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In this paper, the initial state using point of saturated nucleation to produce the Voronoi mesh, let the nucleation rate is 0.1, the mesh size is 400 × 400 matrix, the simulation area is 2mm × 2mm, and the number of generated grain is 400 × 400 × 0.1 = 16000.
Figure 2 shows the simulation of grain structures with different simulation time steps, while Fig. 3 shows the number of grain size by the Linear intercept method count.
From the Fig. 2, it can be found that with the simulation time increases, large grains annexation small and the grains size become large, the number of edges grains tends to be 6, the shapes of grains tend to be hexagonal, in line with the Neumann-Mullins theoretical equation: (4) Where n is the number of edges grains, αn is the grain size, k is the diffusion constant [10].
Grain Growth Kinetics.
The small grain is annexed by large grains, which follows the principle of the decrease of grain boundary energy and the theory of grain growth.

We have developed a coarse-grain model (CG) for PMMA-b-PVP[6].
Using atomistic simulation technique to modeling the PMMA melts at meso-scale is still difficult nowadays because of the huge number degrees of freedom in the system.
A possible solution to this problem is to reduce the number of degrees of freedom through the mapping of an atomistic model onto coarse grained structures.
Fig. 1 Illustration of the coarse graining mapping schemes for PMMA.
Coarse-Grained Simulations The coarse graining mapping schemes for PMMA is shown in Fig.1.The Iterative Boltzmann Inversion (IBI) method is applied in this study to coarse grain the polymers.

The grain boundary surface is the excess energy of the grain boundary as the lattice on one side of the grain is translated relative to the lattice on the other side of the grain.
Grain boundary sliding is characterized by the relative translation of two adjacent grains parallel to the grain boundary plane driven by applied shear stress.
Figure 2 contains two grains jointed along the grain boundary plane.
In Fig. 4, we present the relative strains (percentage) of the interlayer spacing normal to the grain boundary plane as a function of the number of layers away from a given layer for the ]110)[211(3Σ tilt grain boundary before the sliding.
Interlayer strain (percentage) as a function of the number of layers away from a given layer.

The average grain diameters as well as well as the ASTM grain size number (G) were analyzed.
The ASTM grain size number (G) of unalloyed samples were 2.4 while the Cr added samples showed an ASTM grain size of 2.6 to 3.7 and the average grain diameter varied from 151µm to 92µm.
(G) (e) Microstructure of 4 wt. % Cr Figure 2 shows the variation in the ASTM grain size number (G) of the various compositional microstructures.
The strain induced in the 4 wt. % specimen corresponds this reduction in the ASTM grain size number (G).
These particles refined the as cast grain structure as well as found to control the grain growth.

A large number of fine-grained polycrystalline solids, metals, ceramics, and intermetallics exhibit superplasticity at elevated temperatures [1].
The state of a grain is classified according to its number of faces f , or coordination number.
Euler’s equation gives the law of conservation for the topological parameters of a polyhedron, the number of faces F , the number of edges E , and the number of vertices V : 2 F E V� + = (1) The average number of faces for one grain F was about 13.9 in a monodispersed structure with identical grains, and about 13.0 in a polydispersed structure with grain size distribution.
The strain, which is associated with grain switching, is represented by /F N � ��� = � , where F� is the mean cumulative number of grain switching events.
When the cumulative number of grain switching events is smaller than N�� , ��� is less than �.

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).

Bulk metallic materials possessing uniform grain structure with average crystallite sizes below 100 nm and grains limited by high-angle boundaries can be formed by a number of methods [3, 4].
The data of the statistical analysis calculated from the grain size distributions of Nb3Sn layers in composites with different modes of doping with Ti after the first annealing (the set numbered from 1 to 5) are presented in Table 1.
Parameters of these structures after the two-staged annealing are presented as well (denoted by the numbers from 1¢ to 5¢).
In addition, it should be noted that strong relationship between the mean and the standard deviation allows to reduce the number of independent parameters at modeling of the microstructure evolution.
Plasticity and grain boundary diffusion at small grain sizes, Adv.

However, the second phase particles would break off the grain boundary and the grain nucleus more or less during electrolytic polishing.
Furthermore, due to the 3D distribution of grains and nuclei in the samples, the polishing plane or the plane observed impossibility overpass all the grain nucleus, in this way, the number of the nuclei was finite.
If the C content was higher, the number of the TiC particles was larger and more efficient.
The more complex of nucleation and the nuclei the smaller grain
After the refinement of TiC particles, the number of the stable heterogeneous nucleation is not always increased.

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.

Each site or voxel has its own number, Si that represents the crystallographic orientation at that location, so that if adjacent sites have the same number, they are considered to belong to the same grain.
The lattice site energy is given by the following sum over the sites: , (1) where nn is the number of nearest neighbors (26 for cubic lattice), J(Si, Sj) is the grain boundary energy, Si is the orientation of site i, and δij is the Kronecker delta function.
The number of sub-grain inside the near Goss grain was 1, 3, 5, or 7.
One sub-grain means that the near Goss grain contains no sub-grain boundaries.
Fig. 5 shows that the growth rate of the near Goss grain increases as the number of sub-grains increases.