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A stochastic nucleation model is established to calculate the nucleus number as follows: (3) Where N is the nucleus density, ΔT is the undercooling, Ns is the maximum nucleus density, ΔTσ is the standard deviation of the distribution, and ΔTN is the average nucleation undercooling.
The grain growth is based on the KGT equation [18], and the growth speed of the grain tip is described as follows: (5) Where α and β are the coefficients.
Previous studies [11, 15] showed that a concave shaped mushy zone might make the grain convergent, and a convex shaped mushy zone makes the grain divergent.
There are some broken grain in the exhaust edge of the blade.
The grain grew very well in the back side of the blade, and some stray grain nucleate in the listrium of the blade.

It can be found that the average grain size declined gradually, the minimum grain size of 20 nm appeared at time of 15 min, then it increased with the processing time increasing.
It indicates that there are a large number of nano-scale twins (Fig. 4(e)), which are confirmed by the corresponding SAED pattern, plate strips (Fig. 4(f)), and equiaxed sub-grains(Fig. 4(g)) derived from dislocation cells.
The average grain size of surface is around 20 nm.
With further deformation, dislocation cells became sub-grains.
Finally, the refined plate strips, twins and sub-grains transform nanostructured grains with random orientation.

[4] probably were the first to clearly point out that β solidifying alloys offer significant potential for grain refinement, because in the β/α transformation 12 orientations of α precipitates can be obtained from one parent β grain.
Despite this clear result B could also effect grain refinement during solidification, e.g. at higher concentrations of B and Al.
The interdendritic regions contained Borides and in some cases single-phase γ grains.
Similar refined microstructures could also be achieved in this work in quite a number of alloys containing (44 - 45.5) at.% Al, (0.1 - 0.2) at.% B, (5 - 8) at.% Nb and additions of Mo, W or Fe up to 2 at.%.
After this heat-treatment the alloy showed a microstructure consisting of α grains with a size 0.5 - 1 mm (Fig. 3).

In Cu, the trend is similar to Al so that most of the hardness values lie on a constant level, but close examination reveals that the angular variation is less as the number of the revolution increases.
There are grains with irregular configurations of grain boundaries and many dislocation are visible with the grains.
The grain size is reduced to ~1.5 µm and there are few dislocations within grains.
Some grains are present where many dislocations are visible.
Inspection reveals that there are grains with a low density of dislocations as indicated by an arrow whereas some grains contain many dislocations.

Natural nanostructure in polycrystalline silicon was utilised for charging islands and its grain boundaries were modified by a multi-step annealing technique to form grain-boundary tunnelling barriers.
Subsequent annealing increases the grain-boundary tunnelling barrier height and resistance.
Introduction A single-electron transistor (SET) is a novel device where the on-off states can be formed by a single-electron, without any statistical fluctuation in electron number[1,2,3].
( ) �� � � �� � �−�� � � �� � �−�� � � �� � � = Tnk qV Tk qV m Tk qnVI B B B B P 2 sinh exp 2 2 2/1 * π (1) , where q denotes a carrier charge, m* an effective mass, kB the Boltzmann constant, VB a barrier height, T temperature, n number of grain boundaries, and V an applied voltage.
We found that the charging island is made of crystalline grains covered by an oxide GB barrier[4]

On the basis of analyzing space relationship between scanning line and casting boundary, two conditions were gained, 1) the number of grids crossed by scanning line must be even number; 2) any two boundary grids crossed by scanning line must be not neighbored to each other.
For OGFM, any corner of mesh region can directly regarded as a grain to do filling operation.
In the paper, two algorithms to identify the grid nature are put forward, which are named Inner Grain Filling Method (IGFM) and Outer Grain Filling Method (OGFM).
When scanning line crosses through boundary grids, the existing cases can be summerized as: 1) scanning line encounters 1 boundary grid (fig.2 (1)); 2) scanning line crosses through 2 boundary grids (fig.2 (2)); 3) scanning line crosses through more than 3 boundary grids, and the grids number is odd number(fig.2 (3)); 4) scanning line crosses through more than 4 boundry grids(fig.2 (4)), and the grids number is even number; 5) scanning line crosses through a boundary line(fig.2 (5)); 6) scanning line crosses through double boundary grids(fig.2 (6)).
y x Fig.4 The grain filling for two isolated shapes The algorithm of OSGFM is shown in fig.3.

The following results were found: - As larger amount of cold work can be applied by HPTR cold pilgering, a lower number of intermediate heat treatments are required.
The impact on the grain size, precipitates, hardness and texture is investigated.
In contrast, a larger amount of cold work can be applied by HPTR cold pilgering (several cumulated passes) coupled with a lower number of intermediate heat treatments
Grain size measurements are performed by using image analysis (Visilog software).
Several thousand of grains are analyzed in each case.

However, the achievement of full density depends on a number of factors including the shape, size and plasticity of the particles, hydrostatic pressure applied, processing temperature, deformation rate and actual plastic strain experienced by the particles.
In the case of Al, it was thought to be related to easier recrystallisation at room temperature and decreasing number of pores acting to pin grain boundaries at the higher pressure [25].
The stability of the material was attributed to the presence of pores which pinned grain boundaries and thus prevented grain growth.
Although Fig. 5 Compressive stress versus strain curves showing increases in both strength and strain to fracture with increasing number of passes in Al consolidated from nanosized particles by ECAP at 400°C.
A number of promising materials with both enhanced strength and good ductility have been achieved.

Grain size refinement by using equal channel angular Extrusion (ECAE) is an effective way to improve workability and strength of the magnesium alloys.
The deformation mechanism of ECAE for grain refinement is obtained.
Especially, the relationship of material grain refinement and the deformation behavior needs to be studied further.
Five numbered nodes were marked onto the workpiece, these nodes a–e are placed on the cross-section, 1-1, in the die channel, respectively, as shown in Fig. 3, representing different zones in deformation.
Conclusions This paper presents interesting results concerning a dual analysis between finite element simulation and experiments for understanding and predicting grain refinement in ECAE.

In the course of scientific and technological progress, an increasing number of materials have been created that met the necessary operational requirements.
The calculation of the linear density of twin boundaries was carried out by counting the number of intersections of the grid lines applied to the images and with twin borders.
Grains unfavorable for twinning are deformed by sliding.
The content of such grains was small.
Twining was observed in all grains and its morphology weakly depends on the grain orientation.