Figure 2 shows two views each of different cement particles. The images were taken from a VRML browser. The VRML files are automatically generated from the spherical harmonic expansion [2].
Figure 2: Two views each of two CCRL-133 cement particles: (a) equivalent spherical diameter of about 36 µ m, (b) equivalent spherical diameter of the particle of about 15 µ m.
One way to quantitatively analyze the shape of a collection of particles of various sizes is to plot their surface area vs. their volume. For a sphere, the surface area S in terms of the volume V is ≈ 4.84V2/3. Deviations from this curve are a measure of the non-spherical nature of particles. It is known that a sphere has the minimum surface area for a given volume for convex shapes in 3-D [8], so the curve for these cement particles should lie on or above the sphere curve. Figure 3 shows this data, along the theoretical curve for a sphere. A function has been fit to the cement particle data, of the same form as the theoretical function for a sphere: S = aVb. The value of b was found to be 0.64, and the value of a was 8.0. There was about 10 % uncertainty for both of these numbers, and the R2 value of the fit was 0.98. The particles are definitely non-spherical, although the variation of surface area, S, with volume, V, is qualitatively similar to that of a sphere, in terms of the power law, but with a different prefactor. This would imply that the fracture surfaces of these ball-mill crushed particles, at this length scale, are not fractal but are close to Euclidean, since the error bars for the fitted exponents include the Euclidean exponent of 2/3. Therefore, the shape differences seem to show up in the prefactor, not the exponent. Since the reaction of cement particles with water takes place initially at the particle surface, it would seem important to model the shape of the cement particles accurately in any cement hydration program.
Figure 3: True surface area vs. true volume for all 1200 CCRL-133 cement particles analyzed. The solid line is the theoretical relation for a sphere, S = (36π)1/3 V2/3 ≈ 4.84 V2/3 . The dashed line is a power law fit of the form S = 8.0 V0.64 .
Since cement particles are obtained by grinding much larger cement clinker particles, one might think that the smaller particles, which may have been ground more, would tend to be more spherical in shape. This would tend to be true for homogeneous material. On the other hand, small particles may be fragments knocked off a larger particle that were never ground much more due to their small size. Also, portland cement is not homogeneous.
Figure 3 showed an overall shape analysis for all the particles. One way to estimate the non-sphericity of an individual particle is to compute the ratio of the true surface area to the surface area of a sphere having the equivalent spherical diameter. If V is the true volume of a particle, then the equivalent spherical diameter is (6V/π)1/3, and thus the surface area of a sphere with this diameter is S = π 1/3(6V)2/3 . This surface area ratio is of course unity for a sphere. As this surface area ratio increases from one, the particle is increasingly non-spherical. Figure 4 shows this ratio, plotted against the equivalent spherical diameter, which is used as an approximate and convenient measure of particle size. While there is a lot of noise in this data, one can see that the smaller particles seem to have ratios bigger than those of the larger particles and hence are less spherical than the large particles. This can probably be attributed, as was said above, to the in homogeneous nature of portland cement, with different values of hardness for different phases.
Figure 4: A graph of the true surface area of each particle divided by the surface area obtained from the sphere with the equivalent spherical diameter, versus the equivalent spherical diameter.
In the CEMHYD3D model, spherical particles are randomly placed into a unit cell. The diameters of the particles follow an experimentally measured particle size distribution, and the chemical phases in each particle are statistically assigned according to experimentally measured correlation functions in 2-D. Now that we have real particle shapes, at least for the medium and larger particles, it is interesting to place real shaped particles in a model microstructure, and then statistically place within them the chemical phases as measured on real particles by scanning electron microscopy and X-ray point analysis [5 ,9].
Figure 5 shows a polished section of a real cement/epoxy image (left) and on the right shows a slice of a model 2003 voxel microstructure, 0.5 µm/voxel, where real cement particle shapes, as denoted by spherical harmonic coefficients, have been used for model particles. The model microstructure was made by randomly choosing from the 1200 cement particle images, and shrinking or expanding each in size to match a typical experimental cement particle size distribution. Particles were digitized before placement in the matrix. Particles whose voxel volume, after digitization, was five or more voxels, remained as they were. Particles with less than five voxels in volume were assigned randomly oriented shapes, as shown in Figure 6, since in the digitization process the original shape information is lost for these small shapes. For the three voxel and four voxel particles, other shapes are possible and will be investigated for incorporation into the next version of CEMHYD3D. Obviously, for particles this small, there are digital lattice effects.
Figure 5: The left hand side shows an SEM image of real cement particles, while the right hand image is a slice from a simulated 3-D cement particle packing, using real particle shapes.
Figure 6: Particle shapes used for volumes less than 5 voxels.
Finally, we can assign chemical phases just like has been done with spherical particles. The programs that assign phases operate on digital images of particles, and so do not "know" that the particles are spherical or non-spherical, since the particles are just collections of voxels. Figure 7 shows a single slice from the result of performing this operation for CCRL-133 in 3-D. In the figure, white = C3S, black = porosity, and in descending gray level, from white to black, is C4AF, C2S, and C3A (C = CaO, H = H2O, F = Fe2O3, A = Al2O3, and S = SiO2).
Figure 7: (a) Cement microstructure, without chemical information (different slice from Fig. 5), and (b) same image but with the four standard portland cement clinker phases distributed throughout the basic cement particles by using the Virtual Cement and Concrete Testing Laboratory. Note that in several places in the image, two particles appear to be stuck together, which is often seen in real images and which can be seen in the left hand (real) image of Fig. 5. White = C3S, black = porosity, and in descending gray level, from white to black, is C4AF, C2S, and C3A.