When the Al-30Er alloy is cooled from the liquid state to room temperature at different rates, millimeter-sized primary Al₃Er phase blocks are observed on the surface of the samples. A typical surface image of the sample prepared at a cooling rate of 60 °C/h is shown in Figure 2. The red cross marks the X-ray beam position in bulk-sensitive micro-XRD. The dark contrast is the proeutectic Al₃Er phase. Figure 3 shows the XRD patterns of the samples cooled at different rates. It can be seen that only Al and Al₃Er phases are present, and the crystal structure of the Al₃Er phase differs depending on the cooling rate. At a cooling rate of 60 °C/h, only the cP4-structured Al₃Er phase is formed. When the cooling rate decreases to 50 and 40 °C/h, the hR20-structured Al₃Er emerges, coexisting with the cP4-structured Al₃Er phase. At a slower cooling rate of 30 °C/h (and 5 °C/h), only the hR20-structured Al₃Er phase is observed. These results indicate that as the cooling rate decreases, the solidification process approaches equilibrium solidification, and the proportion of the hR20-structured Al₃Er phase increases progressively until it becomes the sole phase.

The high-angle diffraction peaks (2θ > 60°) of the samples cooled at 60 °C/h and 5 °C/h were selected for rescanning at a slower speed and finer step. The resulting diffraction patterns are shown in Figures 4 and 5. Figure 4 shows the (220), (311), (222) and (320) diffraction peaks of the cP4-structured Al₃Er phase, and the (220), (311) and (222) diffraction peaks of the Al matrix. It can be seen that the diffraction peaks of the cP4-structured Al₃Er phase, indexed as (220), (311) and (222), are located close to those of the Al phase. This is attributed to the similarity of the cP4 and fcc structures with low lattice mismatch. Figure 5 shows the (027), (2 0 11), (0 2 19), (220), (3 0 15), (318), (2 2 15) and (2 1 22) diffraction peaks of the hR20-structured Al₃Er phase, and the (111), (200), (220) and (311) diffraction peaks of the Al matrix. As the hR20 structure belongs to the rhombohedral crystal system with low symmetry, whereas cP4 belongs to the cubic crystal system with high symmetry, there is no correlation between the diffraction peak positions.

The lattice constants were obtained by fitting the experimental diffraction peaks using the Pseudo-Voigt and Pearson VII functions^[24] to extract accurate peak positions, which were then used in the Bragg equation.

The lattice constants of Al and cP4-structured Al₃Er were calculated to be 0.4047 ± 0.0002 nm and 0.4212 ± 0.0003 nm, and the lattice mismatch between the two phases was 4.08 ± 0.09%. The lattice constants of the hR20-structured Al₃Er were calculated to be a = 0.6036 ± 0.0006 nm and c = 3.5705 ± 0.0004 nm. The experimental results obtained in this work and in other experiments^[13-15,20] are compared in Table 1. As shown in Table 1, the measured values are generally consistent with the reported values, with differences comparable to the typical spread among independent experimental studies.

At cooling rates of 50 °C/h and 40 °C/h, two different structures of the Al₃Er phase coexist, indicating an ongoing polymorphic transformation and the potential existence of an orientation relationship between these two phases. A site-specific lift-out specimen containing both Al₃Er phase structures was extracted from the sample cooled at 40 °C/h [Figure 7C] using FIB and subsequently examined by TEM. The specimen was lifted out from a specific location identified in the EBSD phase distribution map, as marked by the yellow box in Figure 10A. Figure 10B shows a HRTEM image in which the phase interfaces are clear, as indicated by the white dashed lines. Figure 10C presents the selected-area electron diffraction (SAED) pattern, confirming the presence of three sets of diffraction spots from the three different phases, as indexed in the figure. The orientation relationship between the cP4-structured Al₃Er phase and the hR20-structured Al₃Er phase is $$ [1 \overline{2} 1]_{\mathrm{cP} 4} / /[1 \overline{1} 00]_{\mathrm{hR} 20} $$, (111)_cP4//(0006)_hR20. Given the lower symmetry of the rhombohedral hR20 structure compared with the cubic cP4 structure, multiple crystallographically equivalent orientation variants may exist.

As shown in Figures 3 and 7, the structure of the Al₃Er phase varies with cooling rate, gradually transforming from the cP4 structure to the hR20 structure as the cooling rate decreases. Since slower cooling rates produce crystal structures at room temperature that more closely approach the equilibrium state^[25-27], this experimental result demonstrates that the hR20-structured Al₃Er phase is more stable than the cP4-structured Al₃Er phase. This result does not support the conclusion presented in the Al-Er phase diagrams^[4,5].

Furthermore, the observations in Figure 7 show that the average grain size of the polycrystalline hR20-structured Al₃Er phase obtained at 5 °C/h is 92 μm, which is unexpectedly much smaller than that of the single-crystalline cP4-structured Al₃Er phase obtained at 60 °C/h (211 μm). This finding contradicts the classical solidification principle that a reduction in cooling rate increases grain size^[26]. To clarify whether the hR20-structured Al₃Er phase precipitates directly from the liquid or forms through a polymorphic transformation from the cP4-structured Al₃Er phase, classical nucleation theory calculations were performed^[26,28].

The driving force for nucleation of a phase from the melt can be calculated using^[29]:

(1) $$ \Delta \mathrm{G}_{\mathrm{V}} \approx \frac{\mathrm{~L}_{\mathrm{m}} \cdot \Delta \mathrm{~T}}{\mathrm{~T}_{\mathrm{m}}} $$

Where L_m is the latent heat of fusion per unit volume. ΔT is the undercooling and T_m is the melting point of Al₃Er, which is 1,603 K^[4,5].

The critical nucleation energy and the steady-state nucleation rate are calculated using^[28]:

(2) $$ \Delta \mathrm{G}^{*}=\frac{16 \pi \gamma_{\mathrm{SL}}^{3}}{3\left(\Delta \mathrm {G_{V}}\right)^{2}} $$

(3) $$ \mathrm{J}=\mathrm{N}_{\mathrm{v}} \mathrm{Z} \beta^{*} \exp \left(-\frac{\Delta \mathrm{G}^{*}}{\mathrm{k}_{\mathrm{B}} \mathrm{T}}\right) \exp \left(-\frac{\mathrm{Q}_{\mathrm{d}}}{\mathrm{RT}}\right) $$

where γ_SL is the solid-liquid interfacial energy. N_v is the number of nucleation sites per unit volume. The Zeldovich factor^[28] is given by $$ \mathrm{Z}=\frac{\mathrm{V}_{\alpha}\left(\Delta \mathrm G_{\mathrm V}\right)^{2}}{8 \pi \sqrt{\gamma_{\mathrm{SL}}^{3} \mathrm{k}_{\mathrm{B}} \mathrm{T}}} $$, where V_α is the average atomic volume of Al₃Er^[3]. k_B =1.38 × 10^-23 J/K is Boltzmann’s constant. R = 8.314 J/(mol · K) is the molar gas constant. The rate of atomic attachment to a growing embryo^[28] is $$ \beta^{*}=\frac{16 \pi \gamma_{\mathrm{SL}}^{2} \mathrm{D}_{\mathrm{L}}}{\left(\Delta \mathrm G_{\mathrm V}\right)^{2} a^{4}} $$. D_L is the diffusion coefficient in the liquid. a is the characteristic atomic spacing. Q_d is the activation energy for atomic transfer across the liquid-solid interface^[30].

For three-dimensional equiaxed solidification, the average grain diameter d is related to the steady-state nucleation rate J and the grain growth velocity G by the following relationship^[26,28]:

(4) $$ \mathrm{d} \propto\left(\frac{\mathrm{G}}{\mathrm{J}}\right)^{1 / 4} $$

Under similar undercooling conditions, G is determined by the crystal structure^[26]. Since both cP4 and hR20 are Al₃Er phases, the atomic attachment kinetics at the solid-liquid interface are not expected to differ substantially; it is reasonable to assume G_cP4 ≈ G_hR20. Consequently, the difference in grain size resulting from the direct solidification of the two structures is predominantly governed by the nucleation rate J.

From Equations (1)-(3), the volumetric driving force ΔG_V increases monotonically with undercooling ΔT, the critical nucleation energy ΔG* decreases monotonically with increasing ΔG_V, and the nucleation rate J therefore increases monotonically with ΔT. This directional relationship is independent of any specific parameter values. Combined with Equation (4), a lower nucleation rate necessarily leads to larger grains. However, the experimentally measured grain size of the Al₃Er phase (hR20-structured) cooled at 5 °C/h is unexpectedly smaller than that of the Al₃Er phase (cP4-structured) cooled at 60 °C/h, which directly contradicts the theoretical prediction. The irreconcilable discrepancy rules out the possibility that the hR20-structured Al₃Er phase precipitates directly from the liquid.

Therefore, the plausible transformation pathway is that the primary cP4-structured Al₃Er phase precipitates first from the liquid and grows into a single crystal. Then, during slow cooling, the hR20-structured Al₃Er phase nucleates heterogeneously at the boundaries of the cP4-structured Al₃Er phase through a polymorphic transformation and grows, finally forming a polycrystalline hR20 microstructure.

This conclusion is in perfect agreement with the EBSD results in Figure 7. The cP4- and hR20-structured Al₃Er phases are iso-compositional polymorphs, and the first-principles calculations indicate that the difference in their formation energies is very small^[22,23], implying a limited chemical driving force for the phase transformation. Therefore, rapid cooling suppresses diffusion and inhibits the transformation, whereas slow cooling extends the residence time of the sample in the high-temperature regime, providing sufficient kinetic conditions for atomic rearrangement. Furthermore, the hR20-structured Al₃Er phase nucleates at the boundaries of the cP4-structured Al₃Er particles, i.e., at the cP4-structured Al₃Er/ Al phase interface, which substantially reduces the critical nucleation energy required to initiate the transformation^[26,28].

The hR20-structured Al₃Er phase particles obtained at a slow cooling rate are polycrystalline, with each grain having nucleated independently at the phase boundary of the cP4-structured Al₃Er grain and subsequently grown. The relatively high nucleation density leads to a fine final grain size for the hR20-structured Al₃Er grains. By comparison, the cP4-structured Al₃Er phase particles formed at a fast cooling rate are single crystals, having grown from a single nucleus in the melt. Therefore, the smaller grain size of the polycrystalline hR20-structured Al₃Er phase particles is a consequence of the polymorphic transformation.

The cooling-rate-dependent structural evolution from cP4 to hR20 directly influences the intrinsic mechanical properties of the Al₃Er phase. Beyond this intrinsic difference, the structural transformation has important implications for the strengthening performance of Er-containing aluminum alloys. In these alloys, the strengthening effect of cP4-structured Al₃Er precipitates relies on their coherency with the α-Al matrix. The lattice mismatch between cP4-Al₃Er and the α-Al matrix is only 4.08 ± 0.09%. This relatively small lattice mismatch facilitates the formation of coherent or semi-coherent interfaces that generate the elastic strain fields responsible for effective dislocation pinning. Although the present study investigates primary Al₃Er particles formed during solidification, the structural evolution identified here is also relevant to nanoscale Al₃Er strengthening precipitates, for which interface coherency with the α-Al matrix is critical to precipitation strengthening. In contrast, the hR20-structured Al₃Er precipitates possess rhombohedral symmetry, which is less compatible with the cubic α-Al lattice than the cP4 structure. Therefore, hR20-structured precipitates in the Al matrix would be expected to exhibit reduced coherency with the Al matrix, potentially decreasing their contribution to precipitation strengthening. Furthermore, reduced interface coherency may increase the interfacial energy, thereby decreasing the resistance to precipitate coarsening and potentially reducing creep resistance at elevated temperatures. Consequently, an uncontrolled cP4 to hR20 transformation during solidification or high-temperature service could adversely affect both the room-temperature strength and the high-temperature stability of Er-containing aluminum alloys. Although direct measurements of strengthening in Er-containing aluminum alloys are beyond the scope of the present work, the structural evolution identified here has important implications for alloy performance. The critical cooling rate of 36 °C/h determined in this work therefore provides practical guidance for avoiding such structural evolution during alloy processing.

Based on the above results, the polymorphic transformation from the cP4 structure to the hR20 structure is a thermally activated diffusion process. That is, a critical cooling rate exists at which the hR20-structured Al₃Er phase fully precipitates. The variation of the transformed volume fraction f with time t can be described by the Johnson-Mehl-Avrami-Kolmogorov (JMAK) equation^[28,31]:

(5) $$ \mathrm{f}=1-\exp \left[-\mathrm{K} \mathrm{t}_{\mathrm{eff}}^{\mathrm{n}}\right] $$

where f is the volume fraction of the transformed hR20-structured Al₃Er phase, K is the temperature-dependent rate constant, n is the Avrami exponent (which depends on the transformation mechanism), and $$ \mathrm{t}_{\text {eff }}=\frac{\Delta \mathrm{T}}{\mathrm{v}} $$ is the effective residence time, where v is the cooling rate and ∆T is the temperature interval over which the transformation occurs.

f_hR20 was determined from the area fraction of the hR20-structured phase in the EBSD phase distribution maps in Figure 7. At a cooling rate of 50 °C/h, f_hR20 = 0.4, while at 40 °C/h, f_hR20 =0.9. According to the Al-Er binary phase diagram, the liquidus temperature of the Al-30Er hypereutectic alloy is approximately 860 °C, i.e., 1,133 K^[4,5]. The primary Al₃Er phase precipitates below the liquidus and persists throughout the subsequent cooling process; therefore, ∆T = 1,133 K - 773 K = 360 K.

By fitting the two sets of data using the JMAK equation, the linear regression result shows that n ≈ 6.8 and K ≈ 8.3 × 10^-7. With only two data points, n and K are fully determined but the goodness of fit cannot be assessed. Nevertheless, the derived values provide a reasonable estimate for subsequent calculations. Taking f_crit = 0.99 as the criterion for completion of the transformation, the corresponding constant C = -ln(0.01) ≈ 4.605. The critical time is then calculated as:

(6) $$ \mathrm{t}_{\text {crit }}=\left(\frac{\mathrm{C}}{\mathrm{K}}\right)^{1 / \mathrm{n}}=\left(\frac{4.605}{8.3 \times 10^{-7}}\right)^{1 / 6.8} \approx 10 \mathrm{~h} $$

The corresponding critical cooling rate is:

(7) $$ \mathrm{V}_{\text {crit }}=\frac{\Delta \mathrm{T}}{\mathrm{t}_{\text {crit }}}=\frac{360 \mathrm{~K}}{10.0 \mathrm{~h}}=36 ~ ^{\circ} \mathrm{C} / \mathrm{h} $$

This value is in excellent agreement with the experimental observations. A slow cooling rate of 30 °C/h or 5 °C/h, being below the critical rate, allows the transformation to proceed completely and yields sole hR20-structured Al₃Er phase. A fast cooling rate of 60 °C/h, which exceeds the critical rate, suppresses the transformation and yields the cP4-structured Al₃Er phase. Intermediate cooling rates of 50 °C/h and 40 °C/h result in a two-phase coexistence microstructure. This finding provides strong evidence that the polymorphic transformation requires a sufficiently long high-temperature residence time and is governed by a well-defined critical cooling rate. Slow cooling rates offer sufficient time for the transformation to nucleate and grow, whereas fast cooling rates lead to its arrest.

It should be pointed out that the JMAK analysis is based on only two available data points; the fitted n and k values are therefore used primarily as engineering parameters for the extrapolated estimation of the critical cooling rate. The unusually high Avrami exponent of 6.8 reveals the pronounced self-acceleration character of the transformation. Regarding the determination of the critical cooling rate, its reliability does not rely on the precise value of n but is directly constrained by the two measured data points. No matter what kinetic model is adopted, the fact that the transformation from f_hR20 = 0.4 to f_hR20 = 0.9 is completed within a difference of only about 1.8 h in effective residence time itself demonstrates an extremely high cooling rate sensitivity of the transformation. The critical cooling rate thus estimated is highly self-consistent with the experimental observations, further confirming the reasonableness of the estimate.

Classical polymorphic transformation from a cubic-based structure to a rhombohedral-based structure, such as the fcc to hcp transition in Co-alloy^[32] and Fe₄₀Mn₄₀Co₁₀Cr₁₀^[33], and the 3C to 6H transformation in SiC^[28,34], typically occurs through the glide of Shockley partial dislocations and is classified as a diffusionless displacive phase transformation. However, in this study, the cP4 to hR20 transition exhibits remarkable diffusion-controlled characteristics. The kinetic analysis presented in Section 4.2 reveals a well-defined critical cooling rate of 36 °C/h for the transformation, along with a high Avrami exponent of nearly 6.8, which is characteristic of diffusion-controlled transformations with site-saturated nucleation and three-dimensional growth. These observations indicate that the transformation is not a fault-mediated displacive process but rather a diffusion-displacive coupled phase transformation.

In the cP4 structure, Er atoms are uniformly distributed on every {111} close-packed plane with an Er:Al ratio of 1:3. In contrast, the hR20 structure exhibits a heterogeneous layered arrangement of Er atoms, wherein certain layers are enriched in Er while others consist entirely of Al, resulting in a periodic compositional modulation, as seen in Figure 11A. Therefore, the transformation from the cP4 to the hR20 structure requires not only a change in the stacking sequence but also long-range diffusion of Er atoms between different stacking layers to achieve the site-occupancy reconstruction. This diffusional requirement governs the thermally activated transformation; i.e., within the effective temperature range, the interlayer diffusion of Er atoms governs the transformation rate. As the temperature decreases, the competition of the decreasing diffusion coefficient and the increasing driving force leads to the characteristic C-curve and the critical cooling rate.

SAED pattern in Figure 10 determined the orientation relationship is $$ [1 \overline{2} 1]_{\mathrm{cP} 4} / /[1 \overline{1} 00]_{\mathrm{hR} 20} $$, (111)_cP4//(0006)_hR20. This parallel relationship of close-packed planes is geometrically consistent with the glide of Shockley partial dislocations on {111} planes, indicating the polymorphic transformation involves periodic glide of $$ \frac{1}{6}\langle 112\rangle $$ partial dislocation on the (111) planes. The parallel close-packed planes serve as the crystallographic prerequisite for the displacive component in the diffusion-displacive coupling mechanism that the glide of partial dislocations alters the stacking sequence, while the interlayer diffusion of Er atoms achieves site-occupancy reconstruction, with both processes proceeding cooperatively on the same close-packed plane. Therefore, this orientation relationship provides crystallographic evidence for the involvement of a displacive component in the transformation.

Furthermore, the Avrami exponent of 6.8 is considerably higher than the typical range of 1-4 in classical JMAK theory. This result indicates that during the polymorphic transformation, the synergistic effects of autocatalytic nucleation, the self-assembly of diffusion channels and the transition in growth dimensionality come into play^[35-37].

In the early stages of the cP4 to hR20 transformation, the hR20 phase nucleates primarily at preferred sites on phase boundaries of the cP4-structured phase, and the nucleation rate is relatively low. Once a nucleus forms, a semi-coherent interface develops between the hR20 and cP4 phases, as seen in Figure 10, and this interface is characterized by substantial lattice distortion and a high vacancy concentration, which provides a high diffusion rate channel^[38,39]. When these interfacial regions become interconnected along the cP4-structured Al₃Er/ Al phase interfaces, they act as highly efficient catalytic sites for subsequent nucleation. As the volume fraction of the hR20-structured phase increases, the total interfacial area expands rapidly, and the number of effective nucleation sites grows exponentially, causing the nucleation rate to rise dramatically^[35,36,40]. This positive feedback loop continues until neighboring hR20 particles impinge upon one another, after which the nucleation rate begins to decrease.

The EBSD phase distribution maps in Figure 7 show that in partially transformed samples, the hR20-structured phase is distributed preferentially along the cP4 grain boundaries, forming a continuous or semi-continuous network that surrounds the untransformed cP4 cores, which is a characteristic morphological feature of autocatalytic nucleation. Moreover, the grain size of the polycrystalline hR20-structured phase formed at the lower cooling rate is much smaller than that of the cP4-structured phase formed at the higher cooling rate, indicating a substantially higher nucleation density. A statistical analysis was performed using the EBSD IPF maps in Figure 7. Within the same field of view, the number density of hR20-structured phase in the 40 °C/h sample is about 3.4 times that in the 50 °C/h sample. As the transformation fraction increases from 0.4 to 0.9, the number density increases by a factor of 3.4, exceeding the increase expected from linear growth. Such an increase is characteristic of an autocatalytic nucleation mechanism, in which new nuclei are generated rapidly at the transformation interface. Furthermore, the coefficient of variation of the hR20-structured phase grain size distribution in the 40 °C/h sample is 0.68, larger than the value of 0.41 measured for the 50 °C/h sample. This indicates the coexistence of fine grains and coarse grains, consistent with the broad grain size distribution expected for autocatalytic nucleation. These results provide direct evidence for the autocatalytic nucleation mechanism.

If the nucleation rate increases exponentially with time, i.e., $$ \dot{\mathrm{N}}(\mathrm{t})=\dot{\mathrm{N}}_{0} \exp (\lambda \mathrm{t}) $$, the effective Avrami exponent can be expressed as n_eff = a + bm + λt^*, where λ is the autocatalytic coefficient and t^* is the time at which the transformation is half complete^[41]. When the autocatalytic effect is pronounced, n_eff can substantially exceed 4. It has been reported that when nucleation occurs on grain boundaries and there is an “infection” effect between nucleation sites, the Avrami exponent can reach values as high as 5 to 7^[40].

In addition, the dynamic transition of growth dimensionality further contributes to the elevated Avrami exponent. The formation of the hR20-structured phase can be divided into two stages^[37]. During the initial stage, hR20-structured phase elongates one-dimensionally along phase boundaries of the primary cP4-structured Al₃Er particles, with growth controlled by grain-boundary diffusion, corresponding to a growth dimensionality of 1. When the phase boundaries are completely covered by the hR20-structured phase, the growth mode shifts to two- or three-dimensional thickening into the grain interiors, corresponding to growth dimensionalities of 2 and 3. The schematic diagram of the polymorphic transformation process is shown in Figure 11B. The EBSD micrographs in Figure 7 reveal that the hR20-structured phase grains are elongated at grain boundaries, which is a typical characteristic of one-dimensional growth, while being equiaxed or spherical within the grains, which is a typical characteristic of two- or three-dimensional growth. These results are consistent with the model of the growth-dimensionality transition. This change in growth dimensionality means that the dimensionality parameter evolves during the transformation. The effective Avrami exponent becomes a weighted average of contributions from different stages. Higher-dimensionality growth occurs in the middle and later stages, when the volume fraction increases most rapidly. As a result, its contribution to the overall Avrami exponent is amplified, producing the markedly high apparent Avrami exponent (n) obtained by fitting^[37].

The diffusion-displacive coupled cP4 to hR20 polymorphic transformation manifests itself in the fact that the layer difference in Er atom occupancy necessitates long-range diffusion during the phase transformation, which in turn dictates the existence of a critical cooling rate of 36 °C/h. The autocatalytic nucleation, the self-assembly of diffusion channels and the transition in growth dimensionality during the transformation lead to an anomalously high Avrami exponent of 6.8, which has a clear mathematical foundation and is experimentally supported in this work.

In summary, the cP4 to hR20 polymorphic transformation can be divided into three stages. At the first stage, the primary cP4-structured Al₃Er phase has precipitated from the liquid as a single crystal, and Er atoms at grain boundaries begin interlayer migration through diffusion, with preferential sites becoming potential nucleation sites. The nucleation rate is extremely low during this stage, corresponding to t₂ schematic diagram in Figure 11B. At the second stage, which is the autocatalytic acceleration period, once the first hR20-structure Al₃Er nucleus forms at the grain boundary, its interface becomes a catalytic site, and the nucleation rate increases exponentially. The number density increases significantly. The grains extend one-dimensionally along the grain boundary, as shown in t₃ of Figure 11. At the third stage, after the grain boundaries are covered, growth shifts to thickening toward the grain interior, ultimately forming a polycrystalline hR20-structured Al₃Er microstructure. The essence of this model lies in the coupling of diffusion and displacement. Diffusion controls the transformation rate, which determines the critical cooling rate, while displacement assists in stacking rearrangement.