 26 2|
|C4AF||725||Ref. 37 3|
w/c = 0.4 and T = 21ºC.|
2 Assuming production of monosulfoaluminate phase.
3 w/c = 0.5 and T = 20ºC.
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 versus time, 51 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. 52 The parabolic form of the model, which generally has been found to provide the better fit to the experimental data,14 takes the form
where Au is the ultimate achievable value of the property, t0 an induction time, and k a rate constant. Equation (6) does not attempt to model degree of hydration, , during the early acceleratory period of the cement hydration (0 < >< 0.15), but generally provides an excellent fit to experimental data for a > 0. 15. 52
Equation (6) has been fitted to the experimental nonevaporable water content data using nonlinear regression analysis available in DATAPLOT,53 a graphical analysis software package developed at NIST. Although the 8 h data point for nonevaporable water content, generally corresponding to an in the range 0.10-0.15, is slightly outside the range of application of the dispersion model recommended by Knudsen,52 the fitted lines deviate little from these data values, suggesting that > 0.10 may be a more practical application range for Eq. (6). Figure 8 provides a representative example of the fit of Eq. (6) to the experimental data; for both cements, the fits in general are excellent. Table VII summarizes the results of the regression analysis, including the results generated at all three temperatures investigated in this study. For the 25ºC results, the value of t0 , is relatively constant for a given cement at the three different w/c ratios.
Fig. 7. Experimental results for CCRL Cements 115 and 116 versus time.
|Table VII. Parameters for Knudsen's Parabolic Dispersion Model for Cements 115 and 116|
|w/c||Temperature (ºC)||Au (g H2O)/(g cement))*||k(h-1/2 ) *||t0 (h)*|
|0.30||15||0.180 (0.007)||0.124 (0.014)||7.3 (0.5)|
|0.30||25||0.171 (0.003)||0.218 (0.017)||6.2 (0.5)|
|0.30||35||0.159 (0.002)||0.359 (0.025)||5.4 (0.5)|
|0.40||15||0.215 (0.003)||0.089 (0.004)||6.9 (0.3)|
|0.40||25||0.193 (0.002)||0.154 (0.005)||6.2 (0.2)|
|0.40||35||0.180 (0.002)||0.272 (0.014)||4.8 (0.5)|
|0.45||15||0.218 (0.008)||0.099 (0.012)||7.0 (0.6)|
|0.45||25||0.207 (0.003)||0.145 (0.008)||5.4 (0.5)|
|0.45||35||0.185 (0.002)||0.271 (0.013)||5.0 (0.4)|
|0.30||15||0.196 (0.005)||0.139 (0.011)||7.6 (0.2)|
|0.30||25||0.181 (0.004)||0.299 (0.031)||7.5 (0.2)|
|0.30||35||0.183 (0.001)||0.372 (0.013)||5.2 (0.2)|
|0.40||15||0.232 (0.009)||0.103 (0.012)||7.5 (0.4)|
|0.40||25||0.221 (0.004)||0.197 (0.016)||7.5 (0.2)|
|0.40||35||0.211 (0.004)||0.304 (0.032)||5.4 (0.7)|
|0.45||15||0.247 (0.007)||0.106 (0.009)||7.5 (0.3)|
|0.45||25||0.231 (0.005)||0.187 (0.016)||7.0 (0.4)|
|0.45||35||0.226 (0.006)||0.275 (0.035)||5.3 (0.9)|
|*Numbers in parentheses indicate approximate standard deviation provided by DATAPLOT 53|
Fig. 8. Fit of Knudsen's parabolic dispersion model to experimental nonevaporable water content (g H2O /(g cement) versus time for Cement 115 with w/c = 0.40 and T = 25ºC.