Matthew Voigt
email: mkvoigt@gmail.com

Chris Rasmussen
email: crasmussen@sdsu.edu

Antonio Martinez
email: antonio.estm93@gmail.com

Calculus variations as figured worlds for math identity development

Keywords: Calculus, mathematical identity, figured worlds, course variations

, we showcase the ways in which these course variations act as if they are different calculus worlds that constitute socially organized and produced realms of being. We highlight the ways in which these figured worlds position or fail to position students with the opportunity to refigure themselves and others.

In the United States (US) there is a need to increase the number of students in science, technology, engineering, and mathematics (STEM) to address the nearly 1 million additional STEM degrees needed to support the nation's growing research and technology economy (PCAST, 2012). To address this need, the PCAST report recommended the adoption of empirically validated teaching practices, replacing standard lab courses with discovery-based research courses, addressing the mathematics-preparation gap, and diversify pathways to STEM degrees. Additionally, any efforts to improve the quality of undergraduate STEM education must also attend to fostering an environment that promotes diversity and inclusion in STEM classrooms (NASEM, 2017).

The vision and enactment of creating an equitable robust STEM education is a complex and multifaceted endeavor that will require continued research; however, one such approach in undergraduate mathematics in the US is the tailoring of calculus courses to meet the needs of individual students, which we refer to as course variations. Course variations have the potential for addressing the recommendations from the PCAST report since they can specifically address the preparation gap for students by incorporating prerequisite material in courses, by stretching out the course content, or by infusing labs and standard based teaching in courses tailored for science majors 1 . Such course variations can even provide diverse pathways into STEM for those that have taken a non-traditional math background through transition courses. Rasmussen et al. (in press) documented how these variations to the standard course across the US have been associated with greater rates of passing calculus and put forth a call for future research to examine the ways that these courses may help promote a sense of community and identity development among students in the different course variations. We take up this call for future research by addressing the following research question: How do the structures and activities of three different calculus courses at one university impact the types of possible mathematical identities that emerge for students from those contexts?

Theoretical Background and Literature Review

Mathematical identity is a powerful concept in the analysis of mathematical learning, in part due to the recent social and political turn in education [START_REF] Adiredja | Taking the sociopolitical turn in postsecondary mathematics education research[END_REF]. Identity frameworks in math education have drawn largely from sociocultural perspectives that link identity and learning to one another and arise from social practices. Additionally, this research often utilizes positioning theory to account for identity as constructed through social interactions to construct storylines about who a person is in relation to others in a social context [START_REF] Langer-Osuna | Identity in research on mathematics education[END_REF]. [START_REF] Holland | Identity and agency in cultural Worlds[END_REF] demonstrate how the sociocultural theory of identity and self, known as figured worlds, is a useful perspective for studying identity production in education, and how the context of education allows or does not allow the emergence of certain identities. Figured worlds are "socially and culturally constructed realms of interpretation in which particular characters and actors are recognized, significance is assigned to certain acts, and particular outcomes are valued over others" (Holland et al., 1998, p. 52). Figured worlds are dynamic. They are constantly formed and re-formed in relation to the everyday activities and events that occur within the realm of possible "as if" worlds. Figured worlds are thus situated in a social context and time period and represent a reflexive relationship and negotiation of the possible identities that can be constructed and affirmed in the figured world. [START_REF] Boaler | Identity, agency, and knowing in mathematics worlds[END_REF] utilize the perspective of figured worlds to illustrate how two different types of secondary school classrooms afforded students different identities and storylines as mathematical learners. One such figured world drew on what they referred to as "received knowing", which promoted the belief of doing mathematics as memorization and being able to quickly recall information. In contrast, the other figured world promoted "connected knowing", which emphasized the belief of doing mathematics as making sense of mathematical concepts and procedures and a sense of agency among the learners. This study highlights how the context of education setting and approach to teaching can impact students' identity production as learners and doers of mathematics, which can impact their decision to continue (or not) in a STEM program.

In related work, [START_REF] Solomon | Safety in numbers: Mathematics support centres and their derivatives as social learning spaces[END_REF] examined how mathematics support centres, which were intended to support skill development for engineering students, were dynamically coopted by the students to support the development of group learning strategies which promoted a strong community identity among the participants. This study highlights the way this STEM community of practice, which can often be highly competitive and individualistic, can refigure itself by reflecting on the positional identities that can be challenged in that space by drawing on the physical resources and artifacts to disrupt the available storylines. For instance, the physical space of the tutoring center, allowed students to refigure their relational identity to mathematics as a social endeavor of helping each other succeed.

Methods

The analysis presented draws on student focus group data from one University, which we refer to as Tree Line University (TLU). TLU offers three different calculus courses. In addition to the standard offering, TLU has a coordinated calculus-physics course for advanced students and a life science course, which includes a focus on biology. For each of the calculus offerings (standard, honors studio physics, life sciences), we conducted a single focus group during week 13 of 16 with three to five students currently enrolled in each of the respective courses. We used student focus groups (as opposed to individual interviews) since they offer opportunities to understand the nature of students' socially constructed figured worlds. Students for the focus group were sent an email invitation as well as recruiting efforts done in-person during the course. As such the students who participated represent a self-selected sample of students willing to participate in the focus group, and while this may not be representative of the entire course they highlight the realm of possible mathematical identities afforded in each of the courses. The focus group conversations centered on topics such as who they are, their experiences in the course, how and why they chose this particular course, what happens during a typical class period, how they relate to others in this course as well as to students in different calculus courses. Each of the focus groups utilized the same interview protocol but the extent to which each topic was discussed was driven by the engagement and discussion of the participants.

All focus group interviews were audio recorded and transcribed for subsequent, thematic analysis [START_REF] Braun | Using thematic analysis in psychology[END_REF]. Guided by our theoretical framing of figured worlds, we developed narrative accounts in a collaborative endeavor among the researchers by first producing a descriptive account of the focus group and then using within and cross-case comparison to develop themes related to the research focus. These narrative accounts centered around the themes of students' emerging mathematical identities, sense of community or belonging, and positional relationship to calculus as (ir)relevant to their major and career goals. We present how these themes are enacted as figured worlds in the three calculus courses along with illustrative quotations from the narrative accounts.

Results

Calculus for Life Sciences: A refiguring of productive mathematical identities

Calculus for Life Sciences at TLU functions as a combined differential and integral calculus course without topics in trigonometry. The course was originally designed at the request of the college of life sciences and agriculture for students majoring in the life sciences. The content remains fairly similar to the standard calculus course but has what faculty described as a "lighter approach" that emphasizes concepts and some application of topics. Our focus group in this course included five students enrolled with the same instructor (Dr. B) for the lecture session but who had different teaching assistants for the twice weekly recitation sections.

Students in the focus group conveyed that prior to enrolling in this course they had identities as poor performers in mathematics, which made them anxious to take a university calculus course. One student shared that they had taken precalculus and had gotten a C-in the course, and stated that it, "was the lowest grade I had ever gotten for a college class," and as a result was worried about how well they would do in this class. All of the students in the focus group concurred with this sentiment, with one student stating, "I did so poorly in that class, and I just thought like I am not meant to pass calculus." Other students discussed how the gap between their last math course in secondary school and taking this calculus course made them less prepared, and that they were "nervous going into calculus." Students in the focus group had a personal social history (history-inperson) that positioned them outside of the world of learners and doers of mathematics. For example, one student stated that they were, "someone who is not naturally inclined to math," while another stated, "I am not meant to pass calculus." However, as we will show, the students conveyed that through their experiences in this course, they were able to refigure their identities as productive mathematical learners largely as a result from positive interactions with their instructor.

Students in the focus group conveyed that as a result of this course they now viewed themselves as someone who was capable of learning and doing mathematics. One student said that "I feel like I'm not completely hopeless at all in math anymore." This sentiment was supported by two other students who recognized a shift from their prior conceptions and experience in mathematics. For example, one student said, "I can actually do this, rather than like, in many past courses where I really have no idea what's going on." Students discussed how they were really "understanding" what they were doing rather than memorizing formulas, which aligned with the goal and vision of the course from the faculty perspective. As a result, students were able to refigure their positionality towards learning mathematics, as exemplified in the following quotes: "I'll be able to succeed in other math heavy courses" and it "boosts my confidence in that regard."

One of the contributing factors that helped students refigure their mathematical identities was their relationship with the instructor. "I can't say enough about our professor, this is probably the only math class that's really felt like it made sense in my life." Students described instructional practice that contributed to their positive experience such as the teacher breaking down concepts in a way that made sense, using anonymous polling to see how they were feeling about course concepts, and providing prerequisite information such as the quadratic formula without assuming the students had memorized this information. These practices seemed to convey to the students that the instructor cared about them and their learning, allowing for them to acknowledge their past mathematical identity while being supported in the negotiation of productive mathematical identities. The impact of the individual instructor versus other features cannot be isolated in this study; however, the instructor through pedagogical techniques such as group work and anonymous polling (both on content and affective issues) allowed for the enactment of a figured world that aligned with the goals of the course to have students focus on understanding and connected knowing. There were also ways in which the enactment of the course variation positioned the students outside the world of mathematics learners. For instance, while one of the students mentioned that they were unaware of the difference between calculus for life sciences and the standard calculus course, three of the students mentioned the ways in which it was "low base calculus" or "more basic algebra" compared to "real calculus." One student even described how their friend who was studying physics teased them saying, "you're not taking calculus, calc for life sciences is just like classical math." Additionally, all but one of the students felt that the stated goal of the course to serve life sciences students was too broad. This resulted in students feeling that the course was not tailored to their specific discipline identities, "I'm either getting pushed aside or pushed under the rug with everybody else by just saying, "Oh well, you're in the life sciences major, you got to do this." In this figured world of calculus for life sciences, students were maintaining a strong discipline identity (equine science, zoology) which they viewed as not needing calculus.

Honors Calculus: A collaborative community of academically-minded students

Honors calculus at TLU is a unique course that it is designed to integrate topics in physics and calculus and takes a theoretical approach to the material. It is a two-course sequence that is cotaught in a studio laboratory by a math instructor and a physics instructor. The math instructor for this course was teaching it for the second time. Our focus group consisted of three students majoring in mathematics. Students emphasized the difficulty of this course by the fact that they often have to rely on one another to finish the homework and study for exams. For example, an agreed upon sentiment is that "Collaboration is actually one of the strengths of the class…you know everyone in the class, you feel like you can trust that they're going to put in the effort, and you're going to put in the effort, and you're going to come together if you need to." The word "trust" was often used by the students in this focus group interview. They felt that there was a need to trust each other in order to do well. It is important to note that the objective for these students was clear; it was not to just pass the class, but to do well in the course together.

All of the students in this focus group entered with AP calculus credit2 . They entered into a world where they viewed their peers as equals who enjoyed learning and doing mathematics as much as they did. From the start they described a course that positioned them in the figured world of calculus where they felt accepted and academically challenged. This is reflected by the students' frequent reference to being surrounded by people who are the "same." One student in the focus group reinforced this idea as follows: "In my calculus class, we have students who are all STEM. They are students who have the same mindset". These students are in a space where they are comfortable to acknowledge that they are joined by, "intelligent people who have the same common objective". This highlights how mathematical identity of high-ability and like-mindedness is socially formed and reproduced [START_REF] Holland | Identity and agency in cultural Worlds[END_REF].

Students were able to relate to each other and work together based on the fact that they are all coming in with similar interests, similar class objectives and career goals. During lecture, they were required to work in groups, which was a point of contention at the beginning of the semester. There was reluctance from some students to work with one another because they wanted to "motor through" the activities. However, they came to view group work positively once they created a world where they were able to openly share their ideas. As one focus group member put it, "I get to share my perspective, I get to hear their perspective," which they felt created a class that was more enjoyable. The figured world of honors calculus that the students created for themselves allowed them to grow and form a mathematical identity that centers around succeeding, understanding the material, and supporting others.

The students in the focus group also reported having an extremely strong sense of community that was supported through the development of their figured world, full of high achieving STEM majors. One student explained how close knit they are as follows: "If I have a concern about anything really, I feel like I can go and find someone from the class and talk to them about it and ask them what they think. And, you know, that's something that I think might be more exclusive to the [Honors Calculus]." There was something special about this particular experience compared to their other honors classes. The figured world that they created for themselves, with the help of their instructors, strengthened their identities as capable mathematicians. These high achieving students created a space where they are comfortable to admit when they need help and trust that their peers can support them in their learning process.

Standard Calculus: A realm of disconnected knowing and isolation

The standard calculus sequence at TLU is primarily a service course for engineering majors. The two-course sequence is taught on a rotating basis by experienced instructors. The focus group consisted of three men, with majors in ocean engineering, mechanical engineering, and chemical engineering. The students in the focus group had varied secondary school mathematics experiences where one student took a non-AP calculus course, one took an AP calculus course, and the third student did not take calculus in secondary school.

Although each of the students entered with different levels of math preparation, they each expressed a similar experience in the courseit was fast-paced and disconnected. The one student who had not taken calculus in secondary school described Calculus 1 as fast-paced and not well-organized. He also expressed some personal disconnect with the material when he said, "I didn't know what a derivative, like what is the definition of a derivative, till like two weeks after we had started them." The other two students who had taken calculus in secondary school also felt that the course was fast-paced but were less concerned with the material. In general, the three students positioned themselves as external to calculus, where calculus was something they had to do, as opposed to something that they were excited about learning. For example, one student said, "it's a class and I have to do work for it. That's just normal college stuff" and another student said calculus was a course "they had to take." Thus, upon entering calculus as first year students, none of the three positioned themselves as particularly excited about mathematics or very interested in mathematics. As they progressed from Calculus 1 to Calculus 2, this feeling of being disconnected from mathematics was not refigured, but rather seemed to become entrenched and reified.

In both Calculus 1 and 2, the three students had similar experiences in the lecture portion of the course. One student explained that he felt so disconnected that he stopped going to his assigned lecture and attended a different lecture instead. He recounted that in class he felt, "nobody knows what's going on because you're just up there writing, and you won't answer the questions. So, this is very frustrating." Another student chimed in that "Everything that he just said that happens this semester, happened for me last semester." The feeling of being personally disconnected from their instructors and the course content was amplified in Calculus 2. In contrast to Calculus 1 where they felt the material was more applicable and useful, their experiences in Calculus 2 was on memorization. For example, one student contrasted his experience in Calculus 1 and 2 as follows: "The expectation [in Calculus 1] was that there would be understanding. The latter [Calculus 2] is memorization without any expectation of understanding." This was a common sentiment for all three students. In fact, one student explained that he was told that Calculus 2 is "really advanced math" and so there they are not expected to "understand what we are doing." Even his teaching assistant (TA) positioned the content as something that was not within their reach for understanding. "And like my TA has dropped a line similar to just saying like, 'You don't need to know further, this is what you need in order to do this. So, this is what you're given.'" Thus, their experience in calculus at TLU resonates with the figured world of "received knowing" described by [START_REF] Boaler | Identity, agency, and knowing in mathematics worlds[END_REF].

TLU's no calculator policy seemed to further figure calculus as something that is disconnected from their interests and previous experiences. For example, one student explained that in secondary school their exams had calculator and no calculator parts and he liked the calculator part because "you could actually like finish the problem." The no calculator policy in calculus stood in contrast to how he imagined his future self in the workplace as an engineer. "You're not going to be working in a laboratory somewhere and they're just having you do calculations of derivatives and integrals like, in your head. Like you're going to have a calculator. Especially if you want to do real-world problems." They also contrasted their calculator experience in calculus with that in physics and chemistry, where calculators are used all the time. This positioned mathematics for them as outside the realm of connection with other disciplines.

When asked about the extent to which they felt they had formed bonds or connections with their classmates, the three students agreed that any relationships they formed were not the result of how class was structured or due to any effort on the part of their instructors. Instead, those that they do homework with are either friends or live in the same residence hall. Their ability to work with a wide range of students from different lectures was made possible because TLU has tightly coordinated curriculum, homework, and assessments. As these three students explained, "there's a lot of behind the scenes learning from kids explaining, or students explaining stuff to one another" and "there's a lot of, frankly, bonding over freaking out." Thus, at a system level, the course coordination allowed for considerable peer to peer bonding that otherwise might not have happened and allowed these three students to refigure their relational identities as helping residence hallmates survive calculus.

Discussion

Given the exploratory nature of this work we did not posit any hypothesis regarding how the different course variations would impact student mathematical identities, and instead our aim was to capture the salient features described by the students and how those related to their beliefs about knowing and doing mathematics. The enactment of these figured worlds considers the totality of the lived experience such as the role of the instructor, calculator policies, discipline-based problems, and the structures surrounding entry and pathways into the courses These elements cannot be separated since they are fundamental to the construction of the figured world. For instance, instructors for the calculus for life sciences are selected knowing the course should emphasize mathematical understanding and are aware that most of the students have had negative experiences with mathematics prior to starting the course. This results in assigning instructors who often are more student-centered in their teaching approach. The selection of instructors for the standard calculus is less intentional, but not random. Experienced instructors with good teaching reputations are typically tapped to teach this course because it is a core requirement for all STEM majors.

We also want to stress that the three different figured worlds are not consequences of the course variations themselves, but rather lie in the possibilities that the different course variations offer for how it is enacted and experienced. For example, the role of the instructor to either express care for their learning, to encourage peer collaboration, or to lecture the material at a quick pace was a paramount factor in how the students described their beliefs about being able to learn and do mathematics. The way the instructors approached teaching we speculate is tied with the programmatic features of the course variation. Whereby the standard course is content heavy and puts pressure on instructors to cover the material through lecture, the honors course has more contact hours and was designed with collaborative labs, and the life science course focus on understanding and less on computation which promotes instructor inquiry into student thinking.

AP Calculus is a first course in differential calculus offered through the advanced placement system, which is a program in the US and Canada that offers university level curricula and examinations to secondary school students.


Carlyn R Larosa

Mulaine Shih

Céline Varvenne

Maryam Ghazisaeidi

Solid solution strengthening theories of high-entropy alloys

Solid solution strengthening theories of high-entropy alloys

INTRODUCTION

The term "high entropy alloy" (HEA), coined by Yeh et al. , refers to a relatively new class of multicomponent alloys, usually five or more components, in equal or near equal concentrations [START_REF] Yeh | Nanostructured high-entropy alloys with multiple principal elements: Novel alloy design concepts and outcomes[END_REF]. Instead of the ordered intermetallic phases expected from classical physical metallurgy, some HEA systems strikingly crystalize as single phase solid solutions with simple crystal structures [START_REF] Cantor | Microstructural development in equiatomic multicomponent alloys[END_REF][START_REF] Yang | Microstructure and compressive properties of nbtivtaalx high entropy alloys[END_REF][START_REF] Bei | Multi-component solid solution alloys having high mixing entropy[END_REF][START_REF] Lucas | Magnetic and vibrational properties of high-entropy alloys[END_REF][START_REF] On Senkov | Microstructure and room temperature properties of a high-entropy tanbhfzrti alloy[END_REF]. The initial reasoning for the formation of HEAs was that the high configurational entropy would tend to stabilize solid solution formation; hence the name "high entropy". This concept was very attractive, in the sense that a single phase cubic alloy with high concentrations showed promise to overcome the strength-ductility trade off; the cubic structure provides multiple available slip systems, promoting ductility, while solid solution strengthening in highly concentrated alloys enhances strength, presumably beyond what is possible by dilute additions. Indeed, a remarkable combination of strength, ductility and fracture toughness was observed in the fcc FeCoCrMnNi, stimulating several subsequent studies on this alloy and its quaternary and ternary derivatives [START_REF] Gludovatz | Exceptional damage-tolerance of a medium-entropy alloy crconi at cryogenic temperatures[END_REF][START_REF] Otto | Relative effects of enthalpy and entropy on the phase stability of equiatomic high-entropy alloys[END_REF].

The significance of configurational entropy on mechanical properties -and consequently the number of components and the equimolar composition -have since been critically questioned [START_REF] Miracle | A critical review of high entropy alloys and related concepts[END_REF][START_REF] Li | Metastable high-entropy dual-phase alloys overcome the strength-ductility trade-off[END_REF]. In fact, a new paradigm is being followed, where instead of a search for compositions that favor single phase HEAs, formation of second phases are sought after to take advantage of precipitation hardening [START_REF] Li | Metastable high-entropy dual-phase alloys overcome the strength-ductility trade-off[END_REF][START_REF] Tsao | The high temperature tensile and creep behaviors of high entropy superalloy[END_REF]. However, the idea of a concentrated solid solution is still attractive, since in addition to the single phase HEAs, the different phases of the newer alloys often comprise of concentrated solid solutions. Therefore, understanding solid solution strengthening mechanisms in concentrated solutions is an important topic, which applies to a wide range of systems beyond HEAs, such as other finite-concentration solid solutions, and Ni-and Co-based superalloys.

Solid solution strengthening stems from the interaction of lattice dislocations with solutes.

As a dislocation moves in the crystal, the distortion it induces on the lattice interacts with the distortions around substitutional solutes. Substitutional solutes in cubic materials distort a lattice both geometrically -as a result of size mismatch with atoms of the host lattice -and chemically, by locally introducing a different bonding environment. Consequently, the dislocation/solute interaction energy typically consists of size misfit and more specific chemical core contributions [START_REF] Clouet | The vacancy -edge dislocation interaction in FCC metals: a comparison between atomic simulations and elasticity theory[END_REF][START_REF] Joseph A Yasi | First-principles data for solid-solution strengthening of magnesium: From geometry and chemistry to properties[END_REF][START_REF] Gpm Leyson | Solute strengthening from first principles and application to aluminum alloys[END_REF], where the former results from the interaction of the volumetric strain field around the solute with the elastic stress field of the dislocation and the latter is caused by the different bonding environment within the nonlinear core and stacking fault regions of the dislocations. The motion of a dislocation is hindered by such interactions, requiring additional resolved shear stress for continued plastic deformation.

Theories of solute strengthening in alloys can be broadly divided into two categories of strong-pinning and weak-pinning models. The strong-pinning models, such as the ones proposed by Friedel [15] and Fleischer [START_REF] Fleischer | Solution hardening[END_REF][START_REF] Fleischer | Substitutional solution hardening[END_REF], treat the solutes as individual point obstacles that pin the dislocations. Dislocation then bows out in the regions between the solutes and can break free from these obstacles upon application of additional stress. On the other hand, the weak-pinning models, originally proposed by Mott [START_REF] Nevill | Cxvii. a theory of work-hardening of metal crystals[END_REF] and Labusch [START_REF] Labusch | A statistical theory of solid solution hardening[END_REF][START_REF] Labusch | Statistische theorien der mischkristallhärtung[END_REF], consider the collective effect of a field of randomly-distributed solids on the dislocations. Favorable fluctuations in the solute distribution lower the energy of the dislocations, thereby pinning them, and the dislocation bows out between these pinned regions. Under additional resolved shear stress, the dislocations can escape from these obstacles, resulting in increased yield strength. In both strengthening models, the dislocation is treated as an elastic line, and its bow-out has an energetic cost due to the line tension. More recently, Leyson et al. [START_REF] Gpm Leyson | Solute strengthening from first principles and application to aluminum alloys[END_REF][START_REF] Paul | Quantitative prediction of solute strengthening in aluminium alloys[END_REF] proposed a parameter-free Labusch-type strengthening model for dilute alloys. Contrary to the original Labusch's model, it does not postulate any a priori interaction rangethis typical length scale naturally emerges from the theory -and includes the specific dislocation/solute interaction within the core, that are possibly computed by first principle methods. This model has been successfully applied to predict the solute strengthening of range of fcc [START_REF] Gpm Leyson | Solute strengthening from first principles and application to aluminum alloys[END_REF][START_REF] Paul | Quantitative prediction of solute strengthening in aluminium alloys[END_REF] and hcp [START_REF] Gpm Leyson | First-principles prediction of yield stress for basal slip in Mg-Al alloys[END_REF][START_REF] Buey | Core structure and solute strengthening of secondorder pyramidal c+ a dislocations in Mg-Y alloys[END_REF] slip and even twinning modes [START_REF] Ghazisaeidi | Solute strengthening of twinning dislocations in mg alloys[END_REF], and has shown great agreement with experiments. Moreover, Leyson and Curtin [START_REF] Gpm Leyson | labusch: the strong/weak pinning transition in solute strengthened metals[END_REF] compared the applicability of Friedel vs Labusch-type strengthening models, and concluded that, except for the highly localized dislocation cores, Labusch-type models control the strengthening for concentrations greater than 10 -4 and temperatures greater than 78K, i.e ranges relevant to most engineering alloys.

The above historical models are developed with the assumption of dilute alloys, where a base element makes the host lattice, and small amounts of alloying elements are introduced as solutes. This assumption is clearly not valid in the case of highly concentrated solid solutions. The contrast is even more pronounced for equiatomic alloys, where the definition of solute and solvent breaks down.

Here we review the solid solution strengthening theories for general random alloys with high elemental concentrations. The initial attempts by Toda-Caraballo et al. [START_REF] Toda | Modelling solid solution hardening in high entropy alloys[END_REF] were based on application of the Labusch model, in its original form, and assuming only elastic misfits contributions to strengthening. The only new theory for solid solution strengthening in concentrated solid solutions, is proposed by Varvenne et al. [START_REF] Varvenne | Theory of strengthening in fcc high entropy alloys[END_REF]. This theory assumes an effective average medium as the matrix and individual elemental types as solutes interacting with dislocations into the average medium. In the limit of dilute concentrations, this model reduces to the modified Labusch-type model of Leyson et al. [START_REF] Gpm Leyson | Solute strengthening from first principles and application to aluminum alloys[END_REF][START_REF] Paul | Quantitative prediction of solute strengthening in aluminium alloys[END_REF]. We describe this full theory, some numerical validations, useful model simplifications, and some examples of its successful application, in the following section. Other efforts in modeling solute strengthening in HEAs are listed and discussed afterwards. Finally, we conclude by discussing the influence of several materials inputs of existing models as well as future work directions.

EFFECTIVE-MEDIUM-BASED THEORY OF SOLUTE STRENGTHENING

Varvenne et al. proposed a theory for predicting the yield strength of random fcc alloys with arbitrary compositions [START_REF] Varvenne | Theory of strengthening in fcc high entropy alloys[END_REF]. The theory is based on the assumption that in a general random alloy, each elemental component can be regarded as "solute" in an effective "solvent" representing the average properties of the alloy. The solvent in then formally defined as an effective matrix at the overall average composition (mean field alloy), and solutes correspond to local composition-fluctuation with respect to the overall composition of the effectivematrix reference state [START_REF] Varvenne | Average-atom interatomic potentials for random alloys[END_REF].

Presentation of the full model

First, the reference material is defined with the average lattice parameter a, elastic constants {C ij } and stable and unstable stacking fault energies γ SF and γ U SF , all of which correspond to the composition of the random alloy. The usual {111}(110) slip systems are considered in this effective fcc matrix, where a (110) type dislocation dissociates onto two Shockley partials, bounding a stacking fault on {111} type planes. This setup is shown schematically in Figure 1 taken from Ref. [START_REF] Varvenne | Theory of strengthening in fcc high entropy alloys[END_REF].

For an N-component alloy, the interaction energy for a solute of type n at position (x i , y j ) shows the effective average matrix of the same alloy, with an embedded A "solute" at position (x i , y j , z k ) relative to the dislocation centered at the origin, with interaction energy U n (x i , y j , z k ).

Adapted from Varvenne et al. [START_REF] Varvenne | Theory of strengthening in fcc high entropy alloys[END_REF].

from the center of a straight dislocation and at position z k along the dislocation line is named U n (x i , y j , z k ). Major contributions to this interaction energy are from the elastic interaction of the dislocation's stress field with the misfit strain tensor of the solute and from the chemical misfit resulting from the change in the bonding environment of the solutes in the partial dislocation core geometry and the stacking fault region between them. In a random solute distribution, local fluctuations in the solute concentrations arise where the dislocation is attracted to the locally favorable fluctuations and is repelled by the unfavorable ones. The dislocation potential energy is lowered by finding the favorable solute fluctuations, requiring higher levels of shear stress to free the dislocation from these obstacles.

s n ijk [U n (x i -w, y j , z k ) -U n (x i , y j , z k )], (1)
where s n ijk is an occupancy parameter, which equals 1 if a n-type solute occupies position (x i , y j , z k ) and 0 otherwise. The typical favorable fluctuation in the interaction energy -i.e those that pin the dislocation by lowering its energy -is given by the standard deviation of the total potential energy change

σ ∆Utot (ζ, w) = ∆U 2 tot (ζ, w) -∆U tot (ζ, w) 2 1/2 , (2)
where the brackets are ensemble averages over the occupancy parameters s n ijk . This averaging can be performed analytically along the dislocation line segment z k , neglecting any correlation between the different chemical occupations of atomic sites, which yields

σ ∆Utot (ζ, w) = ζ √ 3b 1/2 ∆ Ẽp (w), (3)
where

∆ Ẽp (w) = i,j c n (( Ū n (x i -w, y j ) -Ū n (x i , y j )) 2 + σ 2 ∆U n ij ) 1/2 . ( 4
)
In the above equation, Ū n (x i -w, y j ) is the average value of U n (x i , y j , z k ) over the sites along the dislocation line z k at in plane position (x i , y j ), and σ 2 ∆U n ij is the associated standard deviation.

While a straight segment of dislocation L reduces its energy by bowing towards the favorable solute interactions, the increase in dislocation line -from straight to wavy configurations -costs energy due to the line tension.

Figure 2 shows the schematic of the original straight dislocation of length L adopting a wavy configuration consisting of straight segments of length 2ζ and of amplitude w as it moves through the random field of solutes. The total energy change corresponding to this event is given by, assuming w ζ

∆E tot (ζ, w) = ∆E LT (ζ, w) -σ ∆Utot (ζ, w) L 2ζ = Γ w 2 2ζ - ζ √ 3b 1/2 ∆ Ẽp (w) L 2ζ , (5)
where Γ is the dislocation line tension. The pinned dislocation geometry is obtained by minimizing the total energy change with respect to both the segment length ζ and the amplitude w. The critical segment length

ζ c is ζ c (w) = 4 √ 3 Γ 2 w 4 b ∆ Ẽ2 p (w) 1/3 . (6)
The subsequent optimization with respect to w needs to be done numerically to obtain the critical amplitude w c . In fact, multiple minima -i.e. multiple pairs of (ζ(w c ), w c ) valuescan be found [START_REF] Gpm Leyson | First-principles prediction of yield stress for basal slip in Mg-Al alloys[END_REF][START_REF] Varvenne | Theory of strengthening in fcc high entropy alloys[END_REF][START_REF] Gpm Leyson | Solute strengthening at high temperatures[END_REF][START_REF] Varvenne | Solute strengthening in random alloys[END_REF], especially when there exist a large dissociation between the partial dislocations, like in HEAs. This is further elaborated in the discussion section.

In the minimum energy configuration, segments of ζ c sit in a local potential energy well.

Approximating this energy well with a sinusoidal function gives the total energy barrier as

∆E b = 1.22 w 2 c Γ∆ Ẽ2 p (wc) b
for the dislocation to escape the energy well, i.e to unpin. The dislocation overcomes this barrier through a thermally activated process, assisted by an applied resolved shear stress. The zero temperature flow stress required to overcome ∆E b is then

τ y0 = π 2 ∆E b bζ c (w c )w c = 1.01 ∆ Ẽ4 p (w c ) Γb 5 w 5 c 1/3 . ( 7
)
At finite temperature, the energy barrier can be overcome with lower stress levels τ y (T, ˙ ) as

τ y (T, ˙ ) = τ y0 1 - kT ∆E b ln ˙ 0 ˙ 2/3 , ( 8
)
where ˙ is the plastic strain rate and ˙ 0 is a reference strain rate value obtained from the dislocation density, Burgers vector, typical slip distance and attempt frequencies from the Orowan relationship [START_REF] Gpm Leyson | Solute strengthening from first principles and application to aluminum alloys[END_REF][START_REF] Varvenne | Theory of strengthening in fcc high entropy alloys[END_REF].

At higher temperature/lower stress regimes [START_REF] Gpm Leyson | Solute strengthening from first principles and application to aluminum alloys[END_REF][START_REF] Gpm Leyson | Solute strengthening at high temperatures[END_REF][START_REF] Labusch | Temperature dependence of the flow stress of solid solutions[END_REF] the flow stress is better represented by

τ y (T, ˙ ) = τ y0 exp - 1 0.51 kT ∆E b ln ˙ 0 ˙ . (9)
In the dilute alloy limit, the present theory is equivalent to the previous predictive model of Leyson et al. [START_REF] Gpm Leyson | Solute strengthening from first principles and application to aluminum alloys[END_REF][START_REF] Paul | Quantitative prediction of solute strengthening in aluminium alloys[END_REF][START_REF] Gpm Leyson | First-principles prediction of yield stress for basal slip in Mg-Al alloys[END_REF][START_REF] Varvenne | Solute strengthening in random alloys[END_REF]. The inputs to the generalized parameter-free model are the Burgers vector b, the dislocation line tension Γ, and the various {U n (x i , y j , z k )}, i.e. the interaction energies of the various "solutes" with the dislocation. All these quantities are evaluated with respect to the effective-medium reference state. To validate some aspects of this model, Varvenne et al. [START_REF] Varvenne | Theory of strengthening in fcc high entropy alloys[END_REF] compared direct molecular statics simulations of the flow stress of short dislocation segments in a series of model (FeNi) 1-x Cr x alloys to the theory predictions, with all inputs computed from atomistics (using an EAM potential [START_REF] Bonny | Interatomic potential to study plasticity in stainless steels: the FeNiCr model alloy[END_REF]).

Note that with these short segments, the dislocations remain straight under applied stress in molecular simulations, and thus model predictions are made without the line tension contribution. A good agreement was obtained for the different investigated alloys, thus validating the simplified description of the potential energy landscape, and the internal length scales controlling the materials strength.

Considering real alloy systems, in the dilute limit, and assuming solutes do not alter the core geometry of the dislocation, the key interaction energies can be computed via direct first principles calculations of the dislocation [START_REF] Joseph A Yasi | First-principles data for solid-solution strengthening of magnesium: From geometry and chemistry to properties[END_REF][START_REF] Buey | Core structure and solute strengthening of secondorder pyramidal c+ a dislocations in Mg-Y alloys[END_REF][START_REF] Dallas R Trinkle | Predicting mg strength from firstprinciples: Solid-solution strengthening, softening, and cross-slip[END_REF][START_REF] Joseph A Yasi | Prediction of thermal cross-slip stress in magnesium alloys from direct first-principles data[END_REF]. However, in HEAs, modeling the dislocation core with first principles calculations is very difficult, in particular due to large simulation sizes necessary to capture chemistry changes along the dislocation line (see discussion section).

Model simplification using elasticity

Alternatively, the dislocation/solute interaction energy can be approximated by a combination of size misfit -interaction of the solute misfit volume with the pressure field of the dislocation -and of chemical misfit -interaction of the solute with stacking faults as a surrogate for the dislocation core -contributions to the interaction energy [START_REF] Varvenne | Solute strengthening in random alloys[END_REF][START_REF] Joseph A Yasi | Prediction of thermal cross-slip stress in magnesium alloys from a geometric interaction model[END_REF]. Beyond the dilute limit, calculations of the size and stacking fault misfit parameters are still challenging and require several simulations of sufficient size and accuracy to capture the fluctuations from the average values caused by high concentrations of multiple elements [START_REF] Ikeda | Ab initio phase stabilities and mechanical properties of multicomponent alloys: A comprehensive review for high entropy alloys and compositionally complex alloys[END_REF]. In light of these complications, Varvenne et al. [START_REF] Varvenne | Theory of strengthening in fcc high entropy alloys[END_REF] presented a reduced version of their theory of strengthening to consider only the main elasticity, i.e the size misfit, contribution to the interaction energy. This interaction energy is a result of the work done by the pressure field of the dislocation p(x i , y j ) on the misfit strain created by inserting an atom of a different size in the host lattice, and is given by

U n el (x i , y j , z k ) = -p(x i , y j )∆V n (x i , y j , z k ), ( 10
)
where ∆V n (x i , y j , z k ) is the change in total volume of the average medium as a solute of type n is inserted at position (x i , y j , z k ).

Representing the dislocation pressure field with p(x i , y j ) = -µ(1+ν) 3π(1-ν) f (x i , y j ), where f (x i , y j ) is a dimensionless pressure field from a distribution of normalized Burgers vectors on the glide plane and µ and ν are the elastic constants, and substituting Equation 10into

Equation 4 results in ∆ Ẽp (w) = µ(1 + ν 3π(1 -ν) ij ∆f 2 ij (w) 1/2 × n c n ∆V 2 n + σ 2 ∆Vn 1/2 . ( 11
)
In the above equation ∆f ij (w) = f (x i -w, y j ) -f (x i , y j ). The term n c n ∆V

τ y0 = 0.051α -1/3 µ 1 + ν 1 -ν 4/3 f 1 (w c ) ×   n c n ∆V 2 n + σ 2 ∆Vn b 6   2/3 , ( 12
) ∆E b = 0.274α 1/3 µb 3 1 + ν 1 -ν f 2 (w c ) ×   n c n ∆V 2 n + σ 2 ∆Vn b 6   1/3 . (13)
Here,

f 1 (w c ) = b wc 5/2 i,j ∆f 2 ij (w c ) 2/3 and f 2 (w c ) = wc b 2 i,j ∆f 2 ij (w c ) 1/3
are the minimized dislocation core coefficients, and the line tension is approximated by the elasticity relation Γ = αµb 2 , where α = 0.123 is a dimensionless number rescaled from direct atomistic simulations [START_REF] Varvenne | Theory of strengthening in fcc high entropy alloys[END_REF]. The elasticity-based reduced version of the theory is fully analytic and easily transferable to any new alloy system: the only inputs to this model are elastic constants, lattice parameters, dislocation core structure, and accurate solute misfit volumes, at the targeted composition. These parameters are either fairly computable by atomistic simulations or can be extracted/evaluated from experiments.

Comparison with experiments

Varvenne et al. [START_REF] Varvenne | Theory of strengthening in fcc high entropy alloys[END_REF] first applied the reduced theory to predict the strength of NiCoCr-FeMn high entropy alloy and some of its derivatives, using only experimental information.

In particular, the average misfit volumes ∆V n are obtained from experimentally measured atomic volumes in various alloys from this family and application of the Vegard's law. Metal HEA [START_REF] Varvenne | Predicting yield strengths of noble metal high entropy alloys[END_REF], with similarly good agreement with experimental predictions.

More recently, Laplanche et al. measured the activation volumes of plastic flow in

CrMnFeCoNi alloy to identify the operative deformation mechanisms [START_REF] Laplanche | Thermal activation parameters of plastic flow reveal deformation mechanisms in the CrMnFeCoNi highentropy alloy[END_REF]. Activation volume corresponds to the volume of the material that is involved in a thermally activated process under the resolved shear stress τ at a certain temperature and is given by tal activation volumes is comparable to the one achieved for dilute alloys [START_REF] Gpm Leyson | Solute strengthening from first principles and application to aluminum alloys[END_REF], showing that the effective-medium theory captures the essential concept of solute strengthening in these alloys and consequently, the compositional complexity of this alloy does not induce any new strengthening mechanism.

V = - ∂∆G(τ ) ∂τ

OTHER MODELING APPROACHES

One of the first attempts to model solute strengthening in HEAs was made by Toda-Caraballo and Rivera [START_REF] Toda | Modelling solid solution hardening in high entropy alloys[END_REF]. The authors start from the original Labusch's approach and consider its extension to dilute multi-component solid solutions [START_REF] Labusch | A statistical theory of solid solution hardening[END_REF][START_REF] Gypen | Multi-component solid solution hardening[END_REF]. The zero-temperature solute strengthening is expressed there as

∆τ ss = n B 3/2 n c n 2/3 , ( 14
)
where c n is the concentration of solute n, B n is the hardening parameter defined as

B n = 3µ 4/3 n Z 0 ; n = η 2 n + α 2 0 δ 2 n 1/2 , ( 15
)
where µ is the shear modulus of the alloy, n a mismatch parameter and Z 0 a fitting constant, i.e. an adjustable parameter. n incorporates the modulus and size misfits η n and δ n as

η n = η n 1 + 0.5|η n | ; η n = 1 µ dµ dc n ; δ n = 1 a da dc n , ( 16
)
where a is the lattice parameter of the alloy and α 0 is an empirical parameter that differentiates between edge and screw dislocation interactions with solutes. The adaptation to HEAs proposed by Toda-Caraballo and Rivera consists in redefining the modulus and size misfits for high concentration materials, actually making an implicit effective medium assumption as all quantities are defined with respect to the average HEA composition. The temperature-dependence of the strength is then introduced with a functional form analogous to Eq. 9, with an energy barrier taken as a constant dependent on the material. Assuming then Vegard's law for the lattice constant and linear variation of shear modulus with composition, misfit parameters and consequently the hardening parameters for the HEAs were calculated and compared with experimental values, and good qualitative agreements were achieved. In particular, the dominant role of the elastic size misfit in FCC materials was identified.

To make connection with the effective medium model presented in the previous section, we proceed from the reduced elasticity version of Eq 12, neglecting the misfit volume standard deviation terms. Then, Eq. 12 for the zero-temperature can be matched to Eq. 14 by

imposing B n = 0.051α -1/3 f 1 (w c ) 1+ν 1-ν µ ∆Vn b 3
4/3 , to be compared with B n = 3α and material quantities, Toda-Caraballo and Rivera's model will not be able to predict the high-temperature strengthening regime that as been discussed elsewhere [START_REF] Varvenne | Theory of strengthening in fcc high entropy alloys[END_REF][START_REF] Gpm Leyson | Solute strengthening at high temperatures[END_REF][START_REF] Varvenne | Solute strengthening in random alloys[END_REF] (see also the discussion section).

More recently, Walbrühl et al. [START_REF] Walbrühl | Modelling of solid solution strengthening in multicomponent alloys[END_REF] proposed an Integrated Computational Materials Engineering (ICME) approach to model solid solution strengthening in multicomponent alloys.

Their perspective is much more pragmatic, and stems from the difficulty to compute the material inputs for a high number of systems in physically-based models. Instead, they follow an empirical approach, still using the Labusch's c 2/3 n dependence of the strength -where c n is again the concentration of element n -but considering a nonlinear composition dependence of the strengthening parameters that are globally fitted in order to directly reproduce the hardness of a database of 895 alloys from the literature. The adopted functional forms are not physically derived, but are meant to account for both substitutional and interstitial solute strengthening and includes the presence of various phases. The solute strengthening part of the yield strength is converted into hardness by applying a constant factor for all materials. After an adjustment procedure, they provide a database of parameters for predicting strengthening in a wide range of multicomponent alloys.

While the authors recognize the lack of physical basis / derivation of their model, they assign the adjustable parameters to various physical mechanisms, i.e. Peierls-like resistance and solute strengthening (the latter embeds both solute and precipitate strengthening in their model). Incidentally they attempt to draw physical conclusions, e.g. in the CoCrFeN-iMn HEA, where only half of the strength would be due to solid strengthening, whereas Peierls stress is almost null in FCC materials, as already shown numerically for random multicomponent alloys [START_REF] Varvenne | Average-atom interatomic potentials for random alloys[END_REF]. This should be proscribed in a fully empirical approach. Next, the agreement between the fitted model and the experimental data for the hardness is within 13%, a quite large deviation for a data interpolation. Part of this is due to uncertainties in experimental measurements, and to the uncontrolled Hall-Petch effect, as the grain size is often not reported in the experimental hardness database, even if its impact on the strength can be important [START_REF] Wu | Temperature dependence of the mechanical properties of equiatomic solid solution alloys with face-centered cubic crystal structures[END_REF][START_REF] Otto | The influences of temperature and microstructure on the tensile properties of a cocrfemnni highentropy alloy[END_REF]. Finally, the transferability of this model to other alloy systems is not ensured, which is a general issue of fitting approaches that require recalibration of the model parameters for investigation of new materials. And the transferability could possibly be lowered by the identified issues on the adjustment hardness database. Consequently, if one is keen on getting pragmatic, adopting machine learning / deep learning approaches, completely leaving physical arguments, and having much flexible functional forms and number of parameters, would be even more appropriate for material design.

DISCUSSION

In this section, the effective-medium theory of strengthening for random alloys, providing a firm theoretical foundation for the observed strengthening trends without any adjustable parameter, is used to discuss the influence of several material inputs, and how they could be computed for specific materials. All along the discussion, we highlight remaining issues and challenges for future work in the area.

Importance of the line tension

While a useful concept to describe the elastic energy cost of a dislocation bow-out [15,[START_REF] Labusch | A statistical theory of solid solution hardening[END_REF][START_REF] Paul | Quantitative prediction of solute strengthening in aluminium alloys[END_REF][START_REF] Varvenne | Theory of strengthening in fcc high entropy alloys[END_REF][START_REF] Zaiser | Dislocation motion in a random solid solution[END_REF][START_REF] Ma | Computationally efficient and quantitatively accurate multiscale simulation of solid-solution strengthening by ab initio calculation[END_REF], the line tension is not an easy quantity to estimate for a given material.

It depends on the dislocation length, dislocation character, dislocation core, and on the anisotropy of the matrix material [START_REF] Barnett | The effects of elastic anisotropy on dislocation line tension in metals[END_REF][START_REF] Proville | Prediction of the kink-pair formation enthalpy on screw dislocations in α-iron by a line tension model parameterized on empirical potentials and first-principles calculations[END_REF][START_REF] Szajewski | Robust atomistic calculation of dislocation line tension[END_REF]. The most common assumption in both historical and more recent models is to assume a unique and fixed value per studied material for the line tension, and to assume isotropic elasticity. In Ref. [START_REF] Varvenne | Theory of strengthening in fcc high entropy alloys[END_REF], a rescaling Γ = αµb 2 was proposed, with α = 0.123 estimated for edge dislocations by comparison with direct line tension measurements in atomistic simulations, performed on a FeNiCr equiatomic alloy, using an EAM potential [START_REF] Bonny | Interatomic potential to study plasticity in stainless steels: the FeNiCr model alloy[END_REF]. This approach presents the advantage of an easy estimate of Γ for any new material of interest.

The importance of such simplifications can be examined through a sensitivity analysis.

A representative example is shown in Fig. 3a (dashed lines), where the strength is predicted with Γ values either twice or half the rescaled value using α = 0.123. Strengths at intermediate temperatures are only weakly dependent on Γ, while below T = 77K, the effect of Γ becomes more visible. This can be rationalized by computing the first derivative of strength with respect to the line tension as ∂τ y (T )/∂Γ = -τ y (T )/3Γ, using Eq.9, and noting that τ y0 ∝ Γ -1/3 and ∆E b ∝ Γ 1/3 . Thus, at intermediate temperatures, when doubling Γ, the effect on τ y0 will be partially compensated by the effect on ∆E b , and the strength will not be modified. On the other hand, for T → 0, ∂τ y (T )/∂Γ -τ y0 /3Γ ∝ Γ -4/3 and then doubling Γ will have much impact on the strength. Consequently, refined line tension measurements would be worth only for the prediction of very low-temperature strengths. Note that similar sensitivity to Γ has been observed [START_REF] Laplanche | Thermal activation parameters of plastic flow reveal deformation mechanisms in the CrMnFeCoNi highentropy alloy[END_REF] for activation volumes due to solute strengthening.

Nevertheless, a detailed methodological work, going beyond this simple analysis, and with a proper investigation of the effect of the dislocation length [START_REF] Varvenne | Solute strengthening in random alloys[END_REF] and elastic anisotropy on the line tension, and thus the predicted alloy solute strengthening quantities, remains to be done.

Role of the dislocation core structure

Accurate knowledge or modeling of the dislocation core structure is usually important for plasticity [START_REF] Rodney | Ab initio modeling of dislocation core properties in metals and semiconductors[END_REF], and in particular has been underlined in the context of solute strengthening in dilute alloys, by various authors [START_REF] Joseph A Yasi | First-principles data for solid-solution strengthening of magnesium: From geometry and chemistry to properties[END_REF][START_REF] Gpm Leyson | Solute strengthening from first principles and application to aluminum alloys[END_REF][START_REF] Paul | Quantitative prediction of solute strengthening in aluminium alloys[END_REF][START_REF] Gpm Leyson | labusch: the strong/weak pinning transition in solute strengthened metals[END_REF][START_REF] Ma | Computationally efficient and quantitatively accurate multiscale simulation of solid-solution strengthening by ab initio calculation[END_REF][START_REF] Patinet | Dislocation pinning by substitutional impurities in an atomic-scale model for Al(Mg) solid solutions[END_REF][START_REF] Proville | Atomic-scale study of dislocation glide in a model solid solution[END_REF]. This motivated first-principles computations of the dislocation core structures in the pure elemental materials of interest, or of the less cumbersome generalized stacking fault energy (GSFE) maps, that can be used as inputs to Peierls-Nabarro models so as to obtain the dislocation core structure.

In HEAs or any concentrated solid solution alloy, the core structure also matters for the strength; this appears in the reduced elasticity model of Eq. 10 for the solute / dislocation interaction, and Eqs. 12 and 13 for the strength. The core structure determines the dislocation pressure field, and then the characteristic lengths (ζ(w c ), w c ) and the minimized core coefficients f 1 (w c ) and f 2 (w c ) of τ y0 and ∆E b , respectively. Its influence on the strength is then analyzed through a parametric study, with the Burgers vector distribution of the (ii) understanding the effect of partial core spreading on the strength, would be beneficial to the community.

The role of the dislocation core structure has been discussed here within with the assumption of isotropic elasticity for the dislocation pressure field, whereas many alloys are elastically anisotropic. A more detailed analysis, accounting for the elastic anisotropy of the HEAs, would be an interesting direction for future work. (b) f 2 (w c ), appearing in Eqs. 12 and 13 for τ y0 and ∆E b , respectively. See main text for details.

Difficulties for DFT computation of model inputs

Introduction of HEAs, where the notion of solute and solvents is to be revisited, stimulated the need to rethink the solute strengthening models. The available physically-based models take as inputs, in the more general version, precise dislocation core geometry, solute/dislocation interaction energies and the dislocation line tension, preferably directly computed with first principles calculations. Such direct calculations are currently prohibitive in case of HEAs; the elasticity reduction of the model, with simplified inputs (Burgers vector, elastic constants, stable and unstable stacking faults, and misfit volumes), is thus valuable and has been shown to capture the main contributions to strengthening, as validated by experiments. The state-of-the art techniques to account for chemical disorder, i.e. the su-percell approach coupled to the concept of special quasi-random structures (SQS) [START_REF] Zunger | Special quasirandom structures[END_REF][START_REF] Van De Walle | Multicomponent multisublattice alloys, nonconfigurational entropy and other additions to the alloy theoretic automated toolkit[END_REF],

or the approach based on the coherent potential approximation (CPA) [START_REF] Johnson | Densityfunctional theory for random alloys: Total energy within the coherent-potential approximation[END_REF][START_REF] Ducastelle | Order and Phase Stability in Alloys[END_REF][START_REF] Vitos | Anisotropic lattice distortions in random alloys from first-principles theory[END_REF][START_REF] Vitos | Total-energy method based on the exact muffin-tin orbitals theory[END_REF], allow to compute these materials quantities, with some care. Using the first approach, large simulations sizes and / or several simulations are necessary to capture the reliable statistics of the "average" response of the alloy and the local fluctuations, causing deviations from the average behavior. This has been demonstrated in the simpler case of stacking fault calculations in Refs [START_REF] Zhao | Stacking fault energies of face-centered cubic concentrated solid solution alloys[END_REF][START_REF] Niu | Magnetically-driven phase transformation strengthening in high entropy alloys[END_REF]. Alternatively, the second approach reproduces a perfectly disordered state very efficiently, but is unable to include the local lattice distortions, which importance depends on the computed quantity. As a consequence, both approaches remain competitive.

Currently, first principles values of b, C ij , γ SF and γ USF have been reported for several HEAs, as nicely summarized in Ref. [START_REF] Ikeda | Ab initio phase stabilities and mechanical properties of multicomponent alloys: A comprehensive review for high entropy alloys and compositionally complex alloys[END_REF]. Misfit volume values are less common; a few computations of average bond lengths, bond length fluctations and elastic size misfit can be found [START_REF] Toda-Caraballo | Interatomic spacing distribution in multicomponent alloys[END_REF][START_REF] Seok Oh | Lattice distortions in the feconicrmn high entropy alloy studied by theory and experiment[END_REF].

Note finally that any progress in more detailed calculations of solute/dislocation interaction energy and dislocation core structure should make the strengthening predictions even more robust.

Short range ordering

An important assumption in all previously discussed strengthening models is a random distribution of atoms in the alloy. This is certainly a good approximation for dilute alloys as well as several HEA systems that are shown to be random experimentally [START_REF] Wu | Temperature dependence of the mechanical properties of equiatomic solid solution alloys with face-centered cubic crystal structures[END_REF][START_REF] Varvenne | Strengthening of high entropy alloys by dilute solute additions: CoCrFeNiAlx and CoCrFeNiMnAlx alloys[END_REF][START_REF] Laplanche | Thermal activation parameters of plastic flow reveal deformation mechanisms in the CrMnFeCoNi highentropy alloy[END_REF][START_REF] Norihiko | Atomic displacement in the crmnfeconi high-entropy alloy-a scaling factor to predict solid solution strengthening[END_REF],

likely due to the so-called "sluggish diffusion" effect in HEAs. However, short range ordering (SRO), i.e. correlation at small distances between chemical occupations of the atomic positions in the alloy, is thermodynamically expected when approaching transition temperatures.

Such SRO arises from interactions between alloy elements, and has been experimentally observed or theoretically predicted for some multicomponent alloys [START_REF] Niu | Spin-driven ordering of Cr in the equiatomic high entropy alloy NiFeCrCo[END_REF][START_REF] Singh | Atomic short-range order and incipient long-range order in high-entropy alloys[END_REF][START_REF] Ding | Tunable stacking fault energies by tailoring local chemical order in CrCoNi medium-entropy alloys[END_REF][START_REF] Tamm | Atomicscale properties of ni-based fcc ternary, and quaternary alloys[END_REF][START_REF] Körmann | Long-ranged interactions in bcc nbmotaw high-entropy alloys[END_REF]. In dilute alloys, the effect of SRO on strengthening has been historically envisioned through the formation of pairs, triplets or clusters, due to solute/solute interactions. Those clusters are then considered as new, independant and randomly distributed species interacting with the dislocation, and applying conservation laws their contribution to strengthening can be included in the framework of the standard strengthening models. Either additional strengthening or softening as compared to equivalent solid solutions having only isolated solutes is then observed [START_REF] Gypen | Multi-component solid solution hardening[END_REF][START_REF] Proville | Atomic-scale models for hardening in fcc solid solutions[END_REF]. SRO also modifies the statistics of dislocation/solute and solute/solute interactions (cf Eq. 4); this would be possible to include in the Labusch-type models but would likely lead to a loss of analyticity in the equations. In HEA systems, the effect of SRO could be included in the generalized theory presented here, with a more comprehensive energetic model and appropriate treatment of statistics, and re-defining the notion of clusters, similarly to what has been done for solutes in random concentrated alloys. The details of addition of SRO effect and an assessment of its relevance to total strengthening, particularly at finite temperatures, remains a challenging task.

Conclusion

The Labusch-type models conbined with an effective-medium approach work well in HEA systems with low Peierls barrier, such as FCC alloys. In materials such as BCC alloys the generally accepted deformation mechanism is kink-pair nucleation for screw dislocation motion. Depending on the effect of solutes on this process, solutes can soften as well as strengthen the dislocation motion [START_REF] Argon | Strengthening mechanisms in crystal plasticity[END_REF][START_REF] Dallas R Trinkle | The chemistry of deformation: How solutes soften pure metals[END_REF]. Therefore, solution strengthening/softening of BCC high entropy alloys await for (i) identification of the relevant operating mechanisms and

(ii) for the development of a related mechanistic theory, dealing with arbitrary compositions.

ACKNOWLEDGMENTS:

FIG. 1 .

 1 FIG. 1. Effective medium concept for dislocation/solute interactions. (a) shows a random 3component alloy, with component types A, B and C, containing a dissociated edge dislocation; (b)

  An important concept in the theory developed by Varvenne et al. is the energy change of a straight segment of dislocation ζ during the glide over a distance w. The relative change in the position of the dislocation and fixed solutes results in a potential energy change given by ∆U tot (ζ, w) = i,j,k,n

FIG. 2 .

 2 FIG. 2. Schematic of a straight dislocation segment of length L, adopting a low-energy wavy configuration as it moves through the random field of solutes. The configuration consists of segments of length 2ζ with amplitude w. The total dislocation energy is optimized with respect to both ζ and w to obtain the critical lengths ζ c and w c . Adapted from Varvenne et al. [27].

2 n + σ 2 ∆Vn

 22 emerges as the main misfit volume quantity, in which ∆V n is the average misfit volume of solute n and σ ∆Vn is the standard deviation. Following the procedure in Equations 5-7 yields τ y0 and ∆E b as

FIG. 3 .

 3 FIG. 3. Comparison between experimental and theoretical prediction of strength. Yield stress vs. temperature are shown for (a) CrFeCoNi and (b) MnCrFeCoNi equiatomic alloys [37, 38], as measured (black symbols) after subtraction of the Hall-Petch contribution to strength and as predicted by the theory (blue symbols) Adapted from Varvenne et al. [27]. Theoretical predictions of flow stress are converted into uniaxial yield strengths σ y (T, ε) by applying the Taylor factor of 3.06.

Figure 3

 3 Figure3shows the comparison between theory and experiments in predicting the strength of a set of equiatomic alloys from the NiCoCrFeMn. The agreement is generally great, particularly at moderate temperatures, given the approximations in the reduced version of the model. At T=77 K, the theory underestimates strength. The authors attribute this to a possible combination of (i) line tension effect -not precisely known -which influence

  τ ) is the stress-dependent Gibbs free energy corresponding to the gliding dislocation overcoming various obstacles (eg. solute atoms, precipitates, forest dislocations). The activation volumes range from the order of b 3 (where b is the magnitude of the Burgers vector) for individual point defects to ≈ 1000b 3 for collective solute fluctuations in dilute alloys. Therefore, measuring activation volumes reveals the intrinsic length scales involved in the thermally activated processes of plasticity, thereby providing insight into the operative deformation mechanisms. Measurements of activation volumes in the CrMnFeCoNi showed that the plastic flow in this alloy is governed by a combination of (i) rate-independent Hall-Petch strengthening (ii) thermally activated solute strengthening and (iii) forest hardening; a behavior typical of other usual FCC alloys. In particular, the thermally-activated solid solution strengthening component was compared to the theoretical predictions of activation volume by the model of Varvenne et al. The agreement between theoretical and experimen-

4 / 3 0 Z 0 µδ 4 / 3 n

 4343 when the modulus misfit is neglected. Both approaches are thus similar, showing the importance of the HEA shear modulus and of the size misfit quantity. However the derivation by Varvenne et al. includes additional material inputs (dislocation core structure, line tension, etc.) -that are embedded into an adjustable parameter by Toda-Caraballo and Rivera -and the full version of the theory includes the solute/dislocation interactions, computable from atomistic simulations, and thus going beyond the elasticity assumptions. In particular, ignoring the details of the core structure and of the energy barrier variation with composition

FCC 2 + e -1 2 ( 2 ,

 222 figurations leading to two different minimized core coefficients for both τ y0 and ∆E b . The first minimized core coefficients f 1 (w c1 ) and f 2 (w c1 ) are associated with the high stress / low-temperature regime, and are insensitive to d, that is related to the stable stacking fault energy (SFE) as d = µb(2 + ν)/(24π(1 -ν)γ SF ), following an isotropic elasticity model. The second minimum energy configuration corresponds to a larger w c value, and is active at a lower stress / higher temperature regime. It is associated with a high energy barrier -typically 3-5 eV -which thus provides a high temperature plateau stress whose value is much more sensitive to d and thus to γ SF . The effect of the partial spreading is finally investigated by comparing the core coefficient values for the first minimum, obtained with σ = 1, 1.5, and 2b (Fig.4). The f 1 (w c1 ) and f 2 (w c1 ) values for large d are entirely fixed by σ, a value roughly scaling with the inverse of the unstable stacking fault energy (USFE), σ ∼ γ USF[START_REF] Bulatov | Computer Simulations of Dislocations. Oxford series on materials modelling[END_REF]; accurate estimate of σ would be required. For smaller separation distances, d and σ both induce variations in f 1 (w c ) and f 1 (w c ), meaning that in these cases the knowledge of both SFE and USFE is important for the strength.

FIG. 4 .

 4 FIG.[START_REF] Bei | Multi-component solid solution alloys having high mixing entropy[END_REF]. Influence of the dislocation core structure on the minimized core coefficients (a) f 1 (w c ) and

This work was supported by the National Science Foundation Grant DMR-1553355. The anonymous referee is acknowledge for useful comments.