Correction of hydraulic radius

Because the correlation structure of the 3-D image produced by rand3d only approximates that of the input 2-D correlation function, a modification is utilized to directly match the surface area fraction (via the hydraulic radius) of the 3-D image to its 2-D counterpart, greatly improving the agreement between the 2-D and 3-D correlations [16]. Typically, the microstructure file produced by rand3d is first analyzed using the program stat3d.c. This C program (listing provided in Appendix B), will determine the phase volume fractions and surface area fractions for each phase present in the 3-D microstructure. The user simply inputs the filename of the 3-D image to be quantified and the name of a file in which to store the results of the analysis. The surface area fractions are reported on a calcium sulfate-free basis, so the user must be sure that the 2-D image has also been analyzed on a sulfate-free basis (i.e., considering only the C₃S, C₂S, C₃A, and C₄AF, as in the first table in Fig. 2). The needed surface area count for a phase can be determined by multiplying the total surface area count by the surface (perimeter) fraction determined in analyzing the 2-D SEM image, as provided in Fig. 2. From this, the appropriate value of the hydraulic radius to be used in the execution of the program sinter3d, R_h, can be determined as:

$$R_h = \frac{6}{4} \times \frac{phase(s)\ volume\ in\ 3-D\ image\ in\ pixels}{needed\ surface\ area\ in\ pixels}.$$ (1)

 

The value of 6/4 is included in the equation to correct for the fact that a digitized 3-D sphere has a surface area of approximately 6 r² as opposed to 4r² [16].

For example, the following output is obtained when executing the program stat3d to quantify the phases present in the 3-D microstructure contained in the image file
cem133wc030n1a.img.

Phase    Volume      Surface     Volume    Surface 
 ID      count        count      fraction  fraction 
  0      569660            0  
  1      338509       380581     0.83237   0.84138
  2           0            0     0.00000   0.00000
  3           0            0     0.00000   0.00000
  4       68173        71747     0.16763   0.15862
Total    406682       452328

  5       23658        23382
  6           0            0
  7           0            0
  8           0            0
  9           0            0
 24           0            0
 25           0            0

Here, one can observe that the surface fraction of silicates (phase 1), 0.84138, is slightly greater than the surface fraction measured on the silicates in the real 2-D composite SEM image (Fig. 1, 0.8255 (the sum of the values for C₃S and C₂S for PERIMETER (SURFACE) in Fig. 2). The requisite hydraulic radius is thus calculated as 6. x 338509. / 4. / (0.8255 x (380581  + 71747)) = 1.36. For the second pass through the filtering process, to segment the silicates into C₃S and C₂S, the equation for R_h would be:

6 x volume C3S in 3 - D image in pixels	(2)
4 x needed surface area fraction of C3S x (surface pixels C3S +  surface pixels C2S)

Note that in this case, the needed surface area fraction of C₃S is the ratio of the PERIMETER fraction for C₃S over the sum of the PERIMETERs for C₃S and C₂S. From Fig. 2, for example, this would be 0.6491/(0.6491+0.1764)=0.7863.

This hydraulic radius is input into the program sinter3d.c (listing provided in Appendix B), which interchanges pixels of two specific phases to obtain the specified hydraulic radius for the first phase. This code was originally developed to simulate the sintering of a ceramic powder by exchanging solid pixels of high curvature with porosity pixels of low curvature [17]. The program is menu driven and the user must first read in the microstructure previously output by execution of rand3d, using menu selection 2. After this, the sintering algorithm (menu selection 4) can be executed to adjust the hydraulic radius. For this algorithm, the following inputs are required:

• the two phase IDs between which to execute the sintering algorithm; the first phase indicates the one whose hydraulic radius will be increased,
• the number of pixels of each phase to exchange in each cycle of the sintering algorithm (default value=200),
• the calculated target hydraulic radius (from equation 1), and
• the radius of the digitized sphere [17] to be used in assessing curvature (default value=3).

For our example, these parameters would take the values:

1 4          (silicates and aluminates) 
200
1.36
3

The sintering algorithm will be iteratively executed until the desired hydraulic radius is achieved or an equilibrium is reached. At this point, the user may elect to output the resulting microstructure to a file using menu selection 5. A file named by the user will be created for this output (for example cem133wc030n1b.img). 2-D slices from the 3-D microstructures obtained after "sintering" are provided in Fig. 6.

  

Figure 6: Two-dimensional slices from 3-D microstructures for cement 133 after "sintering" to correctly adjust surface area fractions (hydraulic radius) following segmentation into silicates and aluminates (left) and following further segmentation 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.

It should be noted that the sintering can only be used to increase the hydraulic radius of a phase, by decreasing its surface area fraction at constant volume fraction. If the desired hydraulic radius for a phase is less than the value present after executing rand3d, the user should simply "reverse" the phases being sintered. For example, in the example outlined above, while the hydraulic radius of phase 1 is being increased by the sintering, that of phase 4 is being decreased. Thus, one can also decrease the hydraulic radius of phase 1 by increasing the hydraulic radius of phase 4. In this case, the first line of input would be 4 1 (instead of 1 4) and the requested hydraulic radius would be the calculated desired value for phase 4 (instead of phase 1).

After the sintering is used to correct the hydraulic radius of the silicates, the
rand3d/stat3d/sinter3d sequence is repeated to separate the silicates into C₃S and C₂S. In this case, the input datafile for rand3d would be:

-188                   negative integer random number seed
1.0                    original phase to be segmented and reassigned (silicates)
2.0                    new phase ID to be assigned to modified pixels (C2S)
'cem133wc030n1b.img'   filename of input 3-D microstructure
'cem133r.c3s'          file containing 1-D correlation function for C3S
0.842194               phase fraction (0.0-1.0) to remain as C3S 
'cem133wc030n1c.img'   filename of output 3-D microstructure

An image from the resultant microstructure is shown in the right side of Fig. 5.

When stat3d is executed on cem133wc030n1c.img, the following output is returned:

Phase    Volume      Surface     Volume    Surface 
 ID      count        count      fraction  fraction 
  0      569660            0  
  1      285898       317785     0.70300   0.70255
  2       52611        55562     0.12937   0.12284
  3           0            0     0.00000   0.00000
  4       68173        78981     0.16763   0.17461
Total    406682       452328

  5       23658        23382
  6           0            0
  7           0            0
  8           0            0
  9           0            0
 24           0            0
 25           0            0

The desired count for the surface count of C₃S would be 0.7863 x (317785 + 55562) = 293563. Thus, the desired R_h to use in the sintering program can be calculated as (6 x 285898) / 4 x 293563) = 1.461.

This value was used in sinter3d and the resultant microstructure output to a file called cem133wc030n1d.img, a 2-D slice from which is shown in the right side of Fig. 6. The above process is then repeated to separate the aluminates (phase 4) into C₄AF (phase 4) and C₃A (phase 3), using the cem133wc030n1d.img file and the correlation file cem133r.c4f as input for rand3d. After the final sintering, the phase fraction statistics are as follows:

Phase    Volume      Surface     Volume    Surface 
 ID      count        count      fraction  fraction 
  0      569660            0  
  1      285898       293384     0.70300   0.64861
  2       52611        79963     0.12937   0.17678
  3       33911        51701     0.08338   0.11430
  4       34262        27280     0.08425   0.06031
Total    406682       452328


  5       23658        23382
  6           0            0
  7           0            0
  8           0            0
  9           0            0
 24           0            0
 25           0            0

Note that all of the volume fractions and surface fractions are very close to the values determined for the real 2-D image and given in the first table in Fig. 2.

Final 2-D slices for the 3-D microstructures for cement 133 for both w/c ratios are shown in Fig. 7, before the addition of one-pixel particles, and in Fig. 8, after this additon. Finally, 3-D images of the central 50 pixel x 50 pixel x 50 pixel region of the initial microstructures can be found in Fig. 9 and Fig. 10, for the w/c=0.3 and w/c=0.45 systems, respectively. These 2-D and 3-D images can be compared directly to the real SEM image shown in Fig. 1.

  

Figure 7: Two-dimensional slices from the final (before addition of any one-pixel particles) 3-D microstructures for cement 133 for w/c=0.30 (left image) and w/c=0.45 (right image). Color assignments are black- porosity, red- C₃S, aqua- C₂S, green- C₃A, yellow- C₄AF, and grey- gypsum. Images are 100 pixels x 100 pixels.

  

Figure 8: Two-dimensional slices from the final 3-D microstructures for cement 133 for w/c=0.30 (left image) and w/c=0.45 (right image) after the addition of the one-pixel particles by the disrealnew program. Color assignments are black- porosity, red- C₃S, aqua- C₂S, green- C₃A, yellow- C₄AF, and grey- gypsum. Images are 100 pixels x 100 pixels.

  

Figure 9: Three-dimensional central image from the final 3-D microstructure for cement 133 for w/c=0.30. Color assignments are black- porosity, red- C₃S, aqua- C₂S, green- C₃A, yellow- C₄AF, and grey- gypsum. Image is 50 pixels x 50 pixels x 50 pixels.

  

Figure 10: Three-dimensional central image from the final 3-D microstructure for cement 133 for w/c=0.45. Color assignments are black- porosity, red- C₃S, aqua- C₂S, green- C₃A, yellow- C₄AF, and grey- gypsum. Image is 50 pixels x 50 pixels x 50 pixels.