Artifacts of particle stimulated nucleation in 2D and 3D—a numerical analysis

Particle stimulated nucleation is considered to be the dominating nucleation mechanism in many general industrially relevant metallic alloys undergoing thermomechanical processing. In metallography, nuclei are typically defined as particle stimulated if they touch or are close to a second phase particle. In the present work, we numerically investigate how much miscategorisation there can be in this type of analysis. This is done by finding how many nuclei that statistically are next to a particle if all the nuclei and particles are randomly distributed in space. Furthermore, as particle stimulated nucleation is typically quantified by 2D experimental methodologies, it is additionally analysed if an area can correctly quantify the effect of the nucleation mechanism in a bulk sample.


Introduction
Nucleation of recrystallisation at second phase particles has been extensively studied and has been found to play a key role in controlling texture and microstructure during thermomechanical processing of materials containing large second phase particles [1][2][3][4][5][6].It is known that deformation zones develop around hard second phase particles during deformation [1].During annealing, if nucleation of recrystallisation takes place at these deformation zones, this is referred to as particle stimulated nucleation (PSN) [1,[7][8][9].
Nuclei are often defined as PSN if they are close to or touching particles [4,10].However, some nuclei may incidentally be found next to a particle.Some of these nuclei might have formed elsewhere and have grown to be close to a particle.This means that some of the nuclei are falsely categorised as PSN.This miscategorisation will be addressed numerically by various simulations in the present work.
Moreover, PSN is mainly documented by experiments carried out in 2D by different means of microscopy [11].However, when using 2D experimental techniques, it is expected that the PSN efficiency is undervalued.A nucleus not observed to touch a particle in 2D, might actually be formed from a particle present above or below the plane characterised [3,12].Therefore, it has been suggested that 3D experiments are important for this kind of investigations [3].We will theoretically investigate, if 2D characterisation can give the same results as found in 3D, and if so, how large an area is needed to describe bulk materials correctly.
In this numerical study, PSN results are analysed by distributing nuclei and particles randomly in a simulated virtual volume.To better understand PSN miscategorisation, parameters, including volume fractions and sizes, are varied to determine how many nuclei touch particles without there being a relationship between the spatial distributions of these.To gain more knowledge about the limitations of 2D experimental techniques, areas, chosen arbitrarily in the simulated volumes, are characterised and compared to the PSN efficiencies obtained from volumetric investigations.

Experimental
To study artifacts concerning PSN statistically, in-house MATLAB scripts were made to simulate PSN in particle-containing bulk metals.To do this, nuclei and second phase particles are distributed in virtual volumes.Both particles and nuclei are represented as spheres.The placement of particles and nuclei were done separately, meaning that the nuclei were placed randomly, using a non-uniform distribution, in a volume and then the particles were placed likewise in a volume of the same size.In this step, the sizes and the number of the spheres could be varied.Overlay of the same kind of spheres was completely avoided when the volume fractions were low, meaning that nuclei did not overlap with other nuclei as well as particles did not overlap with other particles.However, at higher volume fractions, the random placement without overlap of the same kind of spheres cost too much computational time and some overlay was allowed.This means that the intended and actual volume fractions did deviate slightly with the largest deviation being 2.3 %.It was also prioritised that the nuclei or particles were not cut off at the edges of the volume, since in a 3D experimental study these would be excluded from the analysis, as e.g.done in reference [4].The volume containing the nuclei and the volume containing the particles were added to obtain one volume of randomly distributed nuclei and particles.A PSN nucleus is, in this study, defined as a nucleus touching at least one particle, while a PSN particle is defined as a particle touching at least one nucleus.This definition is used, when PSN is quantified in both 2D and 3D.
Since only relative sizes of particles and nuclei are relevant in this study, everything is done in arbitrary units, however one could think of the numbers given as if they were μm.

Miscategorisation of PSN
To study effects of PSN miscategorisation a volume of 100 3 voxels with nuclei and particles was made.Two cases were considered; in the first case, the volume fractions were varied and the particle and nucleus sizes fixed.In the second case, the particle and nucleus sizes were varied and the number of these fixed.
In the first case, when the volume fractions were varied, the radii for nuclei and particles were set to 3 and 5 pixels respectively.As the number of each of these was varied the volume fractions were altered from 0.01 to 0.5 for both nuclei and particles.
In the second case, when the sizes were varied, 100 nuclei and 20 particles were used, meaning that the volume fractions also changed.The nucleus radius was varied from 1 to 10 and the particle radius from 1 to 15 pixels.
The same simulation was run four times for all set-ups to get statistically more relevant values.

2D and 3D quantifications of PSN
The difference between quantifying PSN in 2D and 3D was studied theoretically.Volumes were created as discussed above and arbitrary areas herein were analysed and compared to the 'true' PSN percentage from the volumes, see figure 1a) for this analysis.Radii were set to be 3 pixels for nuclei and 5 for particles, while the volume fractions were set to be 0.1 for both nuclei and particles.The size of the volume was varied, and different numbers of areas were quantified.As the volume size was changed the numbers of the particles and nuclei were scaled with the volume by , where N is the number of nuclei or particles, V is the cubed volume size, V f raction is the chosen volume fraction and s is the size of the nuclei or particles.Six different volumes were made varying from 25 3 to 150 3 voxels, with the largest volume being close to an upper limit in terms of computational time on a standard laptop.
For further investigation, an artificial volume where all nuclei touch at least one particle was introduced, see figure 1b).Here, the particles and nuclei were set to overlap randomly within the size of the radii.In this case, the 3D result will be 100 % PSN.Here, 60 nuclei and 60 particles with a radius of 4 pixels were placed in a volume of 100 3 voxels giving volume fractions of 0.015 each.The PSN percentages found in 2D were compared to the true 100 % PSN efficiency.

Miscategorisation of PSN nuclei and PSN particles
Figure 2a) and 2b) show PSN percentages of nuclei and particles, respectively, found by randomly distributing these in space with varying volume fractions.Since the volume fraction is increased by increasing the number, this can be thought of as nucleation of aluminium nuclei and second phase particles.These PSN percentages are all miscategorised PSN values, since there in the present set-up is no correlation between the position of nuclei and particles, everything is random.
The number of miscategorised PSN nuclei increases drastically with the particle volume fraction, see figure 2a).Small changes in miscategorisation can be observed when you follow the vertical change at 0.2 particle volume fraction.However, since there should not be a change in the probability of a nucleus touching a particle when the particle parameters are fixed, this is attributed to the edge effect having all the spheres complete inside the volume, making the density higher in the middle and lower near the edges of the volume.There is a critical limit of 0.1 particle volume fraction below which the miscategorisation of nuclei is tolerable.This volume fraction of particles is already high when comparing to typical particle-containing metallic alloys [13][14][15][16].Looking at figure 2b), the PSN particle miscategorisation depends on the nucleus volume fraction.The PSN miscategorisation of particles is huge above 0.1 volume fraction recrystallised.This is due to the size relationship chosen between particles and nuclei, with the particles being larger than nuclei.The particle miscategorisation decreases slightly as the volume fraction of the particles increases, this must be due to the overlap of particles at high volume fractions, as discussed earlier.
If the size and number relationship between the particles and nuclei is swapped, the results seen in figure 2a) and 2b) will likewise be swapped.When the volume fractions of nuclei or particles are 0.5, the PSN miscategorisation is expected to be ∼ 100 %, since the spacing between either nuclei or particles is smaller than the size of the opposite kind of sphere.
For further understanding of miscategorisation of PSN, effects of nucleus and particle sizes are shown in figure 3.This figure can be read as growth of both nuclei and particles.The PSN miscategorisation of nuclei and particles increase symmetrically with increasing size of nuclei and particles.By comparing figure 3a) and 3b), it can be observed that at the chosen conditions, the particles are more likely to be miscategorised than the nuclei.This is due to the larger number and thus also larger volume fraction of nuclei.If a given cold rolled aluminium sample is 6 % recrystallised and has a particle volume fraction of 0.005, as observed in [4], a random distribution of nuclei and particles would result in ∼ 10 % of the nuclei being randomly next to a particle, based on this simulation.A miscategorisation of roughly the same magnitude is found based on the sizes being 5 μm in radius for nuclei after annealing and 2.5 μm in radius for particles, as in [4].In other words, based on figure 3a) and 3b), ∼ 10 % of the PSN nuclei would be miscategorised.In [4], 66 % nuclei were found to be PSN.From this set-up, it should then be questioned if a more realistic PSN result, in the mentioned study, is somewhere between 56 % and 66 %.However, the 10 % miscategorisation is an upper-limit based on idealised parameters which are different from the actual experimental study.The exact percentage will depend on the ratio of nuclei to particles, the volume fractions, the shapes and the resolution of the measurements.It is suggested that simulations, as the present, but with sample specific parameters are of relevance to estimate the true PSN fraction found from experimental data.

2D versus 3D
In this section the accuracy of 2D characterisations of PSN is evaluated theoretically.The results presented here can be read as having a sample of a certain (cubic) size with a 'true' PSN efficiency found in 3D.The PSN values from areas chosen arbitrarily inside these volumes then represent PSN values found by characterising different numbers of polished surfaces.
In the first simulation set-up, the nuclei and particles are distributed randomly in space and the results are shown in figure 4. In figure 4a), the PSN quantifications in 3D are shown in the top colour gradient.The 3D quantifications are alternating between 30 % and 40 %, some of the alternations are attributed to the randomness of the simulation.It is expected that the true PSN value at these conditions is found in volumes of ∼ 50 3 voxels and above.When having a sample volume of 50 3 voxels, the number of 2D areas characterised changes the results of the PSN quantification significantly.Characterising 1, 5 and 10 areas of 50 2 pixels increase the PSN results from 9 % to 22 % to 27 %.All of which is indeed lower than the true 35 % PSN found in the 50 3 volume.When characterising areas in volumes larger than 125 3 voxels, the number of areas is less significant for the final 2D result.Following the colour gradient of 1 area, a high PSN efficiency of 29 % is seen at the volume of 75 3 voxels, where in the volume of 100 3 voxels the PSN percentage is found to be 19 %.Alternation is also seen when following the colour gradient of 5 and 10 areas, however to a smaller extend.This is an effect of the few nuclei and particles included in one or few areas which can create scattered results.This clearly illustrates the importance of a large 2D data set.
The effect of number of areas from a 100 3 voxel volume is shown in figure 4b).Here, the 3D PSN percentage is found to be 36 %.The blue points are the PSN percentages found from a certain number of areas in the volume.This figure illustrates that when having a volume of 100 3 voxels, characterising These results suggest that when choosing a random 2D surface within a bulk sample, which is often done in actual experiments [11], the PSN percentages may be underestimated compared to the true PSN percentage in a bulk sample.From this specific case, it is suggested that when doing 2D experiments an area of at least 10×10 4 μm 2 (100 2 pixels ×10 ) should be characterised.However, even when this is done, the true PSN percentage (in 3D) is ∼ 50 % larger than PSN percentage found in 2D.
It should be noted that these idealised results are very case specific and the results highly depend on the present set-up.The nuclei and particle shape, and the direction in which the sample is cut, might influence the PSN counts if the sample has equiaxed nuclei due to deformation [8].

All nuclei formed by PSN.
The above analysis, which is based on a random distribution of nuclei and particles, indicates that by 2D investigations it is very challenging to correctly estimate the PSN fraction.It is thus of interest to simulate the theoretical situation where all nuclei form at particles and then analyse how many PSN nuclei are found in 2D for different numbers of inspected surfaces.To do this, an equal number of nuclei and particles were used, and they all touch at least one of the opposite kind, giving PSN nuclei percentage of 100 %.Two dimensional areas were again characterised, the results from different number of area are presented in figure 5.This figure shows how the PSN percentage found in 2D increase significantly from 1 to 10 areas, and how the value still increases slightly from 20 to 60 areas.However, the PSN percentage found in 2D only reach ∼ 64 %.

Conclusion
In this paper, specific idealised cases of theoretical PSN were analysed by numerical simulations with the purpose of getting insight into the effect of artifacts when studying PSN in 2D and 3D.
To study the importance of statistical miscategorisation of PSN, volumes with randomly distributed spherical nuclei and particles were created.The volume fractions and sizes of nuclei and particles were varied.This specific set-up showed that few nuclei were miscategorised as PSN when the particle volume fraction is below 0.1 and that the miscategorisation of particles is significant at 0.1 volume fractions recrystallised and above.As the sizes of the nuclei and particles were varied, it was clear that the miscategorisation increased symmetrically as the size of nuclei and particles increased.This study clearly shows that miscategorisation of PSN nuclei and PSN particles can play a role in samples which are annealed to medium and high recrystallisation with large volume fractions or large sizes of second 8th International Conference on Recrystallization and Grain Growth Journal of Physics: Conference Series 2635 (2023) 012030 phase particles.However, the results highly depend on the parameters used for the specific numerical simulation.
The results from 2D and 3D theoretical PSN quantifications were compared.PSN was quantified in virtual volumes and in areas inside these volumes.Two situations were set up, first nuclei and particles were distributed randomly and second all nuclei were touching a particle.Both simulations showed that as more areas were quantified the 2D results would stagnate at a PSN percentage which was significantly lower than the PSN percentage found in 3D.These results uncover how 2D investigations may not correctly describe the PSN efficiency in a bulk material and support the assumption that 2D studies undervalue the PSN efficiency.

Figure 1 .
Figure 1.Volumes of 100 3 voxels with an area in the middle shown as dense.Red spheres are particles and blue are nuclei.a) Particles and nuclei are distributed randomly in a volume.The radius of the 100 nuclei is 3 pixels and the radius of the 20 particles is 5 pixels.b) All nuclei and all particles are touching at least one of the other kind, giving a PSN efficiency 100 %.There are 60 particles and 60 nuclei, these both have a radius of 4 pixels.

Figure 2 .
Figure 2. Miscategorisation of PSN dependent on volume fractions of nuclei (i.e.volume fraction recrystallised) and particles.The colours represent the miscategorised PSN percentage.The percentages are shown in the colour bar.The black points show which volume fractions were used.The value at each point was generated 4 times.a) Shows the PSN nuclei miscategorisation and b) shows the PSN particles miscategorisation.

Figure 3 .
Figure 3.The miscategorisation of PSN dependent on sizes of nuclei and particles.The colours represent the miscategorised PSN percentage.The percentages are shown in the colour bar.The black points show exactly which sizes were used.The value at each point was generated 4 times.a) Shows the PSN nuclei miscategorisation and b) shows the PSN particles miscategorisation.

Figure 4 .
Figure 4. Results of quantifying PSN nuclei in volumes and in areas.a) Shows the percentage of PSN nuclei found from a sample with volumes ranging from 25 3 to 150 3 voxels (x-axis) by characterising a number of areas or the whole volume (y-axis).Notice that the colour bar is ranging from 0 to 40 %.b) The 2D and 3D results for a 100 3 voxel volume.

Figure 5 .
Figure 5. 2D and 3D resutls for the situation where all nuclei are simulated to be next to particles.