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## Mathematical Analysis of Cement Images

The final processed images can be analyzed to determine any number of stereological parameters. For cement hydration, parameters of interest include phase area fractions and phase surface perimeter fractions. For an isotropic system, the area fraction of a phase present in a 2-D image will directly correspond to its volume fraction in three dimensions. Similarly, a phase's share of the total perimeter (solid pixels in contact with porosity) will correspond to its share of the total surface area in 3-D. The surface area fractions of the phases are particularly important for cement hydration as the hydration reactions with water occur at the surfaces of the particles. Examples of the quantitative analyses determined using this evaluation procedure are provided in Figures 7 and 8 for cements 135 and 136, respectively. These results are also available in the cement images database at http://ciks.cbt.nist.gov/phpct/database/images.

For an isotropic material, the spatial correlation functions are identical in two and three dimensions, simply being a function of distance, r. Thus, the measured 2-D correlation function for a phase or a combination of phases can be used to reconstruct a 3-D representation of the cement particles [3]. For an M x N image, the two-point correlation function for a phase, S(x,y), is determined as:

 (1)

where I(x,y) is one if the pixel at location (x,y) contains the phase(s) of interest and 0 otherwise. S(x,y) is easily converted to S( ) for distances r in pixels. Because the correlation function implicitly contains information on the volume fraction and specific surface of the phase(s) being analyzed, this function can be employed to reconstruct a three-dimensional representation of the cementitious particles that matches the phase volume and surface area fractions and correlation structure of the 2-D final SEM image. These starting 3-D structures of cement particles in water can then be used as input images for the CEMHYD3D cement hydration and microstructure development computer model [3,4].

The phase compositions estimated using the SEM and image analysis are compared to those calculated based on the cement oxide composition using ASTM C150 [1] in Tables 1 and 2. For both cements 135 and 136, the SEM-measured values for C3S significantly exceed those calculated using ASTM C150, while the values for C 2S and C3 A are generally less than those calculated from the oxide compositions. The calculations presented in ASTM C150 are known to be only approximate [8], with quantitative optical point counting (ASTM C1356M) [1] or X-ray diffraction (ASTM C1365) [1] being preferred methods for performing quantitative phase analysis.

Table 2: Potential Volumetric Phase Compositions for Cement 136
 Phase ASTM C150 composition SEM analysis composition C3 S 59.56 65.6 ± 5.09 C2 S 20.64 18.12 ± 5.21 C3 A 9.46 6.65 ± 0.87 C4 AF 10.34 9.63 ± 0.97

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