In most cases, grain formation is at least partially controlled by minimizing surface energy. Theoretically this should lead to spherical grains, but the volume conservation principle (unless you have void formation) forces the grains to grow into platonic solids which resemble the most to a sphere. These are respectively the dodecahedron (12 faces) and the isocahedron (20 faces). Crystallographic considerations will typically drive the material towards the dodecahedron geometry (at least for metals). The dodecahedron gives quadruple grain boundaries at the most in a 3D visualization. In any 2D cut the dodecahedron shows triple grain boundaries at the most (the forth grain boundary in a quadruple grain boundary location will be oblique to the cut and hence not visible). In any micrograph from a metal (which is a 2D cut), you tend to find only triple grain boundaries and only very exceptionally a quadruple grain boundary (which is a deviation from the dodecahedron principle, but can be due to the local kinematics and dynamics during the grain formation). So if you have a 2D model, triple grain boundaries should be the most representative, in a 3D model you would preferably have a sampling in a dodecahedron geometry, leading to quadruple grain boundaries.

I think 8 is a good figure, which divide space into 8 equal subspaces! 

Possible, you speak about minimal  number of subspaces? We use 3*3*3 = 27 subspaces usually as compromiss between computational costs and reslistic modelling. Serge

I mean considering a thermodynamically reallistic grain growth...

Dear Dr. Petersmann; Image that we have a  hypothetical homogeneous  nucleation, where N  nuclei are uniformly distributed in 3D  space. After isotropic growing there are first  6 faces to be in contact with , and the later only   8 corners to be touched. Hence my logic tells me that only 6 +8= 14 is good figure. Final solid  figure is  the shape of a Brillouin Zone.   Where the Grain Boundary Triple junction junctions can be easily recognized 

In most cases, grain formation is at least partially controlled by minimizing surface energy. Theoretically this should lead to spherical grains, but the volume conservation principle (unless you have void formation) forces the grains to grow into platonic solids which resemble the most to a sphere. These are respectively the dodecahedron (12 faces) and the isocahedron (20 faces). Crystallographic considerations will typically drive the material towards the dodecahedron geometry (at least for metals). The dodecahedron gives quadruple grain boundaries at the most in a 3D visualization. In any 2D cut the dodecahedron shows triple grain boundaries at the most (the forth grain boundary in a quadruple grain boundary location will be oblique to the cut and hence not visible). In any micrograph from a metal (which is a 2D cut), you tend to find only triple grain boundaries and only very exceptionally a quadruple grain boundary (which is a deviation from the dodecahedron principle, but can be due to the local kinematics and dynamics during the grain formation). So if you have a 2D model, triple grain boundaries should be the most representative, in a 3D model you would preferably have a sampling in a dodecahedron geometry, leading to quadruple grain boundaries.

Another approach that has been pursued in terms of modeling the shape and sizes of grains, as well as further geometric relations such as the grain boundary intersection number, is based on stochastic geometry.

One of the models used to describe the growth of grains is the johnson mehl avrami kinetics model. (villa has published a number of papers on this, especially concerning the grain structure in metals http://www.mat.unimi.it/users/villa/publications.htm)

The mathematical nature of this model is quite complicated (see Stochastic Geometry and Its Applications, 3rd Edition, Chiu and Stoyan).

Here are two publications, one more experimental, the other purely theoretical. The introduction of the latter is quite thorough in its coverage of the topic.

Experimental: www.eng.yale.edu/ramirezlab/publications/j17.pdf

Theoretical: eprints.ucm.es/27580/1/Dguez-Adame96libre.pdf

A simpler approach to this problem would be to assume a certain stochastic model for the grain shapes, as is done in neper e.g. a poisson voronoi tesselation of the space in 3D.

This work by tanemura gives distributional properties of these tesselations: www.scipress.org/journals/forma/.../18040221.pdf

I again refer to the compendium by Stoyan and Chiu that contains a few examples of the analysis of grain shapes in metal alloys and iron crystallization.

I believe that the best answer will depend on the material you are looking at and the distributional property that best fits the grain growth or final grain structure.

Best regards

PS: Mr. Tarik Ömer Oğurtani: What you are describing I believe is the Johnson Mehl process. Isotropic circular growth with constant velocity is assumed but will lead to a more complex shape. Of course as Frederik Coghe pointed out crystallographic models need to be taken into account for this as they present a limiting factor for the grains shape.

I think that Frederik Coghe gave the correct answer

http://cmrl.jhu.edu/papers/2008-acta-mat-2.pdf

three grains at triple point boundary.

I think that K.A. Makkuva gave the correct answer. A view of triple joint may be useful.

The only space filling polyhedron without leaving any gaps are cube and truncated octahedron as described in my two comments above. One has 8 faces and other has 14 faces as I stated.  The rest leaves holes which should be filled out by vacancy back flow  [vacancy-solute atom  exchange mechanism]  through  grain boundaries diffusion up to the free surface, which is  followed by surface diffusion to keep the morphology in tact,  Here the main thermodynamics driving  forces are capillary forces in the case of isothermal heating. The application of  the complex stress gradients  having compression  parts  during the isothermal annealing is very effective to accelerate  the filling process.  

Dear Manuel,

Apart from complex models, and considering only a realistic approach,according to the experience in metallography, in a plane (so 2D) tipically you may find 3 or 4 grains around one single point (usually no more), as commented by Frederik Coghe. Now if you add another layer of grains above the previous one, the highest probability is that only one grain would accommodate over the triple (fourth) point, and similarly on the level below.

Then from a realistic point of view you may found that the maximum number of grains sharing a single point would be between 5 and 6 in practice.

Kind regards

Dear Dr. Lukas Mosser;  I  thank you for comments.  You pressed the right points that the  model I have proposed  was very primitive.  But that gave the right answer if we wish to fill the whole space using only one type polyhedron. Actually I have developed very sophisticated irreversible thermodynamic theory  of surfaces and interfaces  (i.e., grain boundaries) evolving  by drift-diffusion   driven by the capillary, electrostatic and elastostatic forces, where anisotropies  in surface free energy as well as in diffusion coefficients  are fully taken into account.  Also we have done extensive computer simulations using this  non-equilibrium theory in various problems that  some how involve surfaces and interfaces.

Best Regards.

Since my commutation facilities are limited  these day, I didn't apply this theory to the  aggregate of particles having different shape and sizes using some statistics for the initial distributions!!   But one of my MS.Student has applied this theory in 2D- version using an aggregate of few spheres with various sizes surprisingly enough he got plausible picture. Where we don't  have to assume any equilibrium di-hedral angles at the triple junctions. System adjust itself spontaneously to reach the non-equilibrium stationary state asymtotically. 

As mentioned by others above, realistic metallographic evidence suggests that in general the maximum number of grains shairing a point in 2D is about 3-4. In general, I would say that the probability of any of these grains vanishing at the triple (or quadruple) point is very small. Therefore, I would say that most likely, there would be one more grain when one of the grains in the layer sharing a triple (or quadruple)  point (acutally a line) terminates. Thus, I think the most likely configuration is 4 or 5 grains sharing a point in an equiaxed microstructure.

Dear Dhriti;  I have check the cube and truncated octahedron, which are the only space filling shapes as I have mentioned above.

I am going  to analyze this  problem from the stability point of view of a given polyhedron in the presence of its nears neighbors,  taking the representative common points or corners into the considerations, and  the number of  facet planes  going through each.   Take a cube, which may be represented by one corner, which is shared by eight cubes. There are 3 mutually perpendicular    axis go through  from each corner. Here,  one has 2  angular degrees of freedom for each axis (, which sums up 1800) 

In the case of truncated octahedron.  Every corner shared by 4  grains,  and there are  3  pot -axis (i.e., triple axis) go through each corner having  2 degrees of freedom each.

At this point of the discussion I have to specify my problem. In my simulations I am concerned about nano-structured grains due to crystallization after amorphization.

Regards

Dear Manuel like the most people you are using a black box without knowing what  type  of  thermodynamic  characteristic functions has  been used for the stability consideration during optimization  procedure. The choice of these  functions such as Helmholtz, Gibbs, energy or enthapy  dependents upon what type of external state variables and constrains you are selecting for the optimization.  You are talking about the transformation strain, which is a tensor quantity, and it would be a state variable if only if you use Helmholtz free energy for the optimization. The stress tensor here is a conjugate variable comes out as a by product if you know the elasticity matrix.   You talk about thermal expansion in your note!  Are you changing the temperature as well as the strain tensor simultaneously or step wise version.  You are not specific while you are trying to describe your problem in the sense of a boundary value problem. For the Helmholtz free energy application one has to use  Dirichlet BC for the elasticity, which means you have to specify the displacement (zero or constant! ) at the boundary. Then we might be the helpful!  Regards.  

Dear Tarik,

I consider the Helmholz free energy which entails the strain energy in my formulation, which i directly calculate using FEM (where i also specify according boundary condtions), but this was not my initial question.

Regards

The answer depends on the definition of a"realistic" microstructure. Most responders assume a homophase, but microstructures can be single phase or multiphase.  In a single phase structure I would expect, at "equilibrium", only four grains at a single point.  If a precipitate phase were present at that point, which could include "impurities" in a nanophase structure, you could say there was only one or, perhaps, 5 grains.  If a sideplate grew from the precipitate, would it not be possible to have 6 grains?

Manuel, grains are not of uniform shape or size, so in reality I found experimentally that the minimum is 2 and the maximum is 4 when the accommodation is ideal to host two other grains after two coupled is important to note the sizes of real grains are far from being of the same size in diameter. 

Dear Manuel; Lord Kelvin (1887) [as I have proposed above] showed that there is one shape made up plane surfaces that will uniformly partition space with less surface area than the rhombic dodeca hedron. This shape is the truncated octahedron, bounded by six square and eight regular hexagonal flat faces, a total 14. which has approximately 1% less surface area  for a given volume than the rhombic dodecahedron. Kelvin also proposed ''minimal tetrakaidecahedron''in which a very slght saddle curvature is given to hexagon faces of the truncated octahedron, the result is approximately 0.103% less surface for a given volume than for the truncated octahedron (Lifshitz). 

The space filling Truncated octahedron has has only one type of vertex, which is 4-connected (4 grains), with alternating angles of  90 and 120 degress.

Note:  tetra kai deca hedron  means: four plus teen faces.

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