Although two-dimensional images of cement particles are useful for characterizing cements, three-dimensional representations are necessary to obtain hydrated microstructures for the computation of percolation, mechanical, and transport properties. Recently, computational techniques have been developed for creating three-dimensional cement particle images that match the following characteristics of the cement of interest: particle-size distribution, phase volume fractions, and phase surface-area fractions. 14 The latter two of these characteristics are determined based on analysis of a two-dimensional cement particle image. In addition, the autocorrelation functions31 for each phase and different groupings of phases are used during the three-dimensional reconstruction process to match the correlation structure of the phases within the three-dimensional cement particles to their two-dimensional counterpart.
Initially, digitized spherical particles matching the particle-size distributions given in Table II, at a resolution of 1 µm/ pixel, are placed from largest to smallest at random locations into a three-dimensional computational volume 100 pixels on a side, using periodic boundary conditions. A fraction of the particles are assigned to be gypsum (to match the gypsum volume fraction of the cement), with the remainder being designated as cement and later separated into distinct phase regions using the algorithm described below. Thus, we are explicitly assuming that the gypsum and the cement particles follow the same particle-size distribution. Because no superplasticizer or water-reducing agent has been used in the experimental studies, after random placement, the particles are flocculated into a single floc structure by randomly displacing their centroids by a distance of one pixel in one of six random directions (± x, ± y, ± z) and moving all contacting particles as a single unit in subsequent iterations of the algorithm.2, 14
To begin the phase segmentation of the three-dimensional particle image, the two-point correlation function is determined for three different phase combinations14 in the final two-dimensional segmented SEM image: the combined silicates, the C3S, and either the C3A or the C4 AF (whichever is the more abundant of the two). This function is evaluated for an M·N image using the following equation:
where, I (x, y) = 1 if the pixel at location (x, y) contains the phase(s) of interest and I (x, y) = 0 otherwise. These values are then converted to S (r) for distances r in pixels by 31
where, for angles t, S (r,t ) = S (r cos t,r sin t ) is obtained by bilinear interpolation from the values of S (x,y).
The two-point correlation function for the silicates is used to separate the cement particles into silicates and aluminates. To do this, each pixel in the three-dimensional cement particle image is assigned a random number following a normal distribution ( N (x, y, z)) generated using the Box-Muller method. 32 This random number image is then filtered using the auto-correlation function ( F (x, y, z)):
The resultant image ( R (x, y, z )) is calculated as
Finally, for those pixels in the resultant image originally assigned to be the phase(s) of interest (cement in this first case), a threshold operation is performed to create the appropriate volume fractions of the two phases. For example, if a cement pixel of interest has an R-value above a critical threshold, it is reassigned to be the aluminate phase. If not, it is assigned to be the silicate phase. The critical threshold value is determined such that, after the threshold operation, the fraction of pixels that has been reassigned corresponds to the desired volume fraction for the reassigned phase (based on analysis of the two-dimensional SEM images).
After this algorithm is executed to separate the cement (non-gypsum) particles into silicates and aluminates, the appropriate volume fractions of these two "phases" exist in the generated three-dimensional image. However, it remains to match the surface-area fractions as well. To do this, a pixel rearrangement algorithm, based on analysis of local three-dimensional curvature, 33, 34 is used. The local curvature is defined simply to be proportional to the fraction of pixels in some local neighborhood (e.g., a 3 x 3 x 3 box or sphere) that is assigned to be porosity. Here, pixels of one solid phase located at high-curvature sites are exchanged with pixels of the other solid phase located at low-curvature sites. This changes the fraction of each phase in contact with the pore space so that the surface-area fractions of each phase can be made to match the perimeter fractions present in the original two-dimensional SEM image.
Once this phase separation is accomplished for converting the "cement" into the silicates and aluminates, the algorithms are executed on the developing three-dimensional image two more times. The silicates are further segmented into C3S and C2 S, whereas the aluminates are further divided into C3A and C4 AF. Figure 4 shows a portion of an initial generated three-dimensional microstructure for Cement 115 at a w/c ratio of 0.40.
Fig. 4. Portion of a reconstructed three-dimensional starting image for Cement 115 with w/c = 0.40 (C3 S is red, C2S is aqua, C3A is green, C4 AF is orange, and gypsum is pale green.