Distribution of Phases in a 3-D Cement Particle Image

Once the particles are placed, the next step is to create multi-phase particles by distributing the phases is such a manner as to match the volume and surface area fractions as estimated from the two-dimensional SEM images. A modification of a technique employed to reconstruct three-dimensional porous media from a two-dimensional image [18,19] is used for this purpose. To begin, the two point correlation function is determined for three different phase combinations in the two-dimensional segmented SEM image: the combined silicates (C₃S and C₂S), the C₃S, and either the C₃A or the C₄AF (whichever is the more abundant of the two). This function is evaluated for an M x N image using the following equation:

$$S(x,y) = \sum_{i=1}^{M-x} \sum_{j=1}^{N-y} \frac{I(i,j)\times I(i+x,j+y)}{(M-x)\times (N-y)}$$ (1)

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:

$$S(r)=\frac{1}{2r+1} \sum_{l=0}^{2r} S(r, \frac{\pi l}{4r})$$ (2)

where S (r, $S(r,\theta)=S(r \cos\theta, r \sin\theta)$) = S(r cos , r sin ) obtained by bilinear interpolation from the values of S(x,y).

The two-point correlation function for the C₃S and C₂S 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 [20]. This random number image is then filtered using the autocorrelation function, F(x,y,z):

$$F(r)=F(x,y,z)=\frac{[S(r=\sqrt{x^2+y^2+z^2})-S(0)\times S(0)]}{[S(0)-S(0)\times S(0)]}$$ (3)

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

$$R(x,y,z)=\sum_{i=0}^{30}\sum_{j=0}^{30}\sum_{k=0}^{30} N(x+i,y+j,z+k)\times F(i,j,k)$$ (4)

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 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.

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 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 3 x 3 x 3 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 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 3-D image two more times. The silicates are further segmented into C₃S and C₂S, while the aluminates are further divided into C₃A and C₄AF. Figure 5 shows a portion of an initial generated 3-D microstructure for Cement 116 at a w/c ratio of 0.4.

 

Figure 5: Portion of initial 3-D image of Cement 116 with w/c=0.4. Phases from brightest to darkest are: C₃A, gypsum, C₄AF, C₃S, C₂S, and porosity.