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2.4 Stereological Parameters

The final processed image 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.

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 [8]. For an M x N image, the two-point correlation function for a phase, S(x,y), is determined as:

 

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, for example, are needed to provide accurate input for three-dimensional cement hydration and microstructural development computer models which predict the development of microstructure and properties of hydrating cement paste with time [8].