Ayn A Salindao

Norton K Salindao

Haron A Mohamad

Noraysa B Kasan

Dr Bernabe G Dela

Modular Distance Learning Gaps: Relationship Between Teachers' and Students' Performances in Grade 9 Science

Keywords: Modular Distance Learning, Student's Performances, Learning Gaps, Science Salindao

The study aimed to determine the relationship of teachers' and students' performances in Grade 9 Science during Modular Distance Learning in Maguindanao II Division. It determined some of the learning gaps experienced by the students in learning science in terms of knowledge, understanding and proficiency. Quantitative research design was used in this study. There were 30 secondary science major teachers and 90 selected grade 9 students' participants. Survey questionnaire was used to gather the data. The data were gathered and analyze using statistical tools such as weighted mean, correlation and standard deviation.The result revealed that the level of performance of Grade 9 students in learning Science in terms of knowledge, understanding and proficiency got the highest mean during Modular Distance Learning. It means that a Grade 9 students are prepared in this new learning modalities. In addition, it was found out that the level of teachers' in teaching science in terms of knowledge, understanding and proficiency got the highest mean. It means that Science teachers performed their task responsibly. Moreover, the investigation showed that there is no significant relationship between the students' mean performance and teachers' mean performance in terms of knowledge, understanding and proficiency.

 (Davis, 2020).

The first step in closing learning gaps is determining where and what those gaps are, as well as which students struggle. Quizzes are a simple and quick approach to formatively assess what students have learnt. These can be mini-quizzes at the end of a topic or simply a few units of a subject. Second, teachers must ensure that a diverse set of questions is used to identify learning gaps. Finally, once the gaps have been identified, teachers must take the appropriate steps and assess the mastery level attained by the students using quizzes. Learning gaps can be addressed and bridged using these procedures [START_REF] Heather | Addressing Learning gaps[END_REF].

Education has a significant impact on the lives of students. The introduction of COVID-19 in the Philippines resulted in significant changes in the educational landscape. One of these is the Department of Education's instructional model. Due to the current economic crisis, most educational systems have been forced to adopt alternatives to face-to-face teaching and learning. Many educational systems transferred activities online to ensure that instruction could continue even while schools were closed (OECD, 2020). It is the transition of school-based teaching and learning to modular distance learning. That is why DepEd officials are constantly looking for solutions to alleviate challenges and equip teachers and school administrators to be more effective in the field of modular distance learning [START_REF] Bagod | Teaching-learning modality under new normal[END_REF].

In accordance with the "Most Essential Learning Competencies," the Education Program Supervisors produced modules in all topics for all grades/year levels throughout four quarters beginning in May 2020. These self-learning modules are already part of a learning package that includes a pre-test, discussion, and a series of evaluation/assessment questions. With the modular learning class schedule, they are dispersed to all students (Bagod,2020).

Academic performance is a metric used to assess a student's progress in several academic areas. Class performance, graduation rates, and test results are commonly used by teachers and education administrators to assess student accomplishment (Brooie, 2014).

When addressing a given activity, at a certain time, a learning approach is the combination of an objective Salindao et al.

3/10 and a strategy. As a result, when a surface approach is used with the goal of achieving excellent grades through the use of acceptable memorizing techniques, the students' performance will be adequate. In terms of the deep approach, it entails self-determined motivation, which entails both effort and enjoyment for the subject under consideration. Students have a good chance of doing well in their academics if they follow this advice. The primary drivers of academic performance are the learning approaches (Barca, and Peralbo 2013).

The Ministry of Basic, Higher and Technical Education (MBHTE) particularly in Maguindanao-II Public Secondary Schools has been monitoring different schools and their students. Any gaps should be addressed at the soonest possible time for the successful mastery of competencies as prescribed in the curriculum. At this pandemic period, as we are embracing the various distance learning modalities, a big challenge for teachers and learners alike is facing our way on how to effectively bridge the identified gaps.

Research Questions

The study aimed to determine the student's performance and teacher's teaching performance in The Department of Education wanted to know which off-school pedagogical modes best suited students based on their location in the archipelago, and they discovered that most parents chose the modular learning modality for their children over other possibilities. This is based on the preliminary results of the Learner Enrollment and Survey Form (LESF), which revealed that 8.8 million parents preferred modular learning to the 3.9 million who preferred blended learning, which is a combination of different modalities such as module, television, and radio with online [START_REF] Bernado | Modular Learning most preferred parents: DepEd[END_REF].

Modular Distance Learning (MDL)

Is a type of distance learning that use Self-Learning Modules (SLM) based on the Department of Education's most essential learning abilities (MELCs).

The modules feature motivation and assessment parts that act as a comprehensive guide to both teachers' and students' desired competencies. When a group or individual education session is not possible, it provides learners with a learning activity [START_REF] Liego | [END_REF]).

Student's Performance

Student performance is described as a student's level of knowledge in a specific area or subject in comparison to the norm, and it is commonly measured using the grade point average [START_REF] Wilcox | Performance as the level of knowledge[END_REF]. Most people think about a person's GPA when they hear the words "academic performance." However, a number of variables represent a student's academic performance. While some students may not be at the top of their class, they may hold positions of leadership in numerous student organizations or perform highly on standardized examinations [START_REF] Williams | Student's academic success[END_REF]. Class performance, graduation rates, and test scores are frequently used by teachers and school officials to assess students' performance [START_REF] Kaiser | What is performance? Kyoshaba[END_REF].

Learning approaches are critical components of academic success because they explain how to combine a goal and a strategy when addressing a certain subject at a specific time. As a result, when there is a superficial approach and the goal of earning good grades utilizing acceptable memorization techniques, the students will do well. The deep approach necessitates a self-determined motivation that involves both effort and enjoyment for the subject under consideration. In this regard, the pupil has a good chance of succeeding in school [START_REF] Barca | The learning approaches as the key determi-nants of academic performance[END_REF].

It's important to note that children' learning styles aren't set in stone, and they can alter over time. A student, on the other hand, can adopt either learning approach depending on the academic task (surface or deep). To put it another way, learning strategies are built on both. "A learning method determines the nature of the relationship between student, circumstance, and task," according to the learner's and teaching setting characteristics [START_REF] Cabigao | Students' individual characteristics and their teaching con-text[END_REF] Academic success is influenced by a variety of characteristics, contexts, and circumstances that should be addressed by empirical research, despite the challenges in its design and methodological strategy (Nieto, 2008). Academic performance refers to a student's, teacher's, or institution's achievement of educational goals. As a result, students' performance is measured by how well they do on coursework-related tests as well as other types of exams (Kyoshaba, 2009).

Factors contributing to Academic Performance

Academic performance is the degree to which a student, instructor, or institution has attained their educational goals. Several studies have been undertaken in various nations to determine the elements that influence students' academic success at various levels. [START_REF] Farooq | Performance of student's at different level[END_REF]

Knowledge

Knowledge acts as a glue, binding information and learning together. We grasp a topic better when we have prior knowledge of it. It is very significant in the lives of students, especially at school. They will have difficulty understanding the text if they do not have relevant knowledge. It is about obtaining, growing, sharing, and using information to have an impact while learning in the process of studying, teaching, and education [START_REF] Kaiser | What is performance? Kyoshaba[END_REF]. Parents are crucial in providing their children with prior information. It is because a child's education begins at home, with their parents serving as their first teachers. Teachers, after parents, play an essential role in a child's earning process. They must comprehend, evaluate, and analyze what the students already know about the subject. One of the most powerful indicators of how effectively students acquire new knowledge connected to the material is what they already know about it. Teachers can involve students in a variety of activities to better grasp their prior knowledge. This will help pique their interest in learning and connecting with the subject. It is easier to create learning with background information [START_REF] Iwu | Parents play avital role in providing knowledge[END_REF].

Understanding

When someone truly understands, they can: (1) put concepts, principles, and processes into their own words, teach it to others, justify their answer, and show

Proficiency

Learning to perform an activity with competence, as measured by ease, speed, and accuracy of execution, is a skill that may be gained via extensive practice.

Motor, perceptual, cognitive, or a mix of these skills may be required.

Teachers Performance

Teachers' performance is a series of process actions carried out by teachers in carrying out their tasks and jobs to the best of their abilities in order to achieve institutional goals. It can be measured by the competency of teachers at the time of their teaching.

Teachers cannot be detached from many difficulties and work issues while performing their duties and jobs, particularly during the COVID-19 pandemic [START_REF] Savtri | What is teacher's performance? Sibanda[END_REF][START_REF] Savtri | What is teacher's performance? Sibanda[END_REF].

Teachers should apply appropriate teaching strategies that best suit specific objectives and competencies to secure and facilitate the process of knowledge

Research Participants

The target participants of this study were the 30 Science major teachers in Secondary Schools of Maguindanao II Division and 90 selected Grade-9 Junior High School Students. I used convenient sampling in choosing 90 Grade-9 Junior High School Students. The availability sampling relies on data collection from the population members who are conveniently available to participate in study.

Research Instrument

The study utilized survey questionnaire as the primary tool in the gathering of data from the respondents. The questionnaire has three parts. The first part was the assessment of performances of students in learning science in Modular Distance Learning. The second part was the assessment of performances of science teacher's in teaching science in Modular Distance Learning. The last part was the learning gaps experienced by the students in terms of knowledge, understanding and proficiency.

Data Gathering Procedures

The researcher performed the following steps in conducting the study. First, she asked permission from the school principal through a letter endorsed by the adviser. Then she asked approval also from the respondents before obtaining the data. After distributing the researcher-made instrument to the participants she collected and extracted the data from the responses of the participants. Then, she utilized the gathered data and used certain statistical tools for the analysis and interpretation of the findings.

Statistical Treatment of Data

The statistical formula used in this study were mean, standard deviation and Correlation Coefficient (Person R). Weighted mean was used to determine the status of grade 9 students' performances in learning science and the status of science teachers' performances in teaching science 9. Standard deviation was used to determine how far the individual responses to a question vary from the mean. Lastly, Correlation Coefficient (Person R) was used to measure the relationship between the students' and the teachers' performances in grade 9 science. The analysis and interpretation of data were presented in details through tables, accompanied by paragraphs of discussion which pointed out important aspect of data using descriptive statistic

Results and Discussion

Relationship Between the Students' and the Teachers' Performance in Grade 9 Science By Subjects in Terms of Knowledge

The data in Table 1 shows the relationship between the Students' and the Teachers' performance in Grade 9 Science by subjects in terms of knowledge.

Table 1 shows that in Biology, there is no significant relationship between the student's mean performance Salindao et al. 7/10 and teacher's mean performance in terms of knowledge. This is indicated by the smaller computed r-value 0.389 as compared to the tabular or critical value ±0.878 at 5% level of significance.

In chemistry subject there is no significant relationship between the student's mean performance and teacher's mean performance in terms of knowledge. This is indicated by the computed r-value 0.802 as compared to the tabular or critical value ±0.878 at 5% level of significance.

Table 1. Relationship Between the Students' and the Teachers' Performance in Grade 9 Science By Subjects in Terms of Knowledge

In Physics subject there is no significant relationship between the student's mean performance and teacher's mean performance in terms of knowledge. This is indicated by the computed smaller r-value 0.308 as compared to the ±0.878 at 5% level of significance.

In Earth Science subject there is no significant relationship between the student's mean performance and teacher's mean performance in terms of knowledge. This is indicated by the computed r-value -0.184 as compared to the tabular or critical value ±0.878 at 5% level of significant tabular or critical value level of significance.

In general, there is no significant relationship between the student's mean performance and teacher's mean performance in terms of knowledge. This is indicated by the computed r-value 0.598 as compared to the tabular or critical value ±0.950 at 5% level of significance. This implies that both student's mean performances and teacher's mean performances are most likely the same.

Relationship Between the Students' and the Teachers' Performance in Grade 9 Science By Subjects in Terms of Understanding

The data in Table 2 shows the relationship between the Students' and the Teachers' performance in Grade 9 Science by subjects in terms of understanding.

In Biology, there is no significant relationship between the student's mean performance and teacher's mean performance in terms of understanding. This is indicated by the computed r-value -0.065 as compared to the tabular or critical value ±0.878 at 5% level of significance In chemistry, there is no significant relationship between the student's mean performance and teacher's mean performance in terms of understanding. This is indicated by the computed r-value 0.273 as compared to the tabular or critical value ±0.878 at 5% level of significance.

In Physics, there is no significant relationship between the student's mean performance and teacher's mean Salindao et al. 8/10 performance in terms of understanding. This is indicated by the computed smaller value r-value -0.226 as compared to the ±0.878 at 5% level of significance.

In Earth Science, there is no significant relationship between the student's mean performance and teacher's mean performance in terms of understanding. This is indicated by the computed r-value 0.021 as compared to the tabular or critical value ±0.878 at 5% level of significant tabular or critical value level of significance.

In general, there is no significant relationship between the student's mean performance and teacher's mean performance in terms of understanding. This is indicated by the computed r-value 0.897 as compared to the tabular or critical value ±0.950 at 5% level of significance. Which means that both student's mean performances and teacher's mean performance are most likely the same.

Relationship Between the Students' and the Teachers' Performance in Grade 9 Science By Subjects in Terms of Proficiency

The data in Table 3 shows the relationship between the Students' and the Teachers' performance in Grade 9 Science by subjects in terms of proficiency. In Biology, there is no significant relationship between the student's mean performance and teacher's mean performance in terms of proficiency. This is indicated by the computed r-value -0.403 as compared to the tabular or critical value ±0.878 at 5% level of significance.

In Chemistry, there is no significant relationship between the student's mean performance and teacher's mean performance in terms of proficiency. This is indicated by the computed r-value -0.099 as compared to the tabular or critical value ±0.878 at 5% level of significance.

In Physics subject there is no significant relationship between the student's mean performance and teacher's mean performance in terms of understanding. This is indicated by the computed smaller r-value 0.155 as compared to the ±0.878 at 5% level of significance.

In Earth Science subject there is no significant relationship between the student's mean performance and teacher's mean performance in terms of understanding. This is indicated by the computed rvalue 0.872 as compared to the tabular or critical value ±0.878 at 5% level of significant tabular or critical value level of significance.

In general, there is no significant relationship between the student's mean performance and teacher's mean performance in terms of understanding. This is indicated by the computed r-value 0.225 as compared to the tabular or critical value ±0.950 at 5% level of significance. This implies that both student's mean performances and teacher's mean performances are most likely the same.

Relationship Between the Students' and the Teachers' Performances in Grade 9 Science in Terms of Knowledge, Understanding, and Proficienc

The correlation coefficient obtained from the association between the student's mean performance and the teacher's mean performance in terms of knowledge, understanding, and proficiency are reflected in Table 4. As shown in Table 4, there is no significant relationship between the student's mean performance and the teacher's performance in terms of knowledge. This is indicated by the smaller computed r-value of 0.598 as compared to the tabular or critical r-value of 0.950 at 5% level of significance.

The computed r-value of 0.598 is high but not sufficient to reject the null hypothesis. However, the coefficient of determination, r² = (0.598)² = 35.76% which indicates that 35.76% of the variations of the students' performance is explained by or associated with the teachers' performance. This means further that 64.24% in the variation of the students' performance is associated with the other factors that are outside or not part of the study.

In the case of "understanding", although the computed r-value of 0.897 is high but failed to reach the critical r-value of 0.950 to be significant at 5% level or significance. The coefficient of determination, r² = (0.897)² = 80.46% indicates that 80.46% of the variation of the students' performance is explained by or associated with the teacher's performance. This means further that 19.54% in the variation of the students' performance is associated with the other factors that are outside or not part of the study.

In the aspect of proficiency, the relationship between the student's mean performance and the teacher's mean performance is not significant. This is indicated by the smaller computed r-value of 0.225 as compared to the tabular or critical r-value of 0.950 at 5% level of significance. This means that in terms of proficiency, the performance of the Grade 9 students in learning Science do not influence the teachers' performance in teaching Science subjects.

Conclusion

The level of performance of grade 9 students in learning science in terms of knowledge, understanding and proficiency has a fair result, and the level of performance of grade 9 science teachers in teaching science in terms of knowledge, understanding and proficiency has an excellent result. Though there are learning gaps in specific science areas, the result is negligible.

Based on the findings of this study, the following recommendations are given: (1) Aside from conducting an in-service training for teachers, the school leaders should provide a regular technical assistance and learning action cell (LAC) session to their teachers to ensure that they are properly practicing the spiral approach in their respective classes.

(2) An intervention program may be developed for students with the aim of improving their academic performance. (3) A similar research may be conducted determining the relationship of teaching method to the academic performance of students and other learning areas. ( 4) Further research studies may be conducted to explore others factors that affect the academic performance of students in science.

  their reasoning; (2) interpret data, text, and experience through images, analogies, stories, and models; (3) apply what they know in new and complex contexts; and (4) demonstrate perspective by effectively seeing the big picture and recognizing patterns (Wiggins and McTighe,2021).

  learning gap is a mismatch between what a pupil has learnt and what they should have learned by a certain point in their schooling. While some learning gaps are tiny and easily remedied (for example, a student missing a single lesson due to illness), others are more serious and result in considerable learning losses or academic setbacks[START_REF] Sullivan | Learning gap is a mismatch of learning[END_REF].

	Closing achievement gaps among these various student
	groups became a focus of federal education
	accountability with the passage of the No Child Left
	Behind Act of 2001, and schools and districts were	A student's academic performance is influenced by his
	required to disaggregate student test scores and other	or her desire in pursuing a subject, co-curricular
	performance data by student characteristics to enable	activities, nationality, and gender. Furthermore,
	better comparisons between groups. This resulted in	regular study, punctuality in school, and self-
	increased awareness of racial disparities as well as	motivation are the most important determinants of
	increased worry about other types of success gaps. The	students' academic achievement (Sibanda, Iwu and
	focus resulted in more tailored interventions for	Olumide).
	diverse groups of pupils, although most achievement
	gaps remained unbridged a decade after the law was	Catherine (2015) discovered that parents'
	established (Ansell, 2011).	socioeconomic position, particularly those with high
		earnings, has a significant impact on students'
		academic achievement. Academic achievement has
		also been linked to a positive classroom atmosphere
		(Nghambi and Osei-Mensah,2012).
		The availability of teaching and learning resources,
		teacher competency, and the environment in which a
		school is located all influence students' academic
		success. Furthermore, students' personality features,
		personal goals and motivation, as well as teacher
		support and degree of expertise, all have a substantial
		impact on their academic achievement (Ulate &
		Carballo, 2011)
		Learning Gaps
		Various educators agreed on a definition of learning
		gaps, which are generally the difference between what
		learners are anticipated to have acquired by a specific
		grade level and what they have learned up to that time.
		These learning gaps are frequently accumulating,
		which means that if they are not addressed promptly,
		learners are likely to fall farther and further behind due
		to the skills and knowledge they have lost. Learning
		gaps occurred when students missed a skill and moved
		on to the next course without remediation (Torres,
		2020).

found that a student's academic performance is influenced by his or her parents' education and socioeconomic level in Pakistan. According to a study conducted in Singapore by (Jayanthi and Balakrishan, 2014).

Identifying learning gaps in writing the outputs of the students in his class. He planned and performed the intervention program from the date obtained. The level of improvement gained was determined by comparing learners' performance before and after the Salindao et al.

5/10 implementation phase

(Cabigao, 2020)

.

A

Table 2 .

 2 Relationship Between the Students' and the Teachers' Performance in Grade 9 Science By Subjects in Terms of Understanding

Table 3 .

 3 Relationship Between the Students' and the Teachers' Performances in Grade 9 Science By Subjects in Terms of Proficiency

Table 4 .

 4 Relationship

	Salindao et al.
	9/10

Between the Students' and the Teacher's Performances in Grade 9 Science in Terms of Knowledge, Understanding, and Proficiency

This study was conducted in selected Secondary Public Schools


E Epifano

D Monceau

D Monceau Corrosion

Ellingham diagram: A new look at an old tool

Keywords:

Ellingham diagrams are thermodynamic tools that allow comparing the relative stability among oxides. They are well established for pure metals, but they can be obtained for any multi-element system if the chemical activity of each component is known. However, chemical activities are very difficult to obtain experimentally and, because of the lack of these data, Ellingham diagrams have never been reported for complex industrial alloys. This work aims to show how nowadays advanced thermodynamic databases can be used to supply for this lack. Herein, the Ellingham diagrams of four wildly used alloys-TiAl 48-2-2, TA6V, Inconel 718 and 304 steel-have been computed. The results are discussed in respect to the oxidation behavior of these alloys and important differences in respect to the Ellingham diagram of pure metals are highlighted.

Introduction

Several industrial fields, such as transportation, materials and energy productions, involve high temperature processes in oxidizing environments. Oxidation resistance is hence a fundamental criterium for alloy design in these fields. The resistance is achieved through the selective oxidation of one element of the alloy, which leads to the formation of a protective, slow growing oxide scale at its surface. The best oxide candidates for this purpose are generally alumina (Al 2 O 3 ), chromia (Cr 2 O 3 ) and silica (SiO 2 ). In order to sustain a selective oxidation, the alloy composition needs to respect several criteria which, as firstly stated by Wagner [START_REF] Wagner | Theoretical analysis of the diffusion processes determining the oxidation rate of alloys[END_REF], are related to thermodynamic and kinetic properties of the alloy and the oxide. Among these criteria, a fundamental one concerns the thermodynamic stability of the desired protective oxide. In this respect, a useful tool is the Ellingham diagram, a graph comparing the stability of oxides as a function of the temperature.

Ellingham diagrams are obtained from basic thermodynamic laws, as they represent the Gibbs free energy change ΔG associated with a reaction of the type: [START_REF] Wagner | Theoretical analysis of the diffusion processes determining the oxidation rate of alloys[END_REF] Where Me is a metallic element, O 2 is a gas molecule of dioxygen and Me a O b is an oxide.

As shown by the Eq. 1, in order to compare the stability of various oxides, reactions are written for the same quantity of oxygen, namely one mole of O 2 (or equivalently for ½ mole of O 2 ). For a practical point of view, this allows answering the question whether one O 2 molecule (the limiting reagent) reacts preferentially with a metal rather than another one.

The Gibbs free energy change of the reaction is generally given in respect to the standard state of the species (°G), which is the stable form of the species at the temperature of interest. It is then: [START_REF]Reducibility of oxides and sulphides in metallurgical processes[END_REF] where Ḡ are the partial Gibbs energy of the species with respect to the standard state. At equilibrium, ΔG = 0 as reactants and products are in equilibrium with each other. The partial Gibbs energy of each species can be rewritten as Ḡ i = RTln(a i ), with a i being the activity, T the absolute temperature and R the molar gas constant. Eq. 2 is then: [START_REF] Saunders | Chapter 14 -Oxidation, hot corrosion and protection of metallic materials[END_REF] If the oxidation reaction involves a pure metal, Me and Me a O b are both pure species in their standard state and by definition their activities are equal to 1. For O 2 , the activity is equal to the partial pressure p(O 2 )/p(O 2 )°, where the standard p(O 2 )°= 1 bar. With these simplifications, the following equation is obtained: [START_REF] David | High temperature oxidation and corrosion of metals[END_REF] Based on the latter equation, Ellingham diagrams [START_REF]Reducibility of oxides and sulphides in metallurgical processes[END_REF] have been computed for pure metals and used to predict the relative stability among oxides [START_REF] Saunders | Chapter 14 -Oxidation, hot corrosion and protection of metallic materials[END_REF][START_REF] David | High temperature oxidation and corrosion of metals[END_REF].

However, experimental evidences on oxidation of alloys are sometimes in contradiction with these predictions. One reason for discrepancy is obviously related to the fact that Ellingham diagrams are merely thermodynamic tools and they do not consider the kinetics: difficult nu-cleation and slow diffusion of the protective alloying element, or large dissolution and fast diffusion of oxygen into the alloy can impede the formation of the most stable oxide at the surface of the alloy, and can lead to internal oxidation [START_REF] Wagner | Theoretical analysis of the diffusion processes determining the oxidation rate of alloys[END_REF]. However, in some cases, the discrepancies between prediction and observation are due to an inexact use of Ellingham's diagram. As herein shown, Eq. 4 is correct only for pure metals, as the activity of pure substances is equal to 1. In alloys, the activity of each component is always < 1 and therefore Eq. 3 becomes: [START_REF] Andersson | Thermo-Calc & DICTRA, computational tools for materials science[END_REF] Experimental values for activities are rare and they are not available for many industrial alloys. Therefore, Eq. 5 is rarely applied, despite the fact that it is very well known that non-ideality of metallic alloys can be large and can lower the stability of the oxide at the surface of the metal [START_REF] David | High temperature oxidation and corrosion of metals[END_REF]. However, the development of thermodynamic databases based on the CALPHAD approach has made possible and easy to obtain activity values for any alloy composition. The aim of this work is to show how to obtain Ellingham's diagrams for multi-metallic systems and to give a few examples of their usefulness. To the best of our knowledge, Ellingham diagram for specific alloys have never been reported in the literature. For this purpose, we have selected four different alloys that are wildly used in industries: the 304 stainless steel alloy, the Inconel 718 nickel-based alloy, the TA6V titanium-based alloy and the TiAl4822 alloy. As we will see, the Ellingham diagrams of these alloys present notable differences compared to that of pure metals. The oxidation behavior of these alloys will be hence discussed in comparison with these new calculations and the advantages and limitations of Ellingham diagrams will be discussed. When studying selective oxidation, it is important to know precisely what is the most stable oxide in contact with the alloy, before to apply kinetic models which will predict if the most stable can form an external scale at the alloy surface.

Method

All the Ellingham diagrams shown in this work have been obtained from thermodynamic calculations performed using the Thermo-Calc software [START_REF] Andersson | Thermo-Calc & DICTRA, computational tools for materials science[END_REF].

For the pure metal, the Ellingham diagram was directly obtained from Eq. 2, i.e. by subtracting the free Gibbs energy of reagents to the Gibbs free energy of the products of the oxidation reaction. The Gibbs free energy of each metal and oxide involved was calculated as a function of the temperature using Thermo-Calc.

For the alloys, Ellingham diagrams were obtained from Eq. 5. In this equation, °G is known from the previous calculation, since it is a state function and its value does not change with the reaction path. The activities of the various metals in the alloy a Me were obtained as a function of the temperature from Thermo-Calc calculations. For these computations, the compositions of the alloys were set as shown in the next sections and the a Me were computed as a function of the temperature T for 20 °C < T < 1500 °C, including all the phases predicted as stable in the calculations (see annex). The standard reference state was set for each element of the alloy, i.e. the stable phase of the element at each computed temperature. For instance, for aluminum, the reference state is the FCC (face-centered cubic) phase from room temperature till 660.3 °C (melting point of Al) and the liquid phase above this temperature.

The following thermodynamic databases were used for the calculations: the TCFE8 database for the 304 alloy, the TCNI9 database for Inconel 718 and the TCTI4 database for TA6V and TiAl4822.

Results

Ellingham's diagram for Me-O systems

Eleven different metals and their oxides were included in this work: Ni, Fe, Mo, Nb, Cr, Mn, Si, V, Ti, Al and Hf. The corresponding binary Me-O phase diagrams were computed for all the elements and they are reported in the annex. Some elements, as for instance aluminum, present rather simple phase diagrams, where only one oxide phase is stable. For example, in the Al-O system, only the α-Al 2 O 3 with corundum structure is stable, despite the existence of several polymorphic forms, that are often observed during the first stages of oxidation, being almost as stable as the main α phase [START_REF] Wriedt | The Al-O (Aluminum-Oxygen) system[END_REF]. Other metals, for example titanium, present complex phase diagrams, with several stable oxide phases, as shown in Fig. 1-(a).

In the case of titanium, one could wonder which oxides should be reported on the Ellingham diagram. Oxidation studies on pure titanium and titanium-based alloys for temperature T < 1000 °C consistently show the formation of a thick rutile TiO 2 scale [START_REF] Wallwork | Oxidation of titanium, zirconium, and hafnium[END_REF][START_REF] Rama Krishna | Thick rutile layer on titanium for tribological applications[END_REF] and sometimes the transient anatase TiO 2 on top of the scale. Rutile TiO 2 is the most stable titanium oxide under usual oxidizing atmospheres (oxygen, air etc.) and one could be interested to compare the stability of this phase with other metallic oxides. Nevertheless, Ellingham diagrams are thermodynamic tools and, according to thermodynamics, TiO 2 cannot be in equilibrium with metallic titanium. Indeed, as shown in Fig. 2, the metallic α-Ti phase can only be in equilibrium with Ti 3 O 2 , for T < 1193 K, and with fcc-TiO 1±x above this temperature. For this reason, the correct reactions to consider for titanium are: which are reported as solid lines, in Fig. 2-(b). All the other reactions, involving Ti and oxides with a higher oxygen molar fraction, are out-of-equilibrium reactions. These are reported as dashed lines in Fig. 2-(b), with the exception of the Magneli phases, that were excluded for sake of simplicity. As expected, these reactions show higher Gibbs free energy values than the oxidation reaction leading to Ti 3 O 2 and fcc-TiO. The oxides richer in oxygen are not directly formed from metallic titanium, but by subsequent reactions involving another oxide and O 2 (gas). For example, according to the computed phase diagram (ignoring the Magneli phases), for T < 732 K, the sequence of reactions is:

Similarly, at higher temperature, fcc-TiO 1±x and Ti 3 O 5 are involved, depending on the temperature range.

One could wonder why these oxides are not systematically observed between the rutile TiO 2 and the metallic titanium in oxidation studies. The reason is likely related to the kinetics of formation/dissolution of these phases and the diffusion of oxygen in the metallic phase, where the solubility of oxygen is very high (33 at%). A recent nano-scale characterization by transmission electron microscopy (TEM) and atom probe tomography (APT) of a Ti62426 alloy oxidized at 650 °C supports this hypothesis: α-TiO and Ti 3 O 2 were observed between the rutile scale and the metallic matrix, but only over lengths of the order of tens nanometers [START_REF] Dupressoire | The role of nitrogen in the oxidation behaviour of a Ti6242S alloy: a nanoscale investigation by atom probe tomography[END_REF].

In summary, for complex Me-O systems with several oxide phases, an Ellingham diagram should report the oxidation reaction involving the oxide with the lowest oxygen fraction, the one that can be in equilibrium with the metal. According to this, the Ellingham diagram of Fig. 2 has been computed. The result is consistent with previously reported Elligham diagrams [START_REF]ASM Handbook[END_REF].

Among the considered metals, hafnium forms the most stable oxide: the hafnia (HfO 2 ). For example, at 1200 °C, the Gibbs free energy change associated with the formation of HfO 2 is equal to -850 kJ/mol, which corresponds to an extremely low oxygen partial pressure of 7 ּ 10 -35 atm. Therefore, for p(O 2 ) < 7 ּ 10 -35 atm, metallic hafnium is stable, while for p(O 2 ) > 7 ּ 10 -35 atm it oxidizes to form hafnia.

Above the Hf/HfO 2 reaction, there are titanium oxides and alumina (Al 2 O 3 ), with very close values. Vanadium oxide follows. It is worth mentioning that the V-O binary system is rather complex (see annex), similarly to the Ti-O one. The oxides observed for vanadium oxidation in common environmental conditions are V 2 O 5 (that is liquid above 690 °C) and VO 2 [START_REF] Keller | The high-temperature oxidation behavior of vanadium-aluminum alloys[END_REF], but the oxide with the lowest oxygen fraction is V 4 O, that is located quite low in the Ellingham diagram. Silica follows and so on. The less stable oxide reported in Fig. 2 is the halite phase NiO. At 1200 °C, the Gibbs free energy change associated with the formation of NiO is equal to -219 kJ/mol, which corresponds to an oxygen partial pressure of 10 -13 atm, almost 22 orders of magnitude higher than the value found for HfO 2 .

Ellingham for industrial alloys

In order to compute the Ellingham diagram for a specific alloy, Eq. 5 can be rewritten as: [START_REF] Wriedt | The Al-O (Aluminum-Oxygen) system[END_REF] to emphasize that pO 2 is the unknown quantity to find. Indeed, °G is equal to the reaction for the pure metal, being a state function, and the activities are obtained from the thermodynamic calculations. It is interesting to remark that the new term RT ln(pO 2 ) is necessarily higher than °G, since all the activities are lower than one and the logarithm is hence a negative term. Therefore, for all the reactions, the curves computed for the metal in the alloys are systematically higher than those of the pure metals, as it is shown in the following. From a practical point of view, this means that the p(O 2 ) of equilibrium between metal/oxide are always higher for the alloys than for the pure metals. In the following, we give a few examples of Ellingham diagrams computed for common commercial structural alloys of different types.

Ti-48Al-2Nb-2Cr (48-2-2 or GE-48) alloy

Ti-Al alloys are a class of materials based mainly on the intermetallic γ-TiAl phase. They are generally slightly alloyed with elements such as Nb, Cr, Mo, Ta or W in order to improve the oxidation resistance and/or their mechanical properties [START_REF] Schmitz-Niederau | The oxidation behavior of several Ti-Al alloys at 900°C in air[END_REF]- [START_REF] Zhu | Microstructure refinement and oxidation behavior improvement in Mo modified TiAl4822 alloy prepared by vacuum sintering[END_REF]. The Ti-48Al-2 Nb-2Cr (48-2-2 or GE-48), whose nominal composition is reported in Table 1, is one of the most used Ti-Al alloys [START_REF] Duan | History and development of γ-TiAl alloys and the effect of alloying elements on their phase transformations[END_REF]. The GEnx™ and LEAP engines used TiAl (alloy 48-2-2) for their low pressure turbine blades [START_REF] Duan | History and development of γ-TiAl alloys and the effect of alloying elements on their phase transformations[END_REF].

The nominal composition of 48-2-2 alloy has been used as input for the thermodynamic computations. The stable phases as a function of temperature are reported in Fig. 3-(a) and the corresponding elemental activities are shown in Fig. 3-(b). According to the computations, the predominant phase (≈ 90 mol.%) of the alloy, from room temperature up to 1400 K is γ-TiAl, as expected. From the binary Ti-Al phase diagram, one could expect a low content of the α 2 -Ti 3 Al phase in equilibrium with γ-TiAl but, for T < 1200 K, an ordered BCC (body centered cubic) phase is predicted, with a maximum quantity of 11 mol.% and decreasing with the temperature. This phase, related to the hightemperature β-phase of titanium, is stabilized by the presence of Nb and Cr in the alloy, according to the calculations. However, it must be noticed that experiments generally show the presence of the α 2 -Ti 3 Al phase in equilibrium with γ-TiAl. This discrepancy between calculations and experimental data for the 48-2-2 alloy has been previously reported by Cakmak et al. [START_REF] Cakmak | A comprehensive study on the fabrication and characterization of Ti-48Al-2Cr-2Nb preforms manufactured using electron beam melting[END_REF], but no explanations were provided for its origin.

The application of the Ti-Al alloys is generally limited at 800 °C and most experimental data on their oxidation behavior do no exceed 900 °C. The Ellingham diagram for the 48-2-2 alloy has been computed in this temperature range and it is shown in Fig. 4, in the form of solid lines, while the values obtained for the pure metal oxides are reported as dashed lines for comparison.

One can notice that all the curves computed for the alloy have higher pO 2 values than those obtained for the pure metals, as expected according to Eq. 6. As for the pure metals, α-Al 2 O 3 and Ti 3 O 2 are still the two most stable oxides among those that can be formed, with almost identical oxygen partial pressures. Several experimental evidences have reported the formation of both α-Al 2 O 3 and rutile TiO 2 on the reactive surfaces of Ti-Al alloys [START_REF] Zhu | Microstructure refinement and oxidation behavior improvement in Mo modified TiAl4822 alloy prepared by vacuum sintering[END_REF], as shown in Fig. 5. Since TiO 2 is less stable than α-Al 2 O 3 on pure metals, previous works [START_REF] Zhu | Microstructure refinement and oxidation behavior improvement in Mo modified TiAl4822 alloy prepared by vacuum sintering[END_REF] have generally imputed the presence of rutile only to kinetic phenomena. However, as previously explained in Section 3.1, a thorough thermodynamic analysis has to take into account all the possible titanium oxides: the formation of Ti 3 O 2 competes with that of Al 2 O 3 and, once Ti 3 O 2 is formed, it can subsequently lead to TiO 2 by further oxidation. The massive formation of rutile generally observed on Ti-Al alloys is hence also in agreement with thermodynamics, as shown by the 48-2-2 Ellingham diagram.

Concerning the other oxides, Fig. 4 shows an inversion in the relative stability between Cr 2 O 3 and niobium oxides in the alloy with respect to the pure metals: Cr 2 O 3 is now more stable than niobium oxides. It is also very interesting and surprising to note that an inversion is also observed between the niobium monoxide and dioxide. According to the Nb-O phase diagram (see annex), NbO is the oxide with the lowest energy of reaction or, equivalently, the lowest partial oxygen pressures (for any temperature of practical interest, from 25 °C till thousands of °C). Therefore, as discussed in 3.1, from a thermodynamic point of view, metallic niobium cannot directly reacts to form NbO 2 , the NbO oxide must be formed in the reaction process. On the contrary, the Ellingham diagram of 48-2-2 alloy indicates that, for the alloy, NbO 2 is more stable than NbO. Mathematically, this inversion is due to the 2a/b exponent of the metal activity in Eq. 6. More intuitively, for the oxidation of pure metals, Ellingham diagram are computed considering the oxygen as the limiting reagent, while metals are available in infinite quantities (Eq. 1). For an alloy, this is not true anymore: the metallic species present in low quantities have very low activities and this can lead to stabilize oxides with lower metallic molar fraction, as for the case of NbO 2 over NbO. This phenomenon can play a key role in the oxidation of alloy but, to the best of our knowledge, nobody has ever highlighted it before from theoretical considerations nor clearly observed it experimentally.

Ti-6Al-4V

The Ti-6Al-4 V alloy, generally named TA6V, is the most used titanium alloy. Its applications are limited to temperatures lower than 400 °C [START_REF] Boyer | Titanium for aerospace: rationale and applications[END_REF] because of creep resistance and oxygen embrittlement. Most of the experimental studies on its oxidation behavior do not exceed 700 °C [START_REF] Frangini | Various aspects of the air oxidation behaviour of a Ti6Al4V alloy at temperatures in the range 600-700 °C[END_REF][START_REF] Vaché | Modeling the oxidation kinetics of titanium alloys: review, method and application to Ti-64 and Ti-6242s alloys[END_REF].

The nominal composition (6 wt% Al, 4 wt% V, Ti balance) has been used to perform the thermodynamic computations using the TCTI3 database from Thermo-Calc (phase fractions and activities are reported in the annex). The Ellingham diagram has been drawn for the temperature range of interest, i.e. up to 700 °C, and it is reported in Fig. 6.

The oxidation reaction energies have been computed for all the binary oxides (see Ti-O and V-O phase diagrams in annex), but only the lowest values, corresponding to the most stable oxides, are reported, for sake of simplicity. From the lowest to the highest RTln(pO 2 ) values, Ti 3 O 2 , α-Al 2 O 3 and VO 4 can be observed. Therefore, contrary to the previous Ti-Al intermetallic alloy, Fig. 6 shows that the formation of tita-Fig. 6. Ellingham diagram for the TA6V alloy (solid lines) compared to the oxides in equilibrium with the pure metals (dashed lines). nium oxides is clearly favored in respect to alumina for the TA6V alloy. Indeed, the low concentration of aluminum implies a low Al activity, which causes a considerable increase of the pO 2 of equilibrium between the alumina and the alloy in comparison to the pure metal. On the contrary, the shift observed for the Ti 3 O 2 oxide is minimum.

From the Ellingham diagram, one can expect the formation of Ti 3 O 2 , eventually followed by other rapid oxidation reactions leading to anatase and rutile TiO 2 . However, experimental studies have shown a more complex oxidation behavior for the TA6V alloy [START_REF] Coddet | Essai d′interprétation du mécanisme d′oxydation de l′alliage de titane TA6V4 par l′oxygène sec entre 700°C et 1000°C[END_REF][START_REF] Du | Air oxidation behavior of Ti-6al-4v alloy between 650-Degrees-C and 850-Degrees-C[END_REF]. Indeed, oxidized TA6V alloys exhibit complex scales formed by an alternance of rutile TiO 2 and Al 2 O 3 layers, with the first being always at the oxide/ substrate interface and the second at the oxide/gas interface [START_REF] Coddet | Essai d′interprétation du mécanisme d′oxydation de l′alliage de titane TA6V4 par l′oxygène sec entre 700°C et 1000°C[END_REF][START_REF] Du | Air oxidation behavior of Ti-6al-4v alloy between 650-Degrees-C and 850-Degrees-C[END_REF]. The micrography of a TA6V alloy oxidized under air, at 600 °C, for 500 h is shown in Fig. 7 [START_REF] Casadebaigt | Etude de l′oxydation à chaud d′échantillons en alliage TA6V élaborés par fabrication additive. Conséquences sur les propriétés mécaniques[END_REF]. The delamination of the oxide scale occurs certainly for the same reason than for pure titanium oxidation. It is generally attributed to the formation of pores and to the stress build-up in the oxide scale due to the large Pilling and Bedworth ratio of Ti [START_REF] Stringer | The oxidation of titanium in oxygen at high temperatures[END_REF]. The special arrangement of alumina and rutile layers can be attributed to diffusion. In the early stage of oxidation, a TiO 2 layer is formed, in agreement with thermodynamics. This oxide scale grows mainly due to inward diffusion of oxygen anions. Nevertheless, both aluminum and titanium cations diffuse outward through the rutile, but the first diffuses much faster than the second [START_REF] Ikeda | Nonequilibrium surface segregation in aluminium-doped TiO2 under an oxidizing potential: effects on redox colorboundary migration[END_REF]. A kinetic demixing phenomenon [START_REF] Schmalzried | Multicomponent oxides in oxygen potential gradients[END_REF] leading to a segregation of aluminum occurs hence at the external interface of the oxide scale, where the precipitation of Al 2 O 3 occurs [START_REF] Du | Air oxidation behavior of Ti-6al-4v alloy between 650-Degrees-C and 850-Degrees-C[END_REF] . Once the oxide scale is detached, a new oxide scale form underneath with the same mechanism. This explanation was reported earlier for a single layer formation during the oxidation of the TiAl [START_REF] Becker | Mechanism of isothermal oxidation of the intel-metallic TiAl and of TiAl alloys[END_REF], Ti 3 Al intermetallic [START_REF] Rakowski | Proceedings of the Second International Conference on the Microscopy of Oxidation Held at Selwyn College[END_REF] and later for TA6V for a multi-layer oxide scale [START_REF] Du | Air oxidation behavior of Ti-6al-4v alloy between 650-Degrees-C and 850-Degrees-C[END_REF][START_REF] Casadebaigt | High temperature oxidation and embrittlement at 500-600 °C of Ti-6Al-4V alloy fabricated by Laser and Electron Beam Melting[END_REF].

Inconel 718

The Inconel 718 (IN718) is the most used nickel-based superalloy that finds applications in numerous domains (aeronautics, nuclear and other energy industries) thanks to good corrosion resistance and excellent mechanical properties. These latter are obtained thanks to the precipitation of γ'-Ni 3 (Al,Ti) and γ''-Ni 3 Nb precipitates in the austenitic matrix, induced by the several alloying elements present in the IN718. The typical composition, which was also adopted for the calculations, is reported in Table 2. The good oxidation resistance of the alloy is mainly related to the high chromium content, which leads to the formation of a scale mostly composed of chromia (Cr 2 O 3 ) [START_REF] Giggins | Oxidation of Ni-Cr-Al alloys between 1000°and 1200[END_REF][START_REF] Greene | Oxidation of inconel 718 in air at high temperatures[END_REF].

The applications of the IN718 are generally limited to 650 °C, close to the γ' solvus, above which a degradation of the mechanical properties occurs [START_REF] Schafrik | Application of Alloy 718 in GE Aircraft Engines: Past, Present and Next Five Years[END_REF]. However, the experimental studies of the oxidation behavior of this alloy have been pushed to higher temperatures, which can be useful for alloy fabrication and heat treatments. Therefore, the Ellingham diagram of the alloy has been computed from room temperature up to 1200 °C and it is shown in Fig. 8, where solid lines represent the results for the alloy and dashed lines represent the oxides in equilibrium with the pure metals.

From lower to higher oxygen partial pressures, the following binary oxides can be predicted for IN718: Al 2 O 3 , titanium oxides, Cr 2 O 3 , NbO 2 , Fig. 7. A TA6V alloy after oxidation under air, at 600 °C, for 500 h [START_REF] Casadebaigt | Etude de l′oxydation à chaud d′échantillons en alliage TA6V élaborés par fabrication additive. Conséquences sur les propriétés mécaniques[END_REF]. The IN718 Ellingham diagram clearly shows the interest to perform such computations for specific alloys, since several differences can be remarked in comparison to the curves obtained for oxides in equilibrium with the pure metals. Indeed, the equilibrium of oxides formed by elements present in low concentration in the alloy are considerably shifted to higher oxygen partial pressures: it is the case for Al 2 O 3 , titanium oxides, SiO 2 and niobium oxides. As logical, these shifts are small for the oxides formed by elements present in high concentration, as for example Cr 2 O 3 , Fe oxides and NiO. In some cases, this can lead to an inversion of stability among oxides. In the IN718, this occurs for SiO 2 and NbO, that are less stable than chromia. Similarly, MoO 2 is in equilibrium with the alloy at higher oxygen pressures than iron oxides, contrary to the corresponding pure metals. Finally, similarly to the case of the TiAl 48-2-2 alloy, some inversions can be remarked in the stabilities of oxides from the same binary systems. The most stable niobium oxide for the IN718 alloy is NbO 2 and not NbO, while the titanium oxide found at lowest pO 2 is Ti 2 O 3 instead of Ti 3 O 2 /TiO.

The Ellingham diagram computed for the IN718 alloy is consistent with several experimental evidences of the oxidation behavior of this alloy, as for instance a recent work which focused on intergranular oxidation at 850 °C [START_REF] Sanviemvongsak | Intergranular oxidation of Ni-base alloy 718 with a focus on additive manufacturing[END_REF][START_REF] Sanviemvongsak | High temperature oxidation of IN 718 manufactured by laser beam melting and electron beam melting: effect of surface topography[END_REF]. An electron microscopy image and its relative EDS cartography from [START_REF] Sanviemvongsak | High temperature oxidation of IN 718 manufactured by laser beam melting and electron beam melting: effect of surface topography[END_REF] are shown in Fig. 9. Below a scale mainly formed by Cr 2 O 3 , this study showed the presence of Al 2 O 3, deep in the grain boundaries, and titanium oxides, still in the grain boundaries, but less deep than alumina [START_REF] Sanviemvongsak | Intergranular oxidation of Ni-base alloy 718 with a focus on additive manufacturing[END_REF]. Therefore, the order of observation of these oxides, from the substrate to the scale (or equivalently from the lowest to the highest oxygen partial pressures), respects their order on the Ellingham diagram. In addition, the same study reported the presence of small particles within the chromia scale, whose measured composition was Ti 0.43 Nb 0.57 O 2 . The presence of a Nb-rich dioxide in contact with Cr 2 O 3 is also consistent with the Ellingham diagram, where the oxide observed above Cr 2 O 3 at 850 °C is NbO 2 .

304 alloy

The 304 stainless steel alloy is a widely used chromia forming alloy applied in several domains, including construction, oil, household appliance, food, chemical and nuclear industries. The nominal composition used for the calculations is reported in Table 3, where for simplicity the carbon content has been excluded.

The 304 alloy is generally used at room temperature or in operating conditions comprised between 300 and 850 °C. Nevertheless, oxidation studies have been conducted at higher temperature, up to 1000 °C [START_REF] Cai | Superior high-temperature oxidation resistance of nanocrystalline 304 austenitic stainless steel containing a small amount of Si[END_REF] and hence the Ellingham diagram has been computed for a wider temperature range, as shown in Fig. 10.

According to the Ellingham diagram of the 304 alloy, from lower to higher oxygen partial pressures, SiO 2 , Cr 2 O 3 , MnO, iron oxides and NiO can be expected. Some big shifts toward higher oxygen partial pressures can be observed for SiO 2 and MnO, as expected considering the low concentrations of these elements and hence their activities in the alloy. For MnO, the shift brings to an inversion of stability with Cr 2 O 3 , while the computed values for SiO 2 are really close to Cr 2 O 3 .

The Ellingham diagram well agrees with the experimental evidences [START_REF] Cai | Superior high-temperature oxidation resistance of nanocrystalline 304 austenitic stainless steel containing a small amount of Si[END_REF][START_REF] Wang | The high-temperature oxidation of bulk nanocrystalline 304 stainless steel in air[END_REF]: SiO 2 is generally observed as an intergranular oxide or as a discontinuous, thin layer between the metal substrate and a Cr 2 O 3 scale. In addition, some manganese-rich oxides are also observed within or on top of the Cr 2 O 3 scale [START_REF] Cai | Superior high-temperature oxidation resistance of nanocrystalline 304 austenitic stainless steel containing a small amount of Si[END_REF][START_REF] Wang | The high-temperature oxidation of bulk nanocrystalline 304 stainless steel in air[END_REF]. These features can be observed in the micrograph from [START_REF] Col | Étude de l′oxydation catastrophique de l′acier 304L: mécanismes et effet d′une pré-déformation[END_REF] reported in Fig. 11.

The non-formation of a continuous SiO 2 scale can be easily understood by computing a second SiO 2 Ellingham curve for a 304 alloy with a slightly lower concentration, as the one shown in Fig. 12. Here, the values computed for a 0.2 wt% concentration of Si have been added (SiO 2 LC) to the Ellingham diagram of the nominal 304 composition. This new curve is above the chromia one for temperatures above 460 °C. Therefore, the concentration (or equivalently the activity) of silicon plays an important role in the 304 alloy, since it determines the relative stability between SiO 2 and Cr 2 O 3 .

The inversions for SiO 2 and MnO may have important consequences for the resistance to high temperature oxidation of steel 304 and of similar chromia-forming alloys. Indeed, if SiO 2 is more stable than chromia, it may form a continuous layer at the chromia/alloy interface [START_REF] Nguyen | Effects of silicon on high temperature corrosion of Fe-Cr and Fe-Cr-Ni alloys in carbon dioxide[END_REF]- [START_REF] Ahmad | STEM analysis of the transient oxidation of a Ni-20Cr alloy at high temperature[END_REF]. Such a scale can act as a diffusion barrier and improve the high temperature oxidation resistance [START_REF] Mikkelsen | The effect of silicon addition on the high temperature oxidation of a Fe-Cr alloy[END_REF]. Nevertheless, it can also decrease the oxide/scale adherence [START_REF] Perez | Oxydation à haute température d′alliages modèles Ni-25Cr, Influence des éléments mineurs Mn et Si sur les mécanismes de croissance des oxydes protecteurs[END_REF]. Moreover, if silica and MnO are more stable than chromia, internal and intergranular oxidation of Si and Mn can occur. If MnO is less stable than chromia, it can form an external scale on top of chromia, and react with chromia to form the spinel MnCr 2 O 4 [START_REF] Nguyen | Effects of silicon on high temperature corrosion of Fe-Cr and Fe-Cr-Ni alloys in carbon dioxide[END_REF][START_REF] Liu | Effects of alloyed Mn on oxidation behaviour of a Co-Ni-Cr-Fe alloy between 1050 and 1250°C[END_REF]- [START_REF] Douglass | Monceau Corrosion Science xxx (xxxx) 111113 mechanism of Ni-20 Cr[END_REF] which is known to protect chromia from volatilization in H 2 O-rich atmospheres [START_REF] Hänsel | Hydrogen Uptake and Hydrogen Profiles in Chromia Scales Formed on Ni25Cr(Mn) in Low pO 2 Test Gases at 1000[END_REF][START_REF] Holcomb | The effect of manganese additions on the reactive evaporation of chromium in Ni-Cr alloys[END_REF]. However, if the manganese content is too high and MnO is more stable than Cr 2 O 3 , the spinel MnCr 2 O 4 can form at the oxide/metal interface. This can decrease the oxide/scale adherence and indeed spallation has been observed in chromia-forming alloys containing more than 2% of Mn [START_REF] Stott | The influence of manganese on the hightemperature oxidation of iron-chromium alloys[END_REF][START_REF] Zurek | Growth and adherence of chromia based surface scales on Ni-base alloys in high-and low-pO(2) gases[END_REF].

The high sensibility of oxidation resistance to the impurity levels is therefore highlighted by this study.

Discussion

Ellingham diagrams are useful tools to evaluate the relative thermodynamic stability among oxides and to provide first indications on the oxidation behavior of metals and alloys. This work shows that modern thermodynamic computations can help improving the potentiality of Ellingham diagrams and avoid some misuses or misinterpretations. More specifically, it is possible to compute Ellingham diagrams for complex multi-element and multi-phase metallic systems, considering the activities of all the present elements. This has been done herein for four different alloys, each of them representative of an important class of materials (Ti-Al, Ti-based, Ni-based alloys and stainless steels). The re- 9. EDS cartography analysis of an IN718 sample oxidized for 1000 h at 850 °C in air [START_REF] Sanviemvongsak | High temperature oxidation of IN 718 manufactured by laser beam melting and electron beam melting: effect of surface topography[END_REF].

C O R R E C T E D P R O O F Fig.

Table 3

Composition of the 304-alloy used in this work for the calculations. sults showed that, in several cases, an inversion in the relative stability of oxides can occur in alloys in comparison to the oxides in equilibrium with the pure metals. For example, SiO 2 is generally considered an oxide more stable than Cr 2 O 3 , on the basis of the "classical" Ellingham diagram, but this is not true for the IN718 and the 304 alloys, which are rich in chromium and poor in silicon. These more accurate Ellingham diagrams can hence help understanding and predicting the oxidation behavior of complex alloys and also the design of new ones. Another interesting point emerged from these thermodynamic computations is the possibility to have an inversion of the relative stabilities among binary oxides belonging to the same Me-O system within an alloy. For example, the computations for IN718 shows that higher oxygen partial pressures are necessary to form NbO rather than the NbO 2 , while it is clearly the opposite for metallic niobium. To the best of our knowledge, even if directly derivable from Eq. 5, this point has never been highlighted before but it could be especially useful for interpretation of intergranular oxidation phenomena, where low oxygen partial pressures could induce these inversions.

Despite the usefulness of Ellingham diagrams for alloys, some important limitations still exist and need to be considered if one wants to use these diagrams to predict the composition of the oxide scale. First of all, from the thermodynamic point of view, there are still approximations and limitations: the solubility of the elements in all the oxides phases is not taken into account and an a priori knowledge is necessary if one wants to include ternary oxides in the calculations. Some alternative thermodynamic calculations can be performed in order to overcome these limitations, as previously shown in the literature [START_REF] Sanviemvongsak | Intergranular oxidation of Ni-base alloy 718 with a focus on additive manufacturing[END_REF]. For example, Fig. 13 reports the result of a thermodynamic computation for 10, one can notice that the same order of appearance is predicted for the binary oxides (SiO 2 , Cr 2 O 3 , Mn and Fe oxides). However, Fig. 13 describes a more complex behavior, with the appearance of several ternary or multielement oxides: a spinel phase containing Fe, Ni, Mn and Cr, a Fe-Mn silicate and a corundum phase dissolving both Cr and Fe. It is clear that these calculations give more information in comparison to an Ellingham diagram. However, these case-by-case computations require more time, knowledge and continuous access to thermodynamic databases, while Ellingham diagrams provide more immediate information.

The second fundamental limit of Ellingham diagrams (or any other thermodynamic calculations) is the lack of any kinetic knowledge which also plays a key role in oxidation and hence needs to be considered for a thorough analysis. It is well known that the thermodynamic condition, i.e. the protective oxide has to be the most stable one, is necessary but not sufficient to form the protective oxide scale [START_REF] Wagner | Theoretical analysis of the diffusion processes determining the oxidation rate of alloys[END_REF] The kinetics aspects of the transition between internal and external oxidation have to be considered. This implies the calculation of the outward flux of the protective alloying element and of the inward flux of oxygen in the alloy [START_REF] Rapp | The transition from internal to external oxidation and the formation of interruption bands in silver-indium alloys[END_REF]. Another related aspect needs to be considered. When an external chromia or alumina scale is formed, this leads often to a depletion of the alloying element below the oxide scale. Therefore, the composition of the alloy in contact with the oxide is different from the bulk composition. If the composition below the oxide scale is known, it is possible to perform a new thermodynamic calculation with this subscale alloy composition, as done in [START_REF] Sanviemvongsak | Intergranular oxidation of Ni-base alloy 718 with a focus on additive manufacturing[END_REF]. For a specific alloy, an Ellingham diagram could also be computed including impurities (O, C) that can slightly change the phase equilibria or for a range of composition, instead of simply the nominal one, in order to have some insight on the evolution of the oxidation behavior. This could be done varying the protective element content (i.e. decreasing Cr and Al content in resp. chromia-and alumina-forming alloys). It could be done also with minor elements, whose contents may vary a lot (Si, Mn, Ti, Al in chromiaforming FeNiCr alloys for example) leading to large variations of their equilibrium oxygen partial pressures.

Conclusions

Ellingham diagrams are useful thermodynamic tools that have been used for the decades to evaluate the relative stability among oxides. This work aims to show how nowadays advanced thermodynamic databases can be used in order to compute Ellingham diagrams for multielement alloys. As herein shown, these often exhibit remarkable differences in comparison to the Ellingham diagram for pure metals, since the activities of the various elements in a complex alloy can be very far from unity. Chemical activities are very difficult to obtain experimentally and the lack of these data explains why Ellingham diagrams were not previously computed for individual alloys. Nowadays, the development of thermodynamic databases allows calculating the chemical activities and hence Ellingham diagrams for specific alloys in a much easier way. This has been shown in this work for four common alloys: TiAl 48-2-2, TA6V, 304 and IN718. Several inversions in the relative stability among oxides in comparison to the "classical" Ellingham diagrams

Fig. 1 .

 1 Fig. 1. (a) Ti-O phase diagram computed using the TCTI1 database from Thermo-Calc. (b) Ellingham diagram comparing the stability of various titanium oxides. Solid lines: reactions corresponding to thermodynamic equilibrium; Dashed lines: out of equilibrium reactions.

Fig. 2 .

 2 Fig. 2. Ellingham diagram for several metal/oxide reactions. Dashed lines represent the oxidation reaction of metastable oxides.

Fig. 3 .

 3 Fig. 3. (a) Phase fractions at equilibrium for the 48-2-2 alloy as a function of the temperature. (b) Ti, Al, Cr and Nb activities in 48-2-2. (Thermodynamic calculations performed using the TCTI4 database from Thermo-Calc).

Fig. 4 .

 4 Fig. 4. Ellingham diagram for the TiAl 48-2-2 alloy (solid lines) compared to oxides in equilibrium with the pure metals (dashed lines).

Fig. 5 .

 5 Fig. 5. TiAl 48-2-2 oxide layer after 100 h at 900 °C based on [17].

C

  O R R E C T E D P R O O F E. Epifano and D. Monceau Corrosion Science xxx (xxxx) 111113

Fig. 8 .

 8 Fig. 8. Ellingham diagram for the IN718 alloy (solid and point lines) compared to the oxides in equilibrium with the pure metals (dashed lines). SiO 2 , iron oxides, MoO 2 and NiO. In complex, multi-element alloys such as IN718, ternary oxides are also likely to form. As a non-exhaustive example, the Ellingham curve of the spinel FeCr 2 O 4 oxide was also computed and it is found between Cr 2 O 3 and Fe 3 O 4 /FeO on the diagram.The IN718 Ellingham diagram clearly shows the interest to perform such computations for specific alloys, since several differences can be remarked in comparison to the curves obtained for oxides in equilibrium with the pure metals. Indeed, the equilibrium of oxides formed by elements present in low concentration in the alloy are considerably shifted to higher oxygen partial pressures: it is the case for Al 2 O 3 , titanium oxides, SiO 2 and niobium oxides. As logical, these shifts are small for the oxides formed by elements present in high concentration, as for example Cr 2 O 3 , Fe oxides and NiO. In some cases, this can lead to an inversion of stability among oxides. In the IN718, this occurs for SiO 2 and NbO, that are less stable than chromia. Similarly, MoO 2 is in equilibrium with the alloy at higher oxygen pressures than iron oxides, contrary to the corresponding pure metals. Finally, similarly to the case of the TiAl 48-2-2 alloy, some inversions can be remarked in the stabilities of oxides from the same binary systems. The most stable niobium oxide for the IN718 alloy is NbO 2 and not NbO, while the titanium oxide found at lowest pO 2 is Ti 2 O 3 instead of Ti 3 O 2 /TiO.The Ellingham diagram computed for the IN718 alloy is consistent with several experimental evidences of the oxidation behavior of this alloy, as for instance a recent work which focused on intergranular oxidation at 850 °C[START_REF] Sanviemvongsak | Intergranular oxidation of Ni-base alloy 718 with a focus on additive manufacturing[END_REF][START_REF] Sanviemvongsak | High temperature oxidation of IN 718 manufactured by laser beam melting and electron beam melting: effect of surface topography[END_REF]. An electron microscopy image and its relative EDS cartography from[START_REF] Sanviemvongsak | High temperature oxidation of IN 718 manufactured by laser beam melting and electron beam melting: effect of surface topography[END_REF] are shown in Fig.9. Below a scale mainly formed by Cr 2 O 3 , this study showed the presence of Al 2 O 3, deep in the grain boundaries, and titanium oxides, still in the grain boundaries, but less deep than alumina[START_REF] Sanviemvongsak | Intergranular oxidation of Ni-base alloy 718 with a focus on additive manufacturing[END_REF]. Therefore, the order of observation of these oxides, from the substrate to the scale (or equivalently from the lowest to the highest oxygen partial pressures), respects their order on the Ellingham diagram. In addition, the same study reported the presence of small particles within the chromia scale, whose measured composition was Ti 0.43 Nb 0.57 O 2 . The presence of a Nb-rich dioxide in contact with Cr 2 O 3 is also consistent with the Ellingham diagram, where the oxide observed above Cr 2 O 3 at 850 °C is NbO 2 .

Fig. 10 .

 10 Fig. 10. Ellingham diagram for the 304 stainless steel (solid lines) compared to the oxides formed on the pure metals (dashed lines).

Fig. 11 .

 11 Fig. 11. 304 steel oxidized for 110 h at 850 °C under O2. Extracted from [37].

Fig. 12 .

 12 Fig. 12. Relative stability of Cr2O3 and SiO2 oxides for 304 alloy. LC = low concentration of Si, equal to 0.2 wt% instead of 0.75 wt%.

Fig. 13 .

 13 Fig. 13. Variation of the phase fraction for the 304 stainless steel composition, with increasing pO 2 , at 850 °C.

Table 1

 1 Nominal composition of the TiAl4822 alloy.

		Ti	Al	Nb	Cr
	Composition (at%)	Balance	48	2	2

Table 2

 2 Composition of the IN718 alloy used for the thermodynamic computations.

		Ni	Fe	Cr	Nb	Mo	Si	Ti	Al
	Composition (wt%)	Balance	24.6	17	4.7	2.8	0.35	0.65	0.2

Data availability

No data was used for the research described in the article.

have been highlighted, some of which can shade lights on the oxidation behavior of these alloys.

Uncited reference [START_REF] Małecka | Transformation and precipitation processes in a metal substrate of oxidized tial-based alloys[END_REF][START_REF] Li | Effects of silicon on the oxidation behavior of ni-base chromia-forming alloys[END_REF][START_REF] Mathieu | On the pre-oxidation treatments of four commercial Ni-based superalloys in air and in Ar-H2O at 950 °C[END_REF][START_REF] Ackermann | Oxidation behaviour of Ni-Cr based alloy containing Si during high temperature application in an oil burner[END_REF].

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix A. Supporting information

Supplementary data associated with this article can be found in the online version at doi:10.1016/j.corsci.2023.111113.


William N Havard

Loann Peurey

Kasia Hitczenko

Alejandrina Cristia

Kasia Hitczenco

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Acknowledgments This work was supported by the J. S. McDonnell Foundation Understanding Human Cognition Scholar Award; European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (ExELang, Grant agreement No. 101001095), Agence Nationale de la Recherche (ANR-16-DATA-0004 ACLEW). This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number OCI-1053575. Specifically, it used the Bridges system, which is supported by NSF award number ACI-1445606, at the Pittsburgh Supercomputing Center (PSC). This publication uses data generated via the Zooniverse.org platform, development of which is funded by generous support, including a Global Impact Award from Google, and by a grant from the Alfred P. Sloan Foundation. The funders had no impact on this study.