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The minimum grain size (85µm) was achieved at 0.4wt.% Zr.
This unique property has led to the development of a number of commercially important zirconium-containing magnesium alloys.
The mean linear intercept technique was used to quantify the grain size with at least 500 grains examined.
However, potent nucleating particles can be effective for grain refinement only if they have sufficiently high particle number density in the melt.
When Zr content is larger than 0.443%, the un-dissolved Zr particles will experience rapid coarsening under intensive melt shearing, resulting in a rapid decrease in particle number density, which in turn gives rise to an increased grain size (Fig. 4).

Introduction The growing interest in the low energy Σ3n (n=1, 2, 3) CSL grain boundaries (GBs) is indicated by the increasing number of studies on the formation of these microstructural elements.
The numbers of cycles were 1, 2, 4 and 8.
The number of mapped points was 30537 at a step size of 2 μm.
Grain size is increasing significantly with the number of iterations as the strain values are decreasing in a manner of ε/m, where m is the number of iterative steps.
For the same reason grain size in specimens with the lower total strain (ε: 1) is growing faster compared to specimens that suffered higher strain as the number of cycles grows.

As the breakdown voltage of a varistor is the sum of the breakdown voltages of all the non-linear (varistor) grain boundaries between the electrodes, it depends on the number of grain boundaries per unit thickness of varistor ceramic, which is inversely proportional to the ZnO grain size.
The anisotropic and exaggerated growth of grains with IBs (nuclei) is caused by a nucleation mechanism for special boundaries.[17] For a smaller number of nuclei a coarsegrained microstructure develops, as the nuclei can grow to a larger size before they collide with each other.
However, if the number of nuclei is large they collide with each other when they are still small, which results in a fine-grained microstructure.
Sufficient amount of Bi2O3-liquid phase at the grain boundaries and inversion boundaries in the ZnO grains promote grain growth.
Amount of added Bi2O3 defines the amount of liquid phase at sintering temperature while the amount of added Sb2O3 affects nucleation of IBs in the ZnO grains at the early stage of sintering and hence number of grains infected by IBs which can grow exaggeratedly.

The measurements were carried out at the acceleration voltage of 20 kV with the step size varying from 50 to 500 nm, depending on grain size (i.e. number of ECAP cycles).
It shows an almost homogeneous microstructure with equiaxed grains separated mostly by high angle grain boundaries.
The grain boundaries are obviously closer to the equilibrium state than grain boundaries in the specimens that underwent a smaller number of ECAP passes.
The ratio KD/Kv is plotted in Fig. 7A as a function of the number of passes.
number of ECAP passes.

The grain size is related to the number of grains in a unit area.
According to the close feature of the binary steel microscopic image, we introduced a filling-and-elimination method to count the number of grains [4].
Counting the number of grains in Fig. 4 by filling-and-elimination counting algorithm, the result is N=46.
Here, the direction number indicates the average value in whole boundary.
The filling-and-elimination counting algorithm can count the number of grains in the image accurately.

Previous research on developing the Interdependence Model assumed that the number density of nucleant particles was constant for the range of alloy compositions considered.
The second term xSd describes the contribution the TiB2 particle number density makes to grain size
Hence, it appears that the main reason for the increase in grain size is not due to changes in xSd due to a change in the number of nuclei that are present, but rather an increase in the value of xnfz.
An analytical model for constitutional supercooling-driven grain formation and grain size prediction.
The Effect of Grain Refinement and Silicon Content on Grain Formation in Hypoeutectic Al-Si Alloys.

The Relationship Between Grain Boundary Energy, Grain Boundary Complexion Transitions, and Grain Size in Ca-doped Yttria Stephanie A.
In the second sample, there is a combination of large grains and small grains.
All comparisons were made between boundaries on the periphery of the large grains (such as between grains labeled 1 and 3 in Fig. 1) and those between smaller grains immediately adjacent to the large grain (such as between grains labeled 2 and 3 in Fig. 1).
The blue line between grain 2 and 3 indicates a boundary between two of the smaller grains, but adjacent to a large grain.
However, this method includes a number of approximations and assumptions.

The PGA may be categorized into three different basic approaches: master-slave, fine grained and coarse grained.
As a multi-population approach, the coarse-grained PGA divides the initial population into several subpopulations (demes) according to the number of processors.
In other words, if the total number of individuals and processors are defined as pop_size and, so the coarse grained model consists of subpopulations that own pop_size/ individuals, and the individuals of each subpopulation evolve independently through generation process.
It can be an integer or a percent commonly describing the number or rate of migrated individuals each time.
This situation may result from the fact that the communication cost will increase with increasing number of demes.

Nc according to their size R and the grain size distribution function ϕi shall then give the fraction of the number of grains belonging to the size class i, i.e. the frequency of the grains i.
The equation of motion for the elementary process of migration of an individual grain boundary µν between the grains number µ and ν can then be written as [5] vµν = m∆Pµν = mγ � Pµν − PDµν� = M � 1 Rµ − 1 Rν − Iµν� (1) Here m is the mobility and γ the energy of the GBs and M = mγ the grain boundary diffusivity.
The statistical system constituted by the grain classes can be characterized by the correlation function wij which gives the fraction of the surface of the i grains shared with j grains.
In order to derive the time evolution of the grain size distribution, the expression for Iij must be calculated for the considered drag mechanism, inserted into Eqs. 6 and 4 and, finally, the resulting dRi/dt must be introduced into the continuity equation which then has to be integrated [3, 4, 5]. 1However this randomness concept on the probability of contact, based on the relative surface for each grain class in the system, establishes very specifical topological correlations between grain size and other microstructural parameters in the grain system (number of corner-grain size, boundary length-grain size, Weaire-Aboav equation, etc. [14, 15])vZ -PZ0 0 -PS0 I III v PS0 II DP PZ0 amPS0 mPS0 vS free vSpin Fig. 1: GB velocity v vs driving force ∆P due to curvature, in the presence of Zener Drag and solute drag. 1/Rj DP Zij 1/Ri+IS0 -aPS0 DP Sij 1/Ri+IZ0 1/Ri-IS0 1/Ri-IZ0 IS0, IZ0 IS0, IZ0 1/Ri -PS0 -1/Ri+IZ -1/Ri+IZ I S,Z III S,Z DPij PS0 aPS0
One sees further (Eq. 13) that dRi/dt and thus the mean pressure ∆Pi on the i grains is obtained by averaging over the pressures due to the grains of range I (consumed by grains i) and the grains of range III (consuming grains i).

It can be seen that for both applied deformation routes the microhardness increases along with the increasing number of passes.
It should be noted that the first pass causes the strongest anisotropy, which becomes reduced along with the increasing number of incremental ECAP passes.
With the increasing number of passes, primary grains become elongated with the deformation direction on the Y and Z planes and gain a low angle grain boundary substructure.
The grains are equiaxial and the significant amount of high angle grain boundaries creates many obstacles to mobile dislocations.
Langdon, The evolution of homogeneity and grain refinement during equal-channel angular pressing: A model for grain refinement in ECAP, Mater.