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

Abstract

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.

1. 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.

Generally, the solidification grain refinement refers to minimization or elimination of columnar grains and refinement of the equiaxed grains. The currently available grain refinement techniques used during steel casting can be categorized as the following three groups:

• Electromagnetic stirring (EMS) [8,9]

This method is effective, but it has long-standing problems related to the mode of stirring, position of the stirrer, and stirring force [10]. In addition, the so called “white band” on a sulfur print of the stirred products, which is associated with negative segregation and positive segregation, is another critical issue [11]. Thus, EMS has limitations.

• 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, Al₂O₃, and Ce₂O₃) 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-Fe₃C 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.

2. 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.

2.1. 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 CeO₂, CeAlO₃, and Ce₂O₂S 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.

2.2. 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.

2.3. 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 La₂O₃, La₂O₂S, 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.

2.4. 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.

2.5. Discussion

To understand the actual roles of REs in grain refinement in different steels, the above-mentioned results are summarized in

Table 1. Published works on grain refinement of as-cast steels through addition of lanthanum or/and cerium.

REs	Steels	Mold Type	Grain Refining Efficiency	Mechanism	Research Type	References	Shortcomings
Ce	00Cr17 high-purity ferrite stainless steel	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.	Heterogeneous nucleation of γ-austenite on cerium oxide inclusions.	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.	Heterogeneous nucleation of primary austenite on either CeO2 or CeAlO3 or Ce2O2S inclusions.	Experimental	[43]
Ce	430 ferrite stainless steel	No information is given	The equiaxed grain zone was enlarged from 3% to 33% and both the columnar and the equiaxed grains were refined.	Both the heterogeneous nucleation on RE sulfide and oxide and the solute effect of cerium contributed to the grain refinement.	Experimental	[47]
La	ASTM A216 steel (plain carbon steel)	No information is given	La refined the grains significantly.	La2O3, La2O2S and LaS acted as the heterogeneous nucleation sites for δ-ferrite and the dissolved lanthanum in the molten steel increased the degree of supercooling.	Experimental	[51]
Ce	1045 carbon steel	Zircon sand mold	Cerium enlarged the equiaxed grain zone and reduced the primary and secondary dendrite arm spacing in the cast alloys.	Cerium solute redistribution during the solidification was the dominant factor in the refinement of the as-cast structure.	Experimental	[52]
Ce	0.20%C–0.02%P steel	Not molded, melted in Al2O3 crucible then quenched in water	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 Ce2O3 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.	Experimental	[25,48,49,50]
30% La and 65% Ce	H13 steel	No information is given	The average grain size was reduced from 50 μm to 25 μm.	Ce2O2S 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.

. 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.

3. Grain Refinement Mechanisms of Rare Earths in Steels

3.1. 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].

3.2. Heterogeneous Nucleation of Ferrite and/or Austenite on RE Inclusions

3.2.1. 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.

3.2.2. 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:

$$\delta =\frac{\left|a_s-a_n\right|}{a_n}$$

(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]:

$${\delta }_{{(hkl)}_n}^{{(hkl)}_s}={\displaystyle \sum _{i=1}^3\frac{\frac{\left|d_{{[uvw]}_s^i}\cos \theta -d_{{[uvw]}_n^i}\right|}{d_{{[uvw]}_n^i}}}{3}\times 100\%}$$

(2)

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 ${(hkl)}_n$, $d_{{[uvw]}_n}$ is the interatomic spacing along ${[uvw]}_n$, $d_{{[uvw]}_s}$ is the interatomic spacing along ${[uvw]}_s$, and θ is the angle between ${[uvw]}_s$ and ${[uvw]}_n$.

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, Ce₂O₃ and La₂O₃, and the primary phases (δ-ferrite and γ-austenite) at 1500 °C. The calculated δ values were: ${\delta }_{\mathrm{ferrite}-\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_3}=$ 4.49%, ${\delta }_{\mathrm{ferrite}-\mathrm{L}{\mathrm{a}}_2{\mathrm{O}}_3}=$ 3.34%, ${\delta }_{\mathrm{austenite}-\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_3}=$ 14.21%, and ${\delta }_{\mathrm{austenite}-\mathrm{L}{\mathrm{a}}_2{\mathrm{O}}_3}=$ 13.17%. In terms of the Bramfitt’s criteria [16], Ce₂O₃ and La₂O₃ could effectively promote heterogeneous nucleation of δ-ferrite, but not for γ-austenite. However, the disregistry between the La₂O₃ 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.

Table 2. Planar misfits between Ce₂O₃, Ce₂O₂S, La₂O₃, La₂O₂S, LaS, and δ-Fe.

${(0001)}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_3}⫽{(100)}_{\mathsf{\delta }-\mathrm{F}\mathrm{e}}$

${(0001)}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_3}⫽{(110)}_{\mathsf{\delta }-\mathrm{F}\mathrm{e}}$

${(0001)}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_3}⫽{(111)}_{\mathsf{\delta }-\mathrm{F}\mathrm{e}}$

Reference

${[uvtw]}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_3}$

$[\overline{1}2\overline{1}0]$

$[10\overline{1}0]$

$[\overline{2}110]$

${[uvtw]}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_3}$

$[\overline{1}2\overline{1}0]$

$[10\overline{1}0]$

$[\overline{2}110]$

${[uvtw]}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_3}$

$[\overline{1}2\overline{1}0]$

$[10\overline{1}0]$

$[\overline{2}110]$

[68]

${[uvw]}_{\mathsf{\delta }-\mathrm{F}\mathrm{e}}$

$[011]$

$[021]$

$[01\overline{1}]$

${[uvw]}_{\mathsf{\delta }-\mathrm{F}\mathrm{e}}$

$[\overline{1}10]$

$[\overline{1}12]$

$[\overline{2}21]$

${[uvw]}_{\mathsf{\delta }-\mathrm{F}\mathrm{e}}$

$[\overline{1}01]$

$[\overline{1}10]$

$[\overline{2}11]$

δ

8.1%

δ

18.3%

δ

37.6%

${(0001)}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_2\mathrm{S}}⫽{(100)}_{\mathsf{\delta }-\mathrm{F}\mathrm{e}}$

${(0001)}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_2\mathrm{S}}⫽{(110)}_{\mathsf{\delta }-\mathrm{F}\mathrm{e}}$

${(0001)}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_2\mathrm{S}}⫽{(111)}_{\mathsf{\delta }-\mathrm{F}\mathrm{e}}$

Reference

${[uvtw]}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_2\mathrm{S}}$

$[\overline{1}2\overline{1}0]$

$[10\overline{1}0]$

$[\overline{2}110]$

${[uvtw]}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_2\mathrm{S}}$

$[\overline{1}2\overline{1}0]$

$[10\overline{1}0]$

$[\overline{2}110]$

${[uvtw]}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_2\mathrm{S}}$

$[\overline{1}2\overline{1}0]$

$[10\overline{1}0]$

$[\overline{2}110]$

[68]

${[uvw]}_{\mathsf{\delta }-\mathrm{F}\mathrm{e}}$

$[011]$

$[021]$

$[01\overline{1}]$

${[uvw]}_{\mathsf{\delta }-\mathrm{F}\mathrm{e}}$

$[\overline{1}10]$

$[\overline{1}12]$

$[\overline{2}21]$

${[uvw]}_{\mathsf{\delta }-\mathrm{F}\mathrm{e}}$

$[\overline{1}01]$

$[\overline{1}10]$

$[\overline{2}11]$

δ

10.3%

δ

17.0%

δ

38.8%

${(0001)}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_2\mathrm{S}}⫽{(100)}_{\mathsf{\delta }-\mathrm{F}\mathrm{e}}$

${(0001)}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_2\mathrm{S}}⫽{(110)}_{\mathsf{\delta }-\mathrm{F}\mathrm{e}}$

${(0001)}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_2\mathrm{S}}⫽{(111)}_{\mathsf{\delta }-\mathrm{F}\mathrm{e}}$

Reference

${[uvtw]}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_2\mathrm{S}}$

$[1210]$

$[2110]$

$[1010]$

${[uvtw]}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_2\mathrm{S}}$

$[1210]$

$[2110]$

$[1010]$

${[uvtw]}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_2\mathrm{S}}$

$[1210]$

$[1100]$

$[2110]$

[30]

${[uvw]}_{\mathsf{\delta }-\mathrm{F}\mathrm{e}}$

$[010]$

$[011]$

$[001]$

${[uvw]}_{\mathsf{\delta }-\mathrm{F}\mathrm{e}}$

$[001]$

$[111]$

$[110]$

${[uvw]}_{\mathsf{\delta }-\mathrm{F}\mathrm{e}}$

$[110]$

$[121]$

$[011]$

δ

55.3%

δ

53.5%

δ

3.5%

${(0001)}_{\mathrm{L}{\mathrm{a}}_2{\mathrm{O}}_3}⫽{(111)}_{\mathsf{\delta }-\mathrm{F}\mathrm{e}}$

${(0001)}_{\mathrm{L}{\mathrm{a}}_2{\mathrm{O}}_2\mathrm{S}}⫽{(111)}_{\mathsf{\delta }-\mathrm{F}\mathrm{e}}$

${(111)}_{\mathrm{LaS}}⫽{(111)}_{\mathsf{\delta }-\mathrm{F}\mathrm{e}}$

Reference

${[uvtw]}_{\mathrm{L}{\mathrm{a}}_2{\mathrm{O}}_3}$

$[2110]$

$[\overline{1}100]$

$[\overline{1}2\overline{1}0]$

${[uvtw]}_{\mathrm{L}{\mathrm{a}}_2{\mathrm{O}}_2\mathrm{S}}$

$[\overline{2}110]$

$[\overline{1}100]$

$[\overline{1}2\overline{1}0]$

${[uvw]}_{\mathrm{LaS}}$

$[\overline{1}01]$

$[\overline{2}11]$

$[\overline{1}10]$

[51]

${[uvw]}_{\mathsf{\delta }-\mathrm{F}\mathrm{e}}$

$[\overline{1}01]$

$[\overline{2}11]$

$[\overline{1}10]$

${[uvw]}_{\mathsf{\delta }-\mathrm{F}\mathrm{e}}$

$[\overline{1}01]$

$[\overline{2}11]$

$[\overline{1}10]$

${[uvw]}_{\mathsf{\delta }-\mathrm{F}\mathrm{e}}$

$[\overline{1}01]$

$[\overline{2}11]$

$[\overline{1}10]$

δ

3.34%

δ

2.6%

δ

0.6%

and

Table 3. Planar misfits between Ce₂O₃, Ce₂O₂S, La₂O₃, La₂O₂S, CeS, and γ-Fe.

${(0001)}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_3}⫽{(100)}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

${(0001)}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_3}⫽{(110)}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

${(0001)}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_3}⫽{(111)}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

Reference

${[uvtw]}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_3}$

$[\overline{1}2\overline{1}0]$

$[10\overline{1}0]$

$[\overline{2}110]$

${[uvtw]}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_3}$

$[\overline{1}2\overline{1}0]$

$[10\overline{1}0]$

$[\overline{2}110]$

${[uvtw]}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_3}$

$[\overline{1}2\overline{1}0]$

$[10\overline{1}0]$

$[\overline{2}110]$

[69]

${[uvw]}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

$[010]$

$[031]$

$[001]$

${[uvw]}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

$[00\overline{1}]$

$[\overline{1}1\overline{1}]$

$[\overline{1}10]$

${[uvw]}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

$[\overline{1}01]$

$[\overline{2}11]$

$[\overline{1}10]$

δ

7.7%

δ

13.3%

δ

21.3%

${(0001)}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_3}⫽{(100)}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

${(0001)}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_3}⫽{(110)}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

${(0001)}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_3}⫽{(111)}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

Reference

${[uvtw]}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_3}$

$[\overline{1}2\overline{1}0]$

$[\overline{1}100]$

$[\overline{2}110]$

${[uvtw]}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_3}$

$[\overline{1}2\overline{1}0]$

$[\overline{1}100]$

$[\overline{2}110]$

${[uvtw]}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_3}$

$[\overline{1}2\overline{1}0]$

$[\overline{1}100]$

$[\overline{2}110]$

[71]

${[uvw]}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

$[010]$

$[031]$

$[011]$

${[uvw]}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

$[100]$

$[\overline{1}12]$

$[\overline{1}10]$

${[uvw]}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

$[\overline{1}01]$

$[\overline{2}11]$

$[\overline{1}10]$

δ

15.8%

δ

13.0%

δ

5.4%

${(0001)}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_2\mathrm{S}}⫽{(100)}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

${(0001)}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_2\mathrm{S}}⫽{(110)}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

${(0001)}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_2\mathrm{S}}⫽{(111)}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

Reference

${[uvtw]}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_2\mathrm{S}}$

$[\overline{1}2\overline{1}0]$

$[10\overline{1}0]$

$[\overline{2}110]$

${[uvtw]}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_2\mathrm{S}}$

$[\overline{1}2\overline{1}0]$

$[10\overline{1}0]$

$[\overline{2}110]$

${[uvtw]}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_2\mathrm{S}}$

$[\overline{1}2\overline{1}0]$

$[10\overline{1}0]$

$[\overline{2}110]$

[69]

${[uvw]}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

$[010]$

$[031]$

$[001]$

${[uvw]}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

$[00\overline{1}]$

$[\overline{1}1\overline{1}]$

$[\overline{1}10]$

${[uvw]}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

$[\overline{1}0\overline{1}]$

$[\overline{2}11]$

$[\overline{1}10]$

δ

10.2%

δ

13.5%

δ

31.1%

${(0001)}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_2\mathrm{S}}⫽{(100)}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

${(0001)}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_2\mathrm{S}}⫽{(110)}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

${(0001)}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_2\mathrm{S}}⫽{(111)}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

Reference

${[uvtw]}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_2\mathrm{S}}$

$[\overline{1}2\overline{1}0]$

$[\overline{1}100]$

$[\overline{2}110]$

${[uvtw]}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_2\mathrm{S}}$

$[\overline{1}2\overline{1}0]$

$[\overline{1}100]$

$[\overline{2}110]$

${[uvtw]}_{\mathrm{C}{\mathrm{e}}_2{\mathrm{O}}_2\mathrm{S}}$

$[\overline{1}2\overline{1}0]$

$[\overline{1}100]$

$[\overline{2}110]$

[71]

${[uvw]}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

$[010]$

$[031]$

$[011]$

${[uvw]}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

$[100]$

$[\overline{1}12]$

$[\overline{1}10]$

${[uvw]}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

$[\overline{1}01]$

$[\overline{2}11]$

$[\overline{1}10]$

δ

14.9%

δ

14%

δ

10.3%

${(0001)}_{{\mathrm{La}}_2{\mathrm{O}}_3}⫽{(100)}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

${(0001)}_{\mathrm{L}{\mathrm{a}}_2{\mathrm{O}}_3}⫽{(110)}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

${(0001)}_{\mathrm{L}{\mathrm{a}}_2{\mathrm{O}}_3}⫽{(111)}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

Reference

${[uvtw]}_{\mathrm{L}{\mathrm{a}}_2{\mathrm{O}}_3}$

$[\overline{1}2\overline{1}0]$

$[\overline{1}100]$

$[\overline{2}110]$

${[uvtw]}_{\mathrm{L}{\mathrm{a}}_2{\mathrm{O}}_3}$

$[\overline{1}100]$

$[\overline{2}110]$

$[\overline{2}110]$

${[uvtw]}_{\mathrm{L}{\mathrm{a}}_2{\mathrm{O}}_3}$

$[\overline{1}2\overline{1}0]$

$[\overline{1}100]$

$[\overline{2}110]$

[72]

${[uvw]}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

$[010]$

$[031]$

$[011]$

${[uvw]}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

$[100]$

$[\overline{1}12]$

$[\overline{1}10]$

${[uvw]}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

$[\overline{1}01]$

$[\overline{2}11]$

$[\overline{1}10]$

δ

15.49%

δ

12.51%

δ

6.7%

${(0001)}_{{\mathrm{La}}_2{\mathrm{O}}_2\mathrm{S}}⫽{(100)}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

${(0001)}_{\mathrm{L}{\mathrm{a}}_2{\mathrm{O}}_2\mathrm{S}}⫽{(110)}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

${(0001)}_{\mathrm{L}{\mathrm{a}}_2{\mathrm{O}}_2\mathrm{S}}⫽{(111)}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

Reference

${[uvtw]}_{\mathrm{L}{\mathrm{a}}_2{\mathrm{O}}_2\mathrm{S}}$

$[\overline{1}2\overline{1}0]$

$[\overline{2}110]$

$[\overline{1}010]$

${[uvtw]}_{\mathrm{L}{\mathrm{a}}_2{\mathrm{O}}_2\mathrm{S}}$

$[\overline{1}2\overline{1}0]$

$[\overline{2}110]$

$[\overline{1}010]$

${[uvtw]}_{\mathrm{L}{\mathrm{a}}_2{\mathrm{O}}_2\mathrm{S}}$

$[\overline{1}2\overline{1}0]$

$[\overline{1}100]$

$[\overline{2}110]$

[73]

${[uvw]}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

$[010]$

$[012]$

$[001]$

${[uvw]}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

$[00\overline{1}]$

$[\overline{1}1\overline{1}]$

$[\overline{1}10]$

${[uvw]}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

$[\overline{1}10]$

$[\overline{2}11]$

$[\overline{1}01]$

δ

5.42%

δ

23.82%

δ

55.69%

${(100)}_{\mathrm{CeS}}⫽{(100)}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

${(100)}_{\mathrm{CeS}}⫽{(110)}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

${(100)}_{\mathrm{CeS}}⫽{(111)}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

Reference

${[uvw]}_{\mathrm{CeS}}$

$[001]$

$[011]$

$[010]$

${[uvw]}_{\mathrm{CeS}}$

$[001]$

$[011]$

$[010]$

${[uvw]}_{\mathrm{CeS}}$

$[001]$

$[011]$

$[010]$

[71]

${[uvw]}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

$[001]$

$[011]$

$[010]$

${[uvw]}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

$[\overline{1}10]$

$[\overline{1}11]$

$[100]$

${[uvw]}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

$[\overline{1}2\overline{1}]$

$[\overline{2}11]$

$[\overline{1}10]$

δ

48.7%

δ

26.5%

δ

30.3%

${(100)}_{\mathrm{CeS}}⫽{(100)}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

${(100)}_{\mathrm{CeS}}⫽{(110)}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

${(100)}_{\mathrm{CeS}}⫽{(111)}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

Reference

${[uvw]}_{\mathrm{CeS}}$

$[001]$

$[011]$

$[010]$

${[uvw]}_{\mathrm{CeS}}$

$[001]$

$[011]$

$[010]$

${[uvw]}_{\mathrm{CeS}}$

$[001]$

$[011]$

$[010]$

[74]

${[uvw]}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

$[001]$

$[011]$

$[010]$

${[uvw]}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

$[\overline{1}10]$

$[\overline{1}11]$

$[100]$

${[uvw]}_{\mathsf{\gamma }-\mathrm{F}\mathrm{e}}$

$[\overline{1}2\overline{1}]$

$[\overline{2}11]$

$[\overline{1}10]$

δ

50.7%

δ

27.3%

δ

31.2%

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:

• Ce₂O₃ is a moderately effective grain refiner for δ-ferrite.
• The heterogeneous nucleation potency of Ce₂O₂S for δ-ferrite is inconsistent. According to reference [68], the efficiency of Ce₂O₂S is moderate. However, in the work of Reference [23], it is prominent.
• All La₂O₃, La₂O₂S and LaS are effective grain refiners for δ-ferrite and the refining potency is in the order of LaS > La₂O₂S > La₂O₃.
• Both Ce₂O₃ and Ce₂O₂S can refine γ-austenite and the potency of Ce₂O₃ is slightly better than that of Ce₂O₂S.
• The heterogeneous nucleation potency of Ce₂O₃ for γ-austenite is inconsistent based on different published works, so is Ce₂O₂S.
• CeS cannot refine γ-austenite.
• Both La₂O₃ and La₂O₂S can act as the heterogeneous nucleation sites of the primary γ-austenite. The efficiency of La₂O₂S is prominent, but the La₂O₃ is moderate.

The data in Table 2 and Table 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 Ce₂O₂S and δ-ferrite calculated by Hao and co-workers [68] was 10.3%, which corresponds to a moderate grain refining efficiency of Ce₂O₂S for δ-ferrite. However, Gao and team [30] calculated the smallest δ-value was 3.5%, implying a high grain refining efficiency of Ce₂O₂S 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:

$$\Delta \zeta =\left|\frac{2({\zeta }_n-{\zeta }_s)}{{\zeta }_n+{\zeta }_s}\right|$$

(3)

where $\Delta \zeta$ is defined as a characteristic parameter, and ${\zeta }_n$ and ${\zeta }_s$ are lattice electron migration potentials of nucleation phase and nucleating agent respectively. Bigger values of $\Delta \zeta$ 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.

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]:

$$f_{\mathrm{r}}=\frac{|r_{\mathrm{M}}-r_{\mathrm{P}}|}{r_{\mathrm{P}}}$$

(4)

$$f_{\mathrm{d}}=\frac{|d_{\mathrm{M}}-d_{\mathrm{P}}|}{d_{\mathrm{P}}}$$

(5)

where $r_{\mathrm{M}}$ and $r_{\mathrm{P}}$ are the interatomic spacing along the matching directions, and $d_{\mathrm{M}}$ and $d_{\mathrm{P}}$ 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, Ce₂O₃, 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, Ce₂O₃, 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 > Ce₂O₃ > 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 LaB₆ as an effective grain refiner for ferritic steels based on the crystallographic study using the E2EM model. A 0.5 wt% addition of LaB₆ significantly refined the grains of the as-cast Fe-4Si alloy. However, due to the low thermodynamic stability, the externally added LaB₆ 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 La₂SO₂ and the solute effect of B. Crystallographic analysis based on the E2EM model indicated the excellence of atomic matching between δ-ferrite and the La₂SO₂. 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 Ce₂O₃/La₂O₃ oxides and the primary phases (δ-ferrite/γ-austenite) using the E2EM model. The calculations showed that both Ce₂O₃ and La₂O₃ 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 Ce₂O₃ and La₂O₃ for δ-ferrite is higher than that for γ-austenite. For δ-ferrite, Ce₂O₃ is more efficient than La₂O₃. However, for γ-austenite, La₂O₃ is more efficient than Ce₂O₃.

3.3. Effects of La and Ce as Solutes

3.3.1. 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].

3.3.2. 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]:

Q = mC₀(k − 1)

(6)

where m is the slope of the liquidus in the binary phase diagram, C₀ 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:

$$Q={\left(\frac{\partial (\mathrm{\Delta }{T}_{\mathrm{cs}})}{\partial f_{\mathrm{s}}}\right)}_{f_{\mathrm{s}}\to 0}$$

(7)

where ΔT_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.

4. 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 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, the value of [RE]/[S], in the molten steel. Typical sulfides are RE₂S₃, RE₃S₄, 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.