Computer simulation is used to generate microstructures by nucleation on a 3D network of Kelvin polyhedra. The Kelvin polyhedra network was used as a model for a polycrystalline network. Information provided by usual descriptors of microstructural evolution: the volume fraction and mean interfacial area density are perhaps insufficient to fully characterize those microstructures. Therefore, microstructural descriptors that are less often used such as the contiguity, the contiguity ratio, the dispersion parameter and the duplex parameter were tested. These parameters showed to be able to increase the understanding of the microstructural changes for Kelvin polyhedra/grain boundary nucleated transformations. The contiguity and the dispersion parameter were particular useful to provide better understanding of the microstructural evolution studied here.

Nucleation and growth transformations are often described by two main stereological parameters: the volume fraction, VV, of the new phase, here denoted the β phase and the mean interfacial area density between the new and the parent phase, SV or SVαβ, where the parent phase is here denoted the α phase. Using these parameters, one may describe the microstructural evolution by plotting the transformation kinetics, i.e. VV as a function of time and the microstructural path (MP) [1], that is, SV as a function of VV. This methodology describes microstructural evolution well when the nucleation sites are uniform randomly distributed in space. The location of the nuclei is clearly very important to phase transformations and it is the subject of study of several researchers [2–4]. In fact, in many cases the nuclei might not be uniform randomly distributed in space. For example, nucleation sites may be located within spherical clusters [5] or on the grain boundaries or grain edges [2]. Under those circumstances the characterization of microstructural evolution using stereological parameters becomes more complex. In those cases, the volume fraction and the microstructural path are not perhaps enough on their own to fully describe the characteristics of the microstructural evolution. For those more complex cases, the usage of additional parameters that are defined below were proposed. It is thought that some of those additional parameters might be helpful to understand microstructural evolution better in those complex situations. One advantage of those parameters is that they are still defined in terms of two easily measurable stereological quantities. The first quantity is the mean interfacial area density between the parent phase, SVαα, that expresses how much of the grain boundaries are consumed by the new phase over transformation time. The second is the mean interfacial area density between the new phase, SVββ, that expresses how much new interface is created. Notice that for, VV→1, SVαα→0, and SVββ→constant.

These two interfacial parameters may be combined with volume fraction and mean interfacial area density between the new and the parent phase so that new useful parameters may be defined [1,6]. First one can define the contiguity, which is a parameter that quantifies the amount of contact between the grains of the same phase, or the degree of adjacency of each phase. The contiguity of either the parent, Cαα, or the new phase, Cββ can be estimated by Eqs. (1) and (2), respectively.

The advantage of using the contiguities is that when VV→1, Cαα→0, and Cββ→1. Another possible parameter is simply the ratio of the contiguities: Cαα/Cββ.

Other two interesting parameters proposed by Hornbogen [6] are the dispersion, δ, and the duplex, Δ, Eqs. (3) and (4), respectively. The dispersion parameter seeks to quantify the scattering of particles of a new phase, which are isolated from each other by the parent phase. On the other hand, the duplex parameter presents a relationship between the interfacial area density of the parent phase and the new phase.

All the above parameters attempt to describe the position of the transformed regions relative to one another. Some of them are especially useful here, as will be seen in what follows.

In this work computer simulation is carried out with the intention of studying the effect of nucleation on the grain boundaries on the microstructural evolution. Specifically, we wish to find out which of the stereological parameters defined above better characterize the evolving microstructures. In the present simulations, the polycrystal is represented by a space-filling network of Kelvin polyhedra [7]. We compare transformations in which nuclei are uniform randomly located within space with transformations in which nuclei are located on the Kelvin polyhedra network interfaces.

2Computer simulation methodologyComputer simulation of the 3D nucleation and growth transformation was carried out using the causal cone method [8–10]. The matrix comprised 304×304×304 cubic cells. To assign physical dimensions to the simulation we regard each cubic cell edge length to be equal to 1μm. Therefore, our “sample” was a cube with a total edge length of 304μm. Of course, periodic boundary conditions used here ensured that the simulation space was “infinite”. Within this cubic matrix a network of Kelvin polyhedra was constructed. Nucleation took place on the interfaces of this network. The absolute number of nuclei, N, used were N=28, N=438, N=1040 and N=3511. These number of nuclei were adopted to obtain final grain sizes between 10 and 150μm, approximately. Site-saturated nucleation was used in all simulations. That is, all nucleation takes place at the start of the transformation so that the number of nuclei remains the same and only growth takes place over time. All growth velocities were kept constant.

3Nucleation and growth kineticsFig. 1 shows two examples of nucleation on the boundaries of the Kelvin polyhedra network. Fig. 2 shows the transformation kinetics, volume fraction against time, for a varying number of nuclei. The solid curves are calculated from Johnson-Mehl, Avrami and Kolmogorov [11–13], the so-called JMAK theory, for homogeneous nucleation and provide a frame of reference to evaluate the effect of nucleation on the boundaries. The kinetic curves fall fairly close to the JMAK theory and significant deviation only starts to be visible when one has N=3511 nuclei on the boundaries. Thus, the kinetics is not very sensitive to the nucleation site except for a large number of nuclei.

Computer simulated microstructures of the early stage of the transformations nucleated on the Kelvin polyhedra network. (a) Low number of nuclei showing that the nuclei behavior approaches the behavior of the uniform nucleation. (b) High number of nuclei showing the Kelvin polyhedra network saturated by nuclei.

Transformation kinetics for the transformation nucleated on the Kelvin polyhedra network. The solid lines serve as a reference and were calculated from JMAK [11–13] theory.

Fig. 3 depicts the microstructural path (MP) of the transformations. The solid curves correspond to the MP for homogeneous site-saturated nucleation [8]:

where NV is the number of nuclei per unit of volume and VV was calculated by JMAK theory. The curves generated by Eq. (5) are used as a reference in Fig. 3. The MP is more sensitive to the nucleation site than the kinetics. The total interface area of a certain new phase region is the sum of two components: the interface area between parent phase/new phase, SVαβ, and the interface area between new phase/new phase, SVββ. We think that the position of nuclei in space affects more significantly the magnitude of each component thus affecting any interface area-based indicator more strongly than the volume fraction. One can see that, already for N=1040 nuclei a significant discrepancy between Eq. (5) and the transformation nucleated at the boundaries can be seen. For N=3511 nuclei the MP of Eq. (5) is far from that produced by the simulation. The discrepancies in the kinetics and in the MP increase as the number of nuclei increases. The higher is the number of nuclei on the grain boundary the higher is the “clustering” effect meaning a less homogeneous nuclei dispersion. Strictly speaking, the kinetic curves deviate from JMAK as soon as the underlying point process is not a Poisson Point Process anymore. Thus, for any N the kinetic curve deviates from JMAK. The fact is that the deviation, although mathematically present, is initially small for visual detection. Obvious visual discrepancy only takes place for large N. Therefore, the sensitivity is more a matter of whether one can visually distinguish the curves rather than a mathematical issue. Of course, if this were only a mathematical issue we could perhaps use the kinetic curves. But the point of the study is to determine which indicator is more appropriated to be used in real microstructures. In those cases, one must consider experimental errors. So, a higher sensitivity is extremely important as it permits the detection of specific characteristics even in the presence of experimental errors. If the sensitivity is too small, say well below the experimental errors then the indicator is not going to be able to show any difference between two situations.Microstructural path for the transformation nucleated on the Kelvin polyhedra network. The solid lines serve as a reference and were calculated from Eq. (5).

Fig. 4 depicts the contiguity of the new phase, Eq. (2). Fig. 5 shows the contiguity ratio: Cαα/Cββ. In Fig. 4 the solid line corresponds to the contiguity of homogeneous nucleation. It is interesting to compare Fig. 4 with Fig. 5 of Rios et al. [14]. In their work, Rios et al. [14] used cellular automata to simulate transformations with nuclei location ranging from periodic to clusters. A periodic nucleation resulted in a contiguity curve below the solid curve that represents uniform nucleation. This suggests that for a small number of nuclei, N=28, the periodic nature of the Kelvin polyhedra network predominated, and one can see in Fig. 4 that the simulated results for N=28 nuclei are located below the curve for uniform nucleation, consistent with the result from Rios et al. [14]. In contrast, the curves for higher number of nuclei are located above the solid curve for uniform nucleation, suggesting that the clustering effect represented by nucleation on the boundaries predominates for higher number of nuclei. Again this is similar to what was observed by Rios et al. [14] for nucleation within clusters. Vandermeer [1] has already remarked this tendency of the behavior of the contiguity for nucleation in clusters.

The contiguity ratio shown in Fig. 5 does not seem to add new information to what was already obtained from the behavior of the contiguity of the new phase shown in Fig. 4. Of course, the contiguity ratio curves are asymptotic to the vertical and horizontal axes. This is so because when VV→0, Cαα→1, and Cββ→0 and when VV→1, Cαα→0, and Cββ→1.

6Dispersion and duplex parametersFigs. 6 and 7 depict the dispersion and duplex parameters, Eqs. (3) and (4) respectively. Both parameters have a similar behavior in this case. The reason for this is that at some point during the transformation the interfaces of the parent phase, SVαα, are consumed by the growing new phase. When the number of nuclei is small, say 28, the nuclei behavior approaches the behavior of the uniform nucleation and the interfaces of the parent phase, SVαα, are fully consumed only toward the very end of the transformation, when VV→1, SVαα→0 so that both δ and Δ→∞. Alternatively, for a large number of nuclei at the boundaries, the boundaries are fully consumed quickly so that SVαα→0 before VV reaches one. In Fig. 6, δ→∞ for VV between 0.6 and 0.7. Fig. 6 shows that in the case of grain boundary nucleated reactions the dispersion parameters are very sensitive to nuclei density on the boundaries. Fig. 7 essentially shows the same information, but the asymptotic behavior appears to take place for higher volume fractions for the same number of nuclei. Therefore, it seems that the dispersion parameter, Eq. (3), is the most useful parameter.

7Summary and conclusionsIn this work, several stereological parameters were tested to see which ones were more useful to describe microstructural evolution of grain boundary nucleated transformations. The main conclusions are:

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The kinetics, volume fraction against time curve, is the least sensitive to nuclei location.

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The microstructural path was significant more sensitive to nuclei location than the kinetics.

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The contiguity of the parent phase was revealed to be quite a useful parameter and exhibited trends that were consistent with those observed in previous work [14]. In contrast, the contiguity ratio did not seem to add new significant information.

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The dispersion parameter was strongly sensitive to nuclei location. In the present case both the dispersion and the duplex parameter behaved similarly but the dispersion parameter appears to be the best choice.

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The contiguity of the new phase, Eq. (2), and the dispersion parameter Eq. (3), showed to be useful parameters to measure in addition to the usual volume fraction and mean interfacial area density between the parent and the new phase.

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In summary, both the dispersion and the contiguity supplied information about microstructural evolution that would be hard to infer without their use.

The authors declare no conflicts of interest.

The authors are grateful to Conselho Nacional de Desenvolvimento Científico e Tecnológico, CNPq, Coordenação de Aperfeiçoamento de Pessoal de Nível Superior, CAPES, and Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro, FAPERJ, for the financial support.

Paper was part of technical contributions presented in the events part of the ABM Week 2017, October 2nd to 6th, 2017, São Paulo, SP, Brazil.