Quantification and prediction of solidification textures under additive manufacturing conditions

Domain: physics
Published: 2025-11-28
Editorial outcome: major_revision

Abstract
Crystallographic textures are a major determinant of the macroscale anisotropic properties of polycrystalline metallic alloys produced in a wide range of additive manufacturing (AM) processes. Here, we introduce a statistical method that can accurately quantify the degree of orientational order of textures despite the large random fluctuations in the orientation of individual grains inherent in AM processes. The method, demonstrated for laser and resolidification of AlSi thin films, extends Z-scoring to a dynamical regime to assess the statistical significance of observed textures compared to randomly generated ones at different stages of solidification. We further show that, combined with phase-field modeling, this method can be used to infer fundamental anisotropic properties of the solid-liquid interface that are essential for texture prediction, and are compared here to the results of atomistic simulations. In addition, phase-field modeling reveals that, even at rapid AM solidification rates, the observed 〈110〉-dominated textures in the AlSi thin films are controlled predominantly by the anisotropy of the interface free-energy and sheds light on the physical mechanism of grain competition. These results significantly enhance both the existing tools for the quantification and prediction of AM crystallographic textures and our basic understanding of their formation.

Introduction
Additive manufacturing (AM), particularly laser-based techniques like Powder Bed Fusion Laser Beam (PBF-LB) and Directed Energy Deposition (DED), has revolutionized the fabrication of complex metallic components, offering unprecedented design freedom and material performance1–3. These processes operate under extreme thermal conditions, including thermal gradients of 103 ~107K/m4,5, solidification velocities ranging from 10−4 to 10m/s4,6, and cooling rates as high as 108K/s3,5,7. These extreme conditions often produce columnar or equiaxed microstructures with complex crystallographic textures that influence mechanical properties and macroscopic anisotropy3,8. However, the mechanisms governing grain texture selection under AM conditions remain poorly understood.

Grain texture evolution during solidification is generally controlled by the growth competition between grains of different crystallographic orientations. This competition selects grains with more favorable orientations at the expense of other grains with less favorable ones9. For columnar dendritic grain structures forming under slow solidification conditions, where the solid-liquid interface can be assumed to remain close to thermodynamic equilibrium, grain textures are primarily governed by the anisotropy of the solid-liquid interfacial free energy (γsl) that determines preferred dendrite growth directions10–13. For commercially important metallic alloys with cubic crystal symmetry, such as Al- or Ni-based alloys, 〈100〉 dendrite growth directions are commonly preferred, leading to 〈100〉 grain textures with 〈100〉 crystal axes preferentially aligned parallel to the heat flow direction. Compositional variations have also been observed to induce dendrite orientation transitions (DOTs) from 〈100〉 to 〈110〉 directions in some Al-based alloys, including AlZn14–16, AlGe17, and AlSm18. These DOTs have been interpreted theoretically using a two-parameter form of γsl predicted by atomistic simulations. While knowledge of preferred dendrite growth directions is an important prerequisite to predict grain textures, it is generally insufficient, since it does not predict which grains survive and which ones are eliminated. The classical minimum undercooling criterion of Walton and Chalmers19 assumes the growth of dendritic grains with the smallest growth undercooling, equivalent to the smallest misorientation between one of the 〈100〉 crystal axes and the heat flow direction, should be favored. However, this criterion alone does not predict how fast grains are eliminated. Several computational studies20–22 in a slow solidification regime have shown that, surprisingly, unfavorably oriented grains can sometimes survive and even outgrow favorably oriented grains. These studies have highlighted the limitation of the minimum undercooling criterion and identified various other factors than growth undercooling influencing grain competition, including neighboring grain bicrystallography20,21 and microstructural length scales of dendritic grain structures22.

For columnar grain structures forming under rapid solidification conditions encountered in AM processes, where the solid-liquid interface can no longer be assumed to remain in local thermodynamic equilibrium, much less is known about what controls grain texture selection. Depending on the alloy type and growth conditions, the microstructure within each grain typically evolves from cellular/dendritic to weakly cellular to planar or banded as the solidification front velocity approaches and exceeds the absolute stability limit23–27. Textures obtained in AM specimens have been previously explained considering the dendritic primary growth direction and the contributions of the thermal gradient, solidification rate and melt pool shape8,28–31. Phase-field (PF) modeling studies at the melt pool scale have either focused on the computationally more tractable planar front case without cellular/dendritic microstructures32,33, or assumed a standard form of γsl with one anisotropy parameter that favors 〈100〉 dendrite growth directions34,35. In addition, to date, comparisons between predicted and observed cellular/dendritic grain textures have remained largely qualitative. As a result, some basic related questions have remained unanswered. Does the anisotropy of γsl, or the kinetic anisotropy known to influence dendrite orientations at rapid rates in undercooled melts36,37 control dendrite orientations under AM conditions? Are two anisotropy parameters essential for interpreting DOTs under slow solidification conditions14,15,17,18 adequate to predict preferred growth directions under rapid solidification? What is the underlying mechanism that governs the grain texture evolution?

Addressing these questions presents a major challenge. For the comparison of predicted and observed textures to be informative of preferred growth directions, the texture analysis need to cope with the inherent stochasticity of the grain competition process, as well as large spatial variations of heat flow direction, which are both especially pronounced in AM processes. In a slow solidification regime where large columnar grains form with 〈100〉 or 〈110〉 crystal axes aligned preferentially with respect to a uniform thermal axis15,17, standard pole figures generally suffice to analyze grain textures and DOTs. However, under AM conditions, grain textures generally consist of a large number of grains with significant variations in growth orientation with respect to a thermal axis that itself varies around the melt pool. Furthermore, the number of grains decreases during solidification. This raises the central question of the present study of whether the comparison of predicted and observed textures can be made sufficiently quantitative to infer anisotropic crystal growth properties.

To address this question, and at the same time answer some of the basic questions raised above, we combine experimental and computational PF studies of pulsed-laser-induced melting and rapid resolidification of thin films of AlSi alloys. By focusing on this quasi-2D geometry, we are able to use in situ imaging with Dynamic Transmission Electron Microscopy (DTEM), complemented by ex situ orientation mapping via TEM-based crystallographic mapping, to measure grain orientations with respect to the local thermal axis around the melt pool at different solidification times. We further introduce a new statistical method based on Z-scoring — Significance Analysis of Solidification Textures (SAST) — to quantify the observed textures. Here, the Z-score represents the number of standard deviations by which the observed grain alignment with the local thermal axis, or the axis perpendicular to it, deviates from its mean value for a distribution of randomly oriented grains. Importantly, the mean and standard deviation of this distribution are calculated as a function of the solid fraction with the same number (N) of actively growing grains as in an experiment or PF simulation, allowing us to assess the statistical significance of the observed textures at different stages of solidification despite the fact that N decreases with increasing solid fraction due to grain competition. Furthermore, using SAST together with PF modeling, we are able to infer both the anisotropic properties of the solid-liquid interface that govern the formation of the intra-grain cellular/dendritic microstructure, and the physical mechanism of the inter-grain growth competition that control the grain texture. The results reveal that the anisotropy of the excess interface free-energy plays a dominant role even at rapid rates typical of AM processes and that grain competition is well-described by the classical Walton and Chalmers19 minimum undercooling criterion.

Results
Pulsed-laser-induced rapid solidification of AlSi thin films by DTEM

In this work, we conducted pulsed-laser-induced rapid solidification experiments on AlSi thin films with homogeneous initial grain sizes and random textures. Thin films approximately 100 nm thick were investigated for three Si compositions: ~1 wt.%, ~3 wt.%, and ~6 wt.%, referred to as Al1Si, Al3Si and Al6Si, respectively. In situ imaging was performed during laser spot melting by DTEM, where the melt pool boundary at different times after the laser was turned off was tracked, as shown in Fig. 1a. Note that solidification does not start at laser shutdown, but the interface remains still for some time (called incubation time) until it starts moving towards the melt pool center. This has been previously reported by McKeown et al.38. Detailed data analysis was performed to estimate the major and minor ellipse semiaxes (a0e and b0e) and instantaneous solidification velocity (V) values with time. Further information regarding DTEM experiments and data analysis is found in Experimental methods. Fig. 1b presents V as a function of the solid fraction (by area) at various Si contents, together with fitted curves. This result demonstrates increasing V as the solid fraction increases, i.e., V increases with solidification time towards the center of the melt pool, consistent with the findings reported by McKeown et al.38. Moreover, lower Si contents result in higher average V values. This trend can be attributed to the corresponding decrease in thermal conductivity, which varies from ≈ 215 Wm−1K−1 for Al1Si to ≈ 160 Wm−1K−1 for Al6Si at room temperature, according to the ThermoCalc® 2022b, TCAL7 database. The thin nature of the DTEM samples effectively eliminates convection, especially when the sample plane is perpendicular to gravity during the experiment. This prevents the formation of convection cells, minimizing disturbances, making solidification conduction-driven39. More detailed information about the DTEM set-up can be found in Ref. 40. Regardless of the level of Si content, average velocities consistently fall within the range of 0.5 ~ 2 m/s. These velocities align with the characteristic range of solidification velocities in AM, ranging from approximately 10−4 to 10 m/s4,6.

Ex situ characterization of grain textures by TEM-based orientation mapping

Ex situ TEM-based orientation mapping was performed to evaluate the crystallographic textures produced by the DTEM experiments; see Experimental methods for additional details regarding the mapping. For this analysis, only face-centered cubic (fcc) Al was considered. Two scans per chemical composition were conducted, where one of them was taken in the as-sputtered region and the other one was taken near the melt pool boundary. The former capture the orientations outside the melt pool edge, in the heat-affected zone, which is characterized by small, equiaxed grains of approximately random orientations. The characterization of the as-sputtered and the heat-affected zone can be found in Supplementary Note 4. An example of the latter mapping is shown by the Inverse Pole Figure (IPF) map corresponding to Al6Si, as shown in Fig. 1c, where grains are defined as regions with a maximum 5∘ inner-grain misorientation. Within the melt pool, the grains that evolve from the edge to the center appear columnar. Notice that at the late stage of solidification, only grains colored red, yellow, and green survive, while the blue grains are eliminated.

To gain a deeper understanding of grain texture evolution during rapid solidification, we analyze the crystallographic mapping data as a function of time (t) and Si amount. For each time step the solid-liquid interface is estimated to be an ellipse, following the procedure outlined in Supplementary Note 2. Since local orientations vary within the grain, the representative orientation at a given time step is determined by averaging the orientations over a 200 nm region from the solid-liquid interface (region with the red dashed line in Fig. 2a). For each filtered grain, a grain coordinate system, (X, Y, Z), is established. Here, X denotes the solidification direction, approximated as the vector perpendicular to the solidification front segment near the grain. The Y-axis represents the in-plane direction perpendicular to X, while Z denotes the out-of-plane direction (Fig. 2b). In addition, we introduce three key directions, ds[100], dp[010], and dp[011], which are illustrated in Fig. 2 and will be further defined in the next section.

Figure 3 shows IPF (out-of-plane direction Z) maps corresponding to varying levels of Si content and different solidification times. These maps feature cubes representing the orientation of each grain, particularly in the region proximal to the solidification front (see Experimental methods for its definition). Adjacent to each map is a set of pole figures representing the mean orientations of filtered grains. In these figures, the grain coordinate system (X, Y, Z) is utilized, whereby the mean grain orientations have been rotated to align the grain solidification direction (X) parallel to the horizontal direction. Spot size in the pole figures is proportional to the grain area, while spot color is determined by the mean grain IPF (out-of-plane direction Z). A more complete set of maps and pole figures, corresponding to a wider range of time-steps, is available in Supplementary Note 5.

The observations from Fig. 3a–c demonstrate the evolution of texture as solidification progresses. Prior to solidification initiation, the texture appears nearly random (Fig. 3a), characterized by a high prevalence of small, equiaxed grains that formed via the sputtering process. As solidification starts, some grains in contact with the melt pool boundary begin to grow epitaxially (IPF map in Fig. 3b), aligning approximately parallel to the heat flux. Even though the texture retains an appearance of randomness (pole figures in Fig. 3b), selective orientations begin to emerge. Towards the end of solidification (Fig. 3c), grain competition becomes evident, leading to the preferential selection of specific crystallographic orientations. The (100) pole figure illustrates that two directions belonging to the 〈100〉 family lie vertically, parallel to Y. Moreover, pole figures corresponding to the (110) and (111) planes show regions of higher intensity, indicating some directionality. Examination of the cube maps on the left indicates not all of the grains grew along their primary [010] directions, as traditionally reported41,42. In contrast, some of them exhibit [011] ∥ X alignment. Note that all of these directions are in-plane, consistent with the sample geometry17. Finally, analysis of these representations of rotated cubes for other Si contents (Fig. 3d, e) confirms the persistence of these trends.

Significance Analysis of Solidification Texture (SAST)

While the pole figures in Fig. 3 offer a qualitative visualization of grain texture, a more rigorous and quantitative approach remains needed to evaluate texture evolution. To address this, we developed a Significance Analysis of Solidification Texture (SAST) methodology (see “Methods”). SAST provides a systematic framework to quantitatively evaluate the alignment of grain orientations and their deviation from random orientations. This robust approach enables a comprehensive assessment of texture evolution across varying experimental conditions, ensuring greater precision and consistency in analysis.

For each grain in the sample plane that is still growing, we first pick (Fig. 2) one of the six principal 〈100〉 directions that has the largest projection onto the Y direction locally parallel to the solidification front, ds[100], and then subsequently pick the one among the 4 remaining 〈100〉 directions in the plane perpendicular to ds[100] that has the largest projection onto the X growth direction, dp[010], and the one among the 4 〈110〉 directions in the same plane that has the largest projection onto the solidification direction, dp[011]. This choice of directions turns out to maximize statistical significance in comparison to other choices discussed in Supplementary Note 6. It is also guided by the pole figures (Fig. 3), which suggest that 〈100〉 directions are oriented preferentially parallel to the solidification front, as expected for typical 〈100〉 textures with secondary dendrite arms perpendicular to the growth direction. This is also visually apparent when examining the preferential orientations of the cubes for each grain (with six faces perpendicular to the 〈100〉 directions) in the left column of Fig. 3. As for the crystallographic direction that is most aligned with the growth direction, the pole figures or the cube orientations are less clear, since they seem to contain a mix of directions close to 〈100〉 and 〈110〉 and other directions in between. This is different than what is expected for standard 〈100〉 textures dominated by primary dendrite arms growing along 〈100〉 directions. While 〈110〉 directions appear more frequently than 〈100〉, it is not clear that this difference is statistically significant. Therefore, while we expect 〈100〉 directions to be preferentially aligned parallel to the solidification front, a statistical analysis is required to confirm this expectation and to determine the preferred crystallographic growth directions.

To perform this statistical analysis, we compare, at different stages of solidification, whether the N grains that are still growing are preferentially aligned with the X and Y axes when compared to N randomly oriented grains. Taking N into account is important because N decreases in time due to grain competition, and statistical significance requires N to be sufficiently large. Since N is largest at the earliest stage of solidification, when grains are randomly oriented, it is not a priori obvious that N is still sufficiently large to reach statistical significance at later stages when the anisotropic growth competition of different grains has selected a texture. Our results below demonstrate there is a large window between the early and final stages of solidification, where grain competition has selected a texture and N is still sufficiently large to quantitatively assess its statistical significance. To measure the alignment of a collection of N grains, we sum over all grains the projections of the crystallographic directions of interest onto the local axes perpendicular and parallel to the solidification front, denoted by Yi and Xi for the ith grain, respectively, which yields the sums

As[100] = ∑Ni=1 ds[100],i ⋅ Yi
Ap[010] = ∑Ni=1 dp[010],i ⋅ Xi
Ap[011] = ∑Ni=1 dp[011],i ⋅ Xi (1)

This approach takes into account the large variation of orientations of the solidification front around the melt pool. For perfectly aligned grains along dα, Aα equals N for each α ∈ {s[100], p[010], p[011]}, while for randomly chosen grains, Aα is randomly distributed. This distribution is generally non-trivial for small N, as illustrated for N = 1 and As[100] = ds[100] ⋅ Y in Fig. 4a; it exhibits a constant frequency for values exceeding 2/2, with a mean of 0.831 and a standard deviation of 0.1 as detailed in Supplementary Note 7. In the large N limit, however, the statistical distribution of Aα’s become Gaussian with a mean value that is proportional to N, albeit much smaller than N since different random projections cancel each other, and a standard deviation √N is obtained as expected from the central limit theorem. This is illustrated in Fig. 4b that shows the statistical distributions of A with increasing N with a perfect fit to a Gaussian distribution (solid black curve) for N = 29, where the values of N correspond to the grain textures of Fig. 3a–c at different stages of solidification.

We can now use the distributions in Fig. 4b to assess the statistical significance of the experimentally observed textures. For this, we superimpose as crosses the computed values of As[100] for the grain textures corresponding to Fig. 3a–c in Fig. 4b, and as pluses the values corresponding to perfect alignment. For the early stage corresponding to Fig. 3a, there are many grains (N = 29) that are randomly oriented. As a result, the measured value of As[100] falls well inside the distribution for randomly oriented grains and is far from the value for perfect alignment. For the intermediate stage of solidification corresponding to Fig. 3b, there are fewer grains left growing (N = 13), but the measured value of As[100] is now a few standard deviations outside the distribution for randomly oriented grains, moving closer to the value for perfect alignment. This trend continues to the later stage of solidification, where the measured value of As[100] remains statistically significant (i.e., outside the distribution for randomly oriented grains), even though the number of grains has decreased further (N = 5).

To further assess the statistical significance of the measured alignments at different stages of solidification, we use standard Z-scoring to quantify how much the experimentally measured Aα values deviate from the mean μα of random orientations, normalized by the standard deviation (σα): zα = (Aα − μα)/σα. The stage of solidification is measured as in Fig. 1b by the solid fraction, which increases in time from zero at the start of solidification to 1 at the end of solidification. The results are shown in Fig. 4c–e that compares the experimentally measured Z-scores to the values for perfect alignment with Aα = N (zs[100],pa, zp[010],pa, zp[011],pa) as a function of solid fraction. Figure 4c shows that zs[100] surpasses the threshold Z-score of statistical significance 1.96 for a large window of solid fraction comprised between the very early and late stages of solidification. This threshold Z-score corresponds to a small p-value of 0.05 for a two-tailed test, an indicator that the observations are not due to random chance. Note that any measurement with a corresponding Z-score for perfect alignment above 1.96 (corresponding to two grains) is statistically representative. This statistical significance confirms the aforementioned expected trend that ds[100] aligns with the solidification front (Y-axis). As solidification progresses, zs[100] approaches zs[100],pa, suggesting that grains with ds[100] aligned with Y increasingly dominate during grain competition. In contrast, Fig. 4d reveals that zp[010] consistently remains below 1.96 throughout solidification, showing no significant alignment trend with the X-axis. Figure 4e highlights pronounced alignment of dp[011] with the X-axis, supported by the zp[011] values that exceed 1.96 for most of the solidification process. However, toward the end of solidification, zp[011] falls below 1.96 due to a reduced number of surviving grains (Fig. 4f). Our analysis reveals clear trends in grain orientation evolution during solidification: [100] directions increasingly align with the Y-axis, while [011] directions align with the X-axis. These findings underscore the role of grain competition in texture evolution and validate the SAST methodology as a robust tool for quantifying texture alignment.

Notably, since the standard deviation of alignment scales with √N, the Z-scores for perfect alignment of 1 and 2 grains are 1.69 and 2.39, respectively. Thus, the minimum number of grains required for statistical relevance is 2. However, this does not explain the decreases in zs[100] and zp[110] at the mid-to-late stage, when the number of grains is larger than 2. To evaluate the alignment of a single grain and to enable comparison of Z-scores across systems with different grain numbers, we normalize the Z-score using the corresponding value for perfect alignment: zα/zα,pa = (Aα − μα)/(N − μα). Supplementary Fig. 5 shows that the Z-score ratios for ds[100] and dp[110] steadily increase during solidification, except in the late stage. The late-stage deviation arises because the grain number decreases substantially below ~13, which is insufficient for reliably selecting preferential orientations during grain competition.

Phase-field modeling of dendritic structures

To elucidate the mechanism underlying experimentally observed 〈011〉 textures, we employed PF modeling to simulate the crystallographic texture evolution with the incorporation of the anisotropies of both the solid-liquid interfacial free energy (as) and atomic attachment kinetics (ak). For a solid-liquid interface with normal direction n, as(n) can be expanded using spherical harmonics Ki(n) (i = 1, 2), which are detailed in Supplementary Note 8:

γsl(n) ≅ as(n) = 1 + ϵ1K1(n) + ϵ2K2(n) (2)
γ0

Here, γ0 denotes the reference solid-liquid interfacial free energy, and ϵ1 and ϵ2 are anisotropy parameters. Previous studies on DOTs under slow solidification conditions for three-dimensional (3D) bulk materials demonstrate that a positive ϵ1 promotes 〈100〉-oriented growth, a negative ϵ2 enhances the 〈011〉 dominance, and a region of positive ϵ1 combined with negative ϵ2 favors a more complicated hyperbranched structure14,17,43. The kinetic anisotropy ak(n), governing interface mobility, is represented by one leading term:

μk(n) ≅ ak(n) = 1 + ϵkK1(n) (3)
μ0k

where μk(n) is the anisotropic kinetic coefficient, μ0k is its isotropic reference value, and ϵk quantifies kinetic anisotropy strength. In 3D bulk materials, the positive ϵk promotes the 〈100〉 growth direction44, although in Al-based alloys at low solidification velocities, its impact is negligible compared with the influence of as14. Conversely, at very high solidification velocities, the effect of kinetic anisotropy becomes significant for dendrite orientations in undercooled melts36,37,45. However, the effects of as and ak in thin-film alloys remain unexplored. While their roles are well-characterized in bulk materials, confinement effects in thin-film geometries may alter the competitive dynamics governing texture evolution.

A challenge in capturing the crystallographic textures from rapid solidification experiments is accurately implementing 3D anisotropies in two-dimensional (2D) simulations. Our AlSi rapid solidification experiments were conducted on thin films with a thickness of approximately 100 nm. However, numerically resolving crystallographic textures at the melt-pool scale in 3D is computationally demanding for PF modeling. Therefore, we approximated the thin films as effectively 2D by assuming that the solid-liquid interface is essentially planar in the direction perpendicular to the film. In standard 2D implementations of γsl anisotropy, existing models reduce the problem to a simplified special case, aligning a [100] direction perpendicular to the sample plane20,23. This approach yields a four-fold symmetry expressed as a function of the misorientation angle α: as = 1 + ϵ4 cos4(θ − α), where ϵ4 is the anisotropy parameter and θ is the angle between the normal direction of the solid-liquid interface and the X-axis. However, this simplification fails to capture the complexity of crystallographic orientations, which are randomly distributed in 3D space. To address this limitation, we calculated as by projecting the interface normal vector n onto the intrinsic crystal coordinates. These projections were directly substituted into the explicit expressions for as(n) and ak(n), with randomly distributed orthogonal crystal axes defined in the laboratory frame (see Supplementary Note 8). This approach capture the full spectrum of possible crystallographic orientations that are essentially a mixture of two-fold and four-fold symmetry terms for the one-parameter anisotropy (Eq. 17 in Supplementary Note 8). Notably, our formulation recovers the conventional four-fold symmetry when [100] is aligned perpendicular to the sample plane and reduces to isotropic as when [111] is perpendicular to it. To simulate the crystallographic textures at the melt pool scale, we first used the commonly used one-parameter form of as, (ϵ1, ϵ2) = (0.06, 0), which is a generalization of the aforementioned simplified as forms. PF simulation results for a quarter melt pool are depicted in Fig. 5 and Supplementary Fig. 12, with the model details provided in the Phase field modeling section of Methods. Figure 5a and Supplementary Movie 1 show representative grain texture evolution characterized by rotated cubes that elucidate grain orientations. The grain texture exhibits a trend where, at the late stage of solidification, one of the 〈100〉 directions tends to align with the solidification direction. This observation can be confirmed by the Discrete Pole Figures (Fig. 5b), Density Pole Figures (Fig. 5c), and Density Pole Figures considering 4 simulations (Fig. 5d) at the solidification time of 15 μs, obtained following the same procedure as in experiments.

Grain texture evolution is further quantitatively analyzed through SAST. Figure 5e–g display three key Z-scores of simulation results, along with the experimentally measured Z-scores for Al1Si (open circles), Al3Si (open square), and Al6Si (open triangles). Figure 5e illustrates most of the zs[100] data points are below 1.96 (p-value of 0.05), suggesting statistically insignificance of ds[100]’s alignment with Y, different from experiments (Fig. 4a). In addition, Fig. 5f exhibits a window of zp[010] exceeding 1.96, in contrast to the experimental measurements of around 0 (Fig. 4b). This alignment indicates dp[011] tends to form a large angle with respect to the X-axis, which explains the negative zp[011] values of around −1.96 (Fig. 5g). In addition, Fig. 5h presents the grain number, which substantially affects Z-score values. This result demonstrates a large initial grain number, followed by a quick decrease to a level that agrees with the experiments.

The inability to replicate the experimentally observed grain texture using the one-parameter γsl anisotropy prompted us to calculate more realistic parameters for AlSi alloys. To achieve this, we conducted atomistic simulations for the solid-liquid interface evolution within a thin slab, using an angular embedded atom method (AEAM) interatomic potential for AlSi46. By employing the capillary fluctuation method (CFM)47, we determined the anisotropy parameters of dilute Al-Si alloys, revealing a shift toward the 〈110〉 region with increasing Si composition (Supplementary Note 9 and Supplementary Fig. 11), consistent with the trend observed in the AlSm alloys18. In this study, we selected (ϵ1, ϵ2) = (0.06, −0.006) as the representative anisotropy parameters.

The PF simulation results are presented in Fig. 6, Supplementary Fig. 13, and Supplementary Movie 2. Figure 6a focuses on a representative grain texture for Al1Si, demonstrating a transition to [011] growth direction at late-stage solidification. The observed grain texture can be validated by the Pole Figures shown in Fig. 6b–d, produced following the same procedure as in experiments. Specifically, in the (100) Pole Figure, a prominent high-density region is observed around the north pole, and in the (110) Pole Figure, a relatively high-density area is at the east. These characteristics underscore ds[100] predominantly aligns with the Y-axis, while dp[011] aligns with the X-axis.

To further quantify the grain texture evolution, we applied the SAST method. Figure 6e–g presents the Z-scores for the projection sum of ds[100] on the Y-axis (Fig. 6e), dp[010] on the X-axis (Fig. 6f), and dp[011] on the X-axis (Fig. 6g), for both PF simulations (blue square) and experiments (gray lines). Additionally, the number of grains in the simulations approaches those in experiments quickly after solidification starts (Fig. 6). Moreover, the Z-score ratio with respect to that for perfect alignment shows an increase for most of the solidification process, and the initial grains outside the melt pool is approximately 205.0 ± 83.7 nm (Supplementary Figs. 14 and 15). These simulation results demonstrate remarkable agreement with the experimental measurements, thus validating our PF model’s capability in capturing the grain texture evolution during rapid solidification of thin-film AlSi alloys.

Compared to γsl anisotropy, kinetic anisotropy plays only a minor role in texture evolution. To illustrate this, we performed two series of PF simulations. In the first, γsl was isotropic while kinetics were anisotropic. Supplementary Fig. 16 show slight increases in Z-scores, but none reaching statistical significance, suggesting near-random orientation and only weak 〈100〉 trends. In contrast, the second series applied γsl anisotropy (ϵ1, ϵ2) = (0.06, −0.006) with isotropic kinetics. Supplementary Fig. 17 reveals significant 〈011〉 alignment, matching the fully anisotropic case in Fig. 6. Together, these results confirm that while kinetic anisotropy promotes a conventional 〈100〉 texture, γsl anisotropy remains dominant in crystallographic texture evolution. Notably, the primary role of γsl anisotropy and the minor influence of kinetic anisotropy remain valid when the solidification velocity follows the experimentally measured profile (Fig. 1b) and the thermal gradient is increased to 5 K/μm (Supplementary Figs. 18, 19, and 20).

The grain textures observed in experiments and polycrystalline PF simulations (Figs. 4–6) can be well explained by the minimum undercooling criterion, which assumes that dendritic grains with the lowest tip undercooling are preferentially selected. To investigate the mechanism of texture formation, we performed single-crystal simulations with the principal axis rotating around the X, Y, and Z-axes (Fig. 7a). For (ϵ1, ϵ2) = (0.06, −0.006), the tip undercooling Δ decreases monotonically for rotations around the Y-axis (orange line), but generally increases for other rotations. To explain more complex orientations in polycrystalline simulations, we further examined sequential rotations: a subsequent X-axis rotation after the initial Y-axis rotation (Fig. 7c) dramatically increases Δ. These behaviors align with the preferred 〈110〉 texture observed experimentally and in polycrystalline simulations at the same anisotropy parameters (Figs. 4, 6). In contrast, for (ϵ1, ϵ2) = (0.06, 0), X-axis rotations (blue line and cube in Fig. 7d) yield the smallest Δ, consistent with the preferred 〈100〉 texture observed in polycrystalline simulations at the same anisotropy parameters (Fig. 5). Notably, the rotation around the Y-axis for Al3Si also exhibit the minimum undercooling (Supplementary Fig. 21), indicating that the Si composition effect does not significantly affect the crystallographic texture evolution in PF simulations.

Methods
Experimental methods

AlSi thin films with Si contents of ≈ 1, 3 and 6 wt. %, exhibiting homogeneous initial grain sizes and random textures, were used in this work. For the sake of brevity, alloys are referred to as Al1Si, Al3Si and Al6Si, respectively. Thin films, approximately 100 nm thick, were deposited onto amorphous, 0.25 × 0.25 mm wide, 50 nm-thick SiN windowed TEM grids by DC magnetron sputtering. Sputtering was conducted in a chamber with a base pressure below 6.7 10−6 Pa, and the process pressure was 0.44 Pa. The main sputter target was Al1Si, run at a power of 300 W, with a secondary Si target to increase the Si content, run at a power of 25–75 W. Further information about the thin film fabrication process can be found in Refs. 38, 49.

We carried out in situ laser spot melting and imaging experiments by movie-mode DTEM at 200 kV50. The thin films were melted by a pulsed Nd:YAG laser (1064 nm wavelength) with a Gaussian beam (1/e2 diameter of ≈ 135 μm), a deposited energy ranging from 2.32 to 3.92 μJ, and a dwell time of 50 ns. As detailed in other works51, after a certain delay time subsequent to the laser shut-off, a 2k x 2k CCD camera with single electron detection capability50 was used to capture nine consecutive elliptical low-magnification multi-beam bright-field TEM images of major and minor axes of ≈ 90 and 80 μm, respectively, and temporal resolution of 1-2.5 μs. Every nine-image series was recorded in a 256 x 256 pixel image file. Therefore, each experiment recorded images corresponding to a maximum duration of 22.5 μs. Because solidification was not always completed after such a period of time, up to three different laser hits were conducted per alloy system, varying the delay times from 0 to 40 μs to capture the entire solidification sequence. More details on the DTEM experimental procedures has been described in previous publications51.

We conducted detailed analyses of DTEM data by tracking the melt pool boundary, as shown in Fig. 1a, to estimate the major and minor ellipse semiaxes (a0e and b0e) as a function of time t. Note that the melt pool was ellipse-shaped, due to the non-perpendicular laser impact on the thin films during the experiments. Using the temporally resolved a0e and b0e, we calculated instantaneous V and plotted them against solid fraction at various Si contents, as shown in Fig. 1b. We performed polynomial fitting of the scattered data in Fig. 1b using second-order polynomials and the least squares method.

We then performed ex situ TEM-based orientation mapping (NanoMEGAS® ASTAR) in a Philips CM300 FEG (S)TEM operated at 300 kV to evaluate the crystallographic textures produced by the DTEM experiments. Orientation maps were obtained by indexing electron diffraction patterns acquired with an electron probe size of 5–10 nm, with average confidence indexes (CI) above 0.25 (see Supplementary Note 4 and Supplementary Note 5 for values corresponding to each scan). Two ASTAR scans were conducted per alloy, in the as-sputtered regions (step size of 5 nm) and near the melt pool boundary (step size of 25 nm); only fcc crystal symmetry was considered. Spot sizes ranging from 3 to 50 nm were used, depending upon the mapped area. ASTAR data analyses were conducted on with Matlab® 2024a and the MTEX toolbox 5.8.052. The analyses of the maps near the melt pool boundary included an initial data cleaning step, in which (a) grains were generated, with a threshold of 1∘; (b) grains with equivalent diameters below 58 nm were deleted; (c) unindexed and deleted pixels were interpolated using the method of nearest neighbor; (d) the process was repeated and grains were recomputed. Subsequently, areas affected by orientation misindexing due to the 180∘ ambiguity problem, described in ref. 53, were manually identified and their orientation was interpolated, where each wrongly indexed pixel was assigned the mean orientation of neighbors belonging to the same grain. After cleaning, grains were generated with a threshold of 5∘.

Phase field modeling

We employed a multi-PF model extending from a previously established model designed for the rapid solidification of binary alloys23, in which far-from-equilibrium effects like solute trapping at the solid-liquid interface are quantitatively reproduced over an extremely wide range interface velocities from μm/s to m/s. Notably, this model distinguishes itself by obviating the need for introducing corrective anti-trapping currents, as typically required in other widely adopted models54,55, to mitigate the spurious solute trapping effect across the solid-liquid interface. Instead of employing corrective anti-trapping currents, our model incorporates a nonlinear solute diffusivity as a function of the PF (ϕ) within the interface region. This approach provides a more accurate representation of the physical phenomena under consideration. Furthermore, the versatility of this multi-PF model allows for the utilization of a larger interface thickness without compromising numerical accuracy. This feature substantially increases the grid spacing achievable during numerical simulations, thereby enhancing computational efficiency and reducing computational resource requirements.

The polycrystalline PF model employed in this study encompasses the evolution of both a Si concentration field, denoted as c, and M = 60 order parameters represented as ϕα with α = 1, 2, ..., N and ϕα ∈ [−1, 1]. To simplify the experimental conditions while retaining the fundamental characteristics of the system, we applied the frozen temperature (T) approximation. This approximation define the temperature, T, as a function of both solidification time, t, and solidification distance, x, and is expressed as follows: T = T0 + G(x − Vpt), where the pulling velocity, Vp, and temperature gradient, G, are aligned with the solidification direction. Here, T0 represent the initial temperature at the melt pool border.

The evolution equations governing the dynamics of ϕα and c

τ(n) ∂ϕα/∂t = ∇·[W(n)2∇ϕα] + [∂i(j∇ϕαj2W(n) ∂W(n)/∂(∂iϕα))] + ϕα − ϕ3α − λg0(ϕα) [c + T − TM − me c∞eb[1 + g(ϕα)]] − ω(1 + 2ϕα) ∑β≠α (1 + 2ϕβ)2
∂c/∂t = ∇·[Dq(ψ)c∇[ln c − bg(ψ)]]

are derived based on the free-energy functional of binary alloys as detailed in Ji et al.23. Here, we take the choices of interface attachment timescale τ(n) = τ0 as(n)2/ak(n), where τ0 = (SW0)2/(Γμ0k), and interface width W(n) = SW0 as(n). These choices are based on the general forms of the interfacial free-energy anisotropy γ(n) = γ0 as(n) and kinetic coefficient anisotropy μk(n) = μ0k ak(n). Within these expressions, γ0 represent the reference value of interfacial free-energy, as(n) denotes the anisotropic strength of the interfacial free-energy, μ0k stands for the reference value of interfacial kinetic coefficient, ak(n) characterizes the anisotropic strength of the kinetic coefficient, n = ∇ϕ/|∇ϕ| represents the unit vector normal to the solid-liquid interface, S serves as a computational coefficient for adjusting interface thickness, W0 denotes the physical interface width (approximately 1 nm), Γ = γ0 Tm/L represents the Gibbs-Thomson coefficient, Tm denotes the melting point, and L stands for the latent heat of melting. Furthermore, the coefficients b and λ are given by b = ln ke/2 and λ = a01/b √me SW0 = [Γ(ke − 1)]−1, where ke is the partition coefficient, a01 = 2√2/3, and me < 0 represents the liquidus slope. In addition, the interpolation function of the energy density is of a common form g(ϕ) = 15(ϕ − 2ϕ3/3 + ϕ5/5)/8, which ensures g0(±1) = g00(±1) = 0 and g(±1) = ±1, thus allowing the free-energy density to attain local minima in the solid (ϕα = 1) and liquid (ϕα = −1) phases.

In this study, we emphasize the anisotropy of the interfacial free energy is critical to grain texture selection during rapid solidification of thin-film alloys. Using an angular embedded atom method (EAM) potential for AlSi in the molecular dynamics simulations, we have calculated the anisotropic parameters of γsl for the Si composition of interest (Supplementary Note 9). It is worth noting that Eq. (2) also applies to ak(n), with the distinction that the second-order term is absent. For comprehensive details regarding the implementation of as(n), please refer to Supplementary Note 8.

In Eq. (4), the term Dq(ψ) represent the solute diffusivity across the solid-liquid interface. Here, D corresponds to the diffusivity in the liquid phase, ψ = −1 + 2∑α(1 + ϕα)/2 serves as a mapping from ϕα to the overall liquid-solid interface. Additionally, the function q(ϕ) = 6(1 − ϕ) − 11(1 − ϕ)2/4 (applicable only for S = 5) is introduced as a nonlinear function to mitigate the spurious excess trapping. Importantly, the variational derivation of the free-energy functional necessitates the replacement of g(ϕα) in Eq. (4) with g(ψ) because the interpolation function g(ϕ) is originally introduced to interpolate free energy between solid and liquid phases. However, when dealing with more than two order parameters, the stability of PF modeling dictates that g(ϕα) should be retained in the evolution equation of ϕα. Otherwise, a stable fixed point emerges at ϕα ≈ −0.5, which is not physically valid, as there is no corresponding phase that exists at this point.

The final term in the evolution equation of ϕα shown in Eq. (4) accounts for the coupling between adjacent grains, with the positive value of ω representing a repulsive force. Assuming that ψ remains constant across the grain boundary, we can derive a unique expression for ω. To obtain this, we first consider the evolution equation for c. At equilibrium, integrating the right-hand side from −∞ to x yields c = c∞eb[1 + g(ψ)]. For the one-dimensional equation of ϕ, we multiply both sides by dϕα/dx and integrate from −∞ to ∞. The resulting relation can then be solved by substituting c = c∞eb[1 + g(ψ)] and using the constraint ψ = 1 + ϕ1 + ϕ2 = 1. Consequently, we obtain

ω = 6λc [2ke + (1 − ke)T − TM / me c∞] (5)

Here c∞ denotes the nominal composition. Notably, this expression is independent of the interface width, and in the limit where the temperature aligns with the liquidus line at Tl = TM + me c∞ for the parameters listed in Table 1, we find that ω = 0.652.

The evolution equations were implemented in C/C++ within the CUDA (Compute Unified Device Architecture) framework to leverage parallel computation. Simulations were performed on NVIDIA V100 or H200 GPUs in single precision to maximize computational speed and accommodate a larger number of phase fields. For a typical system size of 5120 x 5120 grid points, a single simulation on an NVIDIA V100 GPU required ~10 days of wall-clock time. Simulation data are processed using Python (3.9.6) codes.

Significance Analysis of Solidification Texture (SAST)

The elliptic/circular nature of the solid-liquid interface during thin-film spot melting hinder the interpretation of texture development as a function of time, as local solidification directions vary considerably for every grain. Here, we present a novel methodology to analyze the statistics of grain texture development during thin-film solidification: Significance Analysis of Solidification Texture (SAST). Even though the examples given in this section correspond to the experimental data, the same methodology has been applied to the modeling output.

Firstly, the coordinates of the experimental solid-liquid interface as a function of solidification time t were determined, as explained in Supplementary Note 2. Once these coordinates are known, the crystallographic map can be cropped accordingly to analyze the texture data for that specific time t. Fig. 2a shows a schematic diagram for Al3Si, where data within the solid-liquid interface for that time-step and data out of the melt pool boundary has been deleted. In the PF simulations, the solid-liquid interface at a specific time was determined using essentially the same method as in the experiments. Since the melt pool was assumed circular in PF simulations, the calculation can be simplified. In particular, we first identified the dendrite tip position at this specific time and then drew a circle at this position for the grain texture at the final solidification time. Next, for each time-step t, we delimited grains considering a maximum inner grain misorientation of 5∘, as described in refs. 56, 57. In Fig. 2a, grains are delimited by black lines. For a solidification time t, only the grains in contact with the solid-liquid interface, not truncated by the crystallographic map boundaries, were selected for analysis. Details on how filtering was conducted can be found in Supplementary Note 3.

Once grain filtering was finished, we proceeded to analyze the texture data. In the schematic diagram in Fig. 2, some of the definitions introduced in this methodology are provided. Moreover, the green grain delimited by a thick black line in Fig. 2a will be used to define some of the most important variables by the panels on the right, Fig. 2b–d. For every selected grain, we filtered the data-points located at less than 200 nm from the interface, i.e., Di,j ≤ 200 nm (refer to Supplementary Note 3 for a definition of Di,j), see area delimited by a dashed red line in Fig. 2a. Then, we calculated the mean crystallographic orientation (in matrix notation) and centroid coordinates of the selected region: O and (xc, yc), respectively. The reason why the data contained in the whole grain was not considered for the calculation was to avoid artifacts in the calculation associated with experimental orientation gradients, detected as the distance from the melt pool boundary increased, as observed in the same figure by the changes in Inverse Pole Figure (IPF) color within individual grains.

For every grain, the grain coordinate system, (X, Y, Z), was defined. The solidification direction vector X was approximated as the vector perpendicular to the tangent to the section of the solidification front near the grain. The considered section of the solid-liquid interface included any interface point within a 500 nm distance from the centroid (xc, yc). Note that, according to previous works in the literature where thin sample geometries were utilized, dendrites are expected to grow in-plane, as impingement with the sample border forces dendrite arms growing out of the plane to either cease growth or change their growth direction17. The two remaining directions of the grain coordinate system were defined as follows: Y was defined as the in-plane direction perpendicular to X and Z was defined as the out-of-plane direction, see Fig. 2b.

Utilizing this new coordinate system can be useful to estimate the primary and secondary dendrite growth directions. However, to further elucidate trends, a more quantitative and general methodology is needed. In Fig. 2b we defined the secondary [100] and [010] directions (unit vectors d in the crystal coordinate system, denoted by red arrows), which are the growth directions of the secondary arms, as those 〈100〉 crystallographic directions (d<100>) with the largest projection onto the Y-axis in absolute value, i.e., the largest d<100> ⋅ Y. Note that previous investigations have shown that secondary arms of dendrites undergoing DOT are not necessarily perpendicular to the dendrite trunk, which is parallel to the X15,17; confirmation of this assumption would imply additional TEM analysis. Nevertheless, this definition allows for systematic assessment of texture development.

In the plane perpendicular to ds[100], the potential primary [010] and [011] directions (unit vectors d, denoted by blue arrows) are defined as those 〈100〉 directions (d<100>) with the maximum projection onto the X-axis, i.e., the greatest d<100> ⋅ X. Similarly, the primary [011] and [011] (unit vectors d, denoted by orange arrows) are chosen from the 〈011〉 family (d<011>) having the largest projection onto the X-axis, i.e., the greatest d<011> ⋅ X. Considering two possible primary directions allows us to assess which one is more probable, or whether intermediate growth directions may occur. Alternatively, for a specific grain i, these directions can be defined so that their projections onto Xi or Yi satisfy:

ds[100],i ⋅ Yi = max{dα,i ⋅ Yi : α = 〈100〉}

dp[010],i ⋅ Xi = max{dα,i ⋅ Xi : α = 〈100〉 ∖ ds[100],i}

dp[011],i ⋅ Xi = max{dα,i ⋅ Xi : α = 〈011〉 ∖ ds[100],i} (6)

Using these directions, we defined the overall alignment of N grain orientations, illustrated in Eq. (1), quantified the statistical alignment of grains compared to random distribution via Z-scores. Specifically, Z-scores are given by zα = (Aα − μα)/σα, where α ∈ {s[100], p[010], p[011]}, and μα and σα represent the mean and standard deviation, respectively, of Aα for randomly distributed grain orientations. Explicitly,

zs[100] = (As[100] − μs[100])/σs[100],
zp[010] = (Ap[010] − μp[010])/σp[010],
zp[011] = (Ap[011] − μp[011])/σp[011] (7)

A Z-score measures the number of standard deviations a given data point lies from the mean. In this work, the sum of the projections of the selected growth directions have been selected to quantitatively evaluate the grain orientations of a whole texture.

Finally, for every alloy and time step, the Z-scores for perfect alignment (zs[100],pa, zp[010],pa, zp[011],pa) were calculated. This was achieved assuming Y was perfectly aligned with d vector for every grain and X was perfectly aligned with d or d vectors for every grain. These Z-scores were calculated as follows:

zs[100],pa = (N − μs[100])/σs[100]
zp[010],pa = (N − μp[010])/σp[010]
zp[011],pa = (N − μp[011])/σp[011] (8)

Statistics and reproducibility

A large number of random grain orientations (ranging from 100,000 to 1,000,000), each defined by three orthogonal unit vectors (see Supplementary Note 8 for details), were generated to compute alignment distributions for N randomly oriented grains and the corresponding Z-scores (Figs. 4–6). The projections of these orientations were grouped into sets of N data points, discarding any remainder. Within each group, the N projections were summed to obtain A values. The large number of random orientations ensures statistically robust estimates of the mean and standard deviation of A used in Z-score calculations. For the alignment distributions shown in Fig. 4a, b, alignment values were binned into 20 intervals, and the random orientation generation process was repeated ten times to compute mean frequencies and standard deviations for each bin.

The statistical significance is represented by the Z-score, which quantifies the deviation of an experimental or simulated measurement of the alignment from the mean of the random-orientation distributions. The Z-score is therefore not associated with standard error. In Figs. 4–6 and other Supplementary Figures., each Z-score corresponds to a single measurement encompassing all grains within a given texture. Because alignment is computed as the sum of grain-orientation projections, increasing the number of grains reduces stochastic uncertainty. Accordingly, for the four simulations shown in Supplementary Figs. 12A, 13A, and elsewhere in the Supplementary Information, Z-scores were calculated using all grains in the alignment analysis, and thus do not yield standard errors for the reported values.

Data availability

All raw data and codes used to reproduce the results in this study have been deposited in Figshare58. The DOI link of this dataset is https://doi.org/10.6084/m9.figshare.30460382.v2. Source data are provided in this paper.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The manuscript presents a detailed crystallographic orientation analysis of the rapid solidification process in laser-remelted Al–Si alloy thin films. The proposed SATE method demonstrates reasonable accuracy in quantitatively characterizing the evolution of crystallographic textures and shows potential for application in related experimental studies. However, the study predominantly focuses on validating the accuracy and reliability of the measurement method—a point that is already well established—while lacking compelling scientific discoveries. Moreover, the manuscript does not sufficiently explore the underlying physical mechanisms governing texture formation during melt pool solidification. As a result, its implications for alloy design and materials processing remain limited. In addition, the authors should carefully consider and address the following specific issues:\n1. Please define the term \"Z-scores\" upon its first appearance in the manuscript to ensure clarity for readers who may not be familiar with its statistical meaning in this context.\n2. For clarity, the symbols R and R₀ should be explicitly labeled in Figure 1(a).\n3. The initial microstructure of the Al–Si thin film should be described in detail. Additionally, the formation of nano-sized grains outside the melt pool boundary, as shown in Figure 1(c), warrants further explanation. If these grains are formed during rapid solidification near the melt pool boundary—as frequently observed in LPBF-processed Al alloys—this should be explicitly clarified. Moreover, such nano-grains may serve as nucleation substrates for columnar grains, indicating that they play a critical role in texture formation and evolution.\n4. In Lines 240–242, the manuscript discusses a two-dimensional (2D) thin-film scenario, whereas actual melt pool structures are inherently three-dimensional (3D). The authors should provide a clearer analysis of the differences and potential correlations between the 2D experimental setup and realistic 3D solidification behavior. As correctly acknowledged in the manuscript, 2D thin-film solidification experiments cannot fully capture the complexity of 3D crystal growth.\n5. The manuscript is encouraged to include a discussion of the mechanisms behind texture selection, particularly from the perspective of grain competition during solidification. This process plays a key role in determining the final crystallographic texture.\n6. The statistical significance of the conclusions is limited by the relatively small number of analyzed melt pools and grains, especially in the later stages of solidification. This limitation raises concerns about whether the observed preferential orientations can be considered statistically meaningful.\n7. The acronym \"SATE\" (Statistical Analysis of Texture Evolution) may not adequately convey the physical nature of the method. The authors are encouraged to consider renaming it to better reflect the method’s fundamental principles and physical underpinnings.\n8. The potential influence of Si content on crystallographic texture evolution should be analyzed and discussed. Variations in Si concentration may significantly affect the solidification behavior and resulting microstructures.\n9. The manuscript’s writing is occasionally obscure and unnecessarily complex. The authors are advised to revise the text for improved clarity, conciseness, and readability to enhance its accessibility to the broader scientific community.", "reviewer_2": "In the present article, the authors conducted pulsed-laser-induced melting experiments and phase-field simulations to study the crystallographic texture evolution of AlSi thin films during the re-solidification. The authors developed a statistical analysis methodology (SATE) to show that the experimentally observed textures are statistically significant and different from standard <100> textures. In addition, the phase-field (PF) simulations show that a two-parameter anisotropy form is required to obtain experimentally observed textures, and the anisotropy of γsl has a more significant effect on grain textures compared to kinetic anisotropy. The results presented in the paper are useful for quantification of grain textures and theoretical studies on microstructure evolution during AM processes. However, a few important points need to be clarified before publication.\n(1) Indeed, the SATE method enables quantitative evaluation of grain textures and identification of preferred orientations. However, as solidification progresses, the number of grains significantly decreases. As noted by the authors on Page 8, line 221, a larger area with more grains is needed for statistical relevance. Can the authors elaborate on the minimum number of grains required to ensure reliable SATE results?\n(2) In addition, the Z-score decreases with the number of grains, which makes it challenging to directly compare textures across systems with different numbers of grains. Would it be possible to normalize the Z-score on a per-grain basis to provide meaningful comparison between systems with different numbers of grains? This could help better elucidate the evolution of grain textures during solidification.\n(3) In Fig. 6(g), Zp[011] values for the PF model continuously decreases and becomes statistically insignificant, which deviates from experimental observations. Moreover, in Fig. SM10(C) and SM10(D), Zs[100] and Zp[011] data predicted by PF model lack statical significance, which is also inconsistent with experiments. Can the authors comment on these differences?\n(4) Minor suggestion: For Figs. 5, 6, SM9, SM10, SM11, and SM12, including the Si concentration for each PF model curve in the plots or captions would make things clearer and easier for readers to follow.", "reviewer_3": "Overall, the methodology is credible with an innovative way of quantitatively analyzing textures in melt pool based solidification microstructures, which could potentially be applied to metal additive manufacturing microstructures. In general, both the experimental and modelling based methods applied are of high quality and credible.\nOne of the main issues is that the temperature conditions used in the phase field simulations are very qualitative, which might not be clear to a reader not experienced with these modelling techniques. It is likely that the temperature distribution differs significantly in experimental DTEM conditions: in the beginning, the solidification velocity is small and thermal gradient is very large, and the thermal gradient decreases as the velocity increases. I believe the simplified temperature representation decreases the quantitative predictive power of the phase field simulations when compared to the DTEM experiments, including the conclusions made about the role of kinetic anisotropy vs. capillary anisotropy. Specifically, I believe that the thermal gradient is too small for these DTEM melt pools, producing large dendrites with notable side-branching, whereas I would expect the experimental DTEM microstructures to have smaller-scale microsegregation patterning without significant side-branching. How does the microsegregation patterning look like experimentally, in comparison to the phase field simulations? Justify why these results are still applicable even though the temperature distribution is heavily approximated. Performing simulation(s) at different G/V values, e.g. G = 1e7 K/m, would give more confidence to the simulation results.\nMore detailed comments\n• Because the title refers directly to metal AM, but the actual experiments and simulations are performed for thin films, I would highly encourage to modify the title to not mislead the reader. While the thermal gradients and solidification velocities are similar (typically DTEM has higher), DTEM is lacking many of the complexity of metal AM: large three-dimensional melt pools which are overlapping and producing occasionally very large grains, potential effects of melt flow on the solidification, nucleation of new grains, and so on.\n• Line 99: it should be explained what incubation time means.\n• Line 100-102: it is claimed that the solidification velocities decrease with increasing nominal Si content due to decrease in thermal conductivity. I doubt this is the only reason for the difference in velocities.\n• What is the initial grain size in phase field simulations and experiments? Based on Fig. 5h & 6h, it seems that the phase field simulated grain size is very small compared to the experimental grain sizes. Increasing the grain size in the phase field simulations to a more realistic value might increase the discrepancy between experiments and simulations, as the simulated columnar dendrites forming are very large, likely much larger than found experimentally.\n• What do the velocity-dependent k(V) curves look like for each Al-Si alloy in the phase field model?\n• It should be explained how the grain-grain interaction term was derived (eq. 5) for the phase field model.\n• Capillary fluctuation method\no In Supplementary materials it is claimed that Gamma = 2.4e-7 mK, whereas in the main manuscript the Gamma = 1.9e-7 mK, from an experimental source. Clarify what value is used in the phase field simulations.\no Also, it seems that only the solid-liquid anisotropy from CFM is reported and utilized , but CFM also gives the solid-liquid interface energy magnitude. What is the value of the CFM-based solid-interface energy magnitude? Was the CFM-based solid-liquid interface energy magnitude not used in the phase field simulations, and if not, why?\no \"This step was to ensure that the crystal-melt interfaces in the following steps do not merge\" needs clarification. I think what is meant by this is that 'The computed solid and liquid compositions are used in the CFM simulations to ensure correct equilibrium dynamics' or similar.\no MD-derived gamma0 could allow for another estimate of mu as well and kinetic anisotropy for this potential, using Equation 30. Why was this not done, and instead, literature-derived value was used from a different MD potential?\no Minor note: the spelling is Gibbs-Thomson (no “p”)\n• In metal AM there is flow effects. How are the results expected to be changed if metal AM like flow conditions were imposed to the melt? What about the confinement effects that were briefly mentioned?\n• The thin films have a finite (albeit relatively small) thickness, on the order of 50-100 nm. This is far above the typical solutal capillary length for metallic alloys, including aluminum alloys. Justify why the phase field simulations can be carried out in 2D, neglecting the film thickness. How would you expect the results to change if the finite film thickness were resolved in the phase field simulations."}, "author_response": "Response to reviewers:\nQuantification and prediction of solidification textures\nunder additive manufacturing conditions\nMingwang Zhong1,5, Adriana Eres-Castellanos2,3,5, Kaihua Ji1, Alec I. Saville2, Brian R.\nRodgers2, Dan R. Coughlin3, John W. Gibbs3, John D. Roehling4, Joseph T. McKeown4, Amy J. ∗\nClarke2,3,*, Alain Karma1,*\n∗1DepartmentofPhysicsandCenterforInterdisciplinaryResearchonComplexSystems,NortheasternUniversity,\nEngineering, Colorado School of Mines, 920 15th St, Hill Hall, Golden, CO, 80401, USA. 3Sigma Manufacturing\nScienceDivision,LosAlamosNationalLaboratory,P.O.Box1663,LosAlamos,NM,87545,USA.4MaterialsScience\nDivision,LawrenceLivermoreNationalLaboratory,Livermore,CA,94550,USA.5Theseauthorscontributedequally\nto this work. *Corresponding author. Email: aclarke@lanl.gov, amyclarke@mines.edu; a.karma@northeastern.edu\nThe manuscript presents a detailed crystallographic orientation analysis of the rapid solidification\nprocessinlaser-remeltedAl–Sialloythinfilms. TheproposedSATEmethoddemonstratesreasonable\naccuracy in quantitatively characterizing the evolution of crystallographic textures and shows\npotentialforapplicationinrelatedexperimentalstudies. However,thestudypredominantlyfocuses\non validating the accuracy and reliability of the measurement method—a point that is already well\nestablished—while lacking compelling scientific discoveries. Moreover, the manuscript does not\nsufficiently explore the underlying physical mechanisms governing texture formation during melt\npool solidification. As a result, its implications for alloy design and materials processing remain\nlimited.\nWe thank the reviewer for constructive criticisms. In response to the reviewer’s suggestion, we\nperformedadditionalPFsimulationstoexplorethephysicalmechanismgoverningtextureselection.\nThe results, which are discussed in more detail in our answer to point (5) of the reviewer below,\ndemonstrate that the classical Walton and Chalmers minimum undercooling criterion predicts the\nobserved textures remarkably well. While these additional results enlarge the scope of our work,\nwe would like to emphasize that the new proposed Statistical Significance of Solidification Texture\n(SAST) method (see the answer to point (7) below for the change of acronym) is in our view quite\nsignificant in and of itself. In particular, conventional pole figures cannot quantitatively assess the\nstatistical significance of the grain textures observed here and more broadly textures with a high\ndegree of randomness. These textures consist of a large number of grains with significant variations\nin growth orientation relative to a thermal axis that itself varies around the melt pool. As shown\nin Fig. 3d,e, the pole figure data appear noisy and do not reveal a clear texture trend. In contrast,\nthe SAST method yields a Z-score that incorporates all grains with different crystallographic\norientations, allowing extraction of quantitative information from the experimental textures (Fig.\n4). Using this approach, we find that thin-film AlSi alloys exhibit 110 -oriented growth, with one\n⟨ ⟩\n[100] direction aligned perpendicular to the solidification direction.\nFurthermore, with SAST, we are able to infer anisotropic crystal growth properties. Our combined\nexperimental observations and polycrystalline PF simulations demonstrate that the anisotropy of\nthesolid–liquidinterfacialfreeenergyisthedominantfactorgoverninggraintextureevolutioninthe\ncellular/dendriticregimeduringrapidsolidification. Specifically,simulationsusingtheconventional\nanisotropy parameters with a single leading value, (ϵ ,ϵ ) = (0.06,0), produce a 100 -dominated\n1 2\n⟨ ⟩\ntexture(Fig. 5), whereassimulationswith(ϵ ,ϵ ) = (0.06, 0.006), obtainedfromMDsimulations,\n1 2\n−\nyield a 110 -dominated texture (Fig. 6) that agrees remarkably well with the experimental results\n⟨ ⟩\n(Fig. 4). Additionally, simulations with or without the kinetic anisotropy show that it plays only\na minor role in texture selection during rapid solidification.\nThese findings are scientifically meaningful because existing studies have focused on either (1) slow\nsolidification conditions, where γ anisotropy is the critical factor [1–4], or (2) extremely rapid\nsl\nsolidification in deeply undercooled melts or planar front regimes [5–8], where kinetic anisotropy\nbecomesmoreimportant. Therefore,inadditiontopresentinganewmethodtoquantifysolidification\ntextures and explaining their origin (with the new added results in answer to point (5) below), our\nresults clarify the relative roles of energetic and kinetic anisotropies in determining the intragrain\ncellular/dendritic microstructurethat is anothermajor determinant ofthe mechanical propertiesof\npolycrystalline metallic alloys produced in varied AM processes. We revised the bottom paragraph\nof the introduction to summarize our main results and their impact.\n(1) Please define the term ”Z-scores” upon its first appearance in the manuscript to ensure clarity\nfor readers who may not be familiar with its statistical meaning in this context.\nZ-scoring has been explained where it is first introduced in the introduction.\n(2) For clarity, the symbols R and R should be explicitly labeled in Figure 1(a).\nnotation r0 r in the context of the elliptic melt pool geometry, we have replaced it with “solid\n−\nfraction” on the x-axis of the plot. The variables r and r0 no longer appear in the manuscript.\nAdditionally, we remeasured the DTEM melt pool major and minor semiaxes, a0 and b0, and\ne e\nrecomputed the analyses. The velocity plots in Figure 1(b), along with all other affected figures,\nhave been updated. These revisions do not alter the main conclusions of the study. The changes\nin Methods read as follows:\nDetailed data analysis was performed to estimate the major and minor ellipse semiaxes\n(a0 and b0) and instantaneous solidification velocity (V) values with time.\ne e\nFig. 1b presents V as a function of the solid fraction (by area) at various Si contents,\ntogether with fitted curves. This result demonstrates increasing V as the solid fraction\nincreases, i.e., V increases with solidification time t towards the center of the melt pool,\nconsistent with the findings reported by McKeown et al. [9].\nScatter plots of solidification velocity (V) versus the solid fraction (defined in\nSupplementary Note 2) for three solute contents (Al1Si, Al3Si and Al6Si), together\nwith curves fitted to the data for each composition.\nWe conducted detailed analyses of DTEM data by tracking the melt pool boundary,\nas shown in Fig. 1a, to estimate the major and minor ellipse semiaxes (a0 and b0)\ne e\nas a function of time t. Note that the melt pool was ellipse-shaped, due to the\nnon-perpendicular laser impact on the thin films during the experiments. Using the\ntemporally resolved a0 and b0, we calculated instantaneous V and plotted them against\ne e\nsolid fraction at various Si contents, as shown in Fig. 1b. We performed polynomial\nfitting of the scattered data in Fig. 1b using second-order polynomials and the least\nsquares method.\nAdditional edits have been made in the Supplementary Material, although they are not included\nhere for the sake of brevity.\n(3) The initial microstructure of the Al–Si thin film should be described in detail. Additionally, the\nformation of nano-sized grains outside the melt pool boundary, as shown in Figure 1(c), warrants\nfurther explanation. If these grains are formed during rapid solidification near the melt pool\nboundary—as frequently observed in LPBF-processed Al alloys—this should be explicitly clarified.\nMoreover, such nano-grains may serve as nucleation substrates for columnar grains, indicating that\nthey play a critical role in texture formation and evolution.\nWe have clarified that the small nanoscale grains are formed during sputtering of the thin metallic\nfilms on the amorphous silicon nitride substrate before spot melting. We have also clarified that,\nduringtheearlystageorsolidificationafterspotmeltingofthethinfilm, thesmallnanoscalegrains\n3\nat the edge of the melt pool grow competitively. Since their orientation is measured to be close\nto random, the columnar grain texture that develops is selected primarily by the grain growth\ncompetition process and not by a pre-existing texture of the nanoscale grains that form during\nsputtering.\nTo include these results, we have revised the main text in the Result section:\nEx-situ TEM-basedorientationmappingwasperformedtoevaluatethecrystallographic\ntexturesproducedbytheDTEMexperiments; seeExperimentalmethodsforadditional\ndetails regarding the mapping. For this analysis, only face centered cubic (fcc) Al was\nconsidered. Two scans per chemical composition were conducted, where one of them\nwas taken in the as-sputtered region and the other one was taken near the melt pool\nboundary. The former captures the orientations outside the melt pool edge, in the\nheat-affected zone, which is characterized by small, equiaxed grains of approximately\nrandom orientations. The characterization of the as-sputtered and the heat-affected\nzonecanbefoundinSupplementaryNote4. Anexampleofthelattermappingisshown\nbytheInversePoleFigure(IPF)mapcorrespondingtoAl6Si,asshowninFig. 1c,where\ngrains are defined as regions with a maximum 5◦ inner-grain misorientation. Within\nthe melt pool, the grains that evolve from the edge to the center appear columnar.\nNotice that at the late stage of solidification, only grains colored red, yellow, and green\nsurvive, while the blue grains are eliminated.\nWe also revised the text in Methods:\n®\nWe then performed ex-situ TEM-based orientation mapping (NanoMEGAS ASTAR)\nto evaluate the crystallographic textures produced by the DTEM experiments.Two\nASTAR scans were conducted per alloy, in the as-sputtered regions and near the melt\npool boundary; only fcc crystal symmetry was considered.\nFinally,wepresentSupplementaryNote4,inwhichboththeas-sputteredregionsandtheheat-affected\nzones, by the melt pool boundaries, are characterized. Note that Supplementary Fig. SM1 and\nSupplementary Table SM1 have been here reproduced as Fig. R1 and Table R1.\nSupplementaryNote4. Characterizationoftheas-sputteredregionsandtheheat-affected\nzones\nSupplementaryFig. R1(a-c)showsInversePoleFiguremapsoftheas-sputteredregions\nand their corresponding Density Pole Figures. The corresponding experimental grain\nsizes, measured from these maps, are included in Supplementary Table R1. As can\nbe observed, as-sputtered material consists of small grains of 15-20 nm grain size,\nhomogeneouslydistributed. Moreover,theDensityPoleFiguresshowmaximumintensities\nbelow 1.6 Multiples of Random Distribution (MRD), which indicates near-random\ntexture. The heat-affected zones, characterized in Supplementary Fig.R1(d-f) and\nSupplementary Table R1 still show near-random texture. Moreover, significant grain\ngrowth is detected at all Si contents: grain size increases by an order of magnitude,\nmost likely due to thermal effects during metal melting.\n(4) In Lines 240–242, the manuscript discusses a two-dimensional (2D) thin-film scenario, whereas\nactual melt pool structures are inherently three-dimensional (3D). The authors should provide a\nclearer analysis of the differences and potential correlations between the 2D experimental setup\nin (a, d) Al1Si, (b, e) Al3Si and (c, f) Al6Si. Inverse Pole Figure (out-of-plane direction Z⃗) maps\nand corresponding (100) Density Pole Figures, with intensities measured in Multiples of Random\nDistribution (MRD).\nTable R1. Average grain size and standard deviation, estimated in the sputtered\nregions and the heat-affected zones at different chemical compositions\nAlloy Al1Si Al3Si Al6Si\nGrain size in sputtered region (nm) 17.70 12.85 14.86 7.27 15.08 7.16\n± ± ±\nGrain size in heat-affected zone (nm) 301.24 97.38 290.69 101.55 286.91 77.59\n± ± ±\nand realistic 3D solidification behavior. As correctly acknowledged in the manuscript, 2D thin-film\nsolidification experiments cannot fully capture the complexity of 3D crystal growth.\nAs explained in the manuscript, the phase-field simulations attempt to partially capture 3D effects\nin the thin film experiments, which are inherently 3D because the film has a finite thickness of\napproximately 100 nm. This was accomplished by using a more complex form of the anisotropy\nof the interface excess free-energy and the kinetic coefficient that takes into account the fact that\nthe sputtered nanoscale grains have random 3D orientations, and thus not necessarily a 100-axis\nperpendicular to the film as assumed in previous 2D simulation studies that use a simple cos4θ\nform [10, 11]. However, even with this improvement, our 2D simulations are still an approximation\nsince they do not consider the curvature of the interface in the thickness of the film, which could\n5\npotentiallyinfluencegraintextureselection. Wehaveclarifiedthetexttostatethislimitationmore\nexplicitly. We should note that polycrystalline 3D simulations with a film thickness of about 100\nnm are presently not computationally feasible. Furthermore, the interface curvature in the film\nthickness depends on the interfacial excess free energies between both solid and liquid AlSi and\nthe substrate that, together with the solid-liquid interfacial energy, determine the dihedral angle\nof the solid-liquid interface with the substrate. These energies are not precisely known and hard\nto compute by MD simulations, making it difficult to realistically model film-thickness curvature\neffects. However, perhaps more important than realistically modeling these effects is the relevance\nof our quasi-2D results for modeling grain texture selection in applied AM processes such as LPBF,\nwhich involve full 3D melt pools. We believe that the new insights into grain competition under\nAM conditions obtained in a quasi-2D geometry (detailed in the next response to point 5) support\nthe relevance of our results to a wide range of fully 3D AM processes.\nThe clarified text of the limitation in the Discussion reads as follows:\nNotably, these conclusions are drawn from 2D PF simulations, which approximate\nmore realistic three-dimensional (3D) scenarios. The 2D model neglects the curvature\nof the solid-liquid interface through the film thickness, potentially influencing grain\ntexture selection. Full-scale 3D polycrystalline simulations incorporating realistic\nfilm thicknesses ( 100 nm) remain computationally prohibitive. Additionally, the\n∼\ncurvature of the interface along the film thickness is controlled by the interfacial\nexcess free energies between solid/liquid AlSi and the substrate. These, together\nwith the solid-liquid interfacial energy, determine the dihedral angle formed at the\nsolid-liquid-substrate junction. These energies are not precisely known and are\nchallenging to compute using MD simulations, making it difficult to realistically model\nfilm-thickness curvature effects. Despite these limitations, we believe the quasi-2D\nresults presented here provide meaningful insights into grain texture selection relevant\nto applied AM processes, such as LPBF, which involve fully 3D melt pools. The\nfundamental mechanisms of grain competition under AM conditions, revealed through\nour quasi-2D simulations, support the broader applicability of these findings to a wide\nrange of 3D AM processes.\n(5)Themanuscriptisencouragedtoincludeadiscussionofthemechanismsbehindtextureselection,\nparticularly from the perspective of grain competition during solidification. This process plays a\nkey role in determining the final crystallographic texture.\nWe performed additional PF simulations to investigate whether the observed textures could be\nexplainedbytheclassicalWaltonandChalmersminimumundercoolingcriterion[12],whichassumes\nthat the grains that grow with the smallest undercooling (and thus ahead of other grains in a\ntemperature gradient) are preferentially selected. The results (Fig. 7) show that this criterion\npredicts remarkably well the selection of the 100 and 110 dominated textures for different\n⟨ ⟩ ⟨ ⟩\nforms of the anisotropy of the excess solid-liquid interface free-energy. This suggests that the same\ncriterion should be applicable to more general 3D geometries under AM conditions. This result is\ninteresting since several previous studies of grain competition under slow solidification conditions\nhave highlighted some limitations of this minimum undercooling criterion [10, 11, 13, 14]. We\nhope that the reviewer will appreciate that, while we added these results, a full exploration of\nthe mechanisms of grain competition under AM conditions (including grain boundary orientation\nselectionmapsconstructedbyexhaustivebi-crystalsimulationsasinRefs. [10,11,13,14])warrants\na longer exposition in a separate paper.\nmechanisms studied in our work, as follows:\nResults:\nThe grain textures observed in experiments and polycrystalline PF simulations (Figs.\n4-6) can be well explained by the minimum undercooling criterion, which assumes\nthat dendritic grains with the lowest tip undercooling are preferentially selected. To\ninvestigatethemechanismoftextureformation,weperformedsingle-crystalsimulations\nwith the principal axis rotating around the X⃗, Y⃗, and Z⃗ axes (Fig. 7a). For\n(ϵ ,ϵ ) = (0.06, 0.006), the tip undercooling ∆ decreases monotonically for rotations\n1 2\n−\naround the Y⃗ axis (orange line), but generally increases for other rotations. To\nexplain more complex orientations in polycrystalline simulations, we further examined\nsequentialrotations: asubsequentX⃗-axisrotationaftertheinitialY⃗-axisrotation(Fig.\n7c) dramatically increases ∆. These behaviors align with the preferred 110 texture\n⟨ ⟩\nobserved experimentally and in polycrystalline simulations at the same anisotropy\nparameters (Figs. 4,6). In contrast, for (ϵ ,ϵ ) = (0.06,0), X⃗-axis rotations (blue\n1 2\nline and cube in Fig. 7d) yield the smallest ∆, consistent with the preferred 100\n⟨ ⟩\ntextureobservedinpolycrystallinesimulationsatthesameanisotropyparameters(Fig.\n5). Notably, the rotation around the Y⃗ axis for Al3Si also exhibits the minimum\nundercooling (Supplementary Fig. SM21), indicating that the Si composition effect\ndoes not significantly affect the crystallographic texture evolution in PF simulations.\nDiscussion:\nThe minimum undercooling criterion identified inour study for AMconditions assumes\nthat cellular/dendritic grains with the lowest tip undercooling are preferentially\nselected, a mechanism traditionally used to explain grain competition under slow\nsolidification rates [12]. In this framework, differences in dendrite tip undercooling\ncause unfavorable grains to lag behind and be eliminated by favorable grains. Our\nfindings show that this criterion predicts remarkably well the selection of the 100\n⟨ ⟩\nand 110 dominated textures for different forms of the anisotropy of the excess\n⟨ ⟩\nsolid-liquid interface free-energy (Fig. 4-6). This suggests that the same criterion\nmay be applicable to more general 3D geometries under AM conditions. Although\nthe minimum undercooling criterion predicts which grain orientation is favored, it does\nnot predict how quickly a well-oriented grain overgrows neighboring grains with larger\nundercooling. Additional studies, which extend recent studies of grain competition in\na slow solidification regime [11, 13, 14], are needed to address this question.\n(6) The statistical significance of the conclusions is limited by the relatively small number of\nanalyzed melt pools and grains, especially in the later stages of solidification. This limitation\nraises concerns about whether the observed preferential orientations can be considered statistically\nmeaningful.\nThe reviewer raises a good question and answering this question quantitatively is exactly why the\nSAST method was developed. Even though the number of grains decreases with time due to the\nelimination of unfavorably oriented grains, the remaining preferential orientations are statistically\nmeaningful as long as the corresponding Z-score for perfect alignment remains above the threshold\nof statistical significance. This is the case in both experiments and simulations. Note that we have\nchanged the critical value for statistical significance to 1.96. Since Z-scores below -1.96 and above\n1.96arebothconsideredasstatisticallysignificantinourstudy, weuseatwo-tailedtestratherthan\n7\n100\n(a) (b)\n500 nm\n0.67\n0.66\n0.65\nZ X\n0.64\nRotation angle\n(c) 0.70 (d)\n0.67\n0.69\n40\n0.68 0.66\n0.67\n0.66 0.65\n20\n0.65\n0.64\n0.64\n0 20° 40° 0 20° 40°\nRotation angle around X Rotation angle\n0 20 40 60 80 11002\n-0.006). (b)Dendritetipundercooling∆asafunctionofrotationanglesaroundtheX⃗,Y⃗,andZ⃗ axes. Undercooling\nisdefinedas∆=(T −T )/(T −T ),whereT isthetemperatureatthetip,T istheliquidustemperatureatthe\nl tip l s tip l\nnominalcompositionc ,andT isthesolidustemperatureatc . Forbluecirclesbeyond30◦,∆wasobtainedusing\n∞ s ∞\nthe output of smaller-angle simulation as input. The open circle marks the unstable structure shown in panel (a).\n(c) ∆ for a subsequent X⃗-axis rotation after the initial Y⃗-axis rotation (20◦ for light blue lines and 45◦ for bright\nblue lines). The dashed line denotes ∆ without rotation. (d) ∆ versus the rotation angles for (ϵ , ϵ ) = (0.06, 0).\n1 2\none-tailed test: https://www.sfu.ca/personal/archives/richards/Zen/Pages/Chap17.htm#:\n~:text=If%20you%20are%20using%20the,test%20will%20therefore%20be%201.65.. The text\nnow reads as:\nNote that any measurement with a corresponding Z-score for perfect alignment above\n1.96 (corresponding to two grains) is statistically representative.\n(7) The acronym ”SATE” (Statistical Analysis of Texture Evolution) may not adequately convey\nthe physical nature of the method. The authors are encouraged to consider renaming it to better\nreflect the method’s fundamental principles and physical underpinnings.\nWe have replaced the acronym SATE by SAST “Significance Analysis of Solidification Texture”. A\nrelevantanalogyisthewidelyusedstatisticalmethodSAM(“SignificanceAnalysisofMicroarrays”)\nto assess the significance of microarray data in a biological context. We are open to better\nsuggestions.\n(8) The potential influence of Si content on crystallographic texture evolution should be analyzed\nand discussed. Variations in Si concentration may significantly affect the solidification behavior\nand resulting microstructures.\n8\ny=45\ny=20\n( 1, 2)=(0.06, 0.006)\ny\nz\nx\n( 1, 2)=(0.06, 0.006)\nx\nz\ny\n( 1, 2)=(0.06,0)\nWe have used our new Z-score-based method to analyze experimental grain textures for three\ndifferent Si compositions. The results reveal that textures for all compositions show a preferential\n110 texture, although differences across Si content cannot be distinguished in a statistically\n⟨ ⟩\nsignificant way. While the microstructures are expected to be more dendritic at larger composition,\nthe minimum undercooling criterion that selects 110 textures is expected to remain operative. To\n⟨ ⟩\ntestthisinterpretation,weperformedadditionalPFsimulationsasinFig. 7forAl3Si. Supplementary\nFig. SM21(alsoshownasFigureR3below)showsthattheorientationswithrotationaroundtheY⃗\naxis exhibit the minimum undercooling (blue line), consistent with the Al1Si result (Fig. 7).\nThis\nsuggests that even at higher Si compositions, the textures remain predominantly 110 -oriented.\n⟨ ⟩\n0.62\n0.61\n0.60\n0.59\n0.58\n0.57\n0 20° 40°\nRotation angle\nFigure R3. Dendritetipundercooling∆ofvariousorientationsforAl3Si. Solidsymbolsrepresentsdendrites,while\nopen circles correspond to hyperbranched structures.\n(9) The manuscript’s writing is occasionally obscure and unnecessarily complex. The authors\nare advised to revise the text for improved clarity, conciseness, and readability to enhance its\naccessibility to the broader scientific community.\nWe made our best attempt to improve clarity.\nInthepresentarticle,theauthorsconductedpulsed-laser-inducedmeltingexperimentsandphase-field\nsimulationstostudythecrystallographictextureevolutionofAlSithinfilmsduringthere-solidification.\nThe authors developed a statistical analysis methodology (SAST) to show that the experimentally\nobservedtexturesarestatisticallysignificantanddifferentfromstandard 100 textures. Inaddition,\n⟨ ⟩\nthe phase-field (PF) simulations show that a two-parameter anisotropy form is required to obtain\nexperimentally observed textures, and the anisotropy of γsl has a more significant effect on grain\ntexturescomparedtokineticanisotropy. Theresultspresentedinthepaperareusefulforquantification\nof grain textures and theoretical studies on microstructure evolution during AM processes.\nWe appreciate the reviewer’s positive reaction to the article.\nHowever, a few important points need to be clarified before publication.\n(1) Indeed, the SATE method enables quantitative evaluation of grain textures and identification\nof preferred orientations. However, as solidification progresses, the number of grains significantly\ndecreases. As noted by the authors on Page 8, line 221, a larger area with more grains is needed\nfor statistical relevance. Can the authors elaborate on the minimum number of grains required to\nensure reliable SATE results?\nThe reviewer raises an important question. The minimum number of grains required to ensure\nstatistical significance is 2, for which the Z-score of perfect alignment is 2.39—above the critical\nZ-score value of 1.96 for a two-tailed test (https://www.sfu.ca/personal/archives/richards/\nZen/Pages/Chap17.htm#:~:text=If%20you%20are%20using%20the,test%20will%20therefore%\n20be%201.65). Thisvalueisobtainedasfollows. Sincegrainorientationsareindependent,themean\nalignment scales with N and the standard deviation scales with √N: µ = Nµ and σ = √Nσ,\nN N\nwhere µ and σ are the mean and standard deviation of a single grain, respectively. The calculation\nin Supplementary Note 7 gives µ = 0.831 and σ = 0.1, leading to a Z-score of 1.69 for 1 grain\n(below 1.96) and 2.39 for 2 grains (above 1.96).\nHowever, this minimum of 2 grains does not explain the decreases in z and z observed\ns[100] p[110]\nat the mid-to-late stage, when the number of grains exceeds 2. To quantify the alignment of a\nsingle grain and to allow comparison of Z-scores across systems with different grain numbers—as\nthe reviewer suggests in the next comment—we normalize the Z-score by the corresponding value\nfor perfect alignment: z /z = (A µ )/(N µ ). Supplementary Fig. R4 shows that the\nα α,pa α α α\n− −\nZ-score ratios for d⃗ and d⃗ steadily increase during solidification, except at the late stage.\ns[100] p[110]\nThe late-stage decrease arises because the number of grains falls below 13, which is insufficient\n∼\nfor reliably selecting preferential orientations during grain competition.\nIn summary, the SAST method addresses two key questions: 1). Is the observed grain texture\nstatistically significant? 2). If so, how strong is the alignment compared to perfect alignment?\nAccordingly, the sentence “a larger area with more grains is needed for statistical relevance” has\nbeen removed from the Results section, and the following paragraph has been added:\nNotably, since the standard deviation of alignment scales with √N, the Z-scores for\nperfect alignment of 1 and 2 grains are 1.69 and 2.39, respectively. Thus, the minimum\nnumber of grains required for statistical relevance is 2. However, this does not explain\n10\nthe decreases in z and z at the mid-to-late stage, when the number of grains\ns[100] p[110]\nis larger than 2. To evaluate the alignment of a single grain and to enable comparison\nof Z-scores across systems with different grain numbers, we normalize the Z-score\nusing the corresponding value for perfect alignment: z /z = (A µ )/(N µ ).\nα α,pa α α α\n− −\nSupplementary Fig. SM5 shows that the Z-score ratios for d⃗ and d⃗ steadily\ns[100] p[110]\nincrease during solidification, except in the late stage. The late-stage deviation arises\nbecause the grain number decreases substantially below 13, which is insufficient for\n∼\nreliably selecting preferential orientations during grain competition.\n(2) In addition, the Z-score decreases with the number of grains, which makes it challenging to\ndirectly compare textures across systems with different numbers of grains. Would it be possible\nto normalize the Z-score on a per-grain basis to provide meaningful comparison between systems\nwith different numbers of grains? This could help better elucidate the evolution of grain textures\nduring solidification.\nWe normalize the Z-score using the corresponding value for perfect alignment: z /z =\ns[100] s[100],pa\n(A µ )/(N µ ), as detailed in the previous response.\nα α α\n− −\n0.8\n0.6\n0.4\n0.2\n0.0\n0 0.5 1\nSolidified fraction\nap,]001[sz/]001[sz\n(a)\n0.1\n0.0\n0.1 Al1Si\nAl3Si\nAl6Si 0.2\n0 0.5 1\nSolidified fraction\nap,]010[pz/]010[pz\n(b) 0.7\n0.6\n0.5\n0.4\n0.3\n0.2\n0.1\n0.0\n0 0.5 1\nSolidified fraction\nap,]011[pz/]011[pz\n(c)\nFigure R4. RatioofexperimentallymeasuredZ-scorestothoseforperfectalignment. (a)RatioofZ-scoresforthe\nsecondary[100]direction. (b)RatioofZ-scoresfortheprimary[010]direction. (c)RatioofZ-scoresfortheprimary\n[110] direction.\n(3) In Fig. 6(g), Zp[011] values for the PF model continuously decreases and becomes statistically\ninsignificant, which deviates from experimental observations. Moreover, in Fig. SM10(C) and\nSM10(D), Zs[100] and Zp[011] data predicted by PF model lack statical significance, which is also\ninconsistent with experiments. Can the authors comment on these differences?\nThe decrease in z at the late stage of solidification in Fig. 6(g) arises from the reduced number\np[011]\nof grains (N, panel h). This follows from the Z-score definition, z = (A µ )/σ . Since grains\nα α α α\n−\nare independent, both A and µ scale with N, while σ scales with √N. Thus, z increases\nα α α α\n⟨ ⟩\nwith N, leading to insignificant Z-scores when N becomes too small.\nIn Supplementary Fig. SM10 C and D (now Supplementary Fig. SM13), z and z fall\ns[100] p[011]\nbelow statistical significance for two reasons. First, the limited number of simulated orientations\nmeans that some favorable orientations are missing in certain regions of the melt-pool boundary.\nFor example, in the middle-left region of Supplementary Fig. SM13(C) and bottom-right region of\n(D), orientations with the secondary [100] direction perpendicular to the solidification direction are\nabsent, resulting in non-optimal selections. Second, even when favorable orientations appear at the\nboundary, they may be eliminated during grain competition due to microstructural fluctuations.\nDespitetheseeffects, TheZ-scorescalculatedfromfoursimulations(SupplementaryFig. SM13(A))\nshowsthatinthemid-stageofsolidification,whenN issufficientlylarge,thez andz values\ns[100] p[011]\n11\nremain statistically significant, supporting the results shown in Fig. 6.\n(4) Minor suggestion: For Figs. 5, 6, SM9, SM10, SM11, and SM12, including the Si concentration\nfor each PF model curve in the plots or captions would make things clearer and easier for readers\nto follow.\nThank you for the comment. We have made sure that all captions include the Si content of the PF\nsimulation, i.e., Al1Si.\nOverall, the methodology is credible with an innovative way of quantitatively analyzing textures in\nmelt pool based solidification microstructures, which could potentially be applied to metal additive\nmanufacturing microstructures. In general, both the experimental and modelling based methods\napplied are of high quality and credible.\n(1) One of the main issues is that the temperature conditions used in the phase field simulations\nare very qualitative, which might not be clear to a reader not experienced with these modelling\ntechniques. ItislikelythatthetemperaturedistributiondifferssignificantlyinexperimentalDTEM\nconditions: in the beginning, the solidification velocity is small and thermal gradient is very large,\nand the thermal gradient decreases as the velocity increases. I believe the simplified temperature\nrepresentation decreases the quantitative predictive power of the phase field simulations when\ncompared to the DTEM experiments, including the conclusions made about the role of kinetic\nanisotropy vs. capillary anisotropy. Specifically, I believe that the thermal gradient is too small\nfor these DTEM melt pools, producing large dendrites with notable side-branching, whereas I\nwould expect the experimental DTEM microstructures to have smaller-scale microsegregation\npatterning without significant side-branching. How does the microsegregation patterning look\nlike experimentally, in comparison to the phase field simulations? Justify why these results are\nstill applicable even though the temperature distribution is heavily approximated. Performing\nsimulation(s) at different G/V values, e.g. G = 1e7 K/m, would give more confidence to the\nsimulation results.\nWe thank the reviewer for highlighting this important point regarding temperature conditions\nused in the PF simulations. Indeed, the simplified temperature distribution employed in our PF\nsimulations is an approximation of the experimental conditions in DTEM, where the solidification\nvelocity initially is low with a correspondingly high thermal gradient that gradually decreases\nas velocity increases. Despite the simplified temperature approximation resulting in dendritic\nstructures with notable side-branching, we believe the fundamental conclusions regarding the\ndominant role of γ anisotropy remain robust. Our findings primarily focus on grain selection\nsl\nmechanisms and the comparative influences of γ and kinetic anisotropies, which are largely\nsl\nindependent of the exact microsegregation patterns or detailed temperature profiles.\nWhile previous literature (e.g., [15]) suggests higher thermal gradients for these systems, the exact\nmagnitude in experimental Al-Si alloys under DTEM conditions remains uncertain. To investigate\nthe effects of thermal gradient (G) and solidification velocity (V), we performed additional PF\nsimulations with an increased thermal gradient of 5 K/µm, as estimated in Refs. [16–18]. These\nsimulations incorporate a velocity profile directly based on our experimental measurements (Fig.\n1b). In these supplementary simulations, both γ and kinetic anisotropies were set identically to\nsl\nthe parameters used in Fig. 6 of the main manuscript.\nSupplementary Figs. SM18 and SM19 (also reproduced in the main text as Figs. R5 and R6)\nshow that the previously observed crystallographic texture trends are largely preserved at the\nhigher thermal gradient. Although some individual Z-scores occasionally drop below the statistical\nsignificance threshold (1.96), the overall statistical significance of z and z clearly indicates\ns[100] p[110]\nthe persistence of the 110 texture. This is particularly evident when Z-scores are calculated from\n⟨ ⟩\nfour simulations (Supplementary Fig. R6A), where the significance of z and z during the\ns[100] p[110]\nintermediate solidification stages is consistently high.\nthe γ anisotropy turned off (isotropic interfacial energy) but retaining the kinetic anisotropy.\nsl\nSupplementary Fig. SM20 (also reproduced as Fig. R7) reveals only weak development of a 100\n⟨ ⟩\ntexture under these conditions. Taken together, these additional simulation results (Figs. R5,\nR6, and R7) reinforce our conclusion that γ anisotropy remains the dominant influence on grain\nsl\ntexture selection, whereas kinetic anisotropy plays only a minor role, even under conditions with\nelevated thermal gradients.\nWhen the thermal gradient is further increased to 10 K/µm, however, the Z-scores do not indicate\nstatistical significance (results not shown). This suggests that the thermal gradient in experiments\nmay not be as large as 10 K/µm. Notably, previous studies have estimated the thermal gradient\nin a 3D melt pool to be 1.5 K/µm [18], which supports our estimation of the thermal gradient\n≤\nin this study.\nRegarding microsegregation patterns, we recognize the importance of validation via comparison of\nthat comparison of\nthesegregationpatternsofthedendriticstructures. Unfortunately,wehavenotbeenabletoresolve\nthe dendrites in the microstructures or measure the segregation patterns due to two reasons:\n(a) Dendrites cannot be resolved by TEM, on High-angle annular dark-field imaging, HAADF.\nHAADF mode enables the resolution of chemical differences, with increasing contrast as the atomic\nnumberdifferenceishigher. BecauseAlandSi’satomicnumbersare13and14,nochemicalcontrast\nis observed, as can be seen in Fig. R8.\n(b) Because the Al-Si alloys in this work are deposited on SiN substrates, we are not able to\nX\nobtainhigh-qualityquantitativeEnergy-dispersiveX-rayspectroscopy(EDS)data,hencewecannot\nresolve the microsegregation patterning.\n14\n(a) (001) (110) (111)\n(b)\n(c)\n(d)\n2 m\n6\n4\n2\n0\n2\n0 0.5 1\nSolid fraction\n]001[sz\n6\n(e)\n4\n2\n0\n2\n0 0.5 1\nSolid fraction\n]010[pz\n6\n(f) Exp. Al1Si\nExp. Al3Si\nPF model\n2\n0\n2\n0 0.5 1\nSolid fraction\n]110[pz\n100\n(g)\n50\n0\n0 0.5 1\nSolid fraction\nsniarg\nfo\nrebmuN\nY\nX\nMRD\n0 1 2 3 4\n(h)\nFigure R5. SAST results for simulated textures of Al1Si under an experimentally measured\nincreasing solidification velocity (Fig. 1b) and a constant thermal gradient of 5 K/µm. In these\nsimulations, (ϵ ,ϵ ) = (0.06,−0.006) and ϵ = 0.15. The time-dependent velocity profile is given\n1 2 k\nby V =0.838426+0.045579t, where V is in m/s and t is in µs. (a) Representative grain texture\np p\nwith various grain orientations shown by rotated cubes. (b) Pole figures of panel (a) with the\nsolid-liquid interface determined at the same position as the simulation at 15 µs. (c) Density pole\nfigure of panel (a). (d) Density pole figure of 4 simulations. (e) z or Z-score of the projection\ns[100]\nsum of the secondary [100] direction on the Y axis as a function solidification distance. Blue\nsquares correspond to the simulation shown in panel (a). Grey open circles, squares, and triangles\ncorrespond to experimental results for Al1Si, Al3Si, and Al6Si, respectively. (f) z or Z-score\np[010]\nof the projection sum of the primary [010] direction on the X axis. (g) z or Z-score of the\np[011]\nprojectionsumoftheprimary[011]directionontheXaxis. (h)Numberofgrainswithinthemelt\npool shown in panel a.\na\n4\n2\n0\n2\n0 0.5 1\nSolid fraction\n2 m\nC\na\n2 m\nD\na\n2 m\n]001[sz\nb 6\n4\n2\n0\n2\n0 0.5 1\nSolid fraction\n]010[pz\nc\n6\n4\n2\n0\n2\n0 0.5 1\nSolid fraction\n]110[pz\nd 100\n50\n0\n0 0.5 1\nSolid fraction\nsniarg\nfo\nrebmuN\ne\n6\n4\n2\n0\n2\n0 0.5 1\nSolid fraction\n]001[sz\nb 6\n4\n2\n0\n2\n0 0.5 1\nSolid fraction\n]010[pz\nc\n6\n4\n2\n0\n2\n0 0.5 1\nSolid fraction\n]110[pz\nd 100\n50\n0\n0 0.5 1\nSolid fraction\nsniarg\nfo\nrebmuN\ne\n6\n4\n2\n0\n2\n0 0.5 1\nSolid fraction\n]001[sz\nb 6\n4\n2\n0\n2\n0 0.5 1\nSolid fraction\n]010[pz\nc\n6\n4\n2\n0\n2\n0 0.5 1\nSolid fraction\n]110[pz\nd 100\n50\n0\n0 0.5 1\nSolid fraction\nsniarg\nfo\nrebmuN\n6\n4\n2\n0\n2\n4\n0 0.5 1\nSolid fraction\ne\n]001[sz\nA\na\n6\n4\n2\n0\n2\n4\n0 0.5 1\nSolid fraction\n]010[pz\nb\n6\n4\n2\n0\n2\n4\n0 0.5 1\nSolid fraction\n]110[pz\nc\n400\n300\n200\n100\n0\n0 0.5 1\nSolid fraction\nsniarg\nfo\nrebmuN\nd\nFigureR6. SASTresultsforsimulatedtexturesofAl1Siunderanexperimentallymeasured\nincreasing solidification velocity (Fig. 1b) and a constant thermal gradient of 5 K/µm. The\ntime-dependent velocity profile is given by V = 0.838426+0.045579t, where V is in m/s\np p\nand t is in µs. In these simulations, (ϵ ,ϵ ) =(0.06, 0.006) and ϵ =0.15. This figure is a\n1 2 k\n−\ncomplementtoSupplementaryFig. SM18. (A)Z-scores(a-c)andthenumberofgrains(d)\nobtained from 4 simulations. (B)-(D) Grain texture (a), Z-scores (b-d), and the number\nof grains (e) obtained from 3 individual simulations. The remaining simulation used to\nproduce panel A is shown in Fig. 6 of the main text. The dotted lines represent 1.96,\n±\ncorresponding to the p-value of 0.05.\n16\n(a) (001) (110) (111)\n(b)\n(c)\n(d)\n2 m\n6\n4\n2\n0\n2\n0 0.5 1\nSolid fraction\n]001[sz\n6\n(e)\n4\n2\n0\n2\n0 0.5 1\nSolid fraction\n]010[pz\n6\n(f) Exp. Al1Si\nExp. Al3Si\n2\n0\n2\n0 0.5 1\nSolid fraction\n]110[pz\n(g) Individual PF model\nAll\n100\n50\n0\n0 0.5 1\nSolid fraction\nsniarg\nfo\nrebmuN\nY\nX\nMRD\n0 1 2 3 4\n(h)\nFigure R7. SAST results for simulated textures of Al1Si without γ anisotropy and\nsl\nwith kinetic anisotropy remaining at ϵ = 0.15. The solidification velocity aligns with the\nk\nexperimentallymeasuredprofile(Fig. 1b)andthethermalgradientisfixedat5K/µm. The\ntime-dependent velocity profile is given by V = 0.838426+0.045579t, where V is in m/s\np p\nand t is in µs. (a) Representative grain texture with various grain orientations shown by\nrotatedcubes. (b)Polefiguresofpanel(a)withthesolid-liquidinterfacedeterminedatthe\nsame position as the simulation at 15 µs.(c) Density pole figures of panel (b). (d) Density\npolefigureof4simulations. (e)z orZ-scoreoftheprojectionsumofthesecondary[100]\ns[100]\ndirectionontheYaxisasafunctionofsolidificationdistance. Lightbluelinescorrespondto\n4 individual simulations and the blue line is obtained from the data of 4 simulations. Grey\nopen circles, squares, and triangles correspond to experimental results for Al1Si, Al3Si, and\nAl6Si,respectively. ThedottedlinedenotesaZ-scoreof 1.96,correspondingtothep-value\n±\nof 0.05. (f) z or Z-score of the projection sum of the primary [010] direction on the X\np[010]\naxis. (g) z or Z-score of the projection sum of the primary [011] direction on the X\np[011]\naxis. (h) Number of grains that touch the solid-liquid interface.\ndendrites, due to the poor chemical contrast between Al and Si.\n(2) Because the title refers directly to metal AM, but the actual experiments and simulations are\nperformed for thin films, I would highly encourage to modify the title to not mislead the reader.\nWhile the thermal gradients and solidification velocities are similar (typically DTEM has higher),\nDTEM is lacking many of the complexity of metal AM: large three-dimensional melt pools which\nare overlapping and producing occasionally very large grains, potential effects of melt flow on the\nsolidification, nucleation of new grains, and so on.\nWe thank the reviewer for this valuable comment. The title now reads:\nQuantification and prediction of solidification textures under additive manufacturing\nconditions.\nWe believe that this new title clarifies that our results are obtained under solidification conditions\nthat are relevant for AM processes in general without misleading the reader that we investigated a\nspecific 3D AM process like powder bed fusion.\n(3) Line 99: it should be explained what incubation time means.\nThe following sentence has been added to define incubation time:\nNote that solidification does not start at laser shutdown, but the interface remains still\nfor some time (called incubation time) until it starts moving towards the melt pool\ncenter.\n(4) Line 100-102: it is claimed that the solidification velocities decrease with increasing nominal Si\ncontent due to decrease in thermal conductivity. I doubt this is the only reason for the difference\nin velocities.\nAs mentioned in the manuscript, since the DTEM samples are thin, the convection is eliminated\n(see below). Moreover, the experiments take place under vacuum, which eliminates heat transfer\nby radiation.\nThethinnatureoftheDTEMsampleseffectivelyeliminatesconvection,especiallywhen\nthe sample plane is perpendicular to gravity during the experiment. This prevents\nthe formation of convection cells, minimizing disturbances, making solidification\nconduction-driven [19].\nConduction heat transfer is a function of the thermal conductivity and the thermal gradient.\nTherefore, we still believe that the main contributor to the variations in the solidification velocity\nis the thermal conductivity. Note that the thermal gradient is also dependent on the thermal\nconductivity, as it is governed by heat transfer during the melting stage.\n(5) What is the initial grain size in phase field simulations and experiments? Based on Fig. 5h &\n6h, it seems that the phase field simulated grain size is very small compared to the experimental\ngrain sizes. Increasing the grain size in the phase field simulations to a more realistic value might\nincreasethediscrepancybetweenexperimentsandsimulations,asthesimulatedcolumnardendrites\nforming are very large, likely much larger than found experimentally.\nThe reviewer raises an important questions regarding the initial grain size. In response, we have\nmeasured the initial mean grain size in the phase-field simulations. The mean grain size is 205.0\n±\n19\n83.7 nm, with slight run-to-run variations, as shown in Fig. R9. Although this value is smaller\nthan the experimental measurements of 301.24 97.38 for Al1Si, they are still comparable (shown\n±\nin Table R1, or Supplementary Table SM1 in Supplementary Note 4).\nWe have included this result into our manuscript. The revised text in the Result section now reads\nas follows:\nTo further quantify the grain texture evolution, we applied the SAST method. Fig.\n6e-g presents the Z-scores for the projection sum of d⃗ on the Y⃗ axis (Fig. 6e),\ns[100]\nd⃗ on the X⃗ axis (Fig. 6f), and d⃗ on the X⃗ axis (Fig. 6g), for both PF\np[010] p[011]\nsimulations (blue square) and experiments (grey lines). Additionally, the number of\ngrains in the simulations approaches those in experiments quickly after solidification\nstarts (Fig. 6). Moreover, the Z-score ratio with respect to that for perfect alignment\nshowsanincreaseformostofthesolidificationprocess,andtheinitialgrainsoutsidethe\nmelt pool is approximately 205.0 83.7 nm (Supplementary Figs. SM14 and SM15).\n±\nThese simulation results demonstrate remarkable agreement with the experimental\nmeasurements, thusvalidatingourPFmodel’scapabilityincapturingthegraintexture\nevolution during rapid solidification of thin-film AlSi alloys.\n350\n300\n250\n200\n150\n100\n50\n0\n0 100 200 300 400 500 600\nGrain size (nm)\nsniarg\nfo\nrebmuN\nrun #1 350\n205.3 ± 82.9 nm\n250\n200\n150\n100\n50\n0\n0 100 200 300 400 500 600\nGrain size (nm)\nsniarg\nfo\nrebmuN\nrun #2\n205.1 ± 84.7 nm\nTotal number: 2499\n400\n300\n200\n100\n0\n0 100 200 300 400 500 600\nGrain size (nm)\nsniarg\nfo\nrebmuN\nrun #3\n350\n205.4 ± 83.6 nm\n250\n200\n150\n100\n50\n0\n0 100 200 300 400 500 600\nGrain size (nm)\nsniarg\nfo\nrebmuN\nrun #4\n205.1 ± 83.6 nm\nTotal number: 2496\nFigure R9. Initial grain size distributions for each of the PF runs in this work.\n(6) What do the velocity-dependent k(V) curves look like for each Al-Si alloy in the phase field\nmodel?\nThe k(V) curves are shown by the blue lines in Figure R10. Since ϕ = tanh(x/√2W) is a good\n0\n−\napproximation to the full numerical solution [16], the concentration profile can be determined by\nsolvingtheconcentrationevolutionequation(Eqs. 2and5inJietal. [16]). Consequently,thek(V)\ncurves are identical for all dilute Al-Si alloys. For comparison, we also present the k(V) curves for\nthe Al-Cu alloy (orange lines). The results demonstrate that the k(V) curves for Al-Si and Al-Cu\n20\nare nearly identical (with V/V0 as the x-axis).\nd\n1.0\n0.8\n0.6\n0.4\n0.2\n10 3 10 2 10 1 100 101 102\nV/V\nd\n0\n)V(k\nAlSi, CG\nAlSi, PF model\nAlCu, CG\nAlCu, PF model\nFigure R10. k(V)curvesobtainedfromthecontinuousgrowth(CG)modelandtheapproximatesolutionofthePF\n√\nmodel. InthePFmodel,thephase-fieldprofileisassumedtobeϕ =−tanh(x/ 2W),whichisagoodapproximation\n0\nto the full numerical solution.\n(7) It should be explained how the grain-grain interaction term was derived (eq. 5) for the phase\nfield model.\nWe appreciate the reviewer’s suggestion to clarify the derivation of the grain-grain interaction in\nEq. 5 of the main text. We begin by assuming that the phase field ψ remains constant across the\ngrain boundary. In steady state, the one-dimensional solute concentration c(x) satisfies\n(cid:26) (cid:27)\nd d\n0 = D q(ψ)c [lnc bg(ψ)] (1)\nl\ndx dx −\nIntegrating from to x gives\n−∞\nc = c eb[1+g(ψ)] (2)\n∞\nSimilarly, the stationary form of the phase-field equation, including the grain–grain interaction\nterm, is\nd2ϕ (cid:18) T T (cid:19) 1+ϕ (cid:18) 1+ϕ (cid:19)2\n0 = α +ϕ ϕ3 λg′(ϕ ) c+ − M eb[1+g(ϕα)] ω α β (3)\ndx2 α − α − α m − 2 2\ne\n| |\nSubstituting c = c eb[1+g(ψ)], multiplying both sizes by dϕ /dx, and integrating from to\n∞ α\n−∞ ∞\nyields:\n(cid:90) ∞ (cid:18) T T (cid:19) (cid:90) ∞ 1+ϕ (cid:18) 1+ϕ (cid:19)2\nλ g′(ϕ ) c eb[1+g(ψ)]+ − M eb[1+g(ϕα)] dx = ω α β dx (4)\nα ∞\nm − 2 2\n−∞ e −∞\n| |\nFinally, we employ the relationship ψ = 1+ϕ +ϕ 1, i.e., ϕ = ϕ . The above integrals can\n1 2 2 1\n≡ −\nbe evaluated analytically, yielding:\n(cid:20) (cid:21)\nk 1T T\ne M\nω =6λc 2k + − − (5)\n∞ e\nb m c\ne ∞\n| |\nWe have expanded the Methods section to explain how we derived the grain-grain interaction w.\nAssuming that ψ remains constant across the grain boundary, we can derive a unique\nexpression for ω. To obtain this, we first consider the evolution equation for c. At\nequilibrium, integrating the right-hand side from to x yields c = c eb[1+g(ψ)]. For\n∞\n−∞\ntheone-dimensionalequationofϕ,wemultiplybothsidesbydϕ /dxandintegratefrom\nα\nto . The resulting relation can then be solved by substituting c = c eb[1+g(ψ)]\n∞\n−∞ ∞\nand using the constraint ψ = 1+ϕ +ϕ = 1. Consequently, we obtain\n1 2\n(cid:20) (cid:21)\nk 1T T\ne M\nω =6λc 2k + − − (6)\n∞ e\nb m c\ne ∞\n| |\n(8) Capillary fluctuation method (8.1) In Supplementary materials it is claimed that Gamma =\n2.4e-7mK,whereasinthemainmanuscripttheGamma=1.9e-7mK,fromanexperimentalsource.\nClarify what value is used in the phase field simulations.\nIn the phase-field model, we used Γ = 1.96 10−7 mK. The value Γ = 2.4 10−7 mK in\n× ×\nthe supplementary materials was only used to filter out data points that are correlated. To\nmake these values consistent, we re-processed the data using Γ = 1.96 10−7 mK and µ = 0.5\n×\nms-1K-1. There-calculatedanisotropyparametersofinterfacialfreeenergyareshownintheupdated\nSupplementary Fig. SM11, which differs slightly from the previous version of the figure.\n(8.2) Also, it seems that only the solid-liquid anisotropy from CFM is reported and utilized , but\nCFM also gives the solid-liquid interface energy magnitude. What is the value of the CFM-based\nsolid-interface energy magnitude? Was the CFM-based solid-liquid interface energy magnitude not\nused in the phase field simulations, and if not, why?\nThe solid-liquid interface free energies from MD simulations are 109.3 7.7 J/m2 at 926 K,\n±\n100.3 5.9 J/m2 at 900 K, and 97.8 6.7 J/m2 at 880 K. Using the latent heat of pure Al\n± ±\n(9.518 108 J/m3,SupplementaryNote9),wecalculatetheGibbs-Thomsoncoefficienttobe 97.3\n× ∼\nK·m. Single-crystal phase-field simulations using this value (Figure R11) confirm that orientations\nwith their secondary [100] direction normal to the solidification front exhibit minimal undercooling\n(orange curve), consistent with Fig. 7b in the main text. Hence, we believe that the deviation in\nthe Gibbs–Thomson coefficient does not significantly impact the polycrystalline phase-field results.\n(8.3) ”This step was to ensure that the crystal-melt interfaces in the following steps do not\nmerge” needs clarification. I think what is meant by this is that ’The computed solid and liquid\ncompositions are used in the CFM simulations to ensure correct equilibrium dynamics’ or similar.\nThe reviewer’s understanding of this sentence is correct. To improve clarity, we have removed the\noriginal sentence in Supplementary Note 9 and provided a clearer explanation in the subsequent\nparagraph. The revised text now reads as follows:\nThe initial solid and liquid compositions were taken from Table SM3, which guarantees\nthat the two crystal-melt interfaces in a single simulation remain separated and do not\nmerge within 3 ns due to fluctuations.\n(8.4) MD-derived gamma0 could allow for another estimate of mu as well and kinetic anisotropy\nfor this potential, using Equation 30. Why was this not done, and instead, literature-derived value\n22\n0.56\n0.55\n0 20° 40°\nRotation angle\nFigure R11. Undercooling (∆) at the solidification front as a function of rotation angle. The Gibbs–Thomson\ncoefficient is 97.3 K·m, estimated from the solid–liquid interfacial free energy obtained via MD simulations. For all\nshown orientations, the solidification direction points to the right.\nwas used from a different MD potential?\n110[001] 110[11!0]\n(a) (b) (c)\n2)0,k(A\n)t,k(A\n>tixetal/<==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>\">=kCund94uRU3KO2TnKkccsZ8uVL0\"=46esab_1ahs tixetal<\nω/t\n→e\n↑\n1=↓|\n2)0,k\n↑ (A\n2\n|→\n↓|\n|→\n100[010]\n(d) (e) (f)\nFigure R12. Atomistic calculation of the kinetic anisotropy for pure Al. Panels (a)-(c) present the fitting\nof data points versus time obtained from MD simulations, where only the solid circles are used in the curve\nfitting. Panels (d)-(f) show the extracted kinetic coefficients for different orientations of the crystal-melt\ninterface. Theoretically, for a given solid–liquid interface, such as the (110) plane, there is only one kinetic\ncoefficient. Thus, the kinetic anisotropy parameter is calculated as ϵ =(0.904-0.477)/(0.904+0.477)=0.31 or\nk\nϵ =(0.904-0.673)/(0.904+0.673)=0.15.\nk\ncan be measured from MD simulations. Fig. R12 shows the atomistic calculation of the magnitude\nand anisotropy of the kinetic coefficient for pure Al. Hoyt et al. [20] illustrate that the amplitude\nof the Fourier mode, A(k,t), satisfies the relationship:\nA(k,t) A(k,0) 2\n⟨| − | ⟩ = 1 e−t/τ (7)\n⟨| | ⟩\nwhere the decay timescale τ can be related to the kinetic coefficient µ:\n1\nτ = (8)\nΓµk2\nThe above equation is valid only for very dilute alloys, as in more concentrated alloys, solute\ndiffusion can strongly influence dynamic interface fluctuations, leading to [21–23]\nm (c0 c0)\nτ = | | l − s (9)\nD Γk3\nl\nwhere c0 and c0 represent equilibrium liquid and solid concentrations, respectively.\nl s\nTo collect the data of the left hand side of Eq. 7, we used the same geometry as in Supplementary\nNote 9 and ran 10 simulations for each of the 100[100], 110[001], and 110[110] directions at the\nmeltingpoint. Inthecalculation,anydatapointscanbeconsideredast = 0anditscorrelationwith\nalldatapointsatlatertimes(t+dt,t )wereconsidered. Here, dt = 2psdenotestheminimumtime\nn\ninterval between data points output from MD simulations and t = 5.5 ns represents the maximum\nn\ntime. We then fit the data points at the later decaying phase (solid circles shown in panels (a)-(c)\nof Fig. R12) using 1 e−t/τ. Finally, the extracted τ values are used to calculate µ by fitting Eq.\n−\n9 (panels (d)-(f) of Fig. R12). Depending on the directions, µ varies from 0.477 m/s/K to 0.904\nm/s/K and its anisotropy parameter varies from 0.15 to 0.31. Notably, the values we used in the\nPF simulations, µ0 = 0.5 m/s/K and ϵ = 0.15, fall within these ranges.\nk k\nWe have included these results in Supplementary Note 9 to support our choice of parameters.\n(8.5) Minor note: the spelling is Gibbs-Thomson (no “p”)\nThis typo has been corrected.\n(9) In metal AM there is flow effects. How are the results expected to be changed if metal AM like\nflow conditions were imposed to the melt? What about the confinement effects that were briefly\nmentioned?\nFluid flow is expected to be largely suppressed by the very small film thickness in the present\nexperiments. Thisexpectationisconsistentwiththeobservationofuniformmicrostructuresaround\nthe melt pool in the present experiments and previous experiments in other alloys (e.g. AlCu\n[9]). Surface-tension driven Marangoni flow is present in 3D AM processes like LPBF or DED.\nHowever, flow is expected to alter heat and mass transport on the macroscale of the melt pool,\nwhichistypicallyascalemuchlargerthanthethicknessofthediffusionboundarylayeraheadofthe\nsolidificationfront(oforderD/V whereDisthediffusivityandV thegrowthrate). Therefore,while\nit is important to take into account flow effects to quantitatively model macroscopic heterogeneities\n24\nof composition on the melt pool scale, the formation of cellular/dendritic microstructures on the\nscale of the primary spacing is expected to be primarily controlled by the balance of capillary\nforces and solute diffusion, which is the regime investigated in the present thin-film experiments.\nAlready in a slow solidification regime, flow effects on the primary spacing are only present at\nvery low velocity close to the onset of morphological instability (i.e. close to the criticial velocity\npredictedbyconstitutionalsupercoolingofordermicron/sec). Theseeffectsbecomenegligiblewhen\nthe velocity exceeds a few tens of microns per second (see for example Ref. [24]). It is possible\nand even likely that fluid flow will affect grain competition taking place on a scale larger than the\nprimary spacing and therefore the grain texture. Modeling this effect would require to bridge micro\nand macro scales (e.g. coupling PF modeling to flow 3D), which is beyond the scope of the present\nwork. However, the SAST method introduced here would remain applicable to assess the statistical\nsignificance of grain textures influenced by fluid flow.\n(10) The thin films have a finite (albeit relatively small) thickness, on the order of 50-100 nm.\nThis is far above the typical solutal capillary length for metallic alloys, including aluminum alloys.\nJustify why the phase field simulations can be carried out in 2D, neglecting the film thickness. How\nwould you expect the results to change if the finite film thickness were resolved in the phase field\nsimulations.\nAs explained in the manuscript, the phase-field simulations attempt to partially capture 3D effects\nin the thin film experiments, which are inherently 3D because the film has a finite thickness of\napproximately 100 nm. This was accomplished by using a more complex form of the anisotropy\nof the interface excess free-energy and the kinetic coefficient that takes into account the fact that\nthe sputtered nanoscale grains have random 3D orientations, and thus not necessarily a 100-axis\nperpendicular to the film as assumed in previous 2D simulation studies that use a simple cos4θ\nform [10, 11]. However, even with this improvement, our 2D simulations are still an approximation\nsince they do not consider the curvature of the interface in the thickness of the film, which could\npotentiallyinfluencegraintextureselection. Wehaveclarifiedthetexttostatethislimitationmore\nexplicitly. We should note that polycrystalline 3D simulations with a film thickness of about 100\nnm are presently not computationally feasible. Furthermore, the interface curvature in the film\nthickness depends on the interfacial excess free energies between both solid and liquid AlSi and\nthe substrate that, together with the solid-liquid interfacial energy, determine the dihedral angle\nof the solid-liquid interface with the substrate. These energies are not precisely known and hard\nto compute by MD simulations, making it difficult to realistically model film-thickness curvature\neffects. However, perhaps more important than realistically modeling these effects is the relevance\nof our quasi-2D results for modeling grain texture selection in applied AM processes such as LPBF,\nwhich involve full 3D melt pools. We believe that the new insights into grain competition under\nAM conditions obtained in a quasi-2D geometry (detailed in the next response to point 5) support\nthe relevance of our results to a wide range of fully 3D AM processes.\nThe clarified text of the limitation in the Discussion reads as follows:\nmore realistic three-dimensional (3D) scenarios. The 2D model neglects the curvature\nof the solid-liquid interface through the film thickness, potentially influencing grain\ntexture selection. Full-scale 3D polycrystalline simulations incorporating realistic\nfilm thicknesses ( 100 nm) remain computationally prohibitive. Additionally, the\n∼\ncurvature of the interface along the film thickness is controlled by the interfacial\nexcess free energies between solid/liquid AlSi and the substrate. These, together\nwith the solid-liquid interfacial energy, determine the dihedral angle formed at the\nsolid-liquid-substrate junction. These energies are not precisely known and are\nchallenging to compute using MD simulations, making it difficult to realistically model\nfilm-thickness curvature effects. Despite these limitations, we believe the quasi-2D\nresults presented here provide meaningful insights into grain texture selection relevant\nto applied AM processes, such as LPBF, which involve fully 3D melt pools. The\nfundamental mechanisms of grain competition under AM conditions, revealed through\nour quasi-2D simulations, support the broader applicability of these findings to a wide\nrange of 3D AM processes.\n[1] Haxhimali, T., Karma, A., Gonzales, F. & Rappaz, M. Orientation selection in dendritic\nevolution. Nature materials 5, 660–664 (2006).\n[2] Becker, M., Dantzig, J., Kolbe, M., Wiese, S. T. & Kargl, F. Dendrite orientation transition\nin alge alloys. Acta Materialia 165, 666–677 (2019).\n[3] Gonzales, F. & Rappaz, M. Dendrite growth directions in aluminum-zinc alloys. 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Time-resolved in situ measurements during rapid alloy solidification:\nExperimental insight for additive manufacturing. Jom 68, 985–999 (2016).\n[10] Tourret, D. & Karma, A. Growth competition of columnar dendritic grains: A phase-field\nstudy. Acta Materialia 82, 64–83 (2015).\n[11] Dorari, E., Ji, K., Guillemot, G., Gandin, C.-A. & Karma, A. Growth competition between\ncolumnar dendritic grains–the role of microstructural length scales. Acta Materialia 223,\n117395 (2022).\n[12] Walton, D. & Chalmers, u. B. The origin of the preferred orientation in the columnar zone of\ningots. Trans. Metall. Soc. AIME 215, 1 (1959).\n[13] Tourret,D.,Song,Y.,Clarke,A.J.&Karma,A. Graingrowthcompetitionduringthin-sample\ndirectional solidification of dendritic microstructures: A phase-field study. Acta Materialia\n122, 220–235 (2017).\n[14] Pineau, A., Guillemot, G., Tourret, D., Karma, A. & Gandin, C.-A. Growth competition\nbetween columnar dendritic grains–cellular automaton versus phase field modeling. Acta\nMaterialia 155, 286–301 (2018).\n[15] Pinomaa, T. et al. Phase field modeling of rapid resolidification of al-cu thin films. Journal of\nCrystal Growth 532, 125418 (2020).\n[16] Ji, K., Dorari, E., Clarke, A. J. & Karma, A. Microstructural pattern formation during\nfar-from-equilibrium alloy solidification. Physical Review Letters 130, 026203 (2023).\n[17] Ji, K., Zhong, M. & Karma, A. Phase-field model of alloy solidification far from chemical\nequilibrium at the solid-liquid interface. Physical Review Research 7, 033128 (2025).\n[18] Yu, Y. et al. Impact of fluid flow on the dendrite growth and the formation of new grains in\nadditive manufacturing. Additive Manufacturing 55, 102832 (2022).\n[19] McKeown, J. T., Clarke, A. J. & Wiezorek, J. M. Imaging transient solidification behavior.\nMrs Bulletin 45, 916–926 (2020).\n[20] Hoyt, J., Asta, M. & Karma, A. Atomistic simulation methods for computing the kinetic\ncoefficient in solid-liquid systems. Interface Science 10, 181–189 (2002).\n[21] Karma, A. Langevin formalism for solidification. Physical review letters 70, 3439 (1993).\n[22] Karma, A. Fluctuations in solidification. Physical Review E 48, 3441 (1993).\n[23] Asta, M., Hoyt, J. & Karma, A. Calculation of alloy solid-liquid interfacial free energies from\natomic-scale simulations. Physical Review B 66, 100101 (2002).\n[24] Clarke, A. J. et al. Microstructure selection in thin-sample directional solidification of an al-cu\nalloy: In situ x-ray imaging and phase-field simulations. Acta Materialia 129, 203–216 (2017).\n28"}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have addressed the raised issues appropriately, and the revisions adequately improve the quality of the manuscript.", "reviewer_2": "Thank you for the revisions. The authors have addressed the concerns I raised in the previous version, and I find the manuscript acceptable in its current form.", "reviewer_3": "My comments were appropriately addressed and the manuscript is now OK."}, "author_response": ""}]

Decision letter