Figure 1: Measured particle size distribution for
Montalieu cement.
A sample of the Montalieu cement was imaged using SEM/X-ray analysis [2] to obtain a two-dimensional image in which each phase of the cement was uniquely identified. This image, along with the measured particle size distribution for the cement (Figure 1), was used in reconstructing a three-dimensional representation of the cement at each w/c ratio [3,4].
Figure 1: Measured particle size distribution for
Montalieu cement.
The cellular automaton-based 3-D cement hydration and microstructural model has been described in detail in a number of recent publications [3,4, 5]. The model operates as a sequence of cycles, each consisting of dissolution, diffusion, and reaction steps. To calibrate the model kinetics for this particular cement, experimental hydration tests were conducted under saturated conditions for w/c=0.35 at a constant temperature of 25 ºC. Degree of hydration was quantified based on measurement of the non-evaporable water content (WN) using loss on ignition techniques [3]. Figure 2 presents the experimental data and model calibrated results for degree of hydration vs. time. Here, model dissolution cycles were converted to time, t, in hours using the equation:
| t(model, h) = t 0 + β x cycles 2 | (1) |
where t0 is the induction time (7.5 hours) and β is a calibration constant, determined to be 0.0011 ± 0.0001 (one standard deviation) for this particular data set, based on analysis using the parabolic kinetic model of Knudsen [6]. Following this calibration, excellent agreement was observed between the model and experimentally measured degrees of hydration over the full range of the 90 day study.
Figure 2: Comparison of experimental (data points) and simulated (solid lines)
degree of hydration vs. time curves.
For the concretes examined in this study, curing was conducted under sealed conditions. Thus, the microstructural model was also executed under self-desiccating conditions, in which empty pore space is created as the hydration proceeds [4]. Based on the volume stoichiometry of all hydration reactions, the computer model tabulates the current pore volume and the volume of water not yet consumed by reactions, with the chemical shrinkage being the difference between these two quantities. After each cycle of the hydration, the appropriate volume of capillary porosity is converted from being water-filled to being empty, emptying the "pores" from largest to smallest in an attempt to simulate the physical process. For w/c ratios of 0.4 and lower, curing under sealed conditions will result in significantly less hydration being achieved than equivalent curing under saturated conditions, due to the lack of capillary water available to continue the hydration at later ages [7].
To modify the NIST hydration model to operate under adiabatic conditions, the cumulative heat generated by the hydration reactions is tabulated and the heat capacity of the concrete mixture adjusted for the effects of hydration. The heats of reaction used for the individual cement clinker phases and silica fume are provided in Table V, while the heat capacities of various components of a concrete mixture are provided in Table VI. As hydration occurs, the overall heat capacity of the concrete mixture decreases by about 5% as free water is bound into the hydration products.
| TABLE 5 Enthalpy of complete hydration for major phases of cement |
||
|---|---|---|
| Phase | Enthalpy (kJ/kg phase) | Source |
| Tricalcium silicate | 517 | (8)a | Dicalcium silicate | 262 | (8) |
| Tricalcium aluminate | 1144 | (8) |
| Tetracalcium aluminoferrite | 725 | (9)b |
| Silica fume | 780 | (1) |
| a w/c = 0.4 and T = 21ºC | ||
| b w/c = 0.5 and T = 20ºC | ||
| TABLE 6 Heat capacities of concrete components (1) |
|
|---|---|
| Component | Heat Capacity (J/g ºC)) |
| Siliceous Aggregate | 0.75 |
| Limestone Aggregate | 0.84 |
| Cement | 0.75 |
| Silica Fume | 0.75 |
| Water | 4.18 |
| Bound Water | 2.2 |
Knowing the heat released during a given cycle of the hydration and the current heat capacity, an incremental temperature rise from cycle (i-1) to cycle i can be calculated as:
Based on equation 1, the differential time elapsing between cycles (i-1) and i can be determined as:
This increment in time would correspond to a constant reaction temperature of 25 ºC, the temperature at which the model calibration was performed. To use the maturity method to determine the equivalent time at a different temperature, the variation in reaction rate, kT, with temperature is described by an Arrhenius function [10]:
Thus, as the microstructural model executes, equations 2 and 6 are used to update the temperature and elapsed time, respectively. In addition, an induction time and initial temperature must be specified prior to model execution.
To model concretes containing silica fume, it was necessary to further modify the microstructural model to incorporate the pozzolanic reaction between calcium hydroxide (CH) and silica fume (S). Based on the results of Atlassi [11] and Lu et al. [12], the following reaction was used to represent the pozzolanic reaction:
1.1CH + S + 2.8H
C1.1SH
3. 9 | (7) |
Based on molar volumes of 27 cm3/mol for silica, 33.1 cm3/mol for CH, and 18 cm3/mol for water, and assuming a molar volume of 101.81 cm3/mol for pozzolanic C-S-H, one can compute a chemical shrinkage of 0.20 g H2O/g silica fume. This value agrees with the range of values of 0.15-0.24 g H2O/g silica fume measured by Jensen [13]. Lu et al. [12] have observed a reduction in the H/S ratio of the pozzolanic C-S-H from 3.9 to 2.1 in going from 3 days to 28 days. Here, we have elected to use the constant value of 3.9 as it provides a better fit to the 4 to 8 day temperature rise of the concretes investigated experimentally. Also, based on the analysis of experimental data, an activation energy of 83.14 kJ/mol was determined for this pozzolanic reaction, which agrees with the value of approximately 80 kJ/mol measured by Jensen [13]. This activation energy was further validated by using the microstructural model to simulate the temperature-time response of a system containing only calcium hydroxide crystals, silica fume particles, and water, as shown in Figure 3. While some minor differences exist at intermediate times, the overall agreement seems to suggest that a reasonable value is being employed for the activation energy in the model. The increased slope of the experimental temperature curve at intermediate times may be due to the presence of a size distribution of silica fume particles, not accounted for in the simulation.
Figure 3: Comparison of experimental (data points) and simulated (solid lines)
temperature rise curves for system containing only silica fume and CH.
In the microstructural model, silica fume particles are modelled as one pixel (1 µm) elements. Because the silica fume used in this study was only about 90% SiO2, 10% of the silica fume was added as inert particles which do not participate in any chemical reactions. This 90% "efficiency" for the silica fume is also in agreement with the measured long term calcium hydroxide content present in the w/c=0.45 concrete containing a 5% silica fume addition. To incorporate the above pozzolanic reaction into the model, calcium hydroxide crystals which form at any point in the hydration remain soluble, so that they may gradually dissolve over time, generating diffusing calcium hydroxide species which can then react at all silica fume surfaces. When there is no silica fume present in the system or when all of the silica fume present has been consumed, CH diffusing species simply reprecipitate on CH crystals, resulting in the Ostwald ripening of larger crystals at the expense of the smaller ones. The activation energy of the pozzolanic reaction, relative to that for the hydration of the cement, is used to change the probability of a reaction occuring when a diffusing CH species encounters a solid silica fume surface. This probability is adjusted with increasing temperature based on the ratio of the relative increase in the pozzolanic reaction rate to the relative increase in the hydration reaction rate, both calculated based on equation 4. In addition, the solubility (dissolution probability) of CH is adjusted with temperature according to the data provided in [8], which indicates about a 25% decrease in solubility as temperature increases from 25 ºC to 60 ºC.
