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The high number of total lattice points with 250250250 ×× MCUs keeps the number of grains adequately for statistical analysis.
Figure 2: Temporal development of the: a - number of grains vs. grain radius and b - relative number of grains vs. relative grain size.
Fig.3b) so that the number of still existing grains during coarsening is always large enough for statistical data analysis.
Figure 5: a - Number of neighbouring grains vs. relative grain size; b - Volumetric rate of change vs. number of neighbouring grains divided into classes after 500 time steps (the three grey crosses are data that are ignored because the associated grains disappear during the considered time steps).
Acknowledgments The authors would like to thank the Deutsche Forschungsgemeinschaft for financial support under grand number GKMM 828.

Misch metal (Mm) addition to TM Mg-Al-Ca alloys makes precipitates within α-Mg matrix and their number density and size depend on heat-treatment conditions.
The number density and the size of precipitates are possible to control by heat-treatments.
The Cgb is calculated by the following equation; (1) where L is total length of grain boundary measured by TEM photos, N is total number of intermetallic compounds on grain boundaries, and di is the length of each intermetallic compound along the trace of grain boundary.
The number density of the precipitates in Mm added alloys increased after the heat-treatments.
The low temperature heat-treatment is useful to increase the number density of intragranular precipitates; however, the grain boundary strengthening by the network-like compounds decreases significantly.

The number of faces per grain f and the average number of faces of each grain’s nearest neighbors m(f) is counted.
Fig. 2 shows the distribution of the number of grain faces in the system.
Fig. 3 plots the average number of faces of each grain’s nearest neighbors, m(f), vs. the number of faces per grain.
A distribution in the average number of faces of a grain’s neighbors is observed for a given topological class f.
Fig.3 Plot of the average number of faces of the nearest-neighbor grains, m(f), vs. the number of faces of the grains, f.

The samples exhibited a large number of intermetallic particles on polished sections in each case.
No appreciable difference was found in the number density of particles between the base alloy sample and the grain-refined sample.
A total of 63 such particles were captured from a large number of randomly selected viewing fields and probed at 20 kV.
It can be inferred from the Al-Mn phase diagram [7] that this master alloy splatter may contain a large number of metastable intermetallic phase particles such as ε-AlMn.
Therefore, we can make the assumption that the nucleant is ε-AlMn in commercial AZ alloys, a b c d although the number density of these ε-AlMn particles is much lower than for Al8Mn5.

The normalised cumulative number of nuclei/grains, as generated in the simulation, has been plotted as function of the simulated transformed fraction in Fig. 3 b).
For decreasing ratios of SGB V / ˙NO (and the same nucleation rate ˙NV , see above), the same number of nuclei must form on a more and more restricted grain-boundary area and therefore nuclei are more likely to appear close to each other.
This positive correlation of nucleation positions is exhibited in Fig. 1 c) and is the reason for the retardation of the transformation kinetics as observed (Fig. 1 a): The closer to each other nuclei form, the sooner they impinge which leads to slow transformation kinetics and a high number of small grains.
Moreover, for such parent microstructures, the last part of the transformation is further slowed down because there are a number of very large parent grains (which can be transformed only by product grains nucleated at grain boundaries of these grains).
fraction as function of the transformed fraction predikted by the JMAK-like model and b) normalised cumulative number of nuclei/grains during the simulation as function of simulated transformed fraction.

The initial austenite grain size plays an important role in the obtained ferrite nucleation number, and the potential nucleation cells are increased.
Fig.4 shows the ferrite nucleated cell number as a function of austenite grain size.
As shown in this figure, as the initial austenite grain size is increased, the ferrite nucleated number is decreased.
Therefore, as the initial austenite grains size become small, the nucleated ferrite grain numbers large.
Thus, increasing potential nucleation number and improving nucleation probability of each nucleation position can be helpful in obtaining fined ferrite grains.

Briefly, the microstructure to be modeled is discretized, and the orientation number at each grid point is associated with a 3-parameter crystallographic orientation.
Each voxel i in a simple cubic lattice of N=100x100x100=10 6 elements was assigned an orientation number Si=500.
The misorientation distribution (MDF) was quantified by counting the number of voxels adjacent to a different grain and sorting them by misorientation type.
Journal Title and Volume Number (to be inserted by the publisher) 5 (i) (ii) Fig. 3.
Journal Title and Volume Number (to be inserted by the publisher) 7 Figure 6 illustrates the behavior of several systems.

In order to understand why these factors, i.e. the number density and potency of the inoculant particles, affect grain size we need to understand the role of constitutional supercooling (CS).
The important properties of the nucleant particles, whether naturally present in the alloy melt or deliberately added as inoculants, are their nucleation potency, defined by DTn, and their distribution and number density, which define the spacing xSd between nucleants.
A representation showing the relationship between composition as defined by Q and the components that contribute to the grain size. xSd is constant if the number density of nucleant particles does not change with composition as shown in this figure (from [7]).
It has been found that G may decrease as the number of layers increase due to heat build-up in the component.
The ideal conditions for creating an equiaxed grain structure are: • an alloy with solute elements that generate a very high Q value and nanoparticles that slow grain growth and the diffusion rate in the liquid; • a melt inoculated with the most potent particles at a high number density; • a temperature gradient as low as possible; and • optimised processing parameters such as scan speed and any other processing parameters that promote nucleation.

The grain growth behavior depends on the roughening of the interfaces as indicated by the grain and grain boundary shapes.
With increasing impurity content-in particular SiO2-in the alumina powder, abnormal grain growth becomes more pronounced with increasing number of flat grain boundaries.
As shown in Fig. 1, the grain shapes and the grain growth behavior changed with the amount of B added.
The number fraction of the large grains exceeding 2.5 times of the average size appeared to increase with sintering time.
These grain shapes, grain boundary shapes, dihedral angles, and grain size distributions are same as those observed earlier by Park and Yoon [30] and Cho et al. [31] 0 2 4 6 Percentage of Grains (%) 0 8 10 12 0 Normalized Grain Radius 2 4 6 0B 1B 2B 3B 4B 0 2 4 6 Percentage of Grains (%) 0 8 10 12 0 Normalized Grain Radius 2 4 6 0B 1B 2B 3B 4B Fig. 2.

The 3D fractographs illustrated that the numbers of the dimples decrease with the increase of the grain size.
A minimum of 500 grains was measured for the determination of grain diameter for a given annealing condition.
The numbers of the dimples in the fracture surfaces decrease with the increase of the grain size.
The dimple size depends on the grain size.
The 3D fractographs illustrated that the numbers of the dimples decrease with the increase of the grain size.