Filtering of random noise particle image

Once a 3-D image incorporating the desired PSD has been created, the next step is to introduce the appropriate phase volume fractions, phase surface area fractions, and correlation structure into the initially monophase cement (or fly ash) particles. For fly ash particles, two specialized programs for randomly distributing the fly ash phases on either a particle or pixel basis are described in reference [5]. Alternatively, the procedure described below for distributing phases amongst the cement particles could be employed for fly ash particles as well, as long as the spatial statistics (correlations, volume and surface fractions, etc.) for the fly ash particles had been previously determined.

For cement, the phase distribution process is accomplished in a series of steps using one Fortran (or C) program and two programs available only in C. The program available in Fortran or C, rand3d.f or rand3d.c (listings provided in Appendix B), is used to introduce the correct phase volume fraction and the correlation structure measured on the 2-D SEM image (Fig. 1) into the 3-D cement particle image. Starting with an image of random Gaussian noise, generated using the Box-Muller method [14], the measured autocorrelation function for the phase(s) of interest is used to filter the image, introducing the appropriate correlation structure. This correlation structure is then basically overlaid on the cement particle image to introduce the correlation structure and phase fractions into individual cement particles. Each time this program is executed, the user must specify what phase is to be subdivided into two phases and the value to be assigned to the new phase. A typical sequence is to separate the cement into silicates and aluminates, separate the silicates into C₃S and C₂S, and separate the aluminates into C₃A and C₄AF, as illustrated in Fig. 4.

To use rand3d, the user must input:

• a random number seed (negative integer),
• the initial phase ID to be subdivided into two phases,
• the phase ID for the second (new) phase,
• the name of the file containing the current 3-D microstructure to be processed,
• the name of the file containing the correlation data for the phase(s) of interest,
• the phase volume fraction (0.0 to 1.0) of the initial phase that is to retain its original phase value, and
• the name of the file to be created to store the new resultant microstructure (file will contain one pixel/integer per line).

The input for the phase volume fraction can be determined from the 2-D SEM image or from a conventional Bogue analysis of the cement oxide composition [15], being sure to convert from a mass to a volume basis in the latter case. To facilitate this conversion, the specific gravities of the major four clinker phases [15] are provided in Table 4. Due to a few coding changes from the previously available version of rand3d.f, the new version will require only 10 % to 50 % of the CPU time required by the original version to execute a single filtering and phase assignment pass, a significant improvement. The C version is in general 3 to 4 times faster than the Fortran version.

Table 4: Specific Gravities of Major Clinker Phases

Phase	Specific Gravity
C3S	3.21
C2S	3.28
C3A	3.03
C4AF	3.73

The filtering process will be illustrated for cement 133 for the microstructure shown in the left side of Fig. 3. The input datafile to filter the image and separate the cement into silicates and aluminates (for the Fortran version of the code) is as follows:

-1088                  negative integer random number seed
1.0                    original phase to be segmented and reassigned (cement)
4.0                    new phase ID to be assigned to modified pixels
'cem133wc030n1.img'    filename of input 3-D microstructure
'cem133r.sil'          file containing 1-D correlation function for silicates
0.8333                 phase fraction (0.0-1.0) to remain as silicates
'cem133wc030n1a.img'   filename of output 3-D microstructure

  

Figure 4: Flowchart specifying filtering algorithm for assigning pixel phase values to the three-dimensional starting cement microstructure image. Numbers in parentheses indicate phase IDs corresponding to the individual cement compounds.

The value of 0.8333 was chosen for the silicates volume fraction based on the sum of the measured area fractions for C₃S (0.7018) and for C₂S (0.1315) provided in Fig. 2, obtained from analysis of the composite SEM image shown in Fig. 1. The file cem133r.sil (along with cem133r.c3s and cem133r.c4f) was downloaded from the cement images database described earlier. The value of 4.0 was chosen for the new phase assignment, because for this cement, the 3rd correlation file was determined for the C₄AF phase (and not C₃A). This means that the aluminates will subsequently be split into C₄AF (phase ID 4) and C₃A (phase ID 3). By assigning the aluminates an initial phase value of 4.0, simply executing rand3d again with an original phase ID of 4 and a new phase ID of 3 will conveniently and directly create the C₄AF and C₃A phases. If the correlation file had been determined for the C₃A instead of the C₄AF, one would simply use a phase ID value of 3.0 instead of 4.0 on line three of the datafile shown above, as illustrated in the flow diagram in Fig. 4. Finally, it should be noted that the quotation marks around the filenames are only needed when executing the Fortran version of rand3d and must be omitted when using the C version.

A 2-D slice from the resultant 3-D microstructure (e.g., cem133wc030n1a.img) obtained after the execution of rand3d is shown in the left side of Fig. 5. It can be clearly observed that the cement particles have been segmented into silicates and aluminates, as requested.

  

Figure 5: Two-dimensional slices from 3-D microstructures for cement 133 segmented into silicates and aluminates (left) and further segmented into C₃S, C₂S, and aluminates, with w/c=0.30. Color assignments are black- porosity, red- silicates (C₃S), aqua- C₂S, green- aluminates, and grey- gypsum. Images are 100 pixels x 100 pixels.