Prediction of Nanoscale Grain-Boundary Migration in Pure Aluminum Through Assimilation of Time-Resolved Tem-Sped Orientation Maps

1. Introduction

The microstructure of metallic materials strongly influences their mechanical properties, including strength, formability and mechanical anisotropy [1]. Heat treatment is one of the most widely used methods for controlling these properties through tuning microstructural characteristics, such as grain size distribution. The approach has been applied not only to metallic materials processed through conventional manufacturing routes but also to materials produced by advanced techniques, including laser powder bed fusion, which often generates heterogeneous as-built microstructures [2]. To design efficient heat-treatment process, both microstructure characterization and rational simulation are crucial.

Scanning precession electron diffraction (SPED) in transmission electron microscopy (TEM) is well suited to observe the microstructure at the nanometer scale. In SPED, the incident electron beam is precessed around the optic axis at a fixed angle during acquisition. As a result, the recorded diffraction pattern represents an integration over a range of beam incidence directions, substantially reducing dynamical diffraction effects and yielding diffraction patterns that are closer to the kinematical condition [3,4]. Therefore, crystal orientation can be determined at each sampling position by template matching of diffraction spot patterns rather than of Kikuchi bands. This approach enables reliable orientation indexing even in heavily deformed materials [5,6]. Because the measurement is non-destructive, the same area can be re-measured after successive annealing, producing a time-resolved record of boundary migration at the nanoscale.

For predicting the boundary migration during heat treatment it is more effective to integrate the observation results with computational methods than to rely solely on the observation. Multiphase-field (MPF) modeling is widely used to predict microstructural evolution during heat treatment, including dynamic and static recrystallization as well as grain growth [7,8,9,10]. In MPF models, the microstructure is represented by continuous order parameters, allowing topological changes such as grain shrinkage and disappearance to be handled without explicitly tracking the interfaces [9,11]. However, some input parameters are difficult to determine experimentally. For example, the grain boundary mobility, which governs the rate of grain boundary migration in MPF simulations, depends predominantly on temperature and the character of individual grain boundaries [1,12]. Identifying all grain boundary mobilities within an observed area under a given experimental condition is not always possible. Therefore, a rational and practical method for estimating these parameters from experimental observation is required.

Data assimilation (DA) techniques provide an effective method by connecting observation results and simulations. One of the data assimilation techniques, ensemble Kalman filter (EnKF) [13,14], approximates the probability density function of the system state using a finite ensemble, and is therefore applicable to high-dimensional nonlinear systems. This method has been applied to phase-field simulations of grain growth, where anisotropic grain boundary properties were inferred in synthetic microstructure models [15,16]. The other method, Bayesian data assimilation based on an ensemble square root filter, has also been coupled to an MPF model and in situ SEM-EBSD observation of static recrystallization in an aluminum alloy, to estimate the stored-energy distribution of the deformed texture [10]. Although SEM-EBSD records back scattered signals from the specimen surface, the observed surface is influenced not only by the grains at surface but also by the grains beneath the surface. TEM-SPED, on the other hand, records nanoscale electron diffraction patterns from electron-transparent specimens. In most cases, grains do not overlap along the beam direction. Therefore, heat treatment behavior observed by TEM-SPED is likely governed by interactions among visible grains. Considering that TEM-SPED can acquire crystal orientation at the nanoscale, incorporating TEM-SPED data into MPF models would be effective for accurately estimating heat treatment behavior. However, data assimilation methods for SPED have not yet been established.

In this study, we performed data assimilation using SPED datasets and discussed the accuracy of inferred parameters. The same field of view in a 50 % cold-rolled pure aluminum was characterized by TEM-SPED before annealing and after four successive 5min heat treatments at 400 °C. The resulting grain-label maps were assimilated into a multiphase-field model ensemble Kalman filter. Grain boundary mobilities to reproduce the experimentally observed microstructural evolution were subsequently evaluated.

2. Materials and Methods

2.1. Specimen Preparation

A 50 % cold rolled pure aluminum (A1050, Al ≥ 99.5 wt.%) was used as a specimen. Deformation of this magnitude introduces high dislocation density, whose redundant elastic strain energy is released on subsequent annealing [1]. The grain boundary migration observed in this study was therefore driven principally by the stored strain energy.

The specimen was mechanically polished to the thickness of less than 0.1 mm. A disc punch (Gatan, Inc.) was used to obtain a disc specimen with 3 mm in diameter. The specimen was subsequently thinned by twin-jet electropolishing (E.A. Fischione Instruments, Inc.) in a 3:7 mixture of HNO₃ and CH₃OH at −30 °C, 25 V and 40 mA. A reference mark was made on the rim of disk to identify the rolling direction when the specimen was loaded onto a TEM holder. The polished disc is shown in Figure 1(a) and the summary of specimen preparation conditions is listed in Table 1.

2.2. Ex Situ Annealing and Relocation of the Same Field of View

The specimen was observed and heated ex situ, so that every measurement was made at room temperature under stable conditions. The heating was performed using a tubular electric furnace (Asahi Rika Manufacturing Co., Ltd.) at 400 °C for 5 min under an argon atmosphere, followed by air cooling; the 5 min was measured from insertion of the specimen into the pre-heated furnace. The cycle was repeated until cumulative annealing times of 5, 10, 15 and 20 min were reached - five orientation maps were obtained at 0, 5, 10, 15 and 20 min.

Because the specimen was removed from the holder at each step, positional and rotational offsets were unavoidable. To align the region of interest throughout the repeated processing steps, the hole created during electropolishing was used as a fiducial marker for fine translational and rotational correction, as shown in Figure 1(b). The hole maintained a consistent shape throughout the series of measurements, indicating the absence of significant oxidation around it.

2.3. TEM-SPED Acquisition and Orientation Analysis

TEM observation was performed using JEM-ARM200F (JEOL) operated at 200 kV with a double-tilt holder. Electron diffraction patterns were acquired by SPED, over a 5.4 × 5.4 µm area with a step size of 20 nm (270 × 270 points) and a precession angle of 0.5°, using the TopSpin acquisition software (NanoMegas). TEM images of the region of interest were recorded at each step.

Crystal orientation was determined by template matching [5,6] using DiffGen and Index (NanoMegas). Simulated diffraction patterns were generated for the face-centered-cubic aluminum structure an acceleration voltage of at 200 kV on a 1° orientation grid. Image processing incorporated in the Index was performed to the recorded electron diffraction patterns before the template matching, including diffraction pattern centering and camera length calibration, noise reduction and diffraction spot enhancement to maximize the degree of match with the simulated diffraction patterns.

To assign crystal orientations as phase-field variables, the crystal orientations were grouped based on a tolerance of ±2.5° for each component of the Euler angle. Lattice distortion from dislocations perturbs the recorded electron diffraction patterns relative to the simulated ones [5,6], producing isolated mis-indexed points within grain interiors and near grain boundaries. These points were inspected individually by using their electron diffraction patterns and assigned to the adjacent grain (Figure S1), thereby removing indexing artifacts that would otherwise have entered the assimilation as spurious observations.

2.4. Region of Interest and Phase-Field Initialization

A 30 × 30-point square region of interest (ROI), corresponding to an area of 600 × 600 nm² and located 1.45 µm from the edge of the hole, was extracted from the full map as indicated in Figure 2(a). The extracted region contained four groups of orientations, and therefore, four phase-field variables (φ₁ – φ₄) were defined as shown in Figure 2(b). The grain boundary migration proceeded as the heat treatment continued. No nucleation or abnormal grain growth occurred during this study.

3. Numerical Model and Data Assimilation

3.1. Multiphase-Field Model

The microstructure of the region of interest was described by four non-conserved phase-field variables φ_i(r, t), where i = 1, …, N with N = 4 identifies the grain, r = (x, y) is a point on the 30 × 30 square grid of spacing Δx = Δy = 20 nm, and t is time. φ_i equals unity inside grain i and zero outside it, and varies smoothly across a diffuse interface of finite width δ, which in this study was set to 7 grid spacings (140 nm), subject to

$${\displaystyle \sum _{i=1}^N}{\phi }_i\left(\mathit{r},t\right)=1$$

(1)

at every grid point. Because the four fields sum to unity at every point, no point can be left unassigned, and no voids or overlaps can form between grains.

The MPF model used in this study follows the multiphase-field formulation of Koyama and Takaki [7], which is based on the interface-field description of Steinbach and co-workers [11,17]. Because the specimen is a single-phase pure aluminum maintained at a constant temperature, no chemical free-energy difference is assumed between grains; accordingly, the total free energy functional is written as:

$$F={\int }_{\mathsf{\Omega }}\left[\sum _{i<j}\left(-{\displaystyle \frac{{\mathrm{a}}_{ij}}{2}}\nabla {\phi }_i·\nabla {\phi }_j+W_{ij}{\phi }_i{\phi }_j\right)+{\displaystyle \sum _{i=1}^N}{E}_i{\phi }_i\right]d\mathsf{\Omega },0\le {\phi }_i\le 1$$

(2)

Here Ω is the computational domain, a_ij = a_ji is the gradient-energy coefficient of the i–j interface, W_ij = W_ji the height of the double-obstacle potential separating grains i and j, with a_ii = W_ii = 0 and Ei the stored-energy density of grain i. The first term penalizes spatial gradients of the order parameters and the second penalizes the coexistence of two grains at one point; their competition stabilizes an interface of finite width δ and of energy γ_ij. For an equilibrium planar interface of width δ and energy γ_ij the coefficients a_ij and W_ij are given by [7,10,17]

$${\mathrm{a}}_{ij}={\displaystyle \frac{8\delta {\gamma }_{ij}}{{\pi }^2}},W_{ij}={\displaystyle \frac{4{\gamma }_{ij}}{\delta }}$$

(3)

The boundary energy was taken to be the same for every pair, γ_ij = γ = 0.5 J m^-2 [7]. In the dimensionless form defined below this corresponds to γ_ij = 4.47 × 10^-2, and Equation (3) then gives a_ij = 0.253 and W_ij = 2.55 × 10^-2 for every distinct pair.

The evolution of each phase field is obtained by driving the system down the gradient of F [18], with the constraint of Equation (1) accounted for by taking pairwise differences of the variational derivatives [7,11]:

$${\displaystyle \frac{\partial {\phi }_i}{\partial t}}=-\sum _{j\ne i}{\displaystyle \frac{2M_{ij}}{n(\mathit{r},t)}}\left({\displaystyle \frac{\mathsf{\delta }F}{\mathsf{\delta }{\phi }_i}}-{\displaystyle \frac{\mathsf{\delta }F}{\mathsf{\delta }{\phi }_j}}\right)$$

(4)

where M_ij = M_ji ≥ 0 is the mobility coefficient of the i–j interface, referred to below simply as the mobility, and n(r, t) is the number of phase fields that are non-zero at r, so that the pairwise contributions are averaged over the grains that actually coexist at that point: n = 2 along a grain boundary and n = 3 at a triple junction. Evaluating the variational derivatives of Equation (2) gives the working form

$${\displaystyle \frac{\partial {\phi }_i}{\partial t}}=-{\displaystyle \frac{2}{n(\mathit{r},t)}}{\displaystyle \sum _{j=1,j\ne i}^N}{M}_{ij}\left\{{\displaystyle \sum _{k=1}^N}\left[{\displaystyle \frac{1}{2}}\left(a_{ik}-a_{jk}\right){\nabla }^2{\phi }_k+\left(W_{ik}-W_{jk}\right){\phi }_k\right]-{\displaystyle \frac{8}{\pi }}\sqrt{{\phi }_i{\phi }_j}{\mathrm{E}}_{ij}\right\}$$

(5)

in which k runs over all phase fields and ∇2 is the Laplacian. In this study, the driving force E_ij = 8.93 × 10^-3.

Equation (5) was solved in the nondimensional form used by Koyama and Takaki [7], in which all energy densities are scaled by RT/V_m and lengths by Δx, with R = 8.3145 J mol^-1 K^-1 the gas constant, T = 673.15 K the annealing temperature and V_m = 1.0 × 10^-5 m³ mol^-1 the molar volume of aluminum. For the grain-boundary energy this gives γ*_ij = γ_ijV_m /(RTΔx) = 4.47 × 10⁻². The quantities γ*_ij, a_ij, W_ij, M_ij, μ_ij, E_ij and Δt are therefore dimensionless. Equation (5) was integrated explicitly with a fixed dimensionless time step Δt = 1, using a five-point finite-difference Laplacian on the 30 × 30 grid with zero-flux boundaries, and Equation (1) was enforced at every step.

The computational grid was placed in one-to-one correspondence with the measurement points, so that the grid spacing Δx equaled the 20 nm step size and no interpolation was required. Each grid point was initialized as φₖ = 1 for the grain k to which the corresponding measurement point belonged and φⱼ = 0 for j ≠ k. Because the measurement assigned every point wholly to one grain, φ_i changed sharply across grain boundaries, whereas φ_i should be spread over the interface width δ = 7Δx in equilibrium state. The first 150 integration steps were therefore taken up in relaxing the step to the equilibrium profile, during which the boundary positions shifted by up to 2 grid points. At 0 min the four grains occupied 282, 41, 202 and 375 of the 900 grid points, that is 0.1128, 0.0164, 0.0808 and 0.1500 µm², where one grid point corresponded to 4.0 × 10⁻⁴ µm².

The six mobilities M_ij in Equation (5) were the parameters estimated in this study. With four grains there were six independent mobilities, M₁₂, M₁₃, M₁₄, M₂₃, M₂₄ and M₃₄. M_ij is derived from the dimensionless interface mobility μ_ij, as follows [7,17]

$$M_{ij}={\displaystyle \frac{{\pi }^2}{8\delta }}{\mu }_{ij}$$

(6)

3.2. Ensemble Kalman Filter with an Augmented State

The mobilities of the individual boundaries in the observed region were not directly measurable, whereas the positions of those boundaries were recorded at every observation time. Because grain boundary mobility and grain boundary position are strongly correlated, the mobilities can be estimated from the measured boundary positions. The ensemble Kalman filter [13,14] was therefore applied with an augmented state, in which the six mobilities were carried in the state vector alongside the phase fields so that a single analysis step corrected both [14,16]. In this study the augmented state of ensemble member m (m = 1, …, N_e, with N_e = 50), referred to below simply as member m, was

$${\mathit{x}}^{\left(\mathrm{m}\right)}=\left[{\displaystyle \genfrac{}{}{0pt}{}{{\mathit{\phi }}^{\left(\mathrm{m}\right)}}{{\mathit{\theta }}^{\left(\mathrm{m}\right)}}}\right]$$

(7)

where φ^(m) collected the N = 4 phase-field values at each of the N_p = 900 grid points and θ^(m) = (M₁₂, M₁₃, M₁₄, M₂₃, M₂₄, M₃₄)^T collected the six independent mobilities, giving N·N_p + 6 = 3606 components in total.

Every member was initialized by the same 0 min phase-field configuration obtained from the measured grain-label map (Figure 2(b)). The ensemble variance at t = 0 (the ensemble spread) arose entirely from the mobilities and not from the phase fields. For each member and each of the six grain pairs (i, j), an interface mobility μ_ij^(m) was generated independently from a normal distribution of mean μ₀ = 0.75 and standard deviation 0.4 μ₀, resampled until positive, and converted through Equation (6). The prior distribution of μ₀ results in mean and standard deviation of M_ij equals 0.1322 and 0.0529, respectively.

Each member was then integrated forward by Equation (5) for 200 steps, corresponding to the 5 min interval between successive measurements, giving the forecast state

$${\mathit{x}}_n^{\left(m\right)f}=\mathcal{M}\left({\mathit{x}}_{n-1}^{\left(m\right),a}\right),\mathrm{n}=1,\dots ,4$$

(8)

where 𝓜 denotes the 200-step time-stepping operator, n indexes the four assimilation cycles at 5, 10, 15 and 20 min, and the superscripts f and a denote the forecast state and the analysis state, respectively. The forecast state was obtained by integrating from the previous analysis state without using the current observation, and the analysis state was obtained after the observation was assimilated.

The observation vector y_n is the phase-field representation of the measured grain-label map at the n-th observation time, in which the field of the grain occupying a grid point is set to unity and the other three to zero. The observation operator H = [I 0] , where I and 0 are all-one vector and zero vector, respectively, whose the number of columns equal to the dimension of x^(m), extracts all of the phase-field variables of the augmented state and leaves the mobilities unobserved. Each member was updated against a separate noisy realization of the observation, formed by adding an independent random perturbation to the measured map; this keeps the analysis ensemble from being under-dispersed, and is the perturbed-observation form of the filter [14,19],

$${\mathit{x}}_n^{\left(m\right),a}={\mathit{x}}_n^{\left(m\right),f}+{\mathit{K}}_{\mathit{n}}\left({\mathit{y}}_{\mathit{n}}+{\mathit{\epsilon }}_n^{\left(m\right)}-\mathit{H}{\mathit{x}}_n^{\left(m\right),f}\right)$$

(9)

with ε_n^(m) ~$𝒩$(0, R) was generated independently for every member and every cycle, and the observation-error covariance taken to be diagonal, R = σ_o² I with σ_o = 1 × 10^-2, so that measurement errors were treated as uncorrelated between grid points. The Kalman gain is

$${\mathit{K}}_n={\mathit{P}}_n^f{\mathit{H}}^T{\left(\mathit{H}{\mathit{P}}_n^f{\mathit{H}}^T+\mathit{R}\right)}^{-1}$$

(10)

and the forecast covariance was estimated from the ensemble anomalies X’_n, whose m-th column is x_n^(m),f – x̅_n^f with x̅_n^f the ensemble mean:

$${\mathit{P}}_n^f{\mathit{H}}^T={\displaystyle \frac{1}{N_e-1}}{\mathit{X}}_n^{\prime }{\left(\mathit{H}{\mathit{X}}_n^{\prime }\right)}^{\mathrm{T}},{\mathit{H}\mathit{P}}_n^f{\mathit{H}}^T={\displaystyle \frac{1}{N_e-1}}\left(\mathit{H}{\mathit{X}}_n^{\prime }\right){\left(\mathit{H}{\mathit{X}}_n^{\prime }\right)}^{\mathrm{T}}$$

(11)

The anomalies X’_n span both blocks of the augmented state (the 3600 phase-field components and the 6 mobilities), and therefore, the sample covariance P_n^f contains the cross-covariance between them, that is the ensemble covariance of each mobility with each phase-field value. Through Equation (10) this cross-covariance enters K_n, and Equation (9) therefore corrects the boundary positions and the six mobilities in the same operation. After each update the mobilities were constrained to remain positive,

$$M_{ij}\leftarrow \mathrm{m}\mathrm{a}\mathrm{x}\left(M_{ij},{10}^{-6}\right)$$

(12)

which keeps Equation (5) well posed and defines the lower bound of the admissible parameter range. No model error was added during the forecast, and neither covariance inflation [20] nor localization [21] was applied.

Within each cycle the ensemble was first advanced 200 steps, and Equation (9) was then applied. This study defines “DA forecast” as the ensemble state immediately before the update, and “DA analysis” as the ensemble state immediately after the update. The inferred mobilities are defined as the ensemble-mean mobilities after the fourth update. Figure 3 summarizes the sequence and Table 2 lists the settings. It should be noted that annealing time of 5 min corresponded to 200 time-steps in this study, and therefore, 1 time-step in the MPF simulation corresponded to 1.5 s

3.3. Comparison Cases and Evaluation Metrics

In this study, three kinds of simulation were performed to examine the effect of parameter updating by the EnKF. The first case was an MPF simulation without data assimilation, referred to as “Direct”. The second case was an MPF simulation with data assimilation, in which the mobilities were updated during the simulation. The third case, referred to as “DA-init”, was an MPF simulation without data assimilation using the mobilities obtained from the second case. All three simulations started from the same observed 0 min configuration on the same grid and used the same model.

The observation and the simulation were compared point by point, without interpolation or shape matching. At each grid point the grain label was taken as the index of the largest phase-field variable,

$$L\left(\mathit{r},t\right)=\mathrm{a}\mathrm{r}\mathrm{g} \underset{i}{\mathrm{ma}x}{\phi }_i\left(\mathit{r},t\right)$$

(13)

Equation (13) was needed because the measurement assigns each point to one grain, whereas the simulated fields vary continuously across an interface of width δ. At t=5, 10, 15, and 20 min, Equation (13) was applied to the simulated and observed phase-field data to generate grain-label maps.

We calculated ratio of label agreement and the area of grains as evaluation metrics. The ratio of label agreement between the simulation result and the observations was calculated as follows,

$$A\left(t\right)={\displaystyle \frac{1}{N_P}}\sum _{\mathit{r}\in \mathsf{\Omega }}\eta \left[L_{sim}\left(\mathit{r},t\right),L_{obs}\left(\mathit{r},t\right)\right]$$

(14)

where Ω is the region of interest, N_p = 900 is the number of grid points in Ω, and η[a, b] is the Kronecker delta, equal to unity when the two labels coincide and zero otherwise. A(t) = 0, when there is no agreement between the simulation and the observation, and A(t) = 1, when the label matched between the two at all points. The same quantity has been used in the assessment of categorical maps [22], and in the evaluation of image segmentations [23]. This study also calculated (1 – A(t))N_p , which represents the number of disagreeing grid points when the spatial distribution of the error is discussed.

The area of each grain at time t was derived as follows,

$$S_i\left(t\right)={\left(∆x\right)}^2\sum _{\mathit{r}\in \mathsf{\Omega }}\eta \left[L\left(\mathit{r},t\right),i\right]$$

(15)

Equation (15) was evaluated separately for the observation and for each simulation, with (Δx)² = 4.0 × 10^-4 µm² per grid point. The area trajectory S_i^(t) as a function of annealing time provides the rate of change in grain area, and therefore, shows whether the growing and the shrinking grains are reproduced by comparing the observation results and the simulation. This kind of grain-by-grain comparison has been used to distinguish isotropic and anisotropic boundary properties in grain-growth simulations validated against experiment [24].

4. Results

4.1. Observed Microstructural Evolution

Figure 4(a) and Figure 4(b) show TEM images and orientation maps, respectively, of the tracked area at five annealing times. Grain boundary positions changed progressively while the overall grain arrangement was retained, allowing individual grains to be tracked throughout the sequence. The hole maintained a consistent outline throughout the measurement series (Figure 4(a)), indicating that no significant oxidation occurred around it.

Figure 4(c) shows the grain-label map of the region of interest. The four grains behaved very differently; all grain areas below correspond to the portions of the grains contained within the 600 × 600 nm window. Grain φ₂ grew from 0.0164 to 0.1268 µm² (41 to 317 grid points) between 0 and 20 min, representing an increase of 673 %, whereas φ₄ shrank from 0.1500 to 0.0688 µm² (375 to 172 grid points), corresponding to a decrease of 54 %. Grains φ₁ and φ₃ changed much less, from 0.1128 to 0.0892 µm² and from 0.0808 to 0.0752 µm² respectively.

The grain boundary migration was anisotropic. The φ₂/φ₃ and φ₃/φ₄ boundaries migrated significantly, whereas the φ₁/φ₂ and φ₁/φ₃ boundaries showed less pronounced migration. Resultant morphology of φ₂ was approximately rectangular at 20 min. Therefore, the grain boundary mobilities for each pair of grains should be defined independently. Determining six such parameters by hand-tuning until a simulation matched the observations would not be practical in general. The data-assimilation approach makes it possible to estimate the mobilities from the series of observation results, as described in the following sections.

4.2. Mobility Inference Through EnKF

Figure 5(a) shows the variation of the six ensemble-mean mobilities during the data assimilation, and the values obtained at the final time (20 min) are listed in Table 3. Starting from a common prior mean of 0.1322, each M_ij varied substantially during the first and second data assimilation (5 min and 10 min). Most of them subsequently remained nearly constant, whereas M₁₃ decreased after the second update, while the variation of the standard deviation showed roughly constant variation after the second update as indicated in Figure 5(b).

The magnitude of these mobilities was consistent with the observed results. For example, the two larger values, M₂₃ and M₃₄, corresponded with the mobilities of φ₂/φ₃ and φ₃/φ₄ boundary, which sweep through the ROI during the heating. The two smaller values, M₁₄ and M₁₃, on the other hand, corresponded with mobilities of φ₁/φ₄ and φ₁/φ₃ boundary showing less variation. Therefore, each M_ij was corrected so that the simulation results matched with the observation results. The decrease of M₁₃ indicated above may be related to the inconstant evolution of φ₁/φ₃ boundary during heating. Pronounced migration was observed at the early annealing times of 0, 5, and 10 min, whereas little migration was observed at the later annealing times of 15 and 20 min. The later observations therefore favored a lower M₁₃ than that inferred from the early-stage evolution, resulting in additional downward corrections after the second assimilation update.

It should be noted that the grains φ₂ and φ₄ were not in contact each other during the heating, and therefore, M₂₄ was not corrected appropriately and showed the lower bound value. Because there was no φ₂/φ₄ interface throughout the observation, the grain growth proceeded regardless M₂₄. In this case, the Kalman gain in Equation (9) for correcting this component was comparatively small. At the initial update (5 min), the deviation between the simulation result and the observation result (${\mathit{y}}_{\mathit{n}}-\mathit{H}{\mathit{x}}_n^{\left(m\right),f}$ in Equation (9)) was large, and M₂₄ was updated according to Equation (9) even if the Kalman gain was small. The deviation was not significant after the initial update. Therefore, the variation of M₂₄ in Figure 5(a) showed apparently constant after the first update.

4.3. Direct and Data Assimilation

Figures 6(a)-(c) show the observed grain-label maps with those obtained from DA-init and Direct, respectively, where the observed grain-label map was identical with Figure 4(c) and indicated for comparison. DA-init successfully reproduced the observation, as the area fraction of each φ_i was consistent with the observation. The label agreement was 0.9056 at 20 min as shown in Table3. Direct, which used the isotropic mobility, showed poorer agreement with the observation as annealing proceeded. The label agreement was 0.6278 at 20 min as indicated in Table 3. The grain-label maps after each update (DA analysis) are presented in Figure S2.

Figure 7 indicates the variation of the agreement defined in Equation (14). In the figure, the agreement in the Direct decreased at every measurement, from 1.0000 at 0 min to 0.8811, 0.7667, 0.7089 and 0.6278 at 5, 10, 15 and 20 min. The error was directional rather than scattered: the same grains were mis-sized in the same sense at every time. As indicated below, the grain area in Direct significantly deviated from the observation.

4.4. Variation of the Individual Grain Areas

The simulation using inferred mobilities (DA-init) demonstrated good agreement with the observation results also in terms of area of individual grains. Figures 8(a)-(d) shows the variation of the area of individual grains with observation time. The Direct (dashed lines), showed deviation from the observation results for all the grains with proceeding the time. The thin lines labeled “DA” indicate the results obtained by applying DA to the Direct case at the points where the observation results were defined. The sharp variation at the points indicating “Observation” was attributed to the update according to Equation (9). The agreement in the area improved as the data assimilation proceeded. The DA-init case (thick lines), on the other hand, showed continuous variation of the area maintaining the agreement with the observation results. It is worth noting that the variation in DA-init passes between the observation results. Because the observation results contain noise in general, they sometimes show fluctuated trajectory as representatively shown in Figures 8(d). The EnKF-based parameter inference yielded plausible results while suppressing the influence of noise. Note that the spatial distribution of grain-label error between the simulations and the observation is presented in Figure S3.

5. Discussion

In this section, we discuss the influence of the real-time interval on parameter inference. As shown in the previous section, the EnKF-based parameter inference provided reasonable parameter values that successfully reproduced the observation results. During the simulation, the grain growth behavior was corrected according to Equation (9) so that the simulation results matched the observation. However, because the grain growth behavior is inherently nonlinear, as representatively shown in Figure 8(a)-(d), the integration error of the MPF simulation may become non-negligible as the real-time interval increases, because Equation (5) is solved using the finite differential method. Varying the real-time interval may affect not only the simulation results but also the state vector covariance used to calculate the Kalman gain in Equation (10), potentially leading the inferred parameter values. Therefore, the robustness of the EnKF-based parameter inference was examined.

The assimilation interval was changed from 200 dimensionless time steps to 100 and 300 dimensionless time steps, which corresponded to numerical time-step sizes of 3 s and 1 s, respectively, compared with the original value of 1.5 s. The label agreement was summarized in Table 4. In the table, M_ij was multiplied by the dimensionless time-step size so that the effect of the real-time interval was normalized. No clear difference in label agreement was observed among the three cases.

Figure 9 compares the normalized mobilities obtained with the three assimilation intervals. The inferred mobilities were nearly consistent for most cases. In contrast, the mobility M₁₂ showed a large deviation because the limited migration at the φ₁/φ₃ boundary resulted in a less effective update according to Equation (9). However, the label agreement of DA-init at 20 min was nevertheless almost unchanged among the three settings as demonstrated in Table 4. The reproduction of the microstructure was therefore predominantly governed by the larger mobilities. The EnKF-based parameter inference was robust to the real-time interval in the range examined here as long as these dominant mobilities were inferred.

6. Conclusions

Time-resolved TEM-SPED orientation maps of 50% cold-rolled pure aluminum were obtained on the same field of view after cumulative annealing at 400 °C for up to 20 min. The grain growth behavior was observed, and the observation results were assimilated into the MPF model using the EnKF, inferring the six grain-boundary mobilities. The inferred mobilities were effective for reproducing the observation results. This paper also discussed the robustness of the EnKF in terms of the real-time interval, which affected the numerical integration error and, consequently, the state-vector covariance used in the Kalman gain. The principal outcome of the parameter inference was not substantially affected when the number of integration steps varied from half to 1.5 times the baseline setting.

This study demonstrated the data assimilation method for predicting nanoscale heat treatment behavior. Because heat treatment is an essential process for controlling material properties, the method shown in this study would contribute to data-driven process design.