A. Modification during nucleation

The first school of thought on the origins of modification concerns eutectic nucleation, and particularly that of the faceted phase (e.g., Si). In the mid-1960s, Crosley and Mondolfo conducted pioneering metallographic investigations on nucleation phenomena in Al–Si alloys.^{Reference Crosley and Mondolfo47} Their conclusions were 2-fold: (i) Eutectic Al is not affected by trace metal additions, while nucleation of eutectic Si is highly sensitive to the type and amount of modifiers present. In commercial unmodified alloys, eutectic Si is most likely to be nucleated by AlP particles that pre-exist in the melt. That is, the critical undercooling required for the nucleation of eutectic Si by the primary α (Al) phase is higher than that by the AlP particles. We note that P is an unavoidable trace element in Al–Si castings. (ii) Modification of Al–Si by Na results in a “neutralization” of the AlP particles such that easy nucleation of Si is prevented. The evidence for the latter effect is manifested in a depression of the critical nucleation temperature for eutectic Si in cooling curves of the Na-modified alloy. The undercooling required for Si nucleation in the presence of Na was recorded as 6–12 °C, versus 5–7 °C in the unmodified, hypoeutectic Al–Si alloy.^{Reference Crosley and Mondolfo47}

Support for Crosley and Mondolfo’s theory of modified nucleation comes from the recent body of work by Dahle and coworkers.^{Reference Dahle, Nogita, McDonald, Dinnis and Lu46,Reference McDonald, Nogita and Dahle48} Through electron backscatter diffraction (EBSD) of the solidified specimens, they determined that the eutectic Al shares an epitaxial relationship to the primary α (Al) dendrites in unmodified Al–Si alloys, while in Sr- and Sb-modified alloys, the eutectic Al has multiple orientations unrelated to the surrounding dendrites.^{Reference Dahle, Nogita, McDonald, Dinnis and Lu46} This suggests independent nucleation and growth in the Sr- and Sb-modified alloys. Moreover, Dahle and coworkers detected AlP particles located at the center of polyhedral Si. Selected area diffraction patterns in the transmission electron microscope (TEM) show that there is no lattice mismatch between the (111) planes of AlP and the Si crystal. The authors took this result to mean that AlP is a good nucleant for eutectic Si.^{Reference Dahle, Nogita, McDonald, Dinnis and Lu46} Both AlP and Si have cubic crystal structures with nearly identical lattice parameters (5.421 Å^{Reference Straumanis and Aka49} and 5.431 Å,^{Reference Yeh, Lu, Froyen and Zunger50} respectively). The modification of the Al–Si alloy is then thought to be due to the fact that the AlP nuclei are saturated with Sr-containing intermetallics, e.g., Al₂Si₂Sr; this contributes to an increased “nucleation difficulty” for eutectic Si (i.e., higher nucleation undercooling and also lower nucleation frequency).^{Reference Dahle, Nogita, McDonald, Dinnis and Lu46} Further high-resolution evidence was provided by Schumacher and coworkers.^{Reference Li, Hage, Liu, Ramasse and Schumacher51} They report that P, in the form of AlP particles, is located not only at the center of primary Si but also at the interface between eutectic Si and eutectic Al, see Fig. 4.^{Reference Li, Hage, Liu, Ramasse and Schumacher51} During the eutectic reaction, the AlP forms at the surface of Al, and thereby provides favorable conditions for the heterogeneous nucleation of eutectic Si on eutectic Al. Direct experimental verification showing the “poisoning” of these such particles by Sr or Na is still lacking. Nevertheless, it is clear that the potency of AlP as an innoculant must be critically considered in unmodified and modified Al–Si alloys alike.

FIG. 4. (a and b) High-resolution HAADF STEM images; corresponding EELS maps of (c) Al, (d) Si, and (e) P; and (f) line scanning analysis of Al, Si, and P in an Al–18Si–0.03P mater alloy. The AlP particle was observed at the interface between Al and Si. Retrieved with permission from Macmillan Publishers Limited, part of Springer Nature.^{Reference Li, Hage, Liu, Ramasse and Schumacher51}

At low concentrations of P, the AlP particles may also nucleate on oxide bifilms that are folded into the liquid phase, as suggested by Campbell and Tiryakioğlu and others.^{Reference Campbell and Tiryakioğlu19,Reference Pennors, Samuel, Samuel and Doty52} In the absence of any modifier species, the Si phase precipitates on these crumpled bifilms (decorated with the AlP particles). The planar growth of the Si phase straightens the bifilms, thereby creating cracks along the long axis of the Si particles. That is, the Si particles reflect the length of the straightened-out bifilms. From the preceding analysis, Sr deactivates the AlP nucleation sites and hence also the oxide bifilm as a growth substrate for Si. In modified alloys, then, the bifilms are no longer sequestered into Si particles and are now in free suspension in the liquid. To accommodate the solidification shrinkage upon freezing, the bifilm opens and a shrinkage cavity (pore) is initiated. This mechanism of porosity development in modified alloys is predicted by the fact that the oxide bifilm is a double film, unbonded between its two halves.^{Reference Campbell and Tiryakioğlu19}

Through phase-field (PF) simulation linked to CALPHAD databases, Eiken et al. showed that critical threshold amounts of P and Sr are needed for the pre-silicon formation of AlP and Al₂Si₂Sr, respectively.^{Reference Eiken, Apel, Liang and Schmid-Fetzer45} The threshold for P was found to be between 3.2 and 3.8 ppm, where the lower-bound was obtained from PF simulations due to the additional consideration of nucleation undercooling. Thus, the minimum P level to form AlP is indeed well below the commercial purity standards of Al–Si alloys. Meanwhile, thermodynamic calculations revealed a critical Sr threshold of approximately 80 ppm to form Al₂Si₂Sr. In corresponding PF simulations with subcritical Sr levels [Fig. 5(a)], AlP was not neutralized by Al₂Si₂Sr, and thus eutectic Si nucleated with high frequency on AlP with minimal undercooling.^{Reference Eiken, Apel, Liang and Schmid-Fetzer45} The solidification pathway for supercritical Sr levels [Fig. 5(b)] is notably different: Al₂Si₂Sr poisons the AlP particles, which results in a retarded nucleation of eutectic Si. Moreover, the nucleation frequency was found to be significantly less (e.g., one nuclei in the modified alloy versus four per unit volume in the unmodified alloy, see Fig. 5).^{Reference Eiken, Apel, Liang and Schmid-Fetzer45} The refinement and structural modification of the eutectic Si lamellae result from this reduced nucleation rate and will be dealt with below.

FIG. 5. PF simulation results for solidification in AlSi7 + 5 ppm P with two different Sr contents: (a, left column) 50 ppm Sr, and (b, right column) 100 ppm Sr. The first three rows show different time-steps during the microstructural evolution. In (a), eutectic Si nucleated on AlP and grew as plates while in (b) Al₂Si₂Sr first nucleated on AlP, deactivating the nucleation sites of Si. The latter forms at higher growth undercooling in a fibrous morphology. The bottom row shows the phase fractions versus temperature, evaluated from PF (solid lines), and Scheil prediction (dashed lines) for both alloy compositions. Figures adapted with permission from Elsevier.^{Reference Eiken, Apel, Liang and Schmid-Fetzer45}

It should be mentioned that the details surrounding the influence of the modifier species on the nucleation rate are not universally agreed upon. According to Liao and coworkers, the decrease of nucleation temperature in modified alloys only suggests that the required nucleation undercooling is increased.^{Reference Liao, Zhang, Wu, Wang and Sun53} In other words, the addition of the Sr modifier species makes the nucleation process more difficult at higher temperatures. Liao and coworkers comment that the driving force for nucleation increases with the Sr content, and in addition, “other nucleation sites are activated, although they do not have the power to operate at higher temperatures.”^{Reference Liao, Zhang, Wu, Wang and Sun53} Consequently, the higher driving force and the operation of these nucleation sites actually leads to an increased rate of nucleation of the modified eutectic. Thus, the number of nuclei per unit volume per unit time increases with respect to the unmodified alloy. Support for this alternative viewpoint comes from micrographs showing eutectic grain refinement in Al–Si alloys containing Sr and B.^{Reference Liao, Zhang, Wu, Wang and Sun53} Furthermore, not all modifier elements impact the nucleation rate in the same way: For instance, StJohn and coworkers demonstrated that Cu and Mg increased the nucleation density of Al–Si eutectic grains in unmodified and Sr-modified alloys, whereas Fe decreased the nucleation density in the same master alloys.^{Reference Darlapudi, McDonald, Terzi, Prasad, Felberbaum and StJohn54}

What is the influence of the modified nucleation density (either positive or negative) on the resultant Si morphology? In the early 1980s, Flood and Hunt provided one of the pioneering attempts to answer this question.^{Reference Flood and Hunt55} They reasoned that for a given rate of heat extraction, the factor controlling the growth velocity of the eutectic is the solid–liquid interfacial area, which is directly controlled by the nucleation frequency. For a constant rate of heat extraction $\dot{q}$, the growth velocity v will vary inversely with the total solid–liquid surface area A of the system. That is, $\dot{q} = vA{L_{\rm{f}}}$, where L _f is the latent heat of fusion per unit volume. Therefore, the more eutectic grains that nucleate, the larger the interfacial area A and the lower the growth velocity v.^{Reference Flood and Hunt55} In the context of Fig. 5, then, this analysis indicates that growth velocities are higher in the Sr-modified alloy. Additionally, due to kinetic roughening,^{Reference Sunigawa56} Si tends to be nonfaceted at these higher velocities. Flood and Hunt proposed that the morphological transition from plate-like to fibrous Si occurs at the same time as the transition from faceted to nonfaceted growth.^{Reference Flood and Hunt55} However, Dahle and coworkers argued that the increase in velocity in the Sr-modified alloy is not large enough to induce the flake–fiber transition in Si.^{Reference McDonald, Dahle, Taylor and StJohn57}

An alternative explanation has been put forth by StJohn and coworkers to explain the influence of nucleation density on the Si morphology in chemically modified alloys.^{Reference Darlapudi, McDonald, Terzi, Prasad, Felberbaum and StJohn54} During the growth of a eutectic grain, the modifier species is rejected by both eutectic phases into the melt and gradually piles up ahead of the solid–liquid interface. This solute segregation results in the formation of a constitutionally undercooled zone. The constitutional driving force for growth is related to the solute concentration gradient ahead of the interface, dC _l/dz, where C _l is the liquid concentration and z is the distance.^{Reference Kurz and Fisher2} Such constitutional effects in modified alloys can be better understood by considering Fig. 6: As two solid eutectic grains grow toward each other, their solute fields overlap and thus the constitutional driving force for growth decreases. After the driving force begins to decrease, the velocity decreases to a value where the Si morphology changes from fibrous to flake-like. This occurs at a point referred to as the “critical overlap” (see Fig. 6).^{Reference Darlapudi, McDonald, Terzi, Prasad, Felberbaum and StJohn54} Thus, a decrease in the spacing of nuclei (due to a high nucleation rate) causes the solute fields to overlap sooner. In this case, a higher proportion of eutectic grains grow with a fibrous morphology.^{Reference Darlapudi, McDonald, Terzi, Prasad, Felberbaum and StJohn54}

FIG. 6. Schematic illustration showing changes in the concentration gradient (and hence the constitutional supercooling) as two solid eutectic grains impinge on one another during solidification. $C_1^*$ is the equilibrium liquid concentration at the interface (∗), C ₀ is the far-field liquid concentration, z is the distance, and t is the time. At time-step t ₁, the diffusion fields of the solids do not overlap. Eventually, the diffusion fields overlap, see time-step t ₂. Finally, at time-step t ₃, solute accumulation in the overlap region attains a critical value at which the growth velocity decreases to trigger the morphological change to flake-like Si. Figure adapted with permission from Elsevier.^{Reference Darlapudi, McDonald, Terzi, Prasad, Felberbaum and StJohn54}

B. Modification during growth

The second school of thought contends that the modifier species influences growth and not nucleation of the eutectic phases. Among the various “modified growth” theories that have been proposed from the 1960s and onwards, the three that have gained the most traction are (i) the twin plane re-entrant edge (TPRE) mechanism,^{Reference Wagner58,Reference Hamilton and Seidensticker59} (ii) the poisoning of the TPRE mechanism,^{Reference Day and Hellawell60} and (iii) impurity-induced twinning (IIT).^{Reference Lu and Hellawell61} These three mechanisms are thought to be valid under different growth conditions. For instance, at slow cooling rates in unmodified alloys, the TPRE mechanism becomes activated and the Si lamellae assume a plate-like morphology with closely spaced twins aligned parallel to the longer axis. Detailed reviews of TPRE and its variants can be found elsewhere,^{Reference Shahani, Gulsoy, Poulsen, Xiao and Voorhees62,Reference Shahani and Voorhees63} but the key idea is that the growth of new layers of the crystal occurs preferentially at the re-entrant edges, i.e., where the twin boundary intersects the solid–liquid interface. The first account of TPRE was described in faceted Ge dendrites by Wagner and Hamilton and Seidensticker in 1960.^{Reference Wagner58,Reference Hamilton and Seidensticker59} It is worth mentioning that TPRE also occurs in chemically modified alloys, although the twins are no longer parallel to the long axis: According to Shamsuzzoha and Hogan’s viewpoint [Fig. 7(a)], the twins marked AB in the bottom left give rise to branches in the form of twins BC.^{Reference Shamsuzzoha and Hogan64} Further branching produces AB twins on the surface marked C. This process of multiple twinning perpetuates and generates the entire Si fiber, which grows in a net [001] direction.^{Reference Shamsuzzoha and Hogan64} Based on the observations of Wagner and Hamilton and Seidensticker, as well as the concept of surface adsorption, Day and Hellawell advanced the poisoning of the TPRE mechanism in 1968.^{Reference Day and Hellawell60} It was assumed that the modifier retards Si growth by adsorbing at the re-entrant edges, thereby deactivating the TPRE mechanism and forcing the Si to grow in a more isotropic manner. Finally, in 1987, Li and Hellawell proposed that the modifier atoms are adsorbed on the {111} step-surfaces of Si, and the associated change in the stacking sequence facilitates the formation of frequent crystallographic twins, a process they termed IIT.^{Reference Lu and Hellawell61} The IIT mechanism makes two key assumptions: (i) the faceted phase has an FCC crystal structure; and (ii) only modifier elements that meet the ‘ideal’ atomic radius ratio r _i/r ∼ 1.646 enable twin formation. Here, r _i is the atomic radius of the modifier element and r is that of the faceted phase. The arguments made by Lu and Hellawell to reach this conclusion were purely geometric. Yet some elements that do not satisfy this criterion (e.g., Sr and Na) still modify the eutectic Si microstructure to a significant extent by increasing the twin density (see Fig. 3). Thus, there are likely other factors beyond atomic radius that are critical for modified growth. Taken altogether, both poisoning of TPRE and IIT involve modifier adsorption on Si at the growth front; however, the two mechanisms differ in two important aspects: (a) the location of interfacial poisoning (i.e., re-entrant grooves versus facet planes), as well as (b) the consequence of it (twin suppression versus twin formation).

FIG. 7. (a) Schematic of a fully modified Si fiber with an effective [110] growth axis in the Al–14% Si–0.18% Sr alloy. The fiber contains intersecting twins with re-entrant edges in contact with the liquid phase. (b) These edges are “poisoned” by Sr–Al–Si (type-II) co-segregations that prevent the attachment of Si to the growing fiber. (a) Reprinted with permission from Taylor and Francis^{Reference Shamsuzzoha and Hogan64} and (b) with permission from Elsevier.^{Reference Timpel, Wanderka, Schlesiger, Yamamoto, Lazarev, Isheim, Schmitz, Matsumura and Banhart68}

Recently, Schumacher and colleagues have found direct experimental support for the poisoning of TPRE and the IIT mechanisms through high-resolution high angle annular dark field (HAADF) scanning transmission electron microscopy (STEM).^{Reference Li, Hage, Wiessner, Romaner, Scheiber, Sartory, Ramasse and Schumacher65} Their results on the diverse roles of trace Eu additions on the growth of eutectic Si in Al–Si alloys are summarized in Fig. 8. In some cases (see top row), Eu atoms are located approximately between every two Si atomic columns at the TPRE. These images point to the poisoning of the TPRE mechanism. DFT calculations performed by the same team show that structures with mixed Eu–Si occupation at the twin boundary, akin to Fig. 8(a), are energetically preferred (i.e., they possess a more negative segregation energy compared to other configurations).^{Reference Li, Hage, Wiessner, Romaner, Scheiber, Sartory, Ramasse and Schumacher65} On the other hand, Eu atoms can also be located at the intersection of Si facets and twins (see middle row), thereby suggesting that the IIT mechanism is also operative. Finally, Schumacher and colleagues observed the entrainment of continuous layers of Eu within Si (bottom row). The authors propose that during solidification of eutectic Si, Al and Eu will segregate ahead of the growth front (the partition coefficients k _Al < 1 and k _Eu < 1); during continued growth, the {111} planes of Si fold on each other, resulting in solute impingement and entrainment of the segregation fields.^{Reference Li, Hage, Wiessner, Romaner, Scheiber, Sartory, Ramasse and Schumacher65,Reference Li, Albu, Hofer and Schumacher66} No significant twinning was observed in the Si phase adjacent to the entrained Eu layer, indicating that this layer did not result from either the poisoning of TPRE nor the IIT mechanisms.

FIG. 8. High resolution HAADF STEM images and EELS maps of Al, Si, and Eu in an Al–5Si–0.05Eu alloy: (a) Eu atoms are located at the TPRE, indicating that poisoning of the TPRE mechanism is active; (b) Eu-rich atomic columns are located at the intersection of Si facets and twins, indicating that the IIT mechanism is active; (c) Eu atoms are located in a continuous Eu-rich layer, indicating that a solute entrainment occurs within eutectic Si. See text for descriptions of each mechanism. Retrieved with permission from Macmillan Publishers Limited, part of Springer Nature.^{Reference Li, Hage, Wiessner, Romaner, Scheiber, Sartory, Ramasse and Schumacher65}

The poisoning of the TPRE mechanism has also been used recently to explain the EBSD results of Liu et al. on a Sr-modified Al–Si alloy.^{Reference Liu, Zhang, Beausir, Liu, Esling, Yu, Zhao and Zuo67} The authors suggested that modification via poisoning can occur in two distinct ways: Sr can block the Si growth on one initial {111} twin plane and force the formation of (i) new twins with the same misorientation and inclination as the first, or (ii) new twins with different orientations and inclinations. The latter scenario leads to a change in the growth direction, resulting in a more isotropic growth and thus the appearance of “quasi-equiaxed” Si crystals.^{Reference Liu, Zhang, Beausir, Liu, Esling, Yu, Zhao and Zuo67} The authors suppose that due to this Sr-induced deceleration of the Si crystal in the original growth direction, the growth velocities of both eutectic phases become comparable, i.e., Si no longer leads at the growth front. However, this conclusion is impossible to verify through ex situ experimental probes like EBSD. To conclusively determine the structure of the solid–liquid interfaces, one would require nondestructive and 3D imaging of eutectic solidification as it proceeds in real-time. To meet this need, Moniri and coworkers have illustrated through in situ X-ray tomography that the retardation is indeed more dramatic than predicted by Liu et al. The growth velocities are not comparable; instead, it is the eutectic Al phase that leads at the growth front, such that steady-state coupled growth between the eutectic phases can no longer be maintained.^{Reference Moniri, Xiao and Shahani21}

Growth modification may be due to not only single modifier atoms but also intermetallic compounds that pre-exist in the liquid phase, as first suggested by Banhart and coworkers in 2012.^{Reference Timpel, Wanderka, Schlesiger, Yamamoto, Lazarev, Isheim, Schmitz, Matsumura and Banhart68} Their atom probe tomography (APT) investigation revealed that the Sr modifier co-segregates with Al and Si within the eutectic Si phase, as well as at the eutectic Si/Al interface (Fig. 9).^{Reference Timpel, Wanderka, Schlesiger, Yamamoto, Lazarev, Isheim, Schmitz, Matsumura and Banhart68} Interestingly, the chemical compositions of the co-segregations derived from the APT proximograms [see, e.g., Fig. 9(c)] correspond to Al₄Si₃₃Sr and Al₂Si₈₈Sr, and not Al₂Si₂Sr as seen in previous investigations. Nevertheless, both of these former compounds have negative Gibbs free energies of formation at the Al–Si equilibrium eutectic temperature of 577 °C, suggesting that their existence at this temperature is thermodynamically plausible. Such co-segregations interact with the eutectic phases in two distinct ways: (i) By adsorbing at the Si {111} growth steps and promoting a change of stacking sequence of the propagating Si phase; and (ii) by adsorbing along the re-entrant edges and hence preventing a further attachment of Si atoms to the growing crystal.^{Reference Timpel, Wanderka, Schlesiger, Yamamoto, Lazarev, Isheim, Schmitz, Matsumura and Banhart68} The first possibility represents a revision of the IIT mechanism, where it is not Lu and Hellawell’s geometrical size factor that plays a major role in modification but rather the chemistry of the co-segregate.^{Reference Timpel, Wanderka, Schlesiger, Yamamoto, Lazarev, Isheim, Schmitz, Matsumura and Banhart68} Meanwhile, the latter scenario is a generalization of the poisoning of the TPRE mechanism and is depicted schematically in Fig. 7(b). Synchrotron-based X-ray nanodiffraction and X-ray fluorescence (XRF) elemental mapping have also been used to confirm the presence of intermetallic Sr phases in Sr-modified Al–Si alloys.^{Reference Manickaraj, Gorny, Cai and Shankar69} According to Schumacher and coworkers, such nanoscale co-segregates (clusters) are likely subcritical nuclei, and the local ordering of the melt would be that of fluctuating clusters.^{Reference Barrirero, Li, Engstler, Ghafoor, Schumacher, Odén and Mücklich70} Stable clusters would require heterogeneous substrates or high undercooling. Thus, during solidification of eutectic Si, two processes are dominant, that of (a) segregation of Sr and Al out of Si, and (b) heterogeneous nucleation of Sr–Al–Si clusters at the propagating Si growth front.^{Reference Barrirero, Li, Engstler, Ghafoor, Schumacher, Odén and Mücklich70} By contrast, one may view clusters only as an artefact of solute entrainment (described above), and not a leading cause of eutectic modification. Proponents of this viewpoint contend that elements such as Ca and Yb are commonly found in Al₂Si₂Ca and Al₂Si₂Yb phases, respectively, yet these elements do not necessarily modify the eutectic Si morphology.^{Reference Li, Albu, Hofer and Schumacher66} Whether it is single modifier atoms or modifier-containing clusters that induce the multiplication of crystallographic defects remains to be determined. However, both cases have in common the interaction between the modifier and eutectic Si during the growth process.

FIG. 9. APT results of the eutectic Al/Si interface in an Al–10 wt% Si–0.1 wt% Fe alloy modified by 200 ppm Sr. (a) Top and (b) side views of 3D reconstruction of Sr (red) and Al (blue) atom positions in analyzed region-of-interest (ROI) of size 58 × 56 × 93 nm³. Si atoms have been omitted for clarity. The rendered isosurfaces represent 0.17 Sr atoms nm⁻¹ in both views. (c) Proximogram showing Sr, Al, and Si concentration as a function of distance to the Si/Sr–Al–Si co-segregation interface. Figures reprinted with permission from Elsevier.^{Reference Timpel, Wanderka, Schlesiger, Yamamoto, Lazarev, Isheim, Schmitz, Matsumura and Banhart68}

C. Modification during subsequent annealing

The last and most recent school of thought takes the view that eutectic modification occurs in the solid-state, i.e., following nucleation and growth and upon annealing of the fully solidified specimen.^{Reference Albu, Pal, Gspan, Picu, Hofer and Kothleitner71} This is the principal assessment of Kothleitner and coworkers, whose work is procedurally very similar to that of Schumacher et al.: The team used STEM together with DFT simulations to lend support to their theory of solid-state modification. Through STEM of a Sr-modified hypoeutectic Al–Si alloy, they observed that 〈110〉 interstitial Sr columns are located preferentially at the re-entrant groove of the twin boundaries.^{Reference Albu, Pal, Gspan, Picu, Hofer and Kothleitner71} Unlike the poisoning of the TPRE mechanism described above, however, Sr could not be detected along the twin defect in the $\left\langle {1\bar{1}\bar{2}} \right\rangle$ growth direction.

Kothleitner and coworkers also performed ab initio DFT simulations to understand why Sr self-organizes into 〈110〉 interstitial columns and whether this configuration of Sr at the twin boundaries is energetically favorable. Figure 10 shows the variation in the system energy with distance between Sr atoms in the same column when aligned in several different crystallographic directions.^{Reference Albu, Pal, Gspan, Picu, Hofer and Kothleitner71} Both substitutional and interstitial Sr were considered. From these data, the team deduced that a binding energy between Sr atoms exists only when the column is aligned in the 〈110〉 direction. If Sr is substitutional, the strongest binding is in the second nearest-neighbor position; and if Sr is interstitial, the strongest binding occurs in the nearest-neighbor position and hence interstitial Sr columns are composed entirely of Sr.^{Reference Albu, Pal, Gspan, Picu, Hofer and Kothleitner71} This result corroborates their STEM findings mentioned above. The team also compared the system energies of (i) a Si lattice with two successive twins, (ii) the same structure with two 〈110〉 interstitial Sr columns located at the twin boundaries, and (iii) a perfect Si lattice without the twins but with the same Sr columns as in (ii). Out of the three cases, the energetic cost of (ii)—introducing a twin boundary and a Sr column located on it—was the lowest. This indicates that it is thermodynamically favorable for Sr to diffuse to a twin boundary, once the twin has already formed (presumably during solidification, i.e., a growth twin) and the Sr is located away from it. Conversely, the formation of a twin starting exactly from a Sr column is also thermodynamically favorable.^{Reference Albu, Pal, Gspan, Picu, Hofer and Kothleitner71} Despite the elegance of their DFT results, assessing whether such columns are kinetically likely to occur within eutectic Si requires further analysis of nucleation barriers, solid-state diffusion mechanisms, etc.

FIG. 10. Variation of the system energy with distance between Sr atoms in the directions indicated in the legend. The vertical axis represents the excess energy above the pure Si lattice energy. This excess energy is scaled by the long-distance limit for each of the defects, which is 2.296 ± 0.072 eV for (a, top row) the interstitial case and 1.203 ± 0.100 eV for (b, bottom row) the substitutional case. Error bounds result from different k-point grids. The atomic configurations at right show the lowest energy states for interstitial (I) and substitutional (S) 〈110〉 Sr columns. Retrieved with permission from Macmillan Publishers Limited, part of Springer Nature.^{Reference Albu, Pal, Gspan, Picu, Hofer and Kothleitner71}

A. Outlook for experimental studies

External probes such as photons or electrons can be used to interact with matter in the eV and meV energy ranges, respectively, to gain insights that allow for characterizing and identifying materials. The proliferation of advanced characterization tools in recent years for ex situ nanoscale investigations of quenched modified eutectics has revealed some limitations of the classical models of modification and shed new light on the role of various modifier agents.^{Reference Li, Albu, Hofer and Schumacher66,Reference Timpel, Wanderka, Schlesiger, Yamamoto, Lazarev, Isheim, Schmitz, Matsumura and Banhart68,Reference Albu, Pal, Gspan, Picu, Hofer and Kothleitner71} Of relevance is electron energy loss spectroscopy (EELS), which provides the electronic fingerprint of the different constituents of a material and hence can provide atomically resolved mapping of the local composition within a sample.^{Reference Krivanek, Corbin, Dellby, Elston, Keyse, Murfitt, Own, Szilagyi and Woodruff72} For instance, in Eu-modified Al–Si alloys, Schumacher and coworkers observed nanoscale Al₂Si₂Eu intermetallic clusters or continuous Eu-rich layers located ahead of the Si twins via EELS (see above in Sec. III.B).^{Reference Li, Hage, Wiessner, Romaner, Scheiber, Sartory, Ramasse and Schumacher65} The authors also complemented the EELS maps with high-resolution STEM images that display the spatial distribution of Eu atoms in relation to the atomic columns of Si. Altogether, their results support the simultaneous occurrence of the IIT as well as poisoning of the TPRE growth mechanisms.

Given the typical concentrations (<100–1000 ppm) of modifying agents used in Al–Si alloys,^{Reference Sigworth20} the success of analytical methods to investigate the local spatial distribution of the modifying agent relies on some key factors: (i) the chemical identity (atomic number) of the modifier, (ii) the susceptibility of the modifier to electron-beam irradiation, and (iii) the resolution limit of the analytical method to be used (spatial and, if applicable, energy resolutions). Of significance among these factors is the chemical identity of the modifying agent. While Sr has been found to be amenable to a suite of analytical methods such as imaging (direct observation) via STEM,^{Reference Li, Albu, Hofer and Schumacher66,Reference Timpel, Wanderka, Schlesiger, Yamamoto, Lazarev, Isheim, Schmitz, Matsumura and Banhart68,Reference Timpel, Wanderka, Vinod Kumar and Banhart73} elemental analysis via X-ray energy dispersive spectroscopy (EDS),^{Reference Timpel, Wanderka, Schlesiger, Yamamoto, Lazarev, Isheim, Schmitz, Matsumura and Banhart68} and EELS in STEM (STEM-EDS, STEM-EELS), as well as XRF microscopy,^{Reference Nogita, Yasuda, Yoshida, Uesugi, Takeuchi, Suzuki and Dahle74} the detection of trace amounts of Na—first used by Pacz in 1921 and still among the most common modifier species—remains an experimental bottleneck. We note that of the two modifiers, Na is more potent as it produces more uniform fibrous structure at lower concentrations (typically 0.005–0.01%); meanwhile, Sr modification requires somewhat higher concentrations (0.01–0.02%).^{Reference Gruzleski75} Considering only the influence of the modifier species on growth (see Sec. III.B above), this behavior is to be predicted by the IIT mechanism. In other words, Na is predicted to produce more growth twins in Si as compared to Sr.^{Reference Lu and Hellawell61} Only recently was the distribution of Na atoms within eutectic Si in Al–Si observed by laser-assisted APT.^{Reference Li, Barrirero, Engstler, Aboulfadl, Mücklich and Schumacher76} Other considerations for electron microcopy-based techniques include the possible knock-on damage in light elements (e.g., Na) as well as susceptibility to electron-beam irradiation. Another way forward is in situ TEM studies of Al–Si alloys,^{Reference Eswara Moorthy and Howe77,Reference Schneider and Howe78} providing an alternative method to perform future solidification experiments with high chemical and spatial resolutions. Continuing advances in the design of analytical tools for high-resolution characterization of materials containing light elements as well as those prone to beam irradiation will aid further analysis of the atomic distribution of Na in future studies. Of note are recently developed STEM designs^{Reference Sagawa, Hashiguchi, Isabell, Ritz, Simson, Huth, Soltau, Martinez, Nellist and Kondo79} with enhanced annular bright field, a new imaging technique that facilitates the observation of light elements in part by their ability to operate at accelerating voltages as low as 30 kV.

B. Outlook for computational studies

According to the adsorption-and-entrapment model for modified growth discussed in Sec. III.B, the solutal entrapment likely forms an aggregation of Al–Si–X within, as well as in the vicinity of, the eutectic Si phase. Assuming that the chemical composition of these Al–Si–X clusters can be determined using analytical methods, it would in principle be possible to also derive an estimate of the free energy of formation for these intermetallic compounds at the eutectic equilibrium temperature to rationalize their thermodynamic stability, as in Ref. Reference Timpel, Wanderka, Schlesiger, Yamamoto, Lazarev, Isheim, Schmitz, Matsumura and Banhart68. A plausible means to accomplish this task is to use the CALPHAD method, which provides a realistic thermodynamic assessment of alloys based on all available information on the phase equilibria and thermochemical properties of a given system. However, the free energy curves might only be defined over a finite range of composition that may not necessarily coincide with those investigated in solidification studies. Therefore, a problem arises in cases where the free energies of the phases become undefined or discontinuous. We note that, in the phase field community, methods have been recently reported that eliminate such characteristics while largely retaining the free energy values.^{Reference Jokisaari and Thornton80}

Hecht et al. have reviewed the advances in phase field modeling of multiphase solidification.^{Reference Hecht, Gránásy, Pusztai, Böttger, Apel, Witusiewicz, Ratke, De Wilde, Froyen, Camel, Drevet, Faivre, Fries, Legendre and Rex81} Other reviews^{Reference Kulkarni, Kohanek, Tyler, Hanson, Kim, Thornton and Braun82,Reference LLorca and Orera83} and research articles,^{Reference Steinmetz, Hötzer, Kellner, Genau and Nestler84,Reference Zhang, Guo and Xiong85} as well as references therein, provide additional examples of how PF modeling and other theoretical methods can be used to understand the physical mechanisms controlling the microstructural development. However, in a phase field simulation, the interface is diffuse and thus the incorporation of interfacial energy anisotropy requires careful consideration. Such is the case for the eutectic Si constituent considered here, which forms sharp facets with a discrete set of orientations (below its roughening transition). When the anisotropic interfacial energy is sufficiently strong, the unregularized Cahn–Hilliard equations become ill-posed. To remove the ill-posedness, a number of regularization strategies have been introduced, see, e.g., Refs. Reference Eggleston, McFadden and Voorhees86 and Reference Torabi, Lowengrub, Voigt and Wise87.