Roles of Lanthanum and Cerium in Grain Reﬁnement of Steels during Solidiﬁcation

: Reﬁnement of as-cast structures is one of the most effective approaches to improve mechanical properties, formability, and surface quality of steel castings and ingots. In the past few decades, addition of rare earths (REs), lanthanum and cerium in particular, has been considered as a practical and effective method to reﬁne the as-cast steels. However, previous reports contained inconsistent, sometime even contradictory, results. This review summaries the major published results on investigation of the roles of lanthanum or/and cerium in various steels, provides reviews on the similarity and difference of previous studies, and clariﬁes the inconsistent results. The proposed mechanisms of grain reﬁnement by the addition of lanthanum or/and cerium are also reviewed. It is concluded that the grain reﬁnement of steels by RE additions is attributed to either heterogeneous nucleation on the in-situ formed RE inclusions, a solute effect, or the combined effect of both. The models/theories for evaluation of heterogeneous nucleation potency and for solute effect on grain reﬁnement of cast metals are also brieﬂy summarized.


Introduction
Generally, fine-grained microstructure leads to improved formability and to higher strength and toughness of steels. During solidification, microstructural refinement can also minimize the segregation and porosity in steel ingots or castings [1][2][3][4]. Hence, grain refinement has been regarded as an effective approach to improve the quality of steel products. Generally, grain refinement of steels can be achieved either by inoculation treatment of the steel melt to refine the as-cast microstructure or through refining austenite in the solid state. The former is termed as solidification grain refinement and the latter as solid grain refinement. The conventional solid grain refinement consists of adding strong carbide-forming elements, such as Nb, V, and Ti, during the steelmaking to enable carbides and carbonitride to precipitate during the subsequent hot working process, which impedes the growth of austenitic grains. The problem associated with this method is that the carbides or carbonitrides could reduce ductility by enabling surface transverse cracking during hot working if the process was not well-controlled [5]. Advanced solid grain refinement of steels is achieved through thermomechanical processing [6,7], which involves deforming and refining the supercooled austenite prior to transformation, and therefore the fine microstructure is obtained at room temperature. This thermomechanical processing, however, is not obviously suitable for cast steels. Furthermore, because the as-cast microstructure of steels has a direct influence on the subsequent formability of rolling or forging, and then on the mechanical properties of final steel products, control of the solidification is one of the key steps in controlling the quality of the final steel products.
• Fast cooling [12,13] This is another effective approach, but it is limited by the actual casting equipment and the actual size of the castings.
• Chemical grain refiners Grain refinement is promoted through heterogeneous nucleation by the addition of a master alloy containing inoculant particles. This is commonly referred to as inoculation treatment and is considered to be the most cost-effective and practical technique to produce small equiaxed grains in a metal casting process [14,15]. Compared with cast aluminum alloys and cast magnesium alloys, in which the inoculation treatment is widely used in industrial practice, grain refinement is not the primary purpose of melt treatment during the steel making process, except for cast steels. However, in order to improve the quality of steels, researchers and engineers have realized its importance and have tried hard to identify effective grain refiners for steels, especially for cast steels [16][17][18], in the past few years. A number of carbides (TiC, ZrC, and NbC), nitrides (AlN, TiN, and VN), and oxides (CaO, Al 2 O 3 , and Ce 2 O 3 ) have been tested through externally adding the particles into the melts [16,19,20]. Unfortunately, no breakthroughs have been reported.
An alternative approach to grain refinement of steels is to add rare earths (REs) into the steel melt during ingot casting and continuous casting [21]. This includes cast steels [22], carbon steels [23][24][25][26][27], non-oriented electrical steels [28], Fe-6.5 wt% Si alloy [29], and other engineering steels [30,31]. Although additions of REs in these steels led to significant grain refinement, the actual mechanisms of how rare earths refine the grains and their effects on mechanical properties are still unclear. Commonly, it is considered that RE additions lead to either purification of liquid steels [32], modification of inclusions [32][33][34], a microalloying effect [35][36][37][38][39][40][41][42], grain refinement [32,34], or their combinations. Previously reported results about the grain refinement during solidification of steels by adding lanthanum or/and cerium are rather inconsistent. There are no commonly accepted theories/models that can be used to fully explain and understand the roles of the REs. The present work reviews the most recently published results on RE-containing steels, aiming to identify the causes of the inconsistency and to understand the roles and mechanisms of lanthanum or/and cerium on the grain refinement during solidification of steels.
Unlike Al-and Mg-based alloys, the solidification process of steels may include direct formation of δ-ferrite (BCC crystal structure) from the liquid, peritectic reaction (δ + liquid→γ), and direct formation of γ-austenite (FCC crystal structure) from the liquid. According to the Fe-Fe 3 C binary diagram, when the carbon content is below 0.09 wt%, primary BCC δ-ferrite directly forms from the liquid, followed by a phase transformation from δ-ferrite to FCC γ-austenite. When carbon content is between 0.09 wt% and 0.17 wt%, primary δ-ferrite directly forms from the liquid followed by a peritectic reaction (δ + liquid→γ), and then δ-ferrite transforms to γ-austenite. In these two cases, the nucleation and growth of δ-ferrite define the size of austenite. Hence, the δ-ferrite should be refined during solidification. For steels with carbon content between 0.17 wt% and 0.53 wt%, primary δ-ferrite first forms from the liquid followed by a peritectic reaction. However, after the completion of the peritectic reaction, γ-austenite also directly forms from the liquid. For these steels, the nucleation and growth of both δ-ferrite and γ-austenite determine the size of austenite. Hence, both δ-ferrite and γ-austenite need to be refined. When the carbon content is over 0.53 wt%, the γ-austenite directly forms from the liquid. Hence, the solidification grain refinement should focus on refining the γ-austenite.
In practice, the non-equilibrium cooling and solute additions significantly influence the critical carbon content as stated above. However, which primary phase-δ-ferrite or γ-austenite-directly forms from the liquid still depends on the contents of carbon and other solutes. The development of effective techniques for grain refinement of steels should use δ-ferrite and γ-austenite as two different targets.

Effects of RE Addition on Macrostructure and Microstructure of As-Cast Steels
RE additions, particularly lanthanum (La) or/and cerium (Ce), directly lead to changes of macrostructure and microstructure of as-cast steels, which in turn influences the subsequent forming processes and properties of steel products. Noticeable grain refinement due to the addition of Ce/La in various steels has been reported. The grain refinement efficiency was expressed in terms of either a decrease in the proportion of columnar grains or a reduction in the average grain size, or both. However, there is still a lack of comprehensive investigations of the grain refinement mechanism due to the addition of REs. Major publications on this topic are reviewed as follows.

RE Additions in Austenitic Steels
For most austenitic steels, γ-austenite directly forms from the liquid during the solidification process. Such austenite can generally be retained at room temperature due to the high content of γ-stabilizers, such as C, Mn, Ni, etc., and relatively fast cooling. Hence, the microstructure of the steels at room temperature can reflect the grain refining efficiency well. Bartlett and co-workers [31] added a Ce-based inoculant, which contained ≈10.0-14.0%Ce, ≈22.0-26.0%Sc, ≈8.0-12.0%Cr, ≈22.0-27.0% Mn, and 1.0%C (weight percent), into an Fe-Mn-Al-C austenitic cast steel. It was found that a 0.1 wt% Ce addition significantly reduced the primary austenite grain size from ≈10-12 mm to 100 µm. Figure 1 shows the macrostructure of the castings. The grain refinement was attributed to the promoted heterogeneous nucleation of austenite on cerium oxide inclusions formed in the melt. Similarly, Haakosen et al [43] reported a reduction in both fraction of columnar structure and in grain size in a cast austenitic manganese steel with 1.42%C, 19.5%Mn, 2.4%Cr, 0.7%Si, 0.03%Al, and 0.15%Ti (weight percent) after the addition of cerium. Figure 2 shows the variation of austenite grain size with the cerium addition level. A critical addition level of ≈0.04-0.05 wt%, over which the grain size was dramatically reduced, was identified. Accordingly, over this critical addition level, the fraction of columnar grains was also significantly decreased, as shown in Figure 3. The authors considered that the grain refinement resulted from the accelerated heterogeneous nucleation of primary austenite via CeO 2 , CeAlO 3 , and Ce 2 O 2 S particles formed in the liquid steel. However, in both Bartlett's work and Haakosen's work, there was no direct experimental evidence to prove the hypothesis of heterogeneous nucleation on Ce-containing inclusions. Other mechanisms, such as the solute effect, may possibly contribute to the grain refinement. Hence, it is necessary to conduct more comprehensive studies to clarify the roles and mechanisms of REs in austenitic steels.   Macrostructure of an as-cast Fe-Mn-Al-C austenitic steel containing different cerium contents: (a) 0%, (b) 0.05%, and (c) 0.1%, reproduced from Reference [31], with permission of Springer, 2016.

Figure 2.
Relationship between the grain size and cerium content in a cast high manganese austenitic steel, reproduced from Reference [43], with permission of AIS Tech, 2011.

RE Additions in Ferritic Steels
According to either the lattice disregistry model [15] or the edge-to-edge matching model [44,45], grain refinement efficiency via heterogeneous nucleation is dependent on the crystal structure of the nucleation sites and the metals to be formed. Due to the different crystal structure of austenite and  Macrostructure of an as-cast Fe-Mn-Al-C austenitic steel containing different cerium contents: (a) 0%, (b) 0.05%, and (c) 0.1%, reproduced from Reference [31], with permission of Springer, 2016.

Figure 2.
Relationship between the grain size and cerium content in a cast high manganese austenitic steel, reproduced from Reference [43], with permission of AIS Tech, 2011.

RE Additions in Ferritic Steels
According to either the lattice disregistry model [15] or the edge-to-edge matching model [44,45], grain refinement efficiency via heterogeneous nucleation is dependent on the crystal structure of the nucleation sites and the metals to be formed. Due to the different crystal structure of austenite and Figure 3. Range of columnar zone versus cerium content in a cast high-manganese austenitic steel, reproduced from Reference [43], with permission of AIS Tech, 2011.

RE Additions in Ferritic Steels
According to either the lattice disregistry model [15] or the edge-to-edge matching model [44,45], grain refinement efficiency via heterogeneous nucleation is dependent on the crystal structure of the nucleation sites and the metals to be formed. Due to the different crystal structure of austenite and ferrite, the roles of REs may vary with steels. Zhang and co-workers [46] reported that the addition of Ce into a 00Cr17 high-purity ferrite stainless steel resulted in the refinement of the as-cast macrostructure [46]. As shown in Figure 4, although the columnar structure was refined with an increase in Ce addition, no equiaxed grains formed in this steel. Such result indicated that, at least, heterogeneous nucleation did not occur within the melt. However, Xu and co-workers [47] reported an increase in the equiaxed grain zone from 3% to 33% in an as-cast 430 ferritic stainless steel after adding 0.073 wt% Ce as shown in Figure 5 [47]. However, the refinement efficiency was very marginal. Xu and co-workers [47] considered that both the heterogeneous nucleation of ferrite on RE sulfide/oxide particles and the solute effect of Ce contributed to the reduction in columnar fraction in the ingots. Similar to the works on austenitic steels, no direct experimental evidence has been reported to verify these hypotheses. an increase in the equiaxed grain zone from 3% to 33% in an as-cast 430 ferritic stainless steel after adding 0.073 wt% Ce as shown in Figure 5 [47]. However, the refinement efficiency was very marginal. Xu and co-workers [47] considered that both the heterogeneous nucleation of ferrite on RE sulfide/oxide particles and the solute effect of Ce contributed to the reduction in columnar fraction in the ingots. Similar to the works on austenitic steels, no direct experimental evidence has been reported to verify these hypotheses.

RE Additions in Plain Carbon Steels
The most common use of REs was in commercial steels, including plain carbon steels. Because of the different carbon contents, the grain refining efficiency due to the RE additions also varied. Tuttle [25,[48][49][50] investigated the effects of RE additions on the microstructure and properties of 1010 and 1030 steel (plain carbon steels) and found that the appropriate additions of La and Ce refined the as-cast microstructure. In order to clarify the grain-refining mechanism, cast samples with various RE addition levels were characterized in terms of grain size, microstructural morphology, and crystallographic features using a scanning electron microscope (SEM) and transmission electron microscope (TEM) with an energy dispersive spectrometer (EDS). It was concluded that simple RE oxides or sulfides were unable to promote the heterogeneous nucleation of δ-ferrite, but more complex RE oxides and/or oxysulfides possibly acted as the nucleants. Unfortunately, such oxides/oxysulfides were not identified in Tuttle's work. an increase in the equiaxed grain zone from 3% to 33% in an as-cast 430 ferritic stainless steel after adding 0.073 wt% Ce as shown in Figure 5 [47]. However, the refinement efficiency was very marginal. Xu and co-workers [47] considered that both the heterogeneous nucleation of ferrite on RE sulfide/oxide particles and the solute effect of Ce contributed to the reduction in columnar fraction in the ingots. Similar to the works on austenitic steels, no direct experimental evidence has been reported to verify these hypotheses.

RE Additions in Plain Carbon Steels
The most common use of REs was in commercial steels, including plain carbon steels. Because of the different carbon contents, the grain refining efficiency due to the RE additions also varied. Tuttle [25,[48][49][50] investigated the effects of RE additions on the microstructure and properties of 1010 and 1030 steel (plain carbon steels) and found that the appropriate additions of La and Ce refined the as-cast microstructure. In order to clarify the grain-refining mechanism, cast samples with various RE addition levels were characterized in terms of grain size, microstructural morphology, and crystallographic features using a scanning electron microscope (SEM) and transmission electron microscope (TEM) with an energy dispersive spectrometer (EDS). It was concluded that simple RE oxides or sulfides were unable to promote the heterogeneous nucleation of δ-ferrite, but more complex RE oxides and/or oxysulfides possibly acted as the nucleants. Unfortunately, such oxides/oxysulfides were not identified in Tuttle's work.

RE Additions in Plain Carbon Steels
The most common use of REs was in commercial steels, including plain carbon steels. Because of the different carbon contents, the grain refining efficiency due to the RE additions also varied. Tuttle [25,[48][49][50] investigated the effects of RE additions on the microstructure and properties of 1010 and 1030 steel (plain carbon steels) and found that the appropriate additions of La and Ce refined the as-cast microstructure. In order to clarify the grain-refining mechanism, cast samples with various RE addition levels were characterized in terms of grain size, microstructural morphology, and crystallographic features using a scanning electron microscope (SEM) and transmission electron microscope (TEM) with an energy dispersive spectrometer (EDS). It was concluded that simple RE oxides or sulfides were unable to promote the heterogeneous nucleation of δ-ferrite, but more complex RE oxides and/or oxysulfides possibly acted as the nucleants. Unfortunately, such oxides/oxysulfides were not identified in Tuttle's work.
Grain refinement via La addition was also observed by Wang et al. [51] in an ASTM A216 steel (a plain carbon steel). The team calculated the lattice mismatch using the two-dimensional disregistry theory and indicated that the La inclusions of La 2 O 3 , La 2 O 2 S, and LaS could act as the heterogeneous nucleation sites of δ-ferrite. However, these inclusions were not experimentally observed. Similar work was done by Li et al. [52] who added 0.16 wt% cerium into a 1045 carbon steel by in-mold addition and observed a remarkable increase in the equiaxed grain zone from 16 ± 2% to 64 ± 2% and a reduction in the average grain size from 762 µm to 541 µm. Then, the effects of cerium addition on the degree of undercooling in a pure iron and the 1045 steel were investigated using the levitation melting method while the surface tension of the levitated droplets was measured. However, based on the results, it was indicated that cerium solute redistribution during the solidification is the dominant factor contributing to the refinement of the as-cast structure of the 1045 steel.

RE Additions in other Engineering Steels
REs have also been widely used in engineering alloy steels. Wang et al. [53] observed a reduction in proportion of the columnar zone from 90.7% to 58.1% in an advanced low alloy steel (no composition of the steel is indicated in the reference) after adding 0.12 wt% mixture of La and Ce. Guo and Suito [23,24] investigated the influence of dissolved Ce and the primary particles of Ce 2 O 3 and CeS on the as-cast macrostructure and microstructure of a 0.20%C-0.02%P steel. They found that the primary and secondary dendrite arm spacing decreased with an increase in either the cerium content or the amount of Ce 2 O 3 and CeS particles. However, the fraction of equiaxed grains was independent of the soluble cerium content even though it was increased owing to the addition of Ce 2 O 3 or CeS particles.
Gao's work [30] also indicated that adding 0.015 wt% mixed RE (mainly containing 30% La and 65% Ce) led to a noticeable grain refinement of an as-cast H13 steel as shown in Figure 6. However, further addition of RE had little effect on the grain size. Detailed characterization of the as-cast H13 steel using both EDS analysis and TEM electron diffraction identified a Ce oxysulfide, Ce 2 O 2 S, in the steel. The authors further calculated the lattice mismatch between the Ce 2 O 2 S and the ferritic matrix according to the two-dimensional lattice disregistry theory, identifying a low mismatch of 3.5%. Hence it was concluded that the grain refinement was attributed to the heterogeneous nucleation of ferrite on the Ce 2 O 2 S inclusion. However, H13 steel contains 0.32 to 0.45 wt% carbon. During solidification process, it is questionable how much δ-ferrite can directly form from the liquid. In addition, explanation of the constant grain size when a higher amount of RE was added is also a challenge. work was done by Li et al. [52] who added 0.16 wt% cerium into a 1045 carbon steel by in-mold addition and observed a remarkable increase in the equiaxed grain zone from 16 ± 2% to 64 ± 2% and a reduction in the average grain size from 762 μm to 541 μm. Then, the effects of cerium addition on the degree of undercooling in a pure iron and the 1045 steel were investigated using the levitation melting method while the surface tension of the levitated droplets was measured. However, based on the results, it was indicated that cerium solute redistribution during the solidification is the dominant factor contributing to the refinement of the as-cast structure of the 1045 steel.

RE Additions in other Engineering Steels
REs have also been widely used in engineering alloy steels. Wang et al. [53] observed a reduction in proportion of the columnar zone from 90.7% to 58.1% in an advanced low alloy steel (no composition of the steel is indicated in the reference) after adding 0.12 wt% mixture of La and Ce. Guo and Suito [23,24] investigated the influence of dissolved Ce and the primary particles of Ce2O3 and CeS on the as-cast macrostructure and microstructure of a 0.20%C-0.02%P steel. They found that the primary and secondary dendrite arm spacing decreased with an increase in either the cerium content or the amount of Ce2O3 and CeS particles. However, the fraction of equiaxed grains was independent of the soluble cerium content even though it was increased owing to the addition of Ce2O3 or CeS particles.
Gao's work [30] also indicated that adding 0.015 wt% mixed RE (mainly containing 30% La and 65% Ce) led to a noticeable grain refinement of an as-cast H13 steel as shown in Figure 6. However, further addition of RE had little effect on the grain size. Detailed characterization of the as-cast H13 steel using both EDS analysis and TEM electron diffraction identified a Ce oxysulfide, Ce2O2S, in the steel. The authors further calculated the lattice mismatch between the Ce2O2S and the ferritic matrix according to the two-dimensional lattice disregistry theory, identifying a low mismatch of 3.5%. Hence it was concluded that the grain refinement was attributed to the heterogeneous nucleation of ferrite on the Ce2O2S inclusion. However, H13 steel contains 0.32 to 0.45 wt% carbon. During solidification process, it is questionable how much δ-ferrite can directly form from the liquid. In addition, explanation of the constant grain size when a higher amount of RE was added is also a challenge.

Discussion
To understand the actual roles of REs in grain refinement in different steels, the abovementioned results are summarized in Table 1. These results lead to the following conclusions: Figure 6. Variation of average grain sizes of as-cast H13 steel with additions of RE, reproduced from Reference [30], with permission of Springer, 2015.

Discussion
To understand the actual roles of REs in grain refinement in different steels, the above-mentioned results are summarized in Table 1. These results lead to the following conclusions:

•
The addition of REs led to a reduction in either fraction of columnar grains or the grain size of equiaxed grains or both in the as-cast steels.

•
The grain-refining mechanism of RE additions seemed similar in ferritic steels, austenitic steels, and engineering steels even though the grain refining efficiency varies in different steels.

•
In terms of grain refinement, the roles of lanthanum and cerium seemed similar in the as-cast steels.

•
The most generally accepted explanation of the grain refinement was heterogeneous nucleation of either δ-ferrite or γ-austenite on the RE inclusions, including oxides, sulfides and oxysulfides, and solute effects.
These conclusions are also associated with a few questions: • What are the predominant factors that led to the reduction in the fraction of columnar grains, and what are the factors that are responsible for the equiaxed grain refinement? To answer these questions, the roles of the inoculant and solutes in the solidification of δ-ferrite and γ-austenite need to be comprehensively investigated though crystallographic and thermodynamic calculation followed by experimental verification.

•
What are the actual causes of significant difference in the grain-refining efficiency in different steels? For example, Ce addition led to a grain size reduction from 10-12 mm down to 100 µm in the Fe-Mn-Al-C cast steel [31]; however, the grain refining efficiency in another austenitic manganese steel [43] is much smaller. In the 00Cr17 stainless steel [46], Ce addition only led to refinement of the columnar grains, but the grain size was slightly reduced in the 430 ferritic stainless steel [47]. Grain refining efficiency depends on various factors, such as the casting condition and cooling rate. To clarify these inconsistent results, the effects of Ce (or other solutes) additions need to be investigated in different steels under the same casting conditions and cooling rate. In addition, the growth restriction factor of Ce (or other solutes) should be more carefully calculated in various steel systems. • Addition of REs in steels resulted in the formation of various inclusions as mentioned above. The question is: which inclusions are more effective in promoting the heterogeneous nucleation of the BCC δ-ferrite and for the FCC γ-austenite? What are the key factors that govern the formation of different types of RE-containing inclusions? How do the oxygen content, sulfur content, and their relative amount (the ratio of oxygen content to sulfur content) affect the formation of the inclusions? To answer these questions, considerable research work needs to be conducted. For example, special alloys should be designed so that the δ-ferrite formed directly from the liquid at high temperature can be retained at room temperature without subsequent phase transformations. The similar alloys are required to keep the γ-austenite directly formed from the liquid to room temperature. In addition, types of RE inclusions formed in these specially designed alloys containing various level of oxygen and sulfur should be comprehensively investigated through thermodynamic calculation and experimental observation. The graining refining efficiencies of different RE inclusions in both δ-ferrite and γ-austenite can be crystallographically predicted and experimentally confirmed under the same casting conditions. • How does the chemical composition of steels influence the grain-refining efficiency of the RE additions? To answer this question, the grain refinement through RE additions needs to be studied in wide range of steels under the same casting conditions, together with a study to understand the affinity of RE to the alloying elements in the steels in addition to oxygen and sulfur.
In addition to these questions, there is also lack of direct experimental evidence to verify the hypothesis that the RE oxides/sulfides/oxysulfides are effective heterogeneous nucleation sites for either δ-ferrite or γ-austenite. In terms of the solute effect of the REs, the actual growth restriction factors (Q-values) of the REs in both ferrite and austenite are also not available. Hence, due to the importance of REs in steels, it is necessary to conduct comprehensive studies to understand the roles and mechanisms of REs in steels, particularly in grain refinement. No information is given The columnar structure was refined.
-Experimental [46] No explanation was provided. La and Ce An advanced low alloy steel (No composition is indicated) No information is given The columnar grains zone reduced from 90.7% to 58.1%.
-Experimental [53] Ce Fe-Mn-Al-C cast steel Air-set silica sand mold coated with zircon The primary austenite grain size was significantly refined from 10~12 mm to 100 µm.
Experimental [31] Direct experimental evidence is needed to verify these hypotheses.

Ce
Cast austenitic manganese steel Olivine sand mold The fraction of columnar structure decreased and the grains were refined. The primary and secondary dendrite arm spacings were decreased and the equiaxed grain zone was enlarged.
The solute effect of Ce and the heterogeneous nucleation of ferrite on Ce 2 O 3 and CeS particles were responsible for the grain refinement.
Experimental [23,24] La and Ce 1010 and 1030 steel (plain carbon steels) Silica based molding sand mold The RE additions refined the as-cast microstructure.
The complex RE oxides or oxysulfides acted as the heterogeneous nucleation sites for δ-ferrite.
Ce 2 O 2 S could act as the heterogeneous nucleation sites for δ-ferrite.

Experimental [30]
Direct experimental evidence is needed to verify the assumption. In addition, it is also questionable whether δ-ferrite directly formed from the liquid.

State of Rare Earths in Steels
Because of the high affinity with oxygen and sulfur, the externally added rare earth elements tend to chemically react with oxygen first and then with sulfur to form various RE inclusions in steel melts depending on the concentrations of oxygen and sulfur. If the content of REs and other elements is sufficient, REs can also react with P or other impurities to form different compounds. If the oxygen and sulfur contents are low, and/or the amount of REs added is higher, the remaining REs would possibly dissolve in the ferrite/austenite forming solid solution. In most industrial steels, the externally added RE elements highly likely form oxides, oxysulfides, or sulfides, and also possibly exist as solutes in the solid solution [54][55][56]. Hence, it is generally considered that grain refinement of steels by RE additions is attributed to either the heterogeneous nucleation on RE inclusions [25,30,31,43,[48][49][50][51][52], a solute effect [52], or both [47,51].

Formation of Rare Earth Inclusions in Molten Steel
Adding lanthanum and cerium into molten steel leads to the formation of various compounds. Figure 7 shows the equilibrium lines showing the affinity of a few alloying elements to oxygen and sulfur in liquid iron at 1600 • C [33]. It can be seen that except for Ca, La and Ce have the highest tendency to form oxides and sulfides. Generally, oxides form prior to sulfides. Considerable studies [57][58][59] verified the formation sequence as: RE oxides > RE oxysulfides > RE sulfides. The typical oxide is RE 2 O 3 . The oxysulfide is an intermediate product during the transition from deoxygenation to desulfuration by cerium and lanthanum. The variety and distribution of the RE oxysulfides rely on the contents of oxygen and sulfur and the condensate depression of the molten steel. The RE 2 O 2 S commonly forms. The type of RE sulfides is related to the ratio of RE content to S content, i.e., the value of [RE]/[S], in the molten steel. Typical sulfides are RE 2 S 3 , RE 3 S 4 , and RES. Figure 8 is the inclusion precipitation diagram for the Fe-Ce-O-S system at 1627 • C, showing the equilibrium reaction of Ce with oxygen and sulfur and the formation sequence of Ce-containing inclusions [33]. The melting temperatures of all these RE inclusions are generally above the solidification temperature of steel melts. Thus, they have the potential to act as the heterogeneous nucleation sites for the primary phases of steels, leading to grain refinement.

State of Rare Earths in Steels
Because of the high affinity with oxygen and sulfur, the externally added rare earth elements tend to chemically react with oxygen first and then with sulfur to form various RE inclusions in steel melts depending on the concentrations of oxygen and sulfur. If the content of REs and other elements is sufficient, REs can also react with P or other impurities to form different compounds. If the oxygen and sulfur contents are low, and/or the amount of REs added is higher, the remaining REs would possibly dissolve in the ferrite/austenite forming solid solution. In most industrial steels, the externally added RE elements highly likely form oxides, oxysulfides, or sulfides, and also possibly exist as solutes in the solid solution [54][55][56]. Hence, it is generally considered that grain refinement of steels by RE additions is attributed to either the heterogeneous nucleation on RE inclusions [25,30,31,43,[48][49][50][51][52], a solute effect [52], or both [47,51].

Formation of Rare Earth Inclusions in Molten Steel
Adding lanthanum and cerium into molten steel leads to the formation of various compounds. Figure 7 shows the equilibrium lines showing the affinity of a few alloying elements to oxygen and sulfur in liquid iron at 1600 °C [33]. It can be seen that except for Ca, La and Ce have the highest tendency to form oxides and sulfides. Generally, oxides form prior to sulfides. Considerable studies [57][58][59] verified the formation sequence as: RE oxides > RE oxysulfides > RE sulfides. The typical oxide is RE2O3. The oxysulfide is an intermediate product during the transition from deoxygenation to desulfuration by cerium and lanthanum. The variety and distribution of the RE oxysulfides rely on the contents of oxygen and sulfur and the condensate depression of the molten steel. The RE2O2S commonly forms. The type of RE sulfides is related to the ratio of RE content to S content, i.e., the value of [RE]/[S], in the molten steel. Typical sulfides are RE2S3, RE3S4, and RES. Figure 8 is the inclusion precipitation diagram for the Fe-Ce-O-S system at 1627 °C, showing the equilibrium reaction of Ce with oxygen and sulfur and the formation sequence of Ce-containing inclusions [33]. The melting temperatures of all these RE inclusions are generally above the solidification temperature of steel melts. Thus, they have the potential to act as the heterogeneous nucleation sites for the primary phases of steels, leading to grain refinement.

Models/Theories for Evaluation of Heterogeneous Nucleation Potency
In order to facilitate the heterogeneous nucleation, a valid nucleating agent (also termed a nucleant or inoculant) should fulfill the following three essential conditions [48,60]: (1) they are solid at the solidification temperature of the metal melt, (2) they are thermodynamically stable in the melt, and (3) they are wetted by the melt (low wetting angle between the melt and the solid). In addition, according to the classic heterogeneous nucleation theory, to improve the nucleation rate and therefore increase the grain refinement efficiency, the interfacial energy between the nucleating agents and the nucleus formed on the agents is required to be as low as possible. Generally, there are a number of factors influencing the interfacial energy [16], such as crystallographic atomic matching and the difference in electronegativity between the nucleating agents and the nucleation phase. Because it is rather difficult to accurately calculate the interfacial energy, crystallographic atomic matching is usually used to evaluate the nucleation potency of particles. This is based on two assumptions. One is that the coherent or semi-coherent interface between two crystalline solids corresponds to lower interfacial energy than the incoherent interface, and another is that the minimization of the interfacial energy of coherent or semi-coherent interface can be achieved through maximization of atomic matching across the interface. In the past few decades, several models have been developed to examine the nucleation potency and some have been successful in evaluating the grain refinement efficiency. This includes the lattice disregistry approach [16,61], the electrostatic action theory [62,63], and the edge-to-edge matching (E2EM) model [64][65][66].

Lattice Disregistry Approach
The lattice disregistry approach is the earliest approach to evaluate the grain refinement potency from crystallographic point view simply based on the lattice parameters of the nucleation agent and phase to be formed on it. In 1952, Turnbull and Vonnegut [61] proposed that the grain refinement potency of a nucleating agent depended on the lattice disregistry between the nucleating agent and the nucleation phase. The one-dimensional disregistry was defined as follows:

Models/Theories for Evaluation of Heterogeneous Nucleation Potency
In order to facilitate the heterogeneous nucleation, a valid nucleating agent (also termed a nucleant or inoculant) should fulfill the following three essential conditions [48,60]: (1) they are solid at the solidification temperature of the metal melt, (2) they are thermodynamically stable in the melt, and (3) they are wetted by the melt (low wetting angle between the melt and the solid). In addition, according to the classic heterogeneous nucleation theory, to improve the nucleation rate and therefore increase the grain refinement efficiency, the interfacial energy between the nucleating agents and the nucleus formed on the agents is required to be as low as possible. Generally, there are a number of factors influencing the interfacial energy [16], such as crystallographic atomic matching and the difference in electronegativity between the nucleating agents and the nucleation phase. Because it is rather difficult to accurately calculate the interfacial energy, crystallographic atomic matching is usually used to evaluate the nucleation potency of particles. This is based on two assumptions. One is that the coherent or semi-coherent interface between two crystalline solids corresponds to lower interfacial energy than the incoherent interface, and another is that the minimization of the interfacial energy of coherent or semi-coherent interface can be achieved through maximization of atomic matching across the interface. In the past few decades, several models have been developed to examine the nucleation potency and some have been successful in evaluating the grain refinement efficiency. This includes the lattice disregistry approach [16,61], the electrostatic action theory [62,63], and the edge-to-edge matching (E2EM) model [64][65][66].

Lattice Disregistry Approach
The lattice disregistry approach is the earliest approach to evaluate the grain refinement potency from crystallographic point view simply based on the lattice parameters of the nucleation agent and phase to be formed on it. In 1952, Turnbull and Vonnegut [61] proposed that the grain refinement potency of a nucleating agent depended on the lattice disregistry between the nucleating agent and the nucleation phase. The one-dimensional disregistry was defined as follows: δ = |a s − a n | a n (1) where δ is the lattice disregistry and the parameters of a n and a s denote the lattice constant in the low index crystal planes of the nucleus and the nucleating agent, respectively. A smaller value of δ implies a lower interfacial energy, which indicates the higher grain refinement efficiency. Because this model is limited to the case that the nucleus and the nucleation particles have the same atomic configuration on the specific low index plane, it has been rarely used.
In 1970, Bramfitt [16] proposed the two-dimensional disregistry model on the basis of the one-dimensional disregistry. The lattice disregistry is expressed as follows [16]: where δ is the lattice disregistry, (hkl) s is a low-index plane of the nucleating agent, [uvw] s is a low-index direction in the (hkl) s , (hkl) n is a low-index plane of the nucleus, [uvw] n is a low-index direction in (hkl) n , d [uvw]  Bramfitt [16] also proposed a criterion for evaluating the grain refinement potency: the most effective grain refinement can be achieved when the δ-value is below 6%; the agents are moderately effective when the δ-value is in between 6% and 12%; and if the δ-value is over 12%, the agent almost cannot refine the grains.
In the past few decades, this model has been widely used. The grain refinement potency of the RE oxides, oxysulfides, and sulfides for δ-ferrite and/or γ-austenite primary phase during solidification of steels have been examined using the two-dimensional disregistry model [25,30,48,51,[67][68][69]. Pan et al. [70] calculated the two-dimensional disregistry between the potential nucleants, Ce 2 O 3 and La 2 O 3 , and the primary phases (δ-ferrite and γ-austenite) at 1500 • C. The calculated δ values were: δ ferrite−Ce 2 O 3 = 4.49%, δ ferrite−La 2 O 3 = 3.34%, δ austenite−Ce 2 O 3 = 14.21%, and δ austenite−La 2 O 3 = 13.17%. In terms of the Bramfitt's criteria [16], Ce 2 O 3 and La 2 O 3 could effectively promote heterogeneous nucleation of δ-ferrite, but not for γ-austenite. However, the disregistry between the La 2 O 3 and δ-ferrite calculated by Tuttle [25,48] was around 30%, which is much higher than that of the generally accepted level for effective nucleation. Tables 2 and 3 list the published δ-values between various RE inclusions and primary δ-ferrite/γ-austenite. From these data, the following approximate predictions can be made:   The data in Tables 2 and 3 also show that the δ-values closely depend on the selection of the low index planes. Because no clear criteria for selecting the low index planes and directions were specified in the two-dimensional disregistry model, the calculated δ-values are highly scattered. Even for a given system of plane and direction, the δ-values were also inconsistent. For example, the smallest δ-value for Ce 2 O 2 S and δ-ferrite calculated by Hao and co-workers [68] was 10.3%, which corresponds to a moderate grain refining efficiency of Ce 2 O 2 S for δ-ferrite. However, Gao and team [30] calculated the smallest δ-value was 3.5%, implying a high grain refining efficiency of Ce 2 O 2 S for δ-ferrite. Hence, although the two-dimensional lattice disregistry model has been widely used, further improvement of the model is needed. In addition, this model is limited to the system of the nucleant and the metal with simple crystal structures. For the nucleants that have a more complicated crystal structure, the calculation of the lattice disregistry would be inaccurate because atoms may not occupy the lattice points in this type of crystal structure. Furthermore, the proposed criterion also sometimes fails. For example, Liang and co-workers' experimental works [75,76] showed different grain refinement efficiency of various nucleants even though they were associated with the same disregistry.

Electrostatic Action Theory
The electrostatic action theory was proposed in 1969 [62,63] based on the calculation of the electronic work functions of the nucleus, liquid phase, and the nucleation agent. Due to the difficulty and even impossibility regarding the accurate calculation of the electronic work function for crystals with complicated structures, the application of this electrostatic action theory has been very limited. To simplify the theory, Liang and co-workers combined Yu's empirical electron theory of solids and molecules (EET theory) and Cheng's improved TFD (Thomas-Fermi-Dirac) model, and proposed a valence electron model [75,76] for heterogeneous nucleation. This model is composed of the hybridization stable layer for nucleation phase cells, the dual crystal lattice electron layer for nucleation phase and the nucleating agent, and the hybridization stable layer for nucleating agent. Moreover, a criterion for evaluation of the grain refinement potency was constructed according to the following equation: where ∆ζ is defined as a characteristic parameter, and ζ n and ζ s are lattice electron migration potentials of nucleation phase and nucleating agent respectively. Bigger values of ∆ζ implies greater atomic migration rate and more stable combination between the nucleation phase and the nucleating agent, which indicates a higher grain refinement potency of the nucleating agent. Liang et al. analyzed the heterogeneous nucleation potency of VC, NbC, TiC, TaC, ZrC, and CaS particles for the non-equilibrium solidification of an austenite steel [75] and that of ZrC, TiC, TiN, and TiO in the process of Au solidification [76] using the model and criterion. The results agreed well with the experiments. Pan and co-workers [70,77] compared the validity of the two-dimensional disregistry model and the valence electron model in terms of the grain refinement efficiency for the two possible primary phases, δ-ferrite or ã-austenite, in steels, and found very inconsistent results. Hence, further refinement of the valence electron model is necessary.

The Edge-to-Edge Matching (E2EM) Model
The edge-to-edge matching model [64][65][66] was originally developed by M-X Zhang and P.M. Kelly for understanding and predicting the orientation relationships (ORs) and habit plane between two adjacent phases in diffusion-controlled solid-solid phase transformations. This model has been the unique crystallography model that has been successful in interpreting the mechanism of grain refinement in cast metals and identifying new grain refiners in Al, Mg, and their alloys [78][79][80][81].
Like all other crystallographic models [82][83][84][85][86], the E2EM model is also based on the assumption that the crystallographic features of the interface between two phases are defined by the minimization of interfacial energy, which can be achieved through maximization of atomic matching across the interface. The model emphasizes the actual atomic matching between two phases rather than lattice matching specified in the two-dimensional disregistry model. Figure 9 schematically demonstrates an interface that forms in the manner of the edge-to-edge matching [66]. Generally, the primary requirement of the model is the atomic matching along the parallel atomic rows between two phases. The matching atomic rows should be close-packed (CP) or nearly CP directions with low interatomic spacing misfit (f r ). The matching rows can be either straight or zigzag, but the E2EM model requires that straight rows match with straight rows, while zigzag rows match with zigzag rows, as is seen in Figure 10 [64]. Such rows are termed as the matching directions. In addition, it also requires a pair of CP or nearly CP planes that contain the matching directions having a small interplanar spacing misfit (f d ). Such planes are termed as matching planes. If these requirements are fulfilled, a favorable OR between the two phases can be constructed. interface. The model emphasizes the actual atomic matching between two phases rather than lattice matching specified in the two-dimensional disregistry model. Figure 9 schematically demonstrates an interface that forms in the manner of the edge-to-edge matching [66]. Generally, the primary requirement of the model is the atomic matching along the parallel atomic rows between two phases. The matching atomic rows should be close-packed (CP) or nearly CP directions with low interatomic spacing misfit (fr). The matching rows can be either straight or zigzag, but the E2EM model requires that straight rows match with straight rows, while zigzag rows match with zigzag rows, as is seen in Figure 10 [64]. Such rows are termed as the matching directions. In addition, it also requires a pair of CP or nearly CP planes that contain the matching directions having a small interplanar spacing misfit (fd). Such planes are termed as matching planes. If these requirements are fulfilled, a favorable OR between the two phases can be constructed.  Based on the identification of the CP (or nearly CP) rows and the CP (or nearly CP) planes, the interatomic spacing misfit (fr) along the matching directions and the interplanar spacing mismatch (fd) of the matching planes between the two phases can be calculated using the following equations [65]: where M r and P r are the interatomic spacing along the matching directions, and M d and P d are the interplanar spacing of the matching planes between the matrix phase and the precipitate particle, respectively. The empirical criteria of the E2EM model is that the values of fr and fd are both less than 10% [60]. Compared with the lattice disregistry theory, the E2EM model can not only estimate the atomic matching between two phases, but also predict the favorable ORs based on the lattice parameters and the positions of the atoms in the two phases. For systems with a simple crystal structure, the interface. The model emphasizes the actual atomic matching between two phases rather than lattice matching specified in the two-dimensional disregistry model. Figure 9 schematically demonstrates an interface that forms in the manner of the edge-to-edge matching [66]. Generally, the primary requirement of the model is the atomic matching along the parallel atomic rows between two phases. The matching atomic rows should be close-packed (CP) or nearly CP directions with low interatomic spacing misfit (fr). The matching rows can be either straight or zigzag, but the E2EM model requires that straight rows match with straight rows, while zigzag rows match with zigzag rows, as is seen in Figure 10 [64]. Such rows are termed as the matching directions. In addition, it also requires a pair of CP or nearly CP planes that contain the matching directions having a small interplanar spacing misfit (fd). Such planes are termed as matching planes. If these requirements are fulfilled, a favorable OR between the two phases can be constructed.  Based on the identification of the CP (or nearly CP) rows and the CP (or nearly CP) planes, the interatomic spacing misfit (fr) along the matching directions and the interplanar spacing mismatch (fd) of the matching planes between the two phases can be calculated using the following equations [65]: where M r and P r are the interatomic spacing along the matching directions, and M d and P d are the interplanar spacing of the matching planes between the matrix phase and the precipitate particle, respectively. The empirical criteria of the E2EM model is that the values of fr and fd are both less than 10% [60]. Compared with the lattice disregistry theory, the E2EM model can not only estimate the atomic matching between two phases, but also predict the favorable ORs based on the lattice parameters and the positions of the atoms in the two phases. For systems with a simple crystal structure, the Based on the identification of the CP (or nearly CP) rows and the CP (or nearly CP) planes, the interatomic spacing misfit (f r ) along the matching directions and the interplanar spacing mismatch (f d ) of the matching planes between the two phases can be calculated using the following equations [65]: where r M and r P are the interatomic spacing along the matching directions, and d M and d P are the interplanar spacing of the matching planes between the matrix phase and the precipitate particle, respectively. The empirical criteria of the E2EM model is that the values of f r and f d are both less than 10% [60]. Compared with the lattice disregistry theory, the E2EM model can not only estimate the atomic matching between two phases, but also predict the favorable ORs based on the lattice parameters and the positions of the atoms in the two phases. For systems with a simple crystal structure, the disregistry model and the E2EM model may give the same results. However, for systems with complicated crystal structure, the E2EM model has an advantage that it more accurately examines the actual matching between atoms. In addition, the E2EM model emphasizes CP or nearly CP planes/directions. This overcomes the problem of how to specify the low-index planes associated with the lattice disregistry model. Since 2005 [78,79], although the E2EM model has been successfully used to understand the grain refinement mechanism and develop new grain refiners in Al, Mg, and their alloys [80,[87][88][89], and cast Zn and Zn alloys [90,91], its application to grain refinement of steels is very limited. Most recently, Li et al. [92] conducted a comprehensive crystallographic study on the grain refinement efficiency of NbO, CeS, TiN, Ce 2 O 3 , and TiC for δ-ferrite using the edge-to-edge matching (E2EM) model. By calculating the misfit (f r ) values along the CP directions and mismatch (f d ) values between CP planes, it was predicted that NbO, CeS, TiN, Ce 2 O 3 , and TiC can all effectively refine the grains of δ-ferrite. According to the calculated values of f r and f d , the grain refining efficiency of these five compounds is ranked in the sequence of NbO > CeS > TiN > Ce 2 O 3 > TiC, which agrees well with previously reported experimental results. Moreover, based on the E2EM calculations, the orientation relationships (ORs) between δ-ferrite and the five compounds have also been predicted. Some ORs were reported before, such as the Baker-Nutting OR between TiN and ferrite [93,94].
Li et al. [95] also identified LaB 6 as an effective grain refiner for ferritic steels based on the crystallographic study using the E2EM model. A 0.5 wt% addition of LaB 6 significantly refined the grains of the as-cast Fe-4Si alloy. However, due to the low thermodynamic stability, the externally added LaB 6 decomposed into La and B in the steel melt, which led to the formation of various compounds. Further investigations found that the grain refinement was attributed to the combined effects of the heterogeneous nucleation of δ-ferrite on La 2 SO 2 and the solute effect of B. Crystallographic analysis based on the E2EM model indicated the excellence of atomic matching between δ-ferrite and the La 2 SO 2 . The predicted OR was also experimentally verified using the electron backscatter diffraction (EBSD) Kikuchi line diffraction technique.
Recently, Ji et al. [96] calculated the atomic matching and predicted the rough ORs between the Ce 2 O 3 /La 2 O 3 oxides and the primary phases (δ-ferrite/γ-austenite) using the E2EM model. The calculations showed that both Ce 2 O 3 and La 2 O 3 can promote heterogeneous nucleation of δ-ferrite and γ-austenite during solidification and therefore refine the as-cast macrostructure and microstructure of the steels. However, the grain refining potency of Ce 2 O 3 and La 2 O 3 for δ-ferrite is higher than that for γ-austenite. For δ-ferrite, Ce 2 O 3 is more efficient than La 2 O 3 . However, for γ-austenite, La 2 O 3 is more efficient than Ce 2 O 3 .

Solid Solution State of La and Ce in Steels
According to the Fe-La and Fe-Ce phase diagrams, the solid solubility of La and Ce in iron is almost zero even though they can completely dissolve in the Fe melt. Hence, during the solidification of steels, the majority of REs are rejected from the solid and segregates in the liquid at front of the solid-liquid interface. Such segregation may generate a constitutional supercooling zone that potentially leads to grain refinement of the δ-ferrite and/or γ-austenite through solute effect [60,[97][98][99][100][101][102][103]. In addition, some studies indicated that the solid solubility of lanthanum and cerium in ferrite is not absolutely zero [104][105][106]. The measured value was around 10 −6 -10 −5 [105,106] orders of magnitude using the inductively coupled plasma mass spectrometry (ICP-MS) through the low-temperature electrolysis of the samples and fast separation of the rare earth compounds. As the RE atoms tend to segregate at grain boundaries, vacancies, and dislocations in engineering steels, a very small amount of such REs can be detected in the steels.
In spite of such low solid solubility of REs, it has attracted great interest and attention of researchers and engineers in order to understand the roles RE solutes in grain refinement of cast steels [23,47,51,52].

Solute Effect of Grain Refinement in Cast Metals
In addition to the heterogeneous nucleation, the solutes also play crucial roles in the grain refinement of cast alloys. It is generally accepted that the role of the solute is attributed to the solute segregation in the liquid phase ahead of the solid-liquid interface [97,98]. The main points are: (1) the solute segregation can restrict the growth of the previously formed solid phase; (2) the solute segregation can generate a constitutional supercooling (C S ) zone in the liquid phase ahead of the solid-liquid interface, which provides additional driving force to facilitate the subsequently new nucleation within the C S zone; and (3) the newly formed grains restrict the further growth of the previously formed grains. The efficiency of grain refinement by the solute can be quantitatively expressed by the growth restriction factor (GRF or Q-value) [99][100][101][102]. The Q-value represents the establishing rate of the C S zone. The solute with higher Q value corresponds to high grain refinement efficiency. In a binary system, the Q-value can be simply expressed as [101]: where m is the slope of the liquidus in the binary phase diagram, C 0 is the content of the solute element in the alloy (mass%), k is the equilibrium partition coefficient of the solute element, and the unit of Q is K. For the multicomponent systems, the calculation of Q is far more complex. A rigorous approach to calculate Q in multicomponent alloys based on the thermodynamics of the alloy phase equilibria was presented by Schmid-Fetzer and Kozlov [103], which was expressed using Equation (7) as follows: where ∆T cs was the constitutional supercooling and f s was the fraction of solid. However, it is still unknown whether the Equation (7) can be directly applied to the system of Fe-C-M-RE (M represents different alloying elements) in which the solid solution content of RE is extremely small. In the past few years, the concept of growth restriction factor has been widely used in study of the grain refinement of cast light metals and alloys [60,107] and cast zinc and Zn alloys [99,[108][109][110]. Hu et al. [111] reported that the addition of Ce decreased the secondary dendrite arm spacing, enlarged the proportion of equiaxial grains, and refined the equiaxed grains of a low-sulfur 16Mn steel. It was considered that the increase in the equilibrium partition coefficients of carbon and manganese by Ce was responsible for the grain refinement. In addition, Hu and co-workers [112] investigated the influence of Ce on the morphology of austenite in a high-Mn steel using the directional solidification technique. It was found that with an increase in Ce content, the cellular structure gradually changed to dendrites and the columnar region was narrowed. It was considered that the solute Ce increases the partition coefficient of Mn. Zhong et al. [113] reported similar results showing that Ce slightly dissolved in austenite, leading to the increase in the equilibrium partition coefficient of Mn and decrease for Si. In a weather-resistant steel, Guo and co-workers [114][115][116] detected the enrichment of REs ahead of the solid-liquid interface, which corresponded to the increase in the non-equilibrium partition coefficient of various solute elements. According to Equation (6), for binary systems, when the partition coefficient of solute (k) is less than 1, the smaller the k is, the larger the value of Q is, and vice versa. For multicomponent systems, such as steels, the calculation of Q is complicated and the effect of REs on the value of Q needs to be studied in depth.
Ji et al. [117] calculated the Q values of a few solutes (Ti, Zr, Ce, La, Nb, and V) in the Fe-X binary alloy systems based on the calculated binary phase diagrams using the phase diagram module of the FactSage software system. The calculated Q values were ranked in the order of Ti > Zr > Ce > La > Nb > V. This result may explain the good grain refining efficiency of Ti and Zr used in steels. It is worth conducting further experimental work to verify the prediction.

Summary
(1) The addition of lanthanum or/and cerium in steel leads to the increase in the proportion of equiaxed crystal zone or the reduction in the average grain size or both. Commonly, it is considered that the structure refinement is attributed mainly to the heterogeneous nucleation of high melting-point RE oxides, oxysulfides, and sulfides for δ-ferrite or γ-austenite, and possibly to the solute effects of lanthanum or/and cerium. (2) The grain refining efficiency by additions of lanthanum or/and cerium varies in different steels.
Furthermore, it is not clear for which phase, δ-ferrite or γ-austenite, the refining effect of REs is more efficient. (3) The formation sequence of RE-containing inclusions in molten steel is RE oxides > RE oxysulfides > RE sulfides. The typical oxide is RE 2 O 3 . The oxysulfide is an intermediate product during the transition from deoxygenation to desulfuration by cerium and lanthanum. The variety and distribution of the RE oxysulfides rely on the contents of oxygen and sulfur and the condensate depression of the molten steel. The RE 2 O 2 S commonly forms. The type of RE sulfides is related to the ratio of RE content to S content, the value of [RE]/[S], in the molten steel. Typical sulfides are RE 2 S 3 , RE 3 S 4 , and RES. (4) To evaluate the grain refining efficiency, although the two-dimensional disregistry model is widely used, the E2EM model is considered as a more accurate model, particularly for more complicated systems, because there is lack of clear criteria for selecting the low index planes in the two-dimensional disregistry model, which may cause inconsistent results. (5) Compared with other models/theories evaluating the heterogeneous nucleation potency, the E2EM model is the most available and powerful tool for understanding the efficiency of nucleants and discovering effective grain refiners from the point of crystallographic view. The effects of solute on grain refinement are described mainly using the growth restriction factor (Q value) theory. Although the study on the Q value theory of ternary or multivariate iron-base alloy system with addition of rare earth has not been reported, the theory has been widely accepted for its successful explanation of the grain refinement of cast light metals and alloys. The E2EM model and the growth restriction theory can provide theoretical basis and feasible approaches to clarify previous inconsistence, leading to full understanding of the structural refinement mechanism of as-cast steels after adding REs.