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Vol. 3. Issue 4.
Pages 349-353 (October - December 2014)
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Vol. 3. Issue 4.
Pages 349-353 (October - December 2014)
Original Article
DOI: 10.1016/j.jmrt.2014.08.002
Open Access
Stereological analysis of the microstructure of pure iron with random nucleation
Daniel Souto de Souza
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Corresponding author.
, Weslley Luiz da Silva Assis, Paulo Rangel Rios, Gláucio Soares da Fonseca
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Corresponding author.
Escola de Engenharia Industrial Metalúrgica de Volta Redonda, Universidade Federal Fluminense (UFF), Volta Redonda, RJ, Brazil
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Figures (3)
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Tables (4)
Table 1. Analytical formulae employed in the calculations carried out in this work.
Table 2. Measurements of NL and NA in pure iron compared with calculations from approximate methods.
Table 3. Values of NL and NA obtained by cellular automata computer simulation compared with calculations from approximate methods.
Table 4. Values of NL obtained by cellular automata computer simulation compared with those determined for pure iron.
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Microstructural characterization of metallic materials is of paramount importance in the qualification and quantification of their desired final properties. Metallic materials have often been characterized by means of optical and electronic microscopy. Results from these techniques are in most of the cases based on 2D analysis. Stereological methods are then employed to obtain 3D information. However, such methods are based on assumptions and approximations of the real material structure. Therefore, it is essential to know the limitations of these methods. A further complication arises when one wishes to compare real materials with computer simulation results and stereological analytical techniques. In this paper, methods normally applied to real microstructure are applied to microstructures simulated by cellular automata (CA) simulation. Experimental results from microstructural characterization of polycrystalline pure iron were used as the starting point for the simulation. Consequently, we could apply analytical formulae of stereology to experimental data from pure iron and to computer generated microstructures from cellular automata simulation. Comparison of analytical formulae, experimental results and computer simulation provided useful insights on limits of applicability and on the meaning of the stereological analysis.

Cellular automata
Pure iron
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Recrystallization and grain growth are fundamental issues of Materials Science. Therefore, continuous progress is always required in understanding those phenomena in order to achieve better control and consequently increasingly better final properties of polycrystalline materials. In this regard, detailed knowledge of the 3D microstructure of the polycrystal is paramount. Traditional metallographic techniques invariably carry out measurements on a planar section [1,2]. From those measurements, one cannot obtain all 3D microstructural parameters. For example, intrinsic 3D quantities, such as the volume of the individual grains, the number of grains per unit of volume or the Gaussian curvature per unit cannot be obtained from classical 2D quantitative metallographic techniques.

Recently, it has become apparent that there is a transition from 2D to 3D microstructural characterization. Specifically, today one has 3D computer simulation, 3D analytical theories and 3D microstructural characterization techniques. This can be seen in the widespread number of papers, conference sessions and even entire conferences dedicated to the topic of 3D Materials Science.

Still, for a long time simple techniques will persist and one may use the 3D developments to improve the simpler 2D techniques. For example, some assumptions may now be tested against a 3D computer simulation. In this way, one may develop useful approximations that may be very helpful for the Engineer to obtain a reasonably accurate result for a fraction of the cost and the time of a full 3D experiment. Following this reasoning, the present work utilizes experimental measures, analytical methods and computer simulations to revisit earlier results and test new approximate expressions that relate 2D measurements to 3D quantities by means of approximate expressions.

2Methods and materials2.1Experimental methods and materials

Samples of polycrystalline pure iron were used. The composition of the pure iron in ppm was: C-41, Mn-940, P-15, S-20S Si-160, Al-20, N-80, Ti-10, Cu-30, Cr-100, Mo-20, Nb-10, V-10, B-4, O-165. The sample was cold rolled up to 80% then annealed 550°C for 3600s (1h) in a quartz tube containing Ar. Specimens were cut from this sample taking care to avoid deformation during cutting. The specimens were then ground and polished. A final polishing step consisting of polishing in a colloidal silica solution was employed. After this, the specimens were etched with Nital 3% solution for observation under the optical microscope. Stereological measurements were carried out in a Nikon Eclipse optical microscope with image analysis software NIS-Elements D 3.0. The number of grains per unit of area, NA, and the number of grains per unit of length of test line, NL, were measured using standard techniques [3].

2.2Computer simulation

A 3D cellular automata computer simulation was carried out. A cubic mesh with 300×300×300 cubic cells, henceforward also referred to as the “matrix”, was employed. A von Neumann neighborhood was adopted. Simulation details can be found in Ref. [4–9]. The simulation employed a 22,300 nuclei uniform randomly located within the matrix at the start of the transformation. This number of nuclei was chosen according to reasons explained in a previous work [10]. The simulation was carried out in a dual processor Intel Xeon workstation with eight cores in each processor and 48GB of memory. The computer program was written in Fortran 2003 and compiled in an Intel FORTRAN compiler using OPEN MP 3.0 directives for shared memory parallelization. Each simulation run took about 10,000s. In order to assign units to this simulation a length to the edge of a cubic cell has to be chosen. We chose this length in such a way to make the number of nuclei per unit of volume equal to that calculated for pure iron using DeHoff's formula. This was done because DeHoff's result was considered the better approximation in that case. Details are given below and results are in Tables 1 and 2, later in this paper. This choice gave a cubic cell with an edge length equal to about 0.5578μm.

Table 1.

Analytical formulae employed in the calculations carried out in this work.

Approximation method  DeHoff [2]  Voronoi [1]  Sphere  Cube 
NL (mm−1=76NA1/2  =1.205NA1/2  =146πNA  =126NA 
NA (mm−2=0.735NL2  =0.6887NL2  =83πNL2  =23NL2 
NV (mm−3=0.422NL3  =0.3247NL3  =169πNL3  =827NL3 
NV (mm−3=0.670NA3/2  =0.5681NA3/2  =6π6NA3/2  =269NA3/2 
Table 2.

Measurements of NL and NA in pure iron compared with calculations from approximate methods.

Stereological quantities  Pure iron (measured)  DeHoff [2]  Voronoi [1]  Sphere  Cube 
NL (mm−1) from NA  67.3  67.0  69.2  62.4  70.4 
NA (×10−2mm−2) from NL  33.0  33.3  31.2  38.5  30.2 
2.3Analytical expressions

The analytical expressions employed in this study either derived above or obtained from publications are summarized in Table 1.

In addition to NL, NA, and NV obtained by DeHoff [2] and for Voronoi polyhedral [1] we derived expressions for these quantities supposing that the grains could be considered as cubes or spheres.

For spherical polycrystalline grains, using S=4πR2, V=43πR3 and the caliper, D=2R, gives, SV, the grain boundary area by unit of volume

Deriving expressions for NV is straightforward

Eq. (1) together with the well-known expression [3]

give other expressions involving NL and NA may be derived

For the cube, we may use an identical reasoning as above, recalling that the mean area and mean volume of the cube are a2 and a3 respectively, where a is the edge length of the cube. Moreover, from stereology [3], λ=4V/S, where V is volume and S the area, for cubic grains λ=1/NL=3a/2. As a result, for cubic grains

3Results and discussion3.1Measurements of NL and NA in pure iron compared with approximate methods

Fig. 1 shows an optical micrograph of fully recrystallized pure iron. NL and NA were measured for the pure iron. These experimental values were employed to assess the theoretical expressions obtained by DeHoff [2], for the Voronoi polyhedra [1] and for spherical grains, Eqs. (6) and (7) and for the cubic grains Eqs. (10) and (11). In order to do this the experimental value of NL was inserted into the theoretical equation to calculate NA and the experimental value of NA was inserted into the equations to calculate NL. The results obtained were compared with the experimental values. We considered that the best theoretical equation was the one that would give the best agreement of calculated and experimental values. The results of these calculations are summarized in Table 2. Table 2 shows that DeHoff's approximated expression gives the best results. For example, DeHoff's values of NL calculated for experimentally measured NA is very close to the experimental value of NL. The same agreement was observed for values of NA calculated for experimentally measured NL as shown in Table 2. Of the other expressions, Voronoi showed better agreement than the cube, which on its turn gave a better agreement than the sphere approximation.

Fig. 1.

Optical micrograph of fully recrystallized pure iron.


In view of these results, we considered DeHoff's approximation to be the best choice to calculate NV for pure iron. Therefore, the values of NV used for pure iron in what follows were those calculated using DeHoff's approximation.

3.2Values of NL and NA obtained by computer simulation compared with approximate methods

Fig. 2 shows a cellular automata computer simulation of the growth of nuclei uniform randomly located within the matrix with a constant velocity up to a fully transformed “microstructure”. Fig. 2a shows a tridimensional view whereas Fig. 2b shows a planar section of Fig. 2a. The values measured from the planar section were 76.2mm−1 for NL and 3824mm−2 for NA.

Fig. 2.

(a) Tridimensional view of a fully transformed computer simulation of the nucleation and growth reaction. (b) Planar section obtained from the 3D “microstructure” depicted in (a).


Table 3 shows values of NL and NA obtained by cellular automata computer simulation compared with approximate methods. In this situation, one would expect cube to be a good approximation and that is indeed the case. The simulation method adopted: all nuclei already present at t=0, without any further nucleation, and all grains growing with a constant velocity in cellular automata simulation produce a cube shaped growing volume as illustrated in Fig. 3b. This point is discussed further in the next section.

Table 3.

Values of NL and NA obtained by cellular automata computer simulation compared with calculations from approximate methods.

Stereological quantities  Cellular automata  DeHoff [2]  Voronoi [1]  Sphere  Cube 
NL (mm−176.2  72.1  74.5  67.1  75.7 
NA (×10−2mm−238.2  42.7  40.0  49.3  38.7 
aNV (×10−4mm−3) from NL  12.9  18.7  14.4  25.0  13.1 
aNV (×10−4mm−3) from NA  12.9  15.8  13.4  17.1  12.9 

Values of NV in the column “Cellular automata” were those used in the simulation. In the other columns, values of NV obtained from NL and NV from NA were calculated using the analytical formulae of Table 1.

Fig. 3.

Growth of a single isolated region with a constant velocity: (a) Spherical growth and (b) Growth during a cellular automata simulation.

3.3Measurements of NL and NA in pure iron compared with values obtained by computer simulation

Table 4 presents a summary of the quantities measured in pure iron along with those obtained by cellular automata simulation using NV of the simulation equal to that calculated for pure iron by DeHoff's approximation. Therefore, in Table 4 for pure iron and the simulation values are the same. There is some discrepancy between the values of NL and NA. As said above, the cube not DeHoff's approximation gave the best agreement with the simulation owing to the cube shape of the growing regions of the simulation.

Table 4.

Values of NL obtained by cellular automata computer simulation compared with those determined for pure iron.

Stereological quantities  Cellular automata  Recrystallized pure iron 
NL (mm−176.2  67.3 
NA (×10−2mm−238.2  33.0 
NV (×10−4mm−312.9  12.9a 

Calculated using DeHoff's expression.

Difficulties of comparing recrystallization evolution in a single crystal pure iron with cellular automata simulation were discussed in depth by Salazar et al. [5]. The difference in the shape of the growing simulation region (cube) and the shape of the growing recrystallized grain in the case examined by Salazar et al. [5], spherical, was responsible for this difference. This is even more so because of the “stepped” nature of the moving boundary of the simulation. The consequence is that a simulation using the same NV as that of the phase transformation or recrystallization should give a larger interfacial area density per unit of volume, as observed here. In practice, for applications of cellular automata simulation to recrystallization [5], this difference is not problematic because it can be accounted for by a shape factor or correction factor that is independent of time [5].

Such a correction factor allows the quantitative data to match. Nonetheless, one must bear in mind that the cellular automata simulated microstructures do have different interfacial area per unit of volume than the real ones. Therefore, as it can be seen here, absolute values of interfacial areas of simulated and real microstructures do not agree so well. The simulation may be corrected to agree with the experiment because the disagreement is systematic not a random error and may be accounted by a time and volume fraction independent correction factor [5].


  • Stereological measurements carried out in pure iron, NL and NA, were compared with four possible approximations to the (mean) grain. The grain shape proposed by DeHoff gave the best result. The point of this good agreement is that if one only had one experimental measurement of, say NL, DeHoff's equation could be used to estimate NA.

  • Stereological measurements, NL and NA, were also carried out in a cellular automata simulated microstructure, NV was known for the simulation. These measurements were compared with four possible approximations of the grain shape. In this case, the cube approximation gave the best result.

Conflicts of interest

The authors declare no conflict of interests.


This work was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico, CNPq, Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro, FAPERJ and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior, CAPES. Daniel Souto de Souza is grateful to Universidade Federal Fluminense (UFF) for his “scientific initiation” studentship. Help from Dr. Simone Oliveira is gratefully acknowledged.

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Journal of Materials Research and Technology

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