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Introduction The main characteristic of the grain structure is the size of grains.
In 3D context it usually means the mean grain volume Ev or its reciprocal value ENV (the mean number of grains per unit volume).
The most common are the counting procedures - profile and intercept counts giving the observed number of profiles per unit area of the examined observing window NA and number of profile intercepts per unit length of examined test line NL.
Fig. 3 Histogram of grain volumes.
Numbers at point denotes CV a, gray numbers at point denotes mean number of generators in cluster.

Introduction At very small grain sizes, besides the specific energy and mobility of the grain boundary faces, other microstructural features of the polyhedral grain network such as triple lines and quadruple points may have likewise a strong effect on the grain growth kinetics [1].
(1) In the above equation the total surface Si, edge length Li and number of vertices Nvi of the i-th grain are expressed by the grain volume (Vi) equivalent radius , and the number of faces Ni, where we have used the approximate scaling relations , , and based on Euler’s theorem (cf. e.g. [4,5]).
In particular, it is well known [7] that for normal grain growth the number of faces or neighbors Ni is, in a statistical sense, a parabolic function of the reduced grain size , Ni(xi) = a xi2 + b xi + c, where is the average grain size.
Figure 1: Grain size distribution functions of the nine types of grain growth kinetics defined by Eq. (6).
(11) Figure 2: 3D Monte Carlo simulation of triple line mobility limited grain growth [10]: left – Relative grain size distributions in the scaling state of the linear growth regime together with fit of Eq. (10) (x1 = 0.964, x0 = 3.12); right – Number of faces N versus relative grain size x at t = 1000 Monte Carlo steps (MCS).

In order to account for this effect, and to test the idea, in some cases, the number of precipitates and / or their pinning strength was / were suddenly decreased by a given amount to test the stability of pinned grain structures.
Journal Title and Volume Number (to be inserted by the publisher) 3 Results To compare the topology of grains during normal growth and particle controlled growth we analyze the average size of neighboring grains versus the size of the considered grain, for all grains of various configuration.
Fig. 1 introduces this representation for normal growth; in average it is independent on the number of grains in the evolving structure, for instance the one shown in Fig. 2.
However, among a large number of different runs, we have sometimes observed some grains exhibiting a very fast growth and able to reach a size (area) up to 16 times the average grain size and up to 10 times the average size of its neighbors.
In Journal Title and Volume Number (to be inserted by the publisher) 5 one case (Fig.6a) the large grain had clearly an initial advantage in size, although still in the limits of the dispersion shown in Fig. 3b.

The following important deformation (sub-) structure quantities are known for each individual modeled grain: orientation g, total dislocation density ρtot, number of different active slip systems during the entire deformation NGLS, amount of shear relaxation shear relax vMε − in a grain, and IGM value.
Orientations in the spectrum with a number of active slip systems exceeding a critical value were assumed to recover fast and to form viable recrystallization nuclei.
In the absence of other nucleation sites like large particles we assume the total number of nuclei Ntot per unit volume as =++ tot rand GB TB NN NN (1) with Nrand, NGB and Ntrans being the numbers of random nuclei, GB-nuclei and TB-nuclei respectively.
Equivalent relations hold for nGB with a critical number of slip systems c GLSN and for nTB with a critical IGM-value Mc .
Analytical approaches which are typically confined to 2D geometry and which have touched both the kinetics and topology of grain growth have recently experienced a revival in several attempts to find an analytical solution to the 3D problem, in particular a 3D formulation of the so-called von Neumann-Mullins relation which ties the growth rate to the number of faces of a grain, respectively the number of sides in a 2D cellular arrangement [21,22].

Based on the 0.6 percent Nb2O5, full-grown elongated grain obtained by using 0.2 percent CAS illustrates that interface Table 1 Ingredient of Al2O3 ceramics Number Al2O3 /wt% Nb2O5 /wt% CAS /wt% 3Y-TZP /wt% 1# 99.8 0.6 0.2 - 2# 99.6 0.6 0.4 - 3# 99.4 0.6 0.6 - 4# 99.2 0.6 0.8 - 5# 99.8 0.6 0.2 12.0 6# 99.6 0.6 0.4 12.0 7# 99.4 0.6 0.6 12.0 8# 99.2 0.6 0.8 12.0 reaction occurs [7].
More CAS content, more liquid phase was around interfaces so that Nb2O5 cannot control the influence of the interface reaction of CAS, so all of interface energies are approximately equa, and the number of block grains was formed.
Compared with the elongated grain without 3Y-TZP, the slenderness ratio was lesser, but the number of elongated grains was considerable, adapting to elongated grain cooperating with second phase to toughen alumina ceramics.
Based on less Nb2O5 than 1.0 percent, with the its content increment, the deffective number is increasing so as to form finite substitution solid solution between Nb2O5 and Al2O3 interaction.
The difference among coordination number, electrical valence and ionic radius results in the distortion of lattice and cation vacancy by Nb 5+ substituting Al3+.

The present analysis reveals that the grain-size exponent is dependent on the grain size and the grain-boundary viscosity: the exponent becomes unity for small grain sizes and/or high viscosity, while it is three for large grain sizes and/or low viscosity.
The stress-directed diffusion, which occurs through grain and/or along grain boundary, induces grain elongation, grain-boundary sliding and macroscopic strain.
In a steady state, the dissipated energy is supplied by the external work, and an equation of energy balance is given by 2Vσ Ý ε = m�� ,W �� D+ m � � ,W �� S, where V is the grain volume, and m is the number of the boundary facets of the grain.
Compared with these values, Eq. 6 gives a larger numerical constant 14 which is identical to the number of the boundary facets per grain (m).
Assuming a space-filling cuboidal grain instead of a tetrakaidecahedron in the present analysis, we obtain 6 for the numerical constant in Eq. 6, which is also the number of the boundary facets of a cube.

A small number of low angle boundaries were found and also many twin boundaries.
Very large grains were produced by the strain-anneal process but these contained a number of isolated grains as residues of the prior structure, typically 10 to 30µm in diameter.
Using this criterion, histograms of the number of grain boundaries in different disorientation classes were generated as shown in Fig.2 (a) and (b) for the different materials.
A significant number of the axes are clustered near the [111] corner.
Also, there are a significant number of other misorientation angles.

So we can know grain is continuously growing up when the edge number is greater than six.
Otherwise, it is continuously shrink; thus we know that the number of edges can affect the grain changing rate.
When a topology type transfers to another, it should be consistent with Euler's formula[14]: 3ΔV=2ΔE=6ΔF (8) ΔV refers to changes in the number of vertices, ΔE refers to the number of edges of change, ΔF refers to changes in surface or grain number.
After this process, the number of grain, grain boundary edge and threes junctions have not been changed. 2) Grain Annihilation, as shown in Figure 2 (b).
The number of vertex and boundary are not changed.

The fact that grain boundary diffusion (GBD) decreases under the action of grain boundary segregation (GBS) is now generally recognized.
Note here, that NGB is unknown number, and the data are obtained for all atoms in the model, including the grains.
Hence, due to the complexes formation the number of mobile atoms decreases.
Rodin, Grain Boundary Diffusion and Grain Boundary Segregation in Metals and Alloys.
Rodin, New model of grain boundary segregation with formation atomic complexes in grain boundary, Phys.

Therefore, those simulations were terminated to small grains in order to keep the number of grains adequately for statistically analysis.
In the present 3D simulations we choose the number of MCU's of the simulated grain structures as 150x150x150.
Figure 3: a - Simulated GSDF versus x after 500 MCS compared to the analytic approximation Eq. 3. b - Number of neighbouring grains s versus grain size x after 500 MCS.
Figure 4: a - Simulated 3D GSDF versus x after 500 MCS compared to the analytic approximation Eq. 3. b - Number of neighbouring grains s versus grain size x after 500 MCS.
In Fig. 4b the number of neighbouring grains of grains of size x is plotted after 500 MCS.