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Table 3 Equations used in the model to calculate austenite grain size and final ferrite grain size.
Equation and equation number Ref
Table 4 Predicted grain austenite grain sizes during roughing.
Ferrite grain sizes at room temperature: Measured and predicted values for ferrite grain size were as follows: measured grain at web, 10 ± 2 μm and 15 ± 2 μm at flange whereas predicted values were 10.2 μm and 15.3 μm, respectively for web and flange.
Table 5 Predicted austenite grain sizes during finishing.

This suggests the occurrence of grain boundary sliding, probably with the help of tiny recrystallized grains existed in the vicinity around grain boundaries.
However, the macroscopic deformation must be impossible to occur just by the grain boundary sliding, because of the large grain size of the plate-like LPSO-phases.
The annealing of specimen at 400 ˚C for 168 h effectively coarsened the tiny recrystallized grains in the vicinity of the grain boundaries as shown in Fig. 3.
In this evaluation, the average length of long-axis of plate-like LPSO-phase grains was estimated as a grain size d.
The equation indicates that the decrease of orientation factor (Taylor factor) m reduces the k-value, which is brought about by the increase of the number of slip systems [7].

After 25% reduction the grains elongate in the direction of rolling.
The majority of grains contain twins (Fig.1b) which grow through the whole grain interior and interacting with each other refine the microstructure.
With strain the number of the twinned grains increases and the initial grain boundaries become less distinct.
The number of such fragments increases with strain (Fig. 1d), so that after 96% rolling the microstructure of the sheet is mainly homogeneous with the mean size of the fragments of about 0.15 µm (Fig. 1e).
At further strain the size of grains decreases gradually.

The grains grow up when the grains are heated, which can lead the mean grains size become larger.
Because the existence of the instauration bonds and distortion bonds at the site of the grains boundary, the energy of the GB is higher than the energy inside the grains.
So the reduction of the number of the GB can lead to the reduction of the energy of the interconnect system.
Annealing can enlarge the grains size reduce the number of GB and reduce the system energy.
As the result of the reduction of the number of GB, which is an important EM diffusion path, the EM diffusion is restrained.

Nearly pore-free microstructure (99.8%) and narrow grain size distribution (4-10 µm) with an average grain size of 7 µm was obtained for the sample sintered in the vacuum atmosphere, while both inner and inter pores with abnormal grain growth, wider grain size distribution (9-27 µm) with the average grain size of 12 µm were detected in air atmosphere.
This causes abnormal grain growth and trapping of pores inner and inter of the grain.
As the sintering temperature increases, the number of the pores decreases (Fig. 6 and 7) and the relative density and grain size increases (Fig. 8 and 9).
At temperature higher than 1590 ˚C, when the advanced stage of sintering occurs, the difference between the number of residual pores, relative densities, and grain sizes are observed for the specimens sintered under air and vacuum atmospheres.
However, both inner and inter pores with abnormal grain growth, wider grain size distribution (9-27 µm) with average grain size of 12 µm were detected in the air atmosphere sample.

In a previously proposed model [5] it was discussed that the growth rate of crystallographic planes is influenced by the number of nearest neighbours that a metallic adatom encounters at the growing surface.
The crystallographic plane that offers the highest number of nearest neighbours to a metallic adatom will have the highest growth rate.
Calculating the number of nearest neighbours, it was predicted that the [002] direction will be the resulting out-of-plane orientation in case of YSZ (a fluorite structure) [5].
As such, the grains that catch the largest number of metallic adatoms will be able to overgrow the others.
The higher grains will shadow the other grains more efficiently when the substrate is tilted.

Y2O3 could act as an inhibitor to the growth of ZnO grains when working with Sb2O3.
With the increase of Y:Sb ratio, a new phase of Y-Sb-O would be formed, which is a more effective inhibitor to the growth of ZnO grains, so the mean size of ZnO grains further decreased.
And the calculated mean sizes of ZnO grains in the corresponding samples (h).
However, without doping of Sb2O3, due to the abnormal growth of ZnO grains and the segregation of the doped Y2O3 at the grain boundaries, the densification of the samples was inhibited and the number of pores increased quickly.
However, without Sb2O3, the densification of samples decreased, the number of pore in them increased, and lots of the doped Y2O3 would segregate at the grain boundaries, resulting in increased carrier concentration there.

And the number of {111} twinning is far less than the other two types of twinning .
In order to describe the twinning number of different deformed structures quantitatively, the ratio of grain area with twinning and microscopic orientation image area is calculated and defined as twinning fraction.
The twinning grains in Fig. 4(a) and untwinning grains in Fig. 4(b) and (c) are marked by black lines.
Additionally, a number of parallel twinnings are existed in grains during the deformation process which means that more grains participate in the dynamic deformation through twinning behavior.
The number of the three types of twinning is 66, 11 and 39 respectively.

As the concentration of Fe2+ ions increases the number of oxygen vacancies increases, and this increases the sintering rate, lowering the maximum temperature of densification [13].
As can be seen in Fig. 3, the SFO1OOSiC sample (SrFe12O19), appears to have a liquid phase on the grain and also across the grain, as can be observed in other samples, but further studies are needed for confirmation.
In the BFO90 sample there was not any change in grain size, while in the sample SFO100 the dopant made a decrease in size of the grains.
Table 3: Grain size variation of the samples with and without additives, sintered at 1000 °C/2h.
In the BFO100 and BFO15 samples, there was a significant growth in the grains, while in the sample BFO90 the grains remained practically the same size and the sample SFO100 had a decrease in grain size.

When the balls are resonated, the sample surface is impacted by a large number of flying balls over a short period of time.
Key to realize the surface self-nanocrystallization of a sample is the introduction of a large number of defects and interface into surface layer of the sample.
In other words, a grain refinement process need to nanoscale surface while coarse-grained structure matrix is constant.
It is known that nanocrystalline materials possess ultrafine grains with a large number of grain boundaries that may act as fast atomic diffusion channels.
A large number of grain boundaries with various kinds of nonequilibrium defects also constitute a high excess stored energy that may further facilitate their chemical reactivity.