Introduction

During the heat treatment of a polycrystalline material, microstructural evolution takes place through the competitive growth of grains due to the curvature-driven boundary migrations. This metallurgical phenomenon, i.e., grain growth, is of great importance to the development of superior materials, since the physical properties of polycrystals largely depend on the average and distribution of their grain sizes [1,2,3]. Thus, intensive studies have been undertaken to better understand and predict grain growth.

To experimentally observe grain growth, three-dimensional (3D) samples are usually investigated by examining their two-dimensional (2D) cross sections. However, the extent to which the 3D microstructural characteristics can be obtained from the cross sections is not clear. In the area of stereology [4], many efforts have been devoted to the analytical extraction of 3D microstructural information from cross-sectional observations. Meanwhile, some experimental studies have undertaken direct measurements of 3D grain growth using 3D characterization techniques, such as grain separating [5,6,7], serial sectioning [8,9,10,11], and X-ray microtomography [12, 13]. One of the most detailed investigations in this area was reported by Rhines et al. [8], who undertook very laborious experiments to measure and compare the temporal variations in 3D and cross-sectional grain structures. Significant discrepancies between the results for each dimension led them to conclude that 3D microstructures and their evolution kinetics cannot be inferred from cross sections, and instead must be determined in 3D space. On the validity or universality of this conclusion, several researchers have raised questions [1, 2, 14,15,16], suggesting that the annealing time used by Rhines et al. [8] may not have been long enough to remove the initial anisotropy present in the mechanically processed specimens. Currently, however, the theory remains controversial due to issues with the associated experiments, specifically the enormous amounts of labor required for the temporal observation of 3D microstructures, the challenges associated with maintaining the desired specimen conditions, and the issue of sampling bias arising from the limited analysis volume and spatial resolution [17].

Numerical simulation provides a helpful means of addressing the above issues, because it enables us to easily compare grain growth in 3D and its cross sections for arbitrarily controlled systems. In this regard, several numerical studies have been conducted using continuum-based grain growth models, such as the Monte Carlo model [18,19,20], vertex or front-tracking model [21,22,23,24], surface-evolver model [25], and phase-field model [26]. However, even for the simplest (or ‘ideal’) grain growth with isotropic grain boundary properties, a conclusive consensus has yet to be established. For instance, grain size distributions during the steady-state regime, in which the normalized size distribution is time-invariant, appearing in the literature exhibit large discrepancies; this is true for both 3D and cross-sectional size distributions. Moreover, while the Monte Carlo study by Anderson et al. [18] reported that cross sections through an evolving 3D grain structure can be well approximated by a fully 2D simulation, the vertex simulations conducted by Weygand et al. [23] showed clear dissimilarities between the cross-sectional and 2D grain growth. These discrepancies could be attributed to the limitations of the previous simulations [27], namely large noise and lattice anisotropy in the Monte Carlo simulations, grain coalescence caused by a limited number of grain orientations, and a statistically insufficient number of grains in the computational system, all of which prevent reliable grain growth statistics from being obtained.

In recent years, there has been great progress in grain growth simulation using the phase-field model: the multi-phase-field (MPF) models developed in several works [28,29,30,31,32,33] and the active parameter tracking (APT) algorithm adopted by three research groups [27, 34, 35] allowed for the accurate and effective computation of grain growth while avoiding the artificial coalescence of grains. In particular, the MPF model reported by Steinbach et al. [28, 31] exhibits quite a high degree of computational efficiency, since the grain boundary region is clearly defined by using the double-obstacle potential and, thus, the governing equation does not have to be solved outside of the boundary region. Furthermore, rapid developments in high-performance computing have been such that it is now easy to perform large-scale phase-field simulations. Some groups conducted phase-field grain growth simulations with several tens of thousands of grains by employing parallel-computing techniques [36,37,38,39]. Graphics processing units (GPUs), which were originally developed to effectively handle graphical output, have also proven to be extremely useful for handling computations on a greatly enlarged scale and have been successfully applied to computational materials science [40,41,42,43,44,45]. In our previous studies, by enabling massive parallel GPU computing on the GPU-rich supercomputer TSUBAME at the Tokyo Institute of Technology, it proved possible to achieve large-scale phase-field simulations of various solidification processes [46,47,48,49,50,51]. Recently, we applied a parallel GPU computing scheme to the MPF model of Steinbach and Pezzolla [31] and succeeded in simulating 3D ideal grain growth for large-scale polycrystalline systems with 25603 grid points and more than three million initial grains [52]. This computational scale enabled us to observe true steady-state growth with a statistically sufficient number of grains and to elucidate the statistical behaviors of the phenomenon.

In the present study, by focusing on ideal grain growth, we set out to elucidate the correlation between 3D and cross-sectional microstructures via large-scale phase-field simulations. We also wanted to address the question of whether the cross-sectional view of 3D grain growth can be fully approximated by 2D growth. The remainder of this paper is laid out as follows: first, “Simulation methodology” section describes the methodology of the grain growth simulations. Next, based on the simulated data, “Results and discussion” section examines grain growth in the 3D, cross-sectional, and 2D domains. The behaviors of the systems and individual grains with different dimensionalities are compared in detail from both morphological and kinetic viewpoints. Finally, in the same section, we discuss an analytical means of estimating the 3D grain structures from cross-sectional observations.

Simulation methodology

Multi-phase-field model

The MPF model proposed by Steinbach and Pezzolla [31] is employed for simulating the ideal grain growth. This model represents a polycrystalline system including N grains through N phase-field variables ϕi (i = 1, 2,…, N), which take a value of 1 in the ith grain and 0 in the other grains, smoothly changing from 1 to 0 at the grain boundaries. The sum of the phase fields at any spatial point in the system must be conserved as:

$$ \sum\limits_{i = 1}^{N} {\phi_{i} = 1} . $$
(1)

For ideal grain growth without additional driving forces other than the grain boundary energy, the total free energy of the system, F, can be expressed as:

$$ F = \int\limits_{V} {\,\sum\limits_{i = 1}^{N} {\,\sum\limits_{j = i + 1}^{N} {\,\,\left( { - \frac{{a_{ij}^{2} }}{2}\nabla \phi_{i} \cdot \nabla \phi_{j} + W_{ij} \phi_{i} \phi_{j} } \right)\,\,{\text{d}}V} } } , $$
(2)

where aij and Wij are the gradient coefficient and barrier height of the boundary between the ith and jth grains, respectively. The time-evolution equation of the phase field ϕi satisfying Eq. (1) is given for each spatial point by:

$$ \frac{{\partial \phi_{i} }}{\partial t} = \,\, - \frac{2}{n}\,\,\sum\limits_{j = 1}^{N} {s_{i} s_{j} M_{ij}^{\phi } \,\left( {\frac{\delta F}{{\delta \phi_{i} }} - \frac{\delta F}{{\delta \phi_{j} }}} \right)} \,\,, $$
(3)

where n is the number of nonzero phase fields at the point; si is a step function that takes a value of 1 if ϕi > 0 and otherwise vanishes; and \( M_{ij}^{\phi } \) is the phase-field mobility of the boundary between the ith and jth grains. The functional derivative of F [Eq. (2)] can be calculated as:

$$ \frac{\delta F}{{\delta \phi_{i} }} = \sum\limits_{k = 1}^{N} {\,\left( {\frac{{a_{ik}^{2} }}{2}\nabla^{2} \phi_{k} + W_{ik} \phi_{k} } \right)} . $$
(4)

Finally, the time-evolution equation reduces to:

$$ \frac{{\partial \phi_{i} }}{\partial t} = \,\, - \,\,\frac{2}{n}\,\sum\limits_{j = 1}^{N} {s_{i} s_{j} M_{ij}^{\phi } \sum\limits_{k = 1}^{N} {\,\left\{ {\frac{1}{2}\left( {a_{ik}^{2} - a_{jk}^{2} } \right)\nabla^{2} \phi_{k} + \left( {W_{ik} - W_{jk} } \right)\phi_{k} } \right\}} .} $$
(5)

\( M_{ij}^{\phi } \), aij, and Wij are related to the width (δ), energy (σij), and mobility (Mij) of the grain boundary according to the following equations:

$$ M_{ij}^{\phi } = \frac{{\pi^{2} }}{8\,\delta }M_{ij} ,\,\,\,\,a_{ij} = \frac{2}{\pi }\sqrt {2\delta \sigma_{ij} } \,,\,\,\,\,\,W_{ij} = \frac{{4\sigma_{ij} }}{\delta }. $$
(6)

Note that, for accurately performing phase-field simulations, a sufficient number of grid points must be contained within the boundary width δ to resolve the boundary regions. If necessary, δ can be set so as to reproduce a realistic grain boundary width (~ nm) by using fine numerical grids [53]. In the current simulations, however, δ is treated as an artificial quantity and used only to implicitly track the grain boundary positions by the regions where nonzero phase fields coexist. Here, we set δ to six times the grid spacing, which has been reported to be a good compromise between computational accuracy and cost [27].

Computational conditions

To elucidate the similarities or dissimilarities between 3D, cross-sectional, and 2D ideal grain growth, we performed both 3D and 2D simulations. Figure 1a, c shows the computational systems employed for simulating 3D and 2D ideal grain growth, which consist of 3125000 and 3000000 initial grains, respectively, under periodic boundary conditions. Here, the grains are distinguished according to their colors. The cross sections perpendicular to each axis of the 3D system contain an average of approximately 26000 grains per section. An example of the sections (at the center of the z axis) is shown in Fig. 1b. Using square or cubic regular grids, the domains were divided into 2560 × 2560 × 2560 grid points for the 3D system and into 32000 × 32000 grid points for the 2D system. The time increments (Δt3D for the 3D system and Δt2D for the 2D system), grid spacing (Δx), grain boundary energy (σ), and mobility (M) were set as follows, being non-dimensionalized with typical scales of 1 s, 10−6 m, 1 J m−2, and 10−12 m4 J−1 s−1, respectively: Δt3D = 0.075, Δt2D = 0.1125, Δx = 1, σ = 1, and M = 1. To guarantee the statistical reliability of the results, three replicated simulations with different initial structures were performed for both the 3D and 2D cases. For each simulation run, the initial grain structures, of a random shape, were created by growing randomly distributed nuclei using a constant driving force. Here, note that the numerical accuracy of phase-field grain growth simulation is generally determined by the relative grain size compared to the grain boundary width δ, in addition to the grid resolution of grain boundaries (i.e., the number of grids contained within δ) [27]. In the current MPF simulations, δ was set to 6Δx for both the 2D and 3D systems; further, the ratio of the initial average grain radius, 〈R(0)〉, to δ was almost same (〈R(0)〉/δ ≈ 1.7) in both the cases. Therefore, the comparison of the 2D and 3D simulations is fair. We also note that although the above computational conditions for the 3D system were basically same as those used in our previous study [52], some new simulations were performed as part of the present study to analyze the cross-sectional microstructures in detail. The initial structures for the present simulations differ from those used in the previous study [52].

Figure 1
figure 1

Polycrystalline systems used for simulating 3D and 2D ideal grain growth. a 3D system with 2560 × 2560 × 2560 grid points and 3125000 initial grains. b Cross-sectional plane of (a) at the center of the z axis. c 2D system with 32000 × 32000 grid points and 3000000 initial grains. The light panel in (c) shows the overall computational domain, while the right one displays part (2560 × 2560 points) of the overall domain indicated by the blue frame. In all the panels, grains are distinguished by different colors

Full size image

To perform the large-scale simulations, we utilized our own CUDA C code [52] that was specifically developed for parallel GPU computation. The code decomposes an entire computational domain into small subdomains, each of which is assigned to one GPU. The connection of the boundary data of the GPUs is performed via their host CPUs [46], while the internode communication is implemented using the message-passing interface (MPI). To reduce the memory requirements, the APT algorithm [27, 34, 35] proposed by Kim et al. [27] is used, storing only nonzero phase-field variables; the maximum number of stored variables at each grid point is set to seven. All computations were carried out on the GPU-rich supercomputer TSUBAME at the Tokyo Institute of Technology, using 200 GPUs (NVIDIA Tesla P100) for the 3D simulations and 64 GPUs (NVIDIA Tesla K20X) for the 2D simulations.

Results and discussion

Using the above-mentioned computational conditions, we performed 3D simulations for a period of 75000Δt3D (i.e., 5625 in dimensionless time), after which around 20000 grains remained in each system. The required computation time per simulation was about 1 day. The 2D simulations were performed over a longer duration of 200000Δt2D = 22500, since the progress of the 2D grain growth is much slower than that of the 3D growth. After the simulations, roughly 40000 grains remained in each system.

Figure 2 depicts the evolved microstructures obtained for the 3D, cross-sectional, and 2D domains after the simulations, starting from the initial structures shown in Fig. 1. For reference, movies of the evolution processes are provided in the Supplementary Material. In all the panels of Fig. 2, we can see that the grains generally exhibit equi-axed shapes, as would be expected for microstructures formed through ideal grain growth. Below, we present a detailed analysis and discussion of the grain growth with different dimensionalities based on the simulated data. Note that, hereafter, the radius R of a 3D grain, R3D, is defined as its volume (V)-equivalent radius, \( \sqrt[3]{3V/(4\pi )}; \) those of a cross-sectional grain, R3DX, and 2D grain, R2D, are determined from their area (A)-equivalent radius, \( \sqrt {A/\pi } \).

Figure 2
figure 2

Evolved microstructures in 3D and 2D ideal grain growth simulations starting from the initial structures shown in Fig. 1. a 3D microstructure at t = 75000Δt3D. b Cross-sectional plane at the center of the z axis. c 2D microstructure at t = 200000Δt2D

Full size image

Determining the grain characteristics

To analyze the microstructures obtained from phase-field simulations, two different methods have been used, namely simple summation [27] and cluster enumeration [54]. In the simple summation, the volume Vi (or area Ai for 2D) of a given grain i is calculated by simply integrating the corresponding phase field ϕi:

$$ V_{i} = \int\limits_{V} {\phi_{i} \,{\text{d}}V} . $$
(7)

On the other hand, the cluster enumeration evaluates the Vi value based on the binarization processing:

$$ V_{i} = \int\limits_{V} {O_{i} \,{\text{d}}V} , $$
(8)

where Oi (r) is an operator that takes a value of 1 if ϕi has the largest value of all the phase fields at a spatial point r. Otherwise, it vanishes.

Herein, to test the applicability of the above methods to the analysis of 3D, cross-sectional, and 2D grain structures, we calculated the grain size distributions of the simulated microstructures shown in Fig. 2 using both methods. The results are shown in Fig. 3, as normalized grain size histograms (R/〈R〉, where 〈R〉 denotes the average grain radius) which were built using a typical bin width (0.1). In the analysis of the cross-sectional grain structure, we took 20 evenly spaced planes along each coordinate axis (i.e., 60 planes in total) and sampled all the grains appearing on those planes. The same is assumed in the remainder of this paper. Figure 3a shows that the simple summation and cluster enumeration give almost the same results for the evaluation of the 3D grain size distribution. Therefore, both methods are regarded as being valid for this application. In Fig. 3c, we can see slight differences between the analyzed 2D size distributions, in terms of their broadness and peak height. However, the shapes of the distributions are similar, with the difference in the peak heights being minimal (around 5%). In contrast, as shown in Fig. 3b, the results obtained for the cross sections exhibit significant differences: while the grain size distribution obtained from the cluster enumeration produces a smooth curve, that from the simple summation suddenly jumps at the smallest grain size class. This sudden jump is probably a result of the diffuse nature of the phase-field model. That is, when the binarization processing is not applied, a large amount of very small grains, cut near the edges of the diffuse interfaces, can appear in the cross sections. Indeed, a jump in the cross-sectional grain size distribution is not observed in the results obtained with the sharp interface models, including the vertex and front-tracking models [22,23,24].

Figure 3
figure 3

Grain size distributions obtained for microstructures shown in Fig. 2 using simple summation and cluster enumeration. a 3D grains. b Cross sections of 3D grains. c 2D grains. In the analysis of the cross-sectional grain size distributions, a total of 60 evenly spaced planes are taken and all grains appearing on the planes are sampled

Full size image

Based on these results, for the analysis of cross-sectional microstructures described below, we mainly used cluster enumeration; simple summation was used only to calculate the time derivative of the individual grain sizes (dR/dt) in Figs. 7 and 8. On the other hand, the 3D and 2D microstructures were analyzed by simple summation, in the same way as in our previous studies [52, 55].

Statistical aspects of grain growth

This section investigates the statistical behaviors of ideal grain growth with different dimensionalities. First, Fig. 4 shows the temporal variations in the grain size distribution for the 3D, cross-sectional, and 2D domains. The numerical data points are given as the average of three replicated simulations with error bars representing the standard deviation, which is the same in the following figures. Most of the error bars are smaller than the symbols, confirming a high degree of statistical reliability of the results. In Fig. 4, we can see that all the grain size distributions rapidly become broader in the early stages of grain growth. During the later stages, however, the changes in the grain size distribution are rather small. In particular, in the period of t ≥ 35000Δt3D for 3D and the cross sections and that of t ≥ 100000Δt2D for 2D, almost no changes are observed, while the statistical errors are very small, indicating that the systems have reached their steady states. If, in Fig. 4, we compare (a) and (c), we find that the 3D and 2D steady-state size distributions exhibit a similar tendency in terms of their peak height and position, as well as their symmetry. A visible difference between the 2D steady-state size distribution and that of the 3D is the existence of a plateau around the peak in the 2D case, which is consistent with previously reported results [23, 24, 27]. In Fig. 4b, the steady-state shape of the cross-sectional grain size distribution exhibits characteristics that are quite different from those of the 3D and full 2D size distributions, in that it has a lower peak height and right-skewed shape.

Figure 4
figure 4

Temporal variations in grain size distribution for a 3D grains, b cross sections of 3D grains, and c 2D grains. Numeric data points are given as the average of three replicated simulations, with error bars representing the standard deviation. Most of the error bars are smaller than the symbols

Full size image

Here, we should note the Monte Carlo study of ideal grain growth performed by Song et al. [19]; they reported that the transient time required to reach the steady state is much longer in 3D than in the cross sections and, thus, the achievement of the steady state in 3D cannot be inferred from cross-sectional observations. In our simulations, however, there is no significant difference in the duration of the transient time in the 3D and cross-sectional growth (around 35000Δt3D in both cases). This discrepancy is probably a consequence of the computational scale that we employed (a grain number that is 750 times greater than that used by Song et al. [19]), as well as the large amount of noise in the Monte Carlo simulation. Provided the initial grain structure is fully homogeneous, and the analyzed data are free of statistical bias, we believe that the achievement of the steady state in 3D ideal grain growth can be judged from the cross sections.

Further, it is worth mentioning that the symmetric grain size distribution obtained from our 2D MPF simulations, though consistent with other continuum-based simulation results [23, 24, 27], is largely different from the lognormal-like distribution observed in 2D atomistic simulations [56] and experiments for thin metal films [57]. One of the possible sources of this deviation is miscounting of the grains in atomistic simulations and experiments, which is inevitable due to the difficulty in detecting low-angle grain boundaries [58,59,60]. Another possible source is some grain growth-related factor not considered in the current simulations for the ideal grain growth. For example, previous grain growth simulations [54, 55, 61,62,63] demonstrated that grain rotation [61, 62], anisotropic boundary properties [54, 55, 62], and triple junction drag [63] can cause the asymmetry in grain size distribution. In addition, although the conventional MPF models including that employed here can accurately handle the ideal grain growth [27, 30, 64], Tóth et al. [65] pointed out that these models do not fully satisfy the criteria for mathematical and physical consistency. They also simulated 2D grain growth using a fully consistent MPF model [65] and observed more asymmetric grain size distributions compared to those predicted by other MPF models. Further investigation of the origins of the contradictory results will be addressed elsewhere.

Next, we examined the kinetics of the systems. Figure 5 shows the temporal variations in the squared average grain size (〈R〉2) for each dimension, with all error bars being smaller than 0.2% of the mean. Fitted curves for the plots were calculated and are depicted in the figure using the well-known parabolic law of steady-state grain growth [66, 67]:

$$ \langle R(t)\rangle^{2} - \langle R(t_{0} )\rangle^{2} = KM\sigma \left( {t - t_{0} } \right), $$
(9)

where t0 is an initial time and K is the kinetic coefficient. The K values are obtained as 0.478, 0.334, and 0.282 for 3D, the cross sections, and 2D, respectively, by applying the least-square fitting to the steady-state regimes (t ≥ 35000Δt3D in 3D and the cross sections; t ≥ 100000Δt2D in 2D). Figure 5 reveals that 〈R〉2 increases linearly with time in accordance with the parabolic law in all cases. However, the obtained values of K differ greatly in each case, showing that the grain growth in 3D proceeds most rapidly with K = 0.478, followed by those in the cross sections (K = 0.334) and 2D (K = 0.282).

Figure 5
figure 5

Temporal variations in average grain size for a 3D and cross-sectional grains and b 2D grains. In both panels, numeric data points are given as the average of three replicated simulations. Error bars (standard deviations) are all smaller than the symbols

Full size image

The above results allow us to conclude that the size distribution and evolution kinetics of the cross-sectional grains are clearly different from those of fully 2D grains. Furthermore, compared to the 3D space, a higher fraction of relatively small grains and a smaller average size are observed for the cross sections. This difference between the 3D and cross-sectional microstructures probably arises from the fact that the 3D grains can be cut such that they are apparently smaller than their true 3D sizes (i.e., stereological effect [4]). This can be confirmed in Fig. 6, in which we plot the relationship between the cross-sectional and 3D grain sizes in the steady-state regime (at t = 75000Δt3D). The black diamonds indicate the average 3D grain sizes for each cross-sectional size class obtained by averaging three replicated simulations. The light-gray circles indicate the data for individual grains, with 5000 randomly sampled grains from the three simulations being plotted to better visualize the density of the data points. In Fig. 6, we can see that most of the grains have 3D sizes that are larger than their cross-sectionally measured sizes, clearly confirming that the stereological effect described above applies in the case of ideal grain growth.

Figure 6
figure 6

Relationship between cross-sectional and 3D grain sizes at t = 75000Δt3D. Black diamonds indicate the average 3D grain sizes for each cross-sectional size class obtained by averaging three replicated simulations. The error bars (standard deviations) are all smaller than the symbols. The light-gray circles indicate the data for individual grains, with only 5000 grains randomly sampled from the three simulations plotted to better visualize the density of the data points

Full size image

Individual grain behaviors

An essential component of the analytical models of grain growth is a description of the growth kinetics of individual grains [2, 3, 67,68,69]. However, for the cross sections of 3D microstructures, the behavior of the individual grains has not been investigated to date. Herein, focusing mainly on the cross-sectional grains, we examine the individual grain behaviors and test the applicability of the classical grain growth models.

Hillert [67] proposed a mean-field approximation for both the 3D and 2D grain growth, describing the growth rate of a given grain as a function of its size:

$$ R\frac{{{\text{d}}R}}{{{\text{d}}t}} = gM\sigma \left( {\rho \frac{R}{\langle R\rangle } - 1} \right), $$
(10)

where g is a geometrical factor that takes a value of 1 for 3D and 0.5 for 2D; ρ is the mean-field parameter given as 〈R〉2/〈R2〉 for 3D and 1 for 2D, where 〈R2〉 denotes the average squared grain radius. For fully 2D systems, the von Neumann–Mullins law [68, 69] is well known as a mathematically rigorous formula for individual grain behaviors, which describes the growth rate of a given grain as a function of its topology (number of edges per grain, ne):

$$ R\frac{{{\text{d}}R}}{{{\text{d}}t}} = \frac{1}{6}M\sigma \left( {n_{\text{e}} - 6} \right). $$
(11)

We tested the applicability of Eqs. (10) and (11) to cross-sectional grain growth by comparing them with the simulated data in the steady-state regime (at t = 75000Δt3D); the obtained results are given in Figs. 7 and 8, respectively. For comparison, the results for the steady-state 3D and 2D growth are also shown in the figures. Here, since a direct comparison of 3D grain growth behaviors to Eq. (11) is not straightforward, the simulated data for the 3D grains are plotted only in Fig. 7.

Figure 7
figure 7

Growth rates of individual grains as functions of their sizes. a 3D grains. b Their cross sections at t = 75000Δt3D. c 2D grains at t = 200000Δt2D. The dark-colored diamonds indicate the average for each size class obtained by averaging three replicated simulations, with the error bars representing the standard deviation. Most of the error bars are smaller than the symbols. The light-colored circles indicate the data for individual grains. To better visualize the density of the data points, only 5000 grains, randomly sampled from the simulations, are plotted in each panel

Full size image
Figure 8
figure 8

Growth rates of individual grains as functions of their topologies. a Cross-sectional grains in 3D system at t = 75000Δt3D. b 2D grains at t = 200000Δt2D. The dark-colored diamonds indicate the average for each topological class obtained by averaging three replicated simulations, with the error bars representing the standard deviation. Most of the error bars are smaller than the symbols. The light-colored circles indicate the data for individual grains. To better visualize the density of the data points, only 5000 grains, randomly sampled from the simulations, are plotted in each panel

Full size image

As can be seen in Fig. 7a, b, in 3D space there is an almost linear relationship between the growth rate of individual grains and their sizes following the mean-field approximation [Eq. (10)]. Meanwhile, the rate-size relationship in the cross sections of the 3D space is not monotonic, with the growth rate values exhibiting a large dispersion for grains of the same size. This difference may be attributable to the stereological effect that was discussed above (“Statistical aspects of grain growth” section and Fig. 6), as a result of which the cross-sectional grains appear to have radii that are different from their true 3D values. Since the kinetics of grains in a 3D system are essentially determined by their 3D quantities, the lack of a one-to-one relationship between the 3D and cross-sectional grain sizes can lead to a high degree of randomness in the individual grain behaviors in the cross sections.

In Figs. 7c and 8b, the growth kinetics of the fully 2D grains are observed to conform to both the mean-field approximation [Eq. (10)] and the von Neumann–Mullins law [Eq. (11)]; in particular, as can be confirmed from Fig. 8b, the correlation between the growth rate and grain topology in a 2D system is quite strong, exhibiting excellent agreement with the von Neumann–Mullins law. A band-like distribution of the data points observed in the RdR/dt versus R2D/〈R2D〉 plane (Fig. 7c) can be understood as a consequence of the von Neumann–Mullins law. For instance, most of the grains located along a horizontal line RdR/dt = 0 in Fig. 7c belong to the same topological class, ne = 6. In contrast to the case of fully 2D grains, the relationship between the growth rate and topology for cross-sectional grains, shown in Fig. 8a, exhibits a large scatter for those grains belonging to the same topological class. In addition, the curve of the average growth rates for each topological class cannot be described well using the von Neumann–Mullins law. It is clear, therefore, that there is a fundamental difference between full 2D grain growth and the cross-sectional view of 3D growth.

Estimating the 3D grain structure from cross-sectional observations

As demonstrated above, the microstructural characteristics observed in the cross sections are apparently different from those in 3D space due to the stereological effect. In addition, the cross-sectional grain growth cannot be approximated by full 2D growth, in terms of both statistical and individual grain behaviors. Hereafter, based on the simulated results and the stereology, we discuss the estimation of the 3D microstructural characteristics from cross-sectional observations.

First, we focus on the average grain size. From stereological considerations, Fullman [70] showed that if 3D grains are approximated by spheres of a uniform size, the relationship between the 3D average grain size, 〈R3D〉, and cross-sectional average size, 〈R3DX〉, can be expressed as:

$$ \langle R_{{3{\text{D}}}} \rangle = \sqrt {\frac{3}{2}} \langle R_{{ 3 {\text{DX}}}} \rangle \,\, \approx 1.225\langle R_{{ 3 {\text{DX}}}} \rangle . $$
(12)

We tested the applicability of this relationship to ideal grain growth by calculating the average size ratio, 〈R3D〉/〈R3DX〉, using the simulated data shown in Fig. 5a. The results are shown in Fig. 9. We can see that the 〈R3D〉/〈R3DX〉 value suddenly declines over a short duration. Subsequently, the variation in 〈R3D〉/〈R3DX〉 occurs very slowly and takes an almost constant value of around 1.2 in the steady state (t ≥ 35000Δt3D), exhibiting good agreement with the estimation given by Eq. (12). Given that the attainment of the steady state in a 3D space can be judged from the cross sections (see “Statistical aspects of grain growth” section), we can say that the 3D average grain size can be easily estimated with considerable accuracy based only on cross-sectional observations.

Figure 9
figure 9

Temporal variations in the ratio of the average grain sizes in 3D and cross sections, as calculated from the data shown in Fig. 5a

Full size image

Next, we discuss the prediction of the 3D grain size distribution. In the field of stereology, the Saltykov–Schwartz method [71, 72] is frequently employed to convert cross-sectional and 3D grain size distributions from one to the other [4, 73,74,75,76]. Here, we briefly describe this method and propose some modifications. First, let us consider the histograms for 3D and cross-sectional grain size distributions, discretized by m bins. The frequency of the ith size class (within an interval of R3D,i-1– R3D,i) in 3D is denoted as Fi; that in the cross sections (within R3DX,i-1– R3DX,i) is denoted as fi. Considering that a cross section of a given size can arise from 3D grains of various sizes, as demonstrated in Fig. 6, fi (i = 1, 2,…, m) can be expressed by a weighed sum of the series of Fi, that is, as:

$$ \left\{ {\begin{array}{*{20}l} {\,f_{1} = \alpha_{11} F_{1} + \alpha_{12} F_{2} + \cdots + \alpha_{1m} F_{m} ,} \hfill \\ {\,f_{2} = \alpha_{21} F_{1} + \alpha_{22} F_{2} + \cdots + \alpha_{2m} F_{m} ,} \hfill \\ {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \vdots } \hfill \\ {\,f_{m} = \alpha_{m1} F_{1} + \alpha_{m2} F_{2} + \cdots + \alpha_{mm} F_{m} ,} \hfill \\ \end{array} } \right. $$
(13)

where αij is the transformation matrix defined by:

$$ \alpha_{ij} = \int\limits_{{R_{{ 3 {\text{DX}},\,i - 1}} }}^{{R_{{ 3 {\text{DX}},\,i}} }} {} \int\limits_{{R_{{ 3 {\text{DX}},\,i - 1}} }}^{{R_{{ 3 {\text{DX}},\,i}} }} {c\frac{{R_{{ 3 {\text{D}}}} }}{{R_{{3{\text{D}},m}} }} \times \,\,p\,(R_{{3{\text{D}}}} ,\,\,R_{{ 3 {\text{DX}}}} )} \,\,{\text{d}}R_{{3{\text{D}}}} {\text{d}}R_{{ 3 {\text{DX}}}} , $$
(14)

where cR3D/R3D,m indicates the relative probability of cutting a 3D grain of size R3D by a plane, with c being a normalizing constant determined later; p(R3D, R3DX) indicates the probability with which a cross section of size R3DX appears as a result of cutting a 3D grain of size R3D. Using the αij matrix, 3D and cross-sectional grain size distribution histograms can be converted to one another:

$$ f_{i} = \,\alpha_{ij} F_{j} \,\,\,\,\,(i,\,\,j = 1,\,\,2,\,\, \ldots ,\,\,m), $$
(15)
$$ F_{j} = \alpha_{ij}^{ - 1} f_{i} \,\,\,\,\,(i,\,\,j = 1,\,\,2,\,\, \ldots ,\,\,m). $$
(16)

After conversion, the normalizing constant c in Eq. (14) is determined such that the total of the calculated frequencies is equal to 1 (or 100%).

The original Saltykov–Schwartz method models a 3D grain as a sphere. In this case, p(R3D, R3DX) in Eq. (14) is rigorously derived, as follows:

$$ p\,(R_{{3{\text{D}}}} ,\,\,R_{{ 3 {\text{DX}}}} ) = \frac{{R_{{ 3 {\text{DX}}}} }}{{R_{{ 3 {\text{D}}}} \sqrt {R_{{3{\text{D}}}}^{2} - R_{{3{\text{DX}}}}^{2} } }}. $$
(17)

The spherical model gives a relatively simple formula. However, it has been reported that this model is likely to incorrectly estimate the frequency of relatively small grains [77]. Therefore, we propose a modified model which approximates a grain as a Kelvin tetrakaidecahedron, which is well known as a space-filling polyhedron with a minimal surface area and is therefore often employed as a grain model [78,79,80,81,82]. Although a mathematically rigorous expression of p(R3D, R3DX) for the tetrakaidecahedron has not yet been established, Matsuura and Itoh [7] evaluated the size distribution of the cross sections of this polyhedron by cutting it with 5000 arbitrary planes via numerical calculations. We depict their results in Fig. 10, comparing them with the analytical curve for a sphere [Eq. (17)], from which we can see that the cutting of the tetrakaidecahedron generates small cross sections with a higher probability than that for a sphere. We created an approximate formula for the tetrakaidecahedron as a weighted sum of the spherical-model function and uniform distribution function:

$$ p\,(R_{{3{\text{D}}}} ,\,\,R_{{ 3 {\text{DX}}}} ) = w\frac{{R_{{ 3 {\text{DX}}}} }}{{R_{{ 3 {\text{D}}}} \sqrt {R_{{3{\text{D}}}}^{2} - R_{{3{\text{DX}}}}^{2} } }} + \left( {1 - w} \right)\frac{1}{{R_{{ 3 {\text{D}}}} }}, $$
(18)

where w is a weighting parameter, being obtained as w = 0.79 by least-square fitting to the data reported by Matsuura and Itoh (Fig. 10).

Figure 10
figure 10

Size distributions of cross sections of 3D objects. The data points for the tetrakaidecahedron indicate the numerical calculation results reported by Matsuura and Itoh [7]

Full size image

When a sufficiently large number of grains are sampled, the maxima of the 3D and cross-sectional grain sizes are almost the same value [4]. By setting the maximum sizes in each dimensional grain size histogram to the same value, the transformation matrix αij [Eq. (14)] with the spherical model [Eq. (17)] can be easily calculated by Cruz-Olive numerical integration [83,84,85]:

$$ \alpha_{ij} = \left\{ {\begin{array}{ll} {\frac{c}{m}\left\{ {\sqrt {\left( {j - \frac{1}{2}} \right)^{2} - \left( {i - 1} \right)^{2} } - \sqrt {\left( {j - \frac{1}{2}} \right)^{2} - i^{2} } } \right\}} \hfill & {{\text{for}}\,i < j,} \hfill \\ {\frac{c}{m}\sqrt {\left( {j - \frac{1}{2}} \right)^{2} - \left( {i - 1} \right)^{2} } = \frac{c}{m}\sqrt {i - \frac{3}{4}} } \hfill & {{\text{for}}\,i = j,} \hfill \\ 0 \hfill & {{\text{for}}\,i > j.} \hfill \\ \end{array} } \right.$$
(19)

Similarly, with the tetrakaidecahedral model [Eq. (18)] αij is obtained as:

$$ \alpha _{{ij}} = \left\{ {\begin{array}{ll} {\frac{c}{m}\left[ {w\left\{ {\sqrt {\left( {j - \frac{1}{2}} \right)^{2} - \left( {i - 1} \right)^{2} } - \sqrt {\left( {j - \frac{1}{2}} \right)^{2} - i^{2} } } \right\} + \left( {1 - w} \right)} \right]} \hfill & {{\text{for}}\,i < j,} \hfill \\ {\frac{c}{m}\left\{ {w\sqrt {i - \frac{3}{4}} + \frac{1}{2}\left( {1 - w} \right)} \right\}} \hfill & {{\text{for}}\,i = j,} \hfill \\ 0 \hfill & {{\text{for}}\,i > j.} \hfill \\ \end{array} } \right. $$
(20)

By applying the above stereological method to the simulated data, we tested the possibility of the interconversion of the 3D and cross-sectional grain size distributions in ideal grain growth. Here, the 3D and cross-sectional grain size histograms were built from the simulations during the steady state (at t = 75000Δt3D) using a typical bin number of 15; the maximum grain size in the histograms was set to 120Δx, which corresponds to the maximum size of cross-sectional grains at the above time. The results are given in Fig. 11. The colored bars show the simulated results as the average of three replicated simulations. The open circles indicate the 3D/cross-sectional size distributions estimated from the cross-sectional/3D simulated data using the spherical-model-based transformation matrix [Eq. (19)]. The filled diamonds denote those using our proposed tetrakaidecahedral-model-based matrix [Eq. (20) with w = 0.79]. As shown in Fig. 11, the spherical model incorrectly estimates the frequency of the relatively small grains, especially regarding the prediction of the 3D grain size distribution from the cross sections. In contrast, the tetrakaidecahedral model gives rather smooth curves for both the 3D and cross sections, closely matching the simulated data.

Figure 11
figure 11

Estimation of a 3D and b cross-sectional grain size distributions from the measured data in the other dimension. The colored bars show the measured results for the steady-state grain growth (at t = 75000Δt3D) as the average of three replicated simulations. The open circles indicate the 3D/cross-sectional size distributions, estimated from the cross-sectional/3D measured data using the spherical-model-based transformation matrix [Eq. (19)] that is based on the original Saltykov–Schwartz method [71, 72]; the filled diamonds denote those using the tetrakaidecahedral-model-based matrix [Eq. (20) with w = 0.79] proposed in the present study

Full size image

The results presented above allow us to conclude that the average size of the 3D grains in the case of ideal grain growth can be inferred from their cross sections, in line with the classical Fullman theory. Moreover, for the stereological prediction of 3D grain size distribution, although the original Saltykov–Schwartz method based on the spherical grain model exhibits an apparent inaccuracy, its modification using our proposed tetrakaidecahedral model allows for accurately interconverting the 3D and cross-sectional size distributions. Since the average and distribution of 3D grain sizes are dominant factors affecting the physical properties of materials, our findings will be helpful to the application of cross-sectionally observed data to materials design.

Conclusions

The present study was conducted to elucidate the correlations between 3D and cross-sectional microstructures during ideal grain growth. We also attempted to reveal whether the cross-sectional view of 3D grain growth can be modeled by fully 2D grain growth. To this end, 3D and 2D large-scale MPF simulations with approximately three million initial grains were performed using multiple GPUs on a supercomputer. This computational scale allowed us to evaluate 3D, cross-sectional, and 2D grain growth with a reasonably high statistical reliability. Based on the simulated results, we discussed how to estimate the microstructural characteristics of 3D grains from those observed in the cross sections. The main findings are summarized as follows:

  1. 1.

    While the steady-state grain size distributions in 3D and 2D show similar symmetrical shapes, that in cross sections exhibits unique characteristics in terms of its right-skewed shape and lower peak height.

  2. 2.

    The temporal variations in the average grain size follow the well-known parabolic law for all of the 3D, cross-sectional, and 2D cases. However, the coefficient of the parabolic law is closely related to the dimensionality.

  3. 3.

    For the 3D and 2D cases, the growth kinetics of individual grains can be described by the mean-field approximation or the von Neumann–Mullins law. In contrast, the individual grain kinetics in the cross sections exhibit a high level of randomness and are not a good match for the above classical grain growth models.

  4. 4.

    The average grain size in the 3D case during steady-state growth can be estimated as being approximately 1.2 times that observed in cross sections, in line with Fullman’s stereological prediction.

  5. 5.

    The steady-state 3D and cross-sectional grain size distributions can be interconverted by using our modified Saltykov–Schwartz method that is based on the tetrakaidecahedral grain model.

This study examined the fundamental aspects of cross-sectional grain behaviors by focusing on ideal grain growth. However, it should be emphasized that real materials often contain a considerable amount of structural anisotropy arising from deformation, chemical driving forces, and anisotropic grain boundary properties. This anisotropy may affect the relationship between the microstructural characteristics in 3D and cross sections and could be required to be considered for the practical application of the present stereological predictions. Thus, our future work will study 3D and cross-sectional grain growth in more realistic systems, while quantifying the degree to which they are affected by the anisotropy.