Section 3 of Hydration Modelling Chapter

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3. 3-D Reconstruction

While 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 which match the following characteristics of the cement of interest: particle size distribution, phase volume fractions, and phase surface area fractions [10]. The latter two of these characteristics are determined based on analysis of a two-dimensional cement particle image. In addition, the autocorrelation functions for each phase and different groupings of phases are employed during the three-dimensional reconstruction process [17, 18] to match the correlation structure of the phases within the 3-D cement particles to their 2-D counterpart.

Initially, digitized spherical particles following the measured particle size distribution of the cement of interest are placed at random locations in a 3-D computational volume. This volume is typically 100³ pixels, where each pixel is one cubic micrometer in volume. The particles are placed from largest to smallest and not allowed to overlap one another. Periodic boundaries are used to eliminate edge effects; if a portion of a particle extends beyond one or more faces of the 3-D box, the remainder of its volume is protruded into the opposite face. Particles typically range from 3 to 35 pixels (micrometers) in diameter. 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 separated into distinct phase regions using the algorithm described below. In addition to assigning phases, each particle may be assigned a unique particle ID, if one is ultimately interested in examining the setting or percolation of the solids in the cement paste microstructure as it hydrates.

During particle placement, particles may be optionally flocculated or dispersed [2]. To flocculate the particles, after placement of all of the particles, each particle centroid is displaced by a distance of one pixel in one of six random directions (+/- 1 in the x, y, or z direction). If this move causes the current particle to impact another one, the two are flocculated and move as a single unit in all future random displacements. This algorithm is repetitively executed until the user-selected number of flocs is formed. To disperse the particles, say by a minimum separation of two pixels, each particle is placed using a diameter two pixels greater than its actual value. These one-pixel computational shells are then removed, assuring that all cement particles are separated by a distance of at least two pixels. It should be noted that this dispersion algorithm is practical only for w/c ratios greater than about 0.45, as at lower w/c ratios, sufficient space may not be available to place all of the "dilated" cement particles.

To begin the phase segmentation of the 3-D particle image, the two point correlation function is determined for three different phase combinations in the 2-D final segmented image (Figure 5 or 6: the combined silicates, the tricalcium silicate, and either the tricalcium aluminate or the tetracalcium aluminoferrite (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) is one if the pixel at location (x,y) contains the phase(s) of interest and zero otherwise. These values are then converted to S(r) for distances r in pixels by [19]:

where S(r,t)=S(rcos(t),rsin(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 3-D cement particle image is assigned a random number following a normal distribution, N(x,y,z), generated using the Box-Muller method [20]. This random number image is then filtered using the autocorrelation function, F(x,y,z):

The resultant image, R(x,y,z), is calculated as

Finally, for those pixels in the resultant image which were 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 which have been reassigned will correspond to the desired volume fraction for the reassigned phase (based on analysis of the 2-D 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 3-D image. However, it remains to match the surface area fractions as well. To do this, a pixel rearrangement algorithm, based on analysis of local 3-D curvature [21, 22] is employed. The local curvature is simply defined to be proportional to the fraction of pixels in some local neighborhood (e.g., a 3³ box or sphere) which are 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 2-D SEM image.

Once this phase separation is accomplished for converting the "cement" into the silicates and aluminates, the algorithms are executed on the developing 3-D image two more times. The silicates are further segmented into tricalcium and dicalcium silicate, while the aluminates are futher divided into tricalcium aluminate and tetracalcium aluminoferrite. Figure 7 shows a portion of an initial generated 3-D microstructure for a cement at a w/c ratio of 0.4. The three-dimensional structure is also illustrated in the movie "flight through cement" which provides a plane by plane flight through a 3-D cement particle image.

Figure 7. Portion of initial 3-D image for a cement with w/c=0.4. Tricalcium silicate is red, dicalcium silicate is blue, tricalcium aluminate is green, tetracalcium aluminoferrite is orange, and gypsum is pale green.

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