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In the present 3D simulations the number of MCUs are 200200200 ×× .
Figure 3: Temporal development of: a - the standard deviation; b - the relative number of grains.
Therefore, the number of grains (cf.
To demonstrate this we have considered the number of neighboring grains s as a function of the relative grain size x of the enclosed grain (cf.
Within this scaling state the average number of faces s of a given grain of size x can be described by a self-similar time-invariant function of the relative grain size (Fig. 4b).

A large number of studies have been realised concerning the stabilisation of nanocrystalline grain structures in many materials and the number of factors influencing the grain boundary mobility in nanocrystalline alloys, like grain boundary segregation, solute drag, pore drag, second phase (Zener) drag and chemical ordering.
The high number of observed grains makes statistical analyses possible.
Number and arrangement of the lattice points regulate only the accuracy of the calculations.
The high number of small grains implies also a high number of small triple junction distances (compare Fig. 3a).
The data of vs. x for all grains with a certain number of faces s show a strict linear relationship.

These properties are determined by a number of factors and depend on the conditions for creating, processing and use.
Based on the fact that the existing model can not be applied to such tasks required the formulation of a number of other mathematical models that have step by step should be updated and supplemented by the accumulation of research in this area [7, 8].
In Figure 1, the grain boundary layer is gray.
Acknowledgment This work was supported by Russian Foundation for Basic Research, grant number 13-01-00444.
Valiev Grain boundaries in ultrafine grained materials processed by severe plastic deformation and related phenomena Mat.

Volume fraction Specimen Average grain size (µm) Goss 40˚<111> Other 84%, 873k/0.5h 0.78 0.06 0.80 0.13 95%, 873k/0.5h 1.03 0.22 0.56 0.23 Regarding the variant selectivity of the 40˚<111> rotated grains, we determined the volume fraction, number density, and the size of each variant.
It is noted that this ratio is almost the same irrespectively of the prior cold reduction level, though the values of the volume fraction and number density are different corresponding to the progress of grain growth.
Volume fraction, number density, and grain size of each variant.
Specimen Rotation axis of 40˚<111> Volume fraction Number density (mm -2) Grain size (µm) Slip plane normal?
This number ratio is in fairly good agreement with the observed variant selectivity.

Quantum-chemical Simulation of Silicon Grain Boundaries Contaminated by Oxygen And Carbon A.M.
As is known, the majority of silicon polycrystals contain a number of oxygen and carbon atoms (~ 1016 -10 18 cm -3) mostly grouped at or around the grain boundaries.
Si69 cluster with GB Σ5 θ = 37° [001]/(130) consisting of two sub-clusters (grains) I and II separated by tilt GB created by rotation of grains I and II around the common axis [001] (lying in GB plane normally to the scketch plane) at the angle θ = 37°.
The cluster presented in Fig. 1 was chosen as a basis for simulation of different configurations of the contaminant-containing complexes SiOmCn Angle θ Grain I Grain IIformed at GB "core".
The rest of the cluster (nonrelaxed) simulated the influence of the silicon grains surrounding the GB "core" on both sides.

High cube volume fractions can be predicted under a number of conditions, though a small surface energy advantage of just 2% for cube-oriented grains is required to match the texture strengthening to the grain size change.
Each simulation was carried out 5 times, using the same experimental starting microstructure in each case, but using different seed values for the random number generator.
Effect of initial experimental data set size There is a trade-off between the sample area that can be modeled and the number of model grid-points (hence the simulation speed).
If a large map step-size is used (giving a large area but a small number of map grid-points) the real boundary curvature is not accurately represented with many small "grains" modeled as regions of just one or two map-pixels.
For the Ni-tape material, the final grain size after grain growth is similar to the tape thickness (90µm), hence during grain growth the fraction of surface grains (with respect to the total number of grains) increases from ≈ 29% to 100%.

The larger size of island grains is their dominant characteristic, and grains which become island grains may have been incipient abnormal grains.
These grains grow excessively, consuming the matrix grains, resulting in a bimodal grain size distribution.
Of the total number of island grains observed, around 75% possessed either low angle (<20o ) or high angle (>45o ) grain boundaries.
The number of low and high misorientations observed at these island grain boundaries is therefore substantially higher than the number found in the uniform structure.
The existence of a number of island grains with mid-range misorientations (20 o -45o ) to the abnormally large grain may be explained by the presence of a Σ5 (36.9 o <100>) boundary misorientation.

To complete this analysis we need a relationship between the number of faces of a grain of radius RS and the mean grain radius .
This can be determined, again approximately, from the work of Rhines and Craig [12], who serial sectioned single phase aluminum undergoing rapid grain coarsening to determined the frequency of grains with different numbers of faces, fig. 6.
So using eq. 5 we obtain: RF = RS 2 / N 1/2 = 2 / 81/2 = 0.707 (5a) NGC -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0 20 40 60 80 100 Number of Grain Faces (N) sin Ф' .
The variation of sin φφφφ' as a function of the 3D number of neighbors N [16].
The sin φ' curvature determined by the grain neighbor number 10 min 60 min 10 20 1 min 2 min 30 min 40 30 Faces Per Grain Number of Grains/ cm 3 x 104 12 4 8 makes grains with N> 14 grow more slowly than grains with N < 14 shrink.

The elongation of the composite decreased gradually with the number of ARB cycles, became almost zero after 4 cycles.
Results and Discussion Changes in mechanical properties of the composite with equivalent strain (number of ARB cycles) are shown in Fig. 2.
The specimens after 2 and 3 cycles have dislocation cell structures, but the cell size become smaller with the number of cycles.
The change in microstructure with the number of ARB cycles is different from that of the monolithic 6061 Al powder.
The elongation of the composites decreased gradually with the number of ARB cycles, became almost zero after 4 cycles

The grain refinement after processing by ECAP is complex resulting in the formation of a bimodal grain structure with high fraction of fine grains already after the first or second pass.
The mean dislocation density rD calculated from Eq. 1 is plotted in Fig. 3 as a function of the number of ECAP passes.
This results in inhomogeneous microstructure in specimens after different number of rotations.
Consequently, the grain fragmentation in the centre and near Fig. 3 Dislocation density in AZ31 specimens subjected to different number of ECAP passes the edge of the specimen differs with increasing number of HPT rotations.
The difference in grain sizes is smeared out with increasing number of HPT rotations and nearly the same microstructure consisting of grains of the average size of 250 nm is observed in central (Fig. 4c) and peripheral region (Fig. 4d) of the sample after 15 HPT revolutions.