EXPERIMENTAL AND MODELLING APPROACH

The general procedure for modelling the behavior of a cement of interest is as follows. The cement to be modelled is potted in a low viscosity epoxy, the resin is cured, and the sample is polished and viewed in the SEM [15]. In addition to the backscattered electron image, a series of X-ray images are collected for a set of elements, typically, Ca, Si, Al, Fe, and S. This series of images is then processed to determine the mineralogical phase located at each pixel, resulting in a final image such as that shown in Figure 1. For this two-dimensional image, the autocorrelation functions are measured for the following phase combinations: the silicates (tricalcium and dicalcium), the C₃S, and either the C₃A or the C₄AF, whichever is present in the greater volume fraction.

The next step in creating a starting three-dimensional image is to place digitized spherical particles at random locations in the three-dimensional computational volume following the measured PSD for the cement of interest such that the desired water-to-cement (w/c) ratio is obtained. Computationally, the particles are placed from largest to smallest to avoid the problem of not being able to find a location for a larger particle in a system already filled with much smaller ones. In this placement process, the particles can be placed totally at random, flocculated into a user-specified number of floc structures, or dispersed such that all cement particles are separated from one another by a specified distance (1 or 2 pixels). The flocculation and dispersion has major effects on the hydration needed to achieve set for model cement pastes [6], but has little influence on long term diffusion and percolation properties. During this placement process, a fraction of the particles are randomly assigned to be gypsum, with the remaining particles placed as cement, to be later assigned a specific mineralogical phase. Thus, the assumption is being made that the gypsum and cement follow the same PSD.

The final step in creating a realistic three-dimensional image of the cement particles is to distribute the cement clinker phases amongst the particles assigned to be cement during particle placement [8]. The measured autocorrelation functions are used to create a three-dimensional filter that is applied to an image of Gaussian random noise overlaid on the 3-D cement particle image. After the filtering process, the correlated random noise image is thresholded to obtain the requested volume fraction of a specific phase. A curvature assessment algorithm [21] is then employed to match the surface area fraction of the phase to that measured in the real 2-D image. This procedure is first executed to separate the cement into silicates and aluminates. In two further executions of the programs, the silicates are further separated into C₃S and C₂S, and the aluminates into C₃A and C₄AF, resulting in a final 3-D image that is ready to be submitted to the hydration model. A portion of such an image for a typical OPC is given in Figure 2.

  

Figure 2: Portion of a reconstructed three-dimensional starting image for an OPC with w/c=0.40. Aluminates are brightest, followed by gypsum, with silicates being the darkest grey and water-filled porosity black.

The hydration model implements a series of CA rules to simulate the reactions that occur between the starting cement phases and water [7,38]. To accurately simulate these reactions, the densities, molar volumes, and heats of formation of the individual compounds are required. Table I summarizes the values employed for these parameters in the current version of the microstructure model [39]. Using these values, each hydration reaction can be expressed in terms of volumetric units (pixels) for implementation in the model. Because in many cases, the volume occupied by the hydration products is less than that of the reactants, empty pore space may be created within the model 3-D structure to simulate the process of chemical shrinkage and self-desiccation [40], which can have a major influence on kinetics and properties in low w/c ratio systems.

Table 1: Physical Properties of Cementitious Materials

Compound Name	Compound Formula	Density ( Mg / m 3 )	Molar volume ( cm 3 / mole )	Heat of form. ( k J / mole )
Tricalcium silicate	C3S	3.21	71.0	-2927.82
Dicalcium silicate	C2S	3.28	52.0	-2311.6
Tricalcium aluminate	C3A	3.03	89.1	-3587.8
Tetracalcium aluminoferrite	C4AF	3.73	128	-5090.3
Gypsum	$C\bar{S}H_2$	2.32	74.2	-2022.6
Calcium silicate hydrate, C-S-H	C1.7SH4	2.12	108	-3283.
Calcium hydroxide	CH	2.24	33.1	-986.1
Ettringite	$C_6A\bar{S_3}H_{32}$	1.7	735	-17539.
Monosulfate	$C_4A\bar{S}H_{12}$	1.99	313	-8778.
Hydrogarnet	C3AH6	2.52	150	-5548.
Iron hydroxide	FH3	3.0	69.8	-823.9
Water	H	1.0	18.0	-285.83

To calibrate the kinetics of the model, a detailed set of experiments was conducted for two OPCs at each of three different w/c ratios and three different temperatures (15 ^oC, 25 ºC, and 35 ºC) [7]. The experimental and modelling program for this validation exercise is outlined in Figure 3. The two cements were characterized by SEM and PSD analysis to provide the needed model inputs. Experimentally, degree of hydration was quantified in three different fashions: non-evaporable water, heat release, and chemical shrinkage. Excellent correlation was observed between these three measures of the hydration progress [7]. Once the model was calibrated for one of the cements at one w/c ratio, it was able to successfully predict these properties for both of the cements at the three different w/c ratios, as well as the effects of curing under sealed as opposed to saturated conditions.

  

Figure 3: Flow diagram summarizing experimental and modeling program for predicting cement performance.