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
functions^{31}
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
combinations^{14
} in the final two-dimensional segmented SEM image: the combined
silicates, the C_{3}S, and either the
C_{3}A or the C_{4}
AF (whichever is the more abundant of the two). This function is
evaluated for an *M*·*N* image using the following equation:

(1) |

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}

(2) |

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*)):

(3) |

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

(4) |

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 C_{3}S and C_{2
}S, whereas the aluminates are further divided into
C_{3}A and C_{4
}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 (C_{3}
S is red, C_{2}S is aqua,
C_{3}A is green, C_{4}
AF is orange, and gypsum is pale green.