13 Morphological Modelling and Simulation …
453
Two primary crystals A and B were considered and it was assumed that B could
be rotated into A by an arbitrary rotation matrix ˜
R. If the crystal model of A and B
has n S symmetry operations, then the rotation of B into A can also be performed by
additionally applying the n s rotation matrices ˜
R i that perform a symmetry operation.
This was possible because the symmetry operations do not change the appearance
of the shape and thus give n s identically-looking crystals. According to Mackenzie
and Thomson [33], a disorientation angle Θ is the smallest of angles Θ i which can
be calculated for each rotation with
Θ i = arccos
⎛
⎝
tr
˜
R i ˜
R
− 1
2
⎞
⎠
(27)
In this work, potash alum crystals were considered to be octahedral. For an octahedron, there are 24 symmetry operations. Thus, to compute the disorientation angle
between two primary particles, one must first find the rotation matrix ˜
R between
these two primary crystals. This could be easily computed provided a shape fit for
each primary particle. Then, 24 symmetry operations were applied and the 24 angles
Θ i were computed according to Eq. (27). The disorientation angle is the smallest of
these angles. A disorientation angle of 0° means that particles are oriented the same
way.
In order to validate the algorithm, agglomerates were first simulated. For the simulated agglomerates, it was possible to compute the distribution of the disorientation
angles because the rotation matrices of the simulated primary crystals were known
exactly. Figure 14a shows the probability density of the resulting angles together with
the theoretical distribution that would be expected to result from an infinite number
of samples. This theoretical distribution is given in the literature for cubes and also
holds for octahedra [33]. In the next step, simulated 3D images of these agglomerates were created and the disorientation angle was computed based on the shapes
identified in the images. The result is shown in Fig. 14b. The comparison of Fig. 14a,
b shows that the proposed algorithms work well for simulated ideal agglomerates
and yield meaningful disorientation angles.
In a next step, a potash alum crystal population was grown in a lab scale reactor
and sampled at the end of the crystallization. The agglomerates of these samples were
visualized using μCT measurements. The 3D images were then processed with the
algorithms described in Sects. 2.2 and 2.3. The segmentation procedure was adapted
to include user-interaction steps which made the segmentation more accurate. The
shape identification procedure was also adapted to work with asymmetrical crystals
observed in the measurement. Then the disorientation angle of the agglomerates was
calculated. Details on the experimental procedures are provided by Kovaˇ cevi´ c et al.
[4]. Exemplary results are shown in Fig. 15. It was observed, that agglomeration of
potash alum crystals tends to show lower disorientation angles than it was expected if
randomly oriented particles form agglomerates. This expectation is shown by the blue
line in Fig. 15, obtained from theoretical considerations in the literature. Kovaˇ cevi´ c
453
Two primary crystals A and B were considered and it was assumed that B could
be rotated into A by an arbitrary rotation matrix ˜
R. If the crystal model of A and B
has n S symmetry operations, then the rotation of B into A can also be performed by
additionally applying the n s rotation matrices ˜
R i that perform a symmetry operation.
This was possible because the symmetry operations do not change the appearance
of the shape and thus give n s identically-looking crystals. According to Mackenzie
and Thomson [33], a disorientation angle Θ is the smallest of angles Θ i which can
be calculated for each rotation with
Θ i = arccos
⎛
⎝
tr
˜
R i ˜
R
− 1
2
⎞
⎠
(27)
In this work, potash alum crystals were considered to be octahedral. For an octahedron, there are 24 symmetry operations. Thus, to compute the disorientation angle
between two primary particles, one must first find the rotation matrix ˜
R between
these two primary crystals. This could be easily computed provided a shape fit for
each primary particle. Then, 24 symmetry operations were applied and the 24 angles
Θ i were computed according to Eq. (27). The disorientation angle is the smallest of
these angles. A disorientation angle of 0° means that particles are oriented the same
way.
In order to validate the algorithm, agglomerates were first simulated. For the simulated agglomerates, it was possible to compute the distribution of the disorientation
angles because the rotation matrices of the simulated primary crystals were known
exactly. Figure 14a shows the probability density of the resulting angles together with
the theoretical distribution that would be expected to result from an infinite number
of samples. This theoretical distribution is given in the literature for cubes and also
holds for octahedra [33]. In the next step, simulated 3D images of these agglomerates were created and the disorientation angle was computed based on the shapes
identified in the images. The result is shown in Fig. 14b. The comparison of Fig. 14a,
b shows that the proposed algorithms work well for simulated ideal agglomerates
and yield meaningful disorientation angles.
In a next step, a potash alum crystal population was grown in a lab scale reactor
and sampled at the end of the crystallization. The agglomerates of these samples were
visualized using μCT measurements. The 3D images were then processed with the
algorithms described in Sects. 2.2 and 2.3. The segmentation procedure was adapted
to include user-interaction steps which made the segmentation more accurate. The
shape identification procedure was also adapted to work with asymmetrical crystals
observed in the measurement. Then the disorientation angle of the agglomerates was
calculated. Details on the experimental procedures are provided by Kovaˇ cevi´ c et al.
[4]. Exemplary results are shown in Fig. 15. It was observed, that agglomeration of
potash alum crystals tends to show lower disorientation angles than it was expected if
randomly oriented particles form agglomerates. This expectation is shown by the blue
line in Fig. 15, obtained from theoretical considerations in the literature. Kovaˇ cevi´ c