2. Procedures
For the comparison, the x-ray microtomography images of CCRL cement 133 mixed at a nominal water-to-cement ratio (w/c) of 0.45 were selected [7]. An image set obtained 3 h after mixing was used as being representative of the starting microstructure prior to any hydration. An image set obtained after 137 h of hydration (at about 25 ºC) was selected to represent a well-hydrated microstructure. 300 x 300 x 300 voxel subvolumes were extracted from each tomographic image to use for further characterization and analysis. As stated previously, the images were obtained with a voxel dimension of 0.95 µm. Individual phases (porosity, cement particles, and hydration products) were segmented from the raw data image based on analysis of the image´s greylevel histogram, as described previously [7]. The segmented three-dimensional images were then input into the correlation analysis programs.
Simulations were conducted using the latest version of the CEMHYD3D microstructure model being developed in the ongoing Virtual Cement and Concrete Testing Laboratory (VCCTL) consortium. The spatial resolution was set at 1 µm/pixel and simulations of both 100 x 100 x 100 and 200 x 200 x 200 microstructures were completed. The measured particle size distribution for cement 133 [10] was used to create a starting three-dimensional microstructure of cement (and gypsum) particles in water. The hydration models were executed to match both the measured w/c and degree of hydration of the real system, 0.47 and 0.62, respectively [7]. In accordance with the experimental conditions [7], hydration was conducted under sealed (and isothermal) conditions. Initial and final (post hydration) microstructures were analyzed using the same correlation analysis as that performed on the segmented real microstructure images. No segmentation per se was required for the model images, as the CEMHYD3D software outputs an image in which each voxel is already uniquely identified as a single phase.
Two-point correlation functions were computed for single or multiple phases using the following equation for an M x N image:
 | (1) |
where I(i,j)=1 if the voxel at location (i,j) contains the phase(s) of interest and I(i,j)=0 otherwise. S(x,y) was computed over all (100, 200, or 300) of the two-dimensional slices through the thickness of the three-dimensional microstructures. Since there is no inherent anisotropy in either the real or simulated microstructures, a complete three-dimensional correlation analysis was deemed unnecessary. These values were then converted to S(r) for distances r in voxels (or micrometers) by [11]:
 | (2) |
Finally, the normalized two-point correlation function was calculated as:
 | (3) |
Results are presented in terms of
the normalized correlation functions to eliminate the minor influence of
variable volumetric phase fractions between the model and real microstructure
images. It is worth noting that S(0)
simply represents the volume fraction of the phase(s) of interest in the
three-dimensional microstructure and N(0)=1. For large values of r, the
probability of two voxels having the same phase(s) is simply a random process,
with S(r) approaching a value of the
volume fraction of the phase(s) of interest squared. In this case, N(r) approaches a value of 0. The initial slope of the S(r)
function is proportional to the specific surface of the phase(s) of interest.
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