Natacha Stolowy

Aurelie Calabrese
email: aurelie.calabrese@univ-amu.fr

Lauren Sauvan

Carlos Aguilar

Thomas François

Núria Gala

Frédéric Matonti

Eric Castet

Aurélie Calabrèse

The influence of word frequency on word reading speed when individuals with macular diseases read text

Keywords: central field loss, macular degeneration, reading, psycholinguistics, word frequency, lexical simplification

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Age-related Macular Degeneration (AMD) is one of the main causes of low vision in industrialized countries and affects about 20% of the population over 65 years [START_REF] Klein | The prevalence of age-related maculopathy by geographic region and ethnicity[END_REF][START_REF] Wong | Global prevalence of age-related macular degeneration and disease burden projection for 2020 and 2040: a systematic review and meta-analysis[END_REF]. Like other macular diseases such as Stargardt's disease, cone dystrophy or myopic maculopathy, AMD causes a central field loss (CFL) that severely impairs reading, an activity that is usually the main complaint of these patients [START_REF] Brown | Characterizing functional complaints in patients seeking outpatient lowvision services in the United States[END_REF][START_REF] Elliott | Demographic characteristics of the vision-disabled elderly[END_REF][START_REF] Mangione | Identifying the content area for the 51-item National Eye Institute Visual Function Questionnaire: results from focus groups with visually impaired persons[END_REF][START_REF] Wolffsohn | The changing face of the visually impaired: the Kooyong low vision clinic's past, present, and future[END_REF].

Currently-available treatments for AMD cannot always restore central vision and, in that case, patients' support is mainly based on low-vision rehabilitation and the use of visual aids [START_REF] Meyniel | Revisiting Vision Rehabilitation[END_REF]. However, the use of these visual aids, whose principles are essentially based on magnification, is not sufficient to restore reading performance in patients with AMD [START_REF] Chung | Psychophysics of reading XVIII -The effect of print size on reading speed in normal peripheral vision[END_REF][START_REF] Legge | Psychophysics of reading in normal and low vision[END_REF][START_REF] Li | Low vision and methods of rehabilitation: a comparison between the past and present[END_REF]. This is why many efforts have been made to improve reading speed in peripheral vision by manipulating other visual factors such as: [START_REF]d'avoir tenté de les débaucher (corrompre) L'homme est bon par nature, la société ne fait que le dégrader (dépraver) Avec son passé, cela n'a pas été très difficile de le dépraver (dégrader) Cette association compte plus de curieux que de disciples (adhérents) Il faudrait encore leur demander d'appeler les adhérents[END_REF] text presentation mode, e.g. static, scrolling or flashing one word at a time [START_REF] Bowers | Preferred retinal locus and reading rate with four dynamic text presentation formats[END_REF][START_REF] Fine | Benefits of rapid serial visual presentation (RSVP) over scrolled text vary with letter size[END_REF][START_REF] Rubin | Low vision reading with sequential word presentation[END_REF]; (2) characteristics of the text, e.g. font [START_REF] Bernard | A New Font, Specifically Designed for Peripheral Vision, Improves Peripheral Letter and Word Recognition, but Not Eye-Mediated Reading Performance[END_REF]Bragg et al., 2017;[START_REF] Mansfield | Psychophysics of reading XV -Font effects in normal and low vision[END_REF], text orientation [START_REF] Calabrèse | Does Vertical Reading Help People with Macular Degeneration: An Exploratory Study[END_REF], letter, word or interline spacing [START_REF] Bernard | Page mode reading with simulated scotomas: a modest effect of interline spacing on reading speed[END_REF][START_REF] Blackmore-Wright | Enhanced text spacing improves reading performance in individuals with macular disease[END_REF][START_REF] Calabrèse | Small effect of interline spacing on maximal reading speed in low-vision patients with central field loss irrespective of scotoma size[END_REF][START_REF] Chung | The effect of letter spacing on reading speed in central and peripheral vision[END_REF][START_REF] Chung | Reading speed does not benefit from increased line spacing in AMD patients[END_REF][START_REF] Yu | Effect of letter spacing on visual span and reading speed[END_REF]; (3) perceptual learning, i.e. the performance improvement that follows practice of a sensory task [START_REF] Chung | Letter-recognition and reading speed in peripheral vision benefit from perceptual learning[END_REF][START_REF] Chung | Improving reading speed for people with central vision loss through perceptual learning[END_REF]. Mixed results have been obtained in terms of the clinical efficiency of these manipulations.

In this paper, we explore another consequence for CFL patients of the necessity of using their eccentric vision. Words processed in the periphery provide a very ambiguous low-level visual input, due for instance to confusions between letters [START_REF] Pelli | Crowding: a cortical constraint on object recognition[END_REF], which should induce a high influence of lexical inference processes (Legge et al., 1997;Legge et al., 1997). Basically, when faced with an ambiguous percept (say a three-letter string starting with "c" and ending with "r"), a patient would make a guess about the most likely word having induced this percept (probably "car"). Thus, word frequency is expected to influence the speed with which a guess would be made. This is consistent with a recent investigation of eye movements in AMD patients during short-sentences reading that reported clusters of fixations that significantly reduce reading speed (Calabrèse et al., 2016). The clusters in some parts of a sentence indicated that specific words required dozens of fixations before successful identification, while the other ones were read with fewer fixations. The presence/absence of clusters was a significant predictor of reading speed, independently of the perceptual span size [START_REF] Calabrèse | Eye movements and reading speed in macular disease: the shrinking perceptual span hypothesis requires and is supported by a mediation analysis[END_REF]. Interestingly, these clusters were more likely to occur in sentences containing a low-frequency word. This new finding suggested that low-frequency words might cause a local slowing of reading speed (i.e. a dense cluster of fixations) due to more difficult lexical access. However, the authors were not able to quantify the strength of the word frequency effect because their protocol did not allow for a clear determination of which word was being processed when a cluster occurred.

The main goal of the present work is to investigate this question further and study the direct effect of word frequency on word reading speed in individuals with bilateral macular disease.

For normally sighted readers, factors such as word familiarity or word frequency are known to affect reading performance at different perceptual levels. To the best of our knowledge, the quantitative relationship between word frequency and natural reading performance with CFL has not been investigated yet. This contrasts with the wealth of the literature for normally sighted observers [START_REF] Ferreira | Eye movements in reading and information processing: Keith Rayner's 40 year legacy[END_REF][START_REF] Kliegl | Tracking the mind during reading: the influence of past, present, and future words on fixation durations[END_REF][START_REF] Rayner | Eye movements in reading and information processing: 20 years of research[END_REF]. The main goal of the present study is thus to test whether word frequency affects word reading speed in individuals with CFL: it is expected that high-frequency words would lead to shorter reading times than low-frequency words. Two distinct analyses were performed to test the effect of word frequency on reading performance: Analysis 1, using absolute values of frequency (in occurrences/million) and Analysis 2, using frequency categories (low vs. high).

We used a reading presentation mode, known as "self-paced reading", that has never been used before in low-vision reading studies. Self-paced reading has been extensively used in psycholinguistic studies as a proxy for natural reading [START_REF] Aaronson | Performance theories for sentence coding: Some quantitative evidence[END_REF][START_REF] Just | Paradigms and processes in reading comprehension[END_REF][START_REF] Mitchell | The effects of context and content on immediate processing in reading[END_REF]. In this context, natural reading, also called eyemediated reading, refers to the natural process of reading that is used in daily life and requires eye movements; this has to be contrasted with paradigms such as RSVP (Rapid Serial Visual Presentation) requiring few or no eye movements since all words are presented one at a time at the same position [START_REF] Rubin | Reading without saccadic eye movements[END_REF]. With the self-paced reading paradigm, sentences appear as a whole, with all words masked and revealed only one at a time.

Compared to other sequential word presentation modes, the self-paced reading paradigm reduces the amount of meaningful text to a single word, while maintaining the overall spatial layout of the sentence. Given that only one word is available to the reader at any given time, this paradigm makes it possible to identify in real time which word is being read, and thus to accurately measure the actual time spent reading each word.

Our secondary goal was to introduce the technique of text simplification and its potential benefit to facilitate reading performance with CFL. Text simplification is the process of reducing the complexity of a text by modifying the vocabulary or the syntax, while preserving its informational content [START_REF] Shardlow | A survey of automated text simplification[END_REF]. The purpose of this technique is to transform a text into a different one, which will convey the same message, while being easier to read and understand by a broader audience [START_REF] Saggion | Automatic Text Simplification[END_REF]. Text simplification techniques have been applied to dyslexic and aphasic patients and have been shown to improve reading performance [START_REF] Siddharthan | A survey of research on text simplification[END_REF]. However, text simplification has not yet been considered for CFL patients. The present work constitutes a first step towards the investigation of the simplification significance for this specific population. By testing the effect of word frequency on individuals with CFL, we focused on a small part of text simplification: the lexical substitution. Our results on word frequency suggest that lexical simplification techniques in general could be used successfully as a new reading aid for individuals with CFL.

METHODS

Participants

29 participants were recruited from December 2016 to June 2017 from the Low-Vision Clinic of the La Timone Hospital (Marseille, France). Recruited patients had to present a bilateral central scotoma, accompanied or not by an additional peripheral lesion and be fluent French readers. Patients with ophthalmologic disorders other than maculopathy (e.g. glaucoma) or cognitive disorders were not included. There was no minimal visual acuity threshold. Among the 29 patients recruited, 6 were excluded from the study because they could not read an entire block of text due to their low-functional vision. 23 participants (14 females) were included in the present work. They had different etiologies: atrophic AMD (n = 11), exudative AMD (n = 2), Stargardt's disease (n = 4), cone dystrophy (n = 1) and myopic retinopathy (n = 5) [START_REF] Kumar | Insight into high myopia and the macula[END_REF][START_REF] Silva | Myopic maculopathy: a review[END_REF][START_REF] Vongphanit | Prevalence and progression of myopic retinopathy in an older population[END_REF]. Mean age (± SD) was 72 ± 15 125 years and ranged from 32 to 88. Mean best-corrected visual acuity was 0.87 logMAR.

Apparatus and stimuli

Stimuli were displayed on an LCD HP LE2201W monitor with a 60Hz screen refresh rate.

Within the full display area of this monitor (47.4 x 29.6 cm; 1680 x 1050 pixels), a 1400 x 1050 pixel window was used to display our stimuli. At the viewing distance of 40 cm, this window subtended 56° x 42°. Stimuli were created with the PsychoPy library [START_REF] Peirce | PsychoPy--Psychophysics software in Python[END_REF][START_REF] Peirce | Generating stimuli for neuroscience using PsychoPy[END_REF]. Characters were black on a white background. Sentences were aligned to the left and displayed in Courier (non proportional font). Print size was chosen optimally for each participant as the value of his/her Critical Print Size, measured before testing with a French computerized version of MNREAD [START_REF] Calabrèse | Eye movements and reading speed in macular disease: the shrinking perceptual span hypothesis requires and is supported by a mediation analysis[END_REF][START_REF] Calabrèse | Does Vertical Reading Help People with Macular Degeneration: An Exploratory Study[END_REF]. Reading was monocular (eye with better visual acuity) with an appropriate correction for near vision (wide-field Metrovision lenses).

Reading material

Reading material was created in French using ReSyf, a French graded lexicon [START_REF] Billami | ReSyf: a French lexicon with ranked synonyms[END_REF] and Lexique3, a French lexical database [START_REF] New | The use of film subtitles to estimate word frequencies[END_REF]. First, we created hundreds of pairs of synonyms by selecting a word of any frequency along with its higher frequency synonym. We used the frequency of the lemma, which includes all the inflexional variants of a word (e.g. gender, number, tense, etc.). Lemma frequency (in occurrences / million) was estimated from a corpus of French film subtitles. Among the synonym pairs created, we then selected the ones that met the following criteria: (1) same number of characters within a pair (ranging from 5 to 10 characters);

(2) frequency ratio between a high-frequency word and its low-frequency synonym comprised between 5 and 10. A total of 32 pairs of synonyms were selected (Figure 1 -Step 1).

From these 32 pairs of target words, we then created 32 pairs of matching sentences so that each word from a pair could fit at the end of either sentence of the corresponding sentence pair. To this end, pairs of sentences were specifically designed to fit the single common sense of a synonym pair. An example is given in Figure 1, with the polysemic word 'records', that means both 'documentation' and 'discs', and its synonym 'archive'. In the sentence "This is probably one of my favorite records", records and archive convey different senses and cannot be considered interchangeable synonyms anymore. Therefore, acceptable sentences for this synonym pair must refer to the meaning 'documentation' only. Apart from this context constraint, all sentences were created similar in length (41 to 65 characters), with a maximum difference of 5 characters within a pair (Figure 1 -Step 2). Lastly, we created our final reading material by combining these sentence pairs with their matching pairs of synonyms (Figure 1 -Step 3). In Condition 1, the first word of a pair was assigned to the first sentence of the corresponding pair, while the second word was assigned to the second sentence, thus creating 64 full sentences. In Condition 2, the "sentence -word" matching was reversed to create a different set of 64 full sentences. These two experimental conditions allowed us to counterbalance any potential effect of the sentence itself (structure, complexity, predictability) by randomly assigning participants to Condition 1 or 2 [START_REF] Steen-Baker | The effects of context on processing words during sentence reading among adults varying in age and literacy skill[END_REF]. Figure 1 illustrates the full creation process.

Reading procedure and experimental design

Sentences were presented within 4 blocks of 16 trials (8 pairs of sentences) each. Participants were randomly assigned to Condition 1 or 2 and read between two and four blocks, depending on their individual reading speed and level of fatigue. Sentences were displayed with the self-paced reading paradigm and presented randomly within each block (Figure 2). Participants were instructed to read aloud as quickly and accurately as possible as soon as the sentence appeared. At the beginning of each trial, the experimenter triggered the display of a sentence, which appeared on the screen with the first word unmasked and all other words masked by strings of "x". Participants were in charge of revealing the whole sentence, one word at a time, using a keyboard in front of them. They could either unmask forward or backward (to reveal again words they already read) as many times as they wanted. When participants considered they had finished reading the sentence accurately (no matter which word was unmasked at that moment), they said the word "stop" and the experimenter stopped the trial.

A preliminary training phase with short sentences (French proverbs) was performed to familiarize the participant with the task and protocol. When the participant was comfortable with the keys and instructions, the test phase began and the blocks of sentences were presented.

Reading time (in seconds) was programmatically recorded for each word as the time during which it was unmasked. For words unmasked only once, total reading time was equal to the first-pass reading time (i.e. the firstand only -unmasked period). For words unmasked several times, total reading time was the sum of all the unmasked periods. accurately before responding to end the trial.

Statistical analysis

Statistical analyses were carried out using R (Team, 2018) with the following additional Rpackages: ggplot2, tidyr, dplyr, gridextra, lme4, stringr. Analyses 1 and 2 were performed with linear mixed-effects models that allow the modeling of experimental designs with unbalanced repeated measures [START_REF] Bolker | Generalized linear mixed models: a practical guide for ecology and evolution[END_REF][START_REF] Cheng | Real longitudinal data analysis for real people: building a good enough mixed model[END_REF]. For each analysis, several models were first constructed with total reading time and accuracy of each target word as dependent variables. Reading time and word frequency value were transformed in natural logarithm (ln) units. The continuous variables were centered around their mean. In a second step, the Akaike Information Criterion (AIC) and likelihood-ratio tests were used to assess an optimal random-effects structure [START_REF] Zuur | A protocol for data exploration to avoid common statistical problems[END_REF]. Once an optimal model structure had been chosen, significance of the fixed effects was estimated using t-values. Usually, the number of degrees of freedom for the t-values of the fixed effects are not exactly known with mixed-effects models [START_REF] Baayen | Mixed-effects modeling with crossed random effects for subjects and items[END_REF][START_REF] Pinheiro | Immediate and delayed effects of word frequency and word length on eye movements in reading: a reversed delayed effect of word length[END_REF]. However, given the large number of observations in the present study, the t-distribution converges to a normal distribution. Therefore, following standard recommendations, t-values (i.e. estimate/standard error) that were larger than 2 were considered significant, which corresponds to a significance level of 5% in a two-tailed test [START_REF] Baayen | Mixed-effects modeling with crossed random effects for subjects and items[END_REF][START_REF] Gelman | Data analysis using regression and multilevel/hierarchical models[END_REF]. Finally, assumptions underlying the models were visually checked with diagnostic plots of residuals [START_REF] Cohen | Applied multiple regression/correlation analysis for the behavioral sciences[END_REF][START_REF] Pinheiro | Immediate and delayed effects of word frequency and word length on eye movements in reading: a reversed delayed effect of word length[END_REF]. In the Results section fixed-effects estimates are reported along with their t-values and 95% confidence intervals [START_REF] Bates | Fitting Linear Mixed-Effects Models Using lme4[END_REF].

RESULTS

Analysis 1: Effect of word frequency value on reading performance

Here we tested the effect of frequency value of the target words (in occurrences/million) on both reading accuracy and total reading time. On average, target words were read accurately 93% of the time, with individual variations ranging from 69 to 100% depending on participants. We found that the effect of word frequency on reading accuracy was not significant (estimate = 0.005, 95% CI = [-0.003; 0.01]), probably due to ceiling effects. Figure 3 shows the target word total reading time of as a function of target word frequency, for all trials and participants. This relationship was further analysed for correct trials only with a linear mixed-effects model containing the following factors as fixed effects: target word frequency, word length (in number of characters), age, low-vision rehabilitation history (yes or no) and "still reading" (yes or no). The random effects were the following: subjects, per subject frequency slopes, target words and condition (nested within target words). The mixedeffects analysis showed a significant effect of word frequency on reading time, as represented by the black solid line in Figure 3. Results of the model are shown in Table 2. The main factor of interest, word frequency, had a significant effect with a regression coefficient estimate of -0.10 (95% CI = [-0.138; -0.06]). As both reading time and frequency are expressed in natural log units, this means that multiplying frequency (in original units) by 10 multiplies reading time (in original units) by 0.79 (10 ^ -0.10), i.e. a 21 % decrease.

Estimate

Similarly, we can also express the effect's amplitude in the following way: multiplying frequency (in original units) by 1000 (from 0.5 to 500, i.e. the range where most of our values lie) multiplies reading time (in original units) by 0.5 (1000 ^ -0.10), i.e. a 50 % decrease. It is notable that low-frequency words are on average longer than high-frequency words, so that the frequency effect could have been induced by the confound between word length and word frequency [START_REF] Kliegl | Regression analyses as a tool for studying reading processes: comment on Just and Carpenter's eye fixation theory[END_REF][START_REF] Rayner | Lexical complexity and fixation times in reading: effects of word frequency, verb complexity, and lexical ambiguity[END_REF]) -the correlation was -0.17 in the present study. This possibility was however eliminated here by partialling out the effect of word length in the model. Finally, we did not find any interaction between word frequency and word length (estimate = -0.002, 95% CI = [-0.02; 0.02]. This result is not consistent with previous studies of natural reading in normally sighted skilled readers based on gaze duration [START_REF] Hyönä | Eye fixation patterns among dyslexic and normal readers: effects of word length and word frequency[END_REF][START_REF] Kliegl | Tracking the mind during reading: the influence of past, present, and future words on fixation durations[END_REF]. Because age is a predictor of reading speed, even for people with normal visual acuity and self-reported healthy vision (Calabrèse et al., 2016;[START_REF] Lott | Reading performance in older adults with good acuity[END_REF][START_REF] Warrington | Effects of aging, word frequency, and text stimulus quality on reading across the adult lifespan: Evidence from eye movements[END_REF], we controlled for its effect by including this factor in our model. Although age was found to be non significant, it must be controlled for because it correlates with other factors, such as motor or attention deficits, that influence reading speed [START_REF] Sass | Low-vision reading speed: influences of linguistic inference and aging[END_REF]. The last factor with a statistically significant effect on reading time was the binary "still reading" (estimate = -0.558, 95% CI = [-0.984; -0.132]). According to the model, participants who still read, even a few minutes each day, are expected to have their word reading time multiplied by 0.57 (exp[-0.558]), i.e. a decrease of 43%. Note that similar results were found when using first-pass reading time as the dependent variable, instead of total reading time (Figure 2).

Analysis 2: Effect of word frequency category on reading performance

Within each pair of synonyms, we tested the effect of frequency category (low vs. high) of the target words on both reading accuracy and total reading time. First we found that frequency category did not have a significant effect on target word accuracy within pairs (estimate = -0.009, 95% CI = [-0.04; 0.02]): low-frequency synonyms were not read less accurately than their higher frequency counterpart. To analyse the effect of frequency category on target word total reading time, we removed all pairs containing one or two inaccurately read words.

The linear mixed-effects model was the same as in Analysis 1 except for two points. First, frequency category (low vs. high) was used as a fixed effect (instead of the continuous frequency value factor). Second, the random structure was different to accommodate for the within-pair analysis so that the optimal model contained: the subject factor, the pair factor and the by-pair category frequency factor as random effects. The results of the model are shown in Within-pair frequency category was found to be a significant predictor of target word total reading time (estimate = -0.13; 95% CI= [-0.23; -0.04]). This means that high-frequency words have a reading time that is multiplied by 0.87 (exp[-0.133]) compared to their lowfrequency synonyms (i.e. a 13 % decrease). This effect is represented in Figure 4 by thick solid lines. The dashed lines in this figure show the mean values for each participant, emphasizing that the amplitude of the frequency effect at the population level (solid lines) is quite small compared to the between-subjects differences. Despite being statistically significant, the 13% decrease in reading time induced by higher frequencies does not seem impressive in terms of clinical relevance. This relatively modest decrease is most likely due to the limited frequency ratio assigned to synonyms of each pair. Indeed, the within-pair frequency ratio was set between 5 and 10 in the present work (Figure 1 -step1), with an average of 7.07. Using results from Analysis 1 to predict the amplitude of the frequency category effect without by-pair grouping, we find that total reading time for high-frequency words would be multiplied by 0.82 (7.07 ^ -0.10 -see Table 2) compared to their lowfrequency synonyms. This 18% decrease, predicted from Analysis 1, is thus very close to the 13% reduction found in Analysis 2. We hypothesize that increasing the ratio of frequencies within pair of synonyms should boost the frequency category effect up to a clinically relevant improvement. Finally, we found that the "still reading" factor had a significant effect on total reading time (Figure 4 & Table 3), similar to the one reported in Analysis 1.

DISCUSSION

The general objective of this work was to investigate whether and how the degraded visual input of individuals with central field loss alters the processes linking low-level visual analysis of words to lexical access. More precisely, our first purpose was to assess the effect of word frequency on word reading time when individuals with bilateral maculopathy are engaged in natural reading. The dependent variable in our study was the reading time of target words (included within sentences) whose visual and linguistic properties are accurately controlled. This has never been studied before in visually impaired patients during natural reading (i.e. eye-mediated reading).

We used the method of self-paced reading as it allowed us to measure the reading time of individual words either for first-pass reading or for second-and nth-pass reading. Thus, when a word is unmasked for long periods (e.g. 10 sec), we can safely assume that many fixations have been made and constitute a cluster as previously described (Calabrèse et al., 2016).

Results obtained with this method can be fruitfully compared to those from eye movement studies of natural reading [START_REF] Ferreira | Eye movements in reading and information processing: Keith Rayner's 40 year legacy[END_REF][START_REF] Pinheiro | Immediate and delayed effects of word frequency and word length on eye movements in reading: a reversed delayed effect of word length[END_REF][START_REF] Rayner | Eye movements in reading and information processing: 20 years of research[END_REF]. In this huge literature, one important dependent variable is "gaze duration", classically defined as the total fixation time on word n before gaze position moves on to word n+1 for the first time:

this might correspond to one or several fixations (when several fixations are made, they are called "refixations" on the same word). Another measure is "total fixation duration" (sum of all fixations on a word n, including those induced by regressive saccades from words to the right of word n). Gaze duration is typically interpreted as reflecting earlier processes than those measured by total fixation duration. In our study, we assume that our two dependent variables, first-pass reading time and total reading time, are relevant proxies of gaze duration and total fixation duration (the only approximation being that durations of saccades are ignored). It is worth noting that in normal reading, short connector words and articles are often skipped entirely by gaze position, whereas the self-paced reading paradigm forces subjects to fixate on each word of the sentence, possibly modifying their reading behavior.

However, this point is not relevant in the present study becasue none of the target words were short connector words or articles.

Our results show a clear-cut effect of word frequency on word reading time (whether firstpass reading time or total reading time) even after partialling out the effect of word length [START_REF] Kliegl | Regression analyses as a tool for studying reading processes: comment on Just and Carpenter's eye fixation theory[END_REF][START_REF] Rayner | Lexical complexity and fixation times in reading: effects of word frequency, verb complexity, and lexical ambiguity[END_REF]. This result is consistent with recent findings reporting that reading speed was significantly reduced by the presence of clusters of fixations and that these clusters seemed located near the words with the lowest frequency (Calabrèse et al., 2016). This result is also in line with the effect of word frequency for subjects with normal vision: higher frequency words yield faster reading speed. However, the effect of word frequency found in our CFL participants is much greater than what has been reported before for normally sighted subjects. In an influential natural reading study with skilled readers, it was reported that gaze duration decreases from 240 to 207 ms when word frequency is multiplied by 1000 -from 1 to 4 log 10 units [START_REF] Kliegl | Tracking the mind during reading: the influence of past, present, and future words on fixation durations[END_REF]. Despite some differences in methodology between this work and the present study, the 14 % decrease in gaze duration reported in [START_REF] Kliegl | Tracking the mind during reading: the influence of past, present, and future words on fixation durations[END_REF] can be compared to the much larger 50 % decrease that we report in Analysis 1 for the same range of word frequencies. Such difference suggests that low-vision individuals rely more on lexical inference than normally sighted subjects.

Our work might be related to a previous study investigating the effect of word frequency on lexical decision accuracy with normally sighted observers [START_REF] Lee | Is word recognition different in central and peripheral vision?[END_REF]. The task was to decide whether stimuli of different durations and seen either with foveal or peripheral vision were words or pseudo-words. The authors reported that significant frequency effects occurred for the shortest durations in central vision, 25-50 ms, whereas significant frequency effects did not occur in peripheral vision until 100 ms. Although it is difficult to predict how these effects relate to a natural reading task, this study was important because it raised the major question of lexical access in peripheral vision. This important issue was also addressed in "Mr

Chips", an implemented ideal-observer model of low-vision reading [START_REF] Legge | Mr. Chips 2002: new insights from an ideal-observer model of reading[END_REF]Legge et al., 1997). Because this model integrates visual and lexical information to predict the pattern of eye fixations required to read a given text, lexical access constraint is expected to induce frequency effects on fixation duration.

Our results do not show any evidence for an effect of word length on word reading time although this effect is commonly reported in eye movement research on natural reading: word reading time increases with word length, mainly as a result of the increasing number of "refixations" [START_REF] Hyönä | Eye fixation patterns among dyslexic and normal readers: effects of word length and word frequency[END_REF][START_REF] Kliegl | Length, frequency, and predictability effects of words on eye movements in reading[END_REF][START_REF] Kliegl | Tracking the mind during reading: the influence of past, present, and future words on fixation durations[END_REF][START_REF] Kliegl | Regression analyses as a tool for studying reading processes: comment on Just and Carpenter's eye fixation theory[END_REF][START_REF] Vitu | Fixation location effects on fixation durations during reading: an inverted optimal viewing position effect[END_REF][START_REF] Vitu | Optimal landing position in reading isolated words and continuous text[END_REF]. Typically, normally sighted readers would make one single fixation on 5-character words and two fixations on 10-character words. This apparent discrepancy about the word length effect between CFL and normally sighted readers is probably due to diverse factors. First, the range of word lengths used in our study is probably too small (relative to the high variability in the dependent variable) to uncover an effect of word length whose amplitude and robustness are known to be smaller than the frequency effect for instance. Word length goes from 5 to 10 characters in the present study, whereas the range of lengths can be as high as 2 to 20 in some studies [START_REF] Kliegl | Tracking the mind during reading: the influence of past, present, and future words on fixation durations[END_REF].

Second, it is possible that the effect of word length is absent, or difficult to measure, in our study because people with CFL systematically make many fixations on nearly all the words of a sentence (Calabrèse et al., 2016;[START_REF] Calabrèse | Eye movements and reading speed in macular disease: the shrinking perceptual span hypothesis requires and is supported by a mediation analysis[END_REF][START_REF] Scherlen | Page mode reading with simulated scotomas: oculo-motor patterns[END_REF]. This high base level would make any subtle difference such as the word length effect more difficult to extract.

The secondary purpose of our work was to consider the potential of text simplification as a valuable tool allowing to improve reading performance of visually impaired patients. One of the many possibilities offered by text simplification is to replace complex words by their simpler synonyms [START_REF] Saggion | Automatic Text Simplification[END_REF], a transformation which requires agreement on a definition of word complexity. In this work, we decided to use word frequency as a surrogate of word complexity. Our second statistical analysis was thus performed on pairs of target word synonyms and showed an effect of relative frequency category (low vs. high) on word reading time: participants read target words of higher relative frequency faster than their lower frequency synonyms.

Importantly, the amplitude of this frequency category effect could be approximated from the mixed-effects model run in Analysis 1, which did not take into account the by-pair grouping.

This result is encouraging because it shows that a moderate frequency effect can be observed between synonyms having a relatively moderate difference in frequency (the frequency ratio ranged from 5 to 10 in the present study, with an average of 7) over a very large range of frequencies. In other words, our results do not apply only to the low frequency words of the French language. They apply to a large range of frequencies (say between 0.5 and 50 occurrences per million -Figure 3) and thus concern a very large number of words. As long as the numerous words lying within this range have synonyms of slightly higher frequency (ratio > about 7), a benefit in reading time is predicted by our statistical analysis. Of course, if synonyms with a higher ratio than 10 exist, they should be chosen preferentially by text simplification algorithms as the predicted reading time decrease would be larger.

Recent advances in the domain of natural language processing, at the frontier between linguistics computer science and artificial intelligence, should allow a large-scale implementation of text simplification methods. Among the current possibilities offered by text simplification, lexical simplification based on word frequency, among other factors, is easy to implement in automated algorithms and could be applied to various contexts (in electronic books, web sites, etc.). Hopefully, reducing the complexity of lexical units in text, without changing meaning, should improve overall reading performance of low-vision readers. We feel that the most important aspect of this improvement would not be the increased performance (e.g. in reading speed) per se but the possibility of keeping a certain level of motivation for reading activities. What our results clearly show is that people who keep on reading are those who read faster. This correlation between the motivation to read and word reading speed may create either a vicious circle or a virtuous one. It seems therefore important to help low-vision persons fight against this pattern. Our hope is that text simplification might encourage low-vision readers to read more and more often, which would, in turn, maintain a certain level of practice preventing them from stopping reading activities altogether.

CONCLUSION

This study showed a robust and large amplitude effect (+ 50%) of word frequency on word reading time when participants with macular diseases read text. The strength of this effect suggests that lexical access in CFL readers heavily relies on inferential processes compared to normally sighted observers. Our study also highlighted the potential benefits of lexical simplification when replacing a word by a synonym of higher frequency. More generally, the long-term challenge of this work is to investigate what aspects of visual, lexical and sentence processing can be simplified in order to significantly improve reading performance of lowvision patients.

One asset of our study is that the amplitude of the frequency effect is quantified by a regression coefficient while controlling for other relevant factors. This should allow future research to quantitatively compare whether the frequency effect is more capable of improving reading speed than other factors. Similarly, regression analyses should also help decide if the effects of frequency or other factors are the same for different categories of reading deficits (e.g. dyslexia, aphasia). If differential effects were found, this would suggest that methods of text simplification should be adapted to the specific needs of different reading disorders.

126

  Additional information was collected for each individual: lens status, duration of 127 maculopathy, low-vision rehabilitation history and reading habits. The research was approved 128 by the Ethics Committee of the French Society of Ophthalmology (IRB 00008855 Société 129 Française d'Ophtalmologie IRB#1) and carried out in accordance with the Code of Ethics of 130 the World Medical Association (Declaration of Helsinki). Informed consent was obtained 131 from all participants after complete explanation of the nature and possible consequences of 132 the study.

  Visual acuity and lens status are given for the tested eye only. Participants with central scotoma only are highlighted in yellow. Participants with central scotoma and peripheral lesions are highlighted in blue. Participants with a reading time different from 0 were considered as "still reading" (cf Analyses 1 and 2). Missing values are represented by the symbol NA.

Figure 1 :

 1 Figure 1: Creation of the reading material

Figure 2 :

 2 Figure 2: Self-paced Reading paradigm

Figure 3 :

 3 Figure 3: Scatterplot of target word total reading time as a function of word frequency value

Figure 4 :

 4 Figure 4: Effect of frequency category on target word reading time for both the "still reading"

Table 1 : Participants' demographic, visual and reading characteristics.

 1 Table 1 summarizes the participants' demographic, visual and reading

	133
	134	characteristics.
	135
	Partici pants	Gen der	Age (years)	Etiology	Disea se onset (years)	Low-vision rehabi litatio n	Visual acuity (logMAR)	Lens status	Daily reading time (min/day)
	1	M		Atrophic AMD	9	Yes	0.5	Phakic	0
	2	M		Atrophic AMD	2	No	1	Phakic	0
	3	F		Atrophic AMD	16	Yes	1	Phakic	0
	4	F		Atrophic AMD	7	No	NA	Pseudophakic	45
	5	F		Atrophic AMD	5	Yes	0.8	Phakic	5
	6	F		Atrophic AMD	6	Yes	1	Phakic	25
	7	M		Atrophic AMD	2	Yes	0.8	Pseudophakic	15
	8	F		Atrophic AMD	12	Yes	0.5	Phakic	10
	9	F		Atrophic AMD	1	Yes	0.7	Pseudophakic	0
	10	F		Atrophic AMD	15	No	0.7	Pseudophakic	60
	11	M		Atrophic AMD	22	Yes	0.6	Phakic	NA
	12	M		Exudative AMD	5	Yes	0.7	Pseudophakic	60
	13	F		Exudative AMD	1.5	Yes	0.5	Pseudophakic	20
	14	M		Stargardt's disease	24	No	1.2	Phakic	360
	15	F		Stargardt's disease	35	Yes	1.3	Phakic	20
	16	M		Stargardt's disease	37	No	1.2	Phakic	60
	17	F		Stargardt's disease	16	No	1	Phakic	120
	18	F		Cone dystrophy	30	No	1	Phakic	NA
	19	M		Myopic retinopathy	15	Yes	1	Phakic	3

Table 2 : Results of the mixed-effects model from Analysis 1: fixed effects estimates. The dependent variable is target word total reading time. The reference level of categorical binary factors is italicized. Factors showing a significant effect on reading time are in bold font.

 2

Table 3 .

 3

		Estimate	Standard error t-value	95% Confidence interval
	intercept	2.21	0.27	8.2	[1.7; 2.7]
	Frequency category	-0.133	0.046	-2.882	[-0.227; -0.043]
	Low frequency
	Word length (n. of characters)	0.051	0.034	1.498	[-0.016; 0.118]
	Age (years)	0.003	0.007	0.501	[-0.009; 0.016]
	Low vision rehabilitation	-0.188	0.214	-0.88	[-0.588; 0.211]
	No
	Still reading	-0.505	0.24	-2.104	[-0.953; -0.057]
	No

Table 3 : Results of the mixed-effects model from Analysis 2: fixed effects estimates. The dependent variable is the target words total reading time. The reference level of categorical binary factors is italicized. Factors with a significant effect on reading time are in bold font.

 3

ACKNOWLEDGMENTS

This work was supported by the Fondation de France and by the Belgian FNRS. The authors thank Dr Frédéric Chouraqui and Dr Alban Comet for their help in recruiting the participants.

This work was supported by the Fondation de France and by the Belgian FNRS.


Guillaume Bracq

Jean-Claude Crivello
email: crivello@icmpe.cnrs.fr

Mathilde Laurent-Brocq

Jean-Marc Joubert

Ivan Guillot

What is the enthalpy contribution to the stabilization of the Co-Cr-Fe-Mn-Ni faced-centered cubic solid solution?

Keywords: Multi-component alloys, High entropy alloys, DFT, SQS, solid solution

The aim of this paper is to investigate the contribution of the mixing enthalpy to the stability of the faced-centered cubic (f cc) phase of the Co-Cr-Fe-Mn-Ni system. For this purpose, systematic first principles calculations on Special Quasirandom Structures (SQS) were performed in a comprehensive manner on the f cc solution in every 5 2 = 10 binary, 5 3 = 10 ternary, 5 4 = 5 quaternary and the quinary systems. The magnetic contributions have been considered carefully i.e. including non-collinear spin polarization. An analysis of the different system order interactions to the enthalpy of mixing is detailed. This concludes that, at equiatomic composition, there is no special "cocktail" effect with an increase of components and that the thermodynamic properties can be well anticipated from the lower order systems, essentially binary and ternary systems.

Introduction

Since the report of Yeh [START_REF] Yeh | Nanostructured high-entropy alloys with multiple principal elements: Novel alloy design concepts and outcomes[END_REF] and Cantor [START_REF] Cantor | Microstructural development in equiatomic multicomponent alloys[END_REF] in 2004, the high entropy alloys (HEA) have received a substantial interest from the scientific community. They are multi-component alloys forming a solid solution and in which all components are very concentrated. A large and growing body of literature has shown that these alloys have promising properties, such as good combination of strength and ductility [START_REF] Senkov | Mechanical properties of nb25mo25ta25w25 and v20nb20mo20ta20w20 refractory high entropy alloys[END_REF][START_REF] Schuh | Mechanical properties, microstructure and thermal stability of a nanocrystalline CoCr-FeMnNi high-entropy alloy after severe plastic deformation[END_REF][START_REF] Gludovatz | A fractureresistant high-entropy alloy for cryogenic applications[END_REF][START_REF] Gali | Tensile properties of high-and medium-entropy alloys[END_REF][START_REF] Lilensten | Study of a bcc multi-principal element alloy: Tensile and simple shear properties and underlying deformation mechanisms[END_REF][START_REF] Bracq | Combining experiments and modeling to explore the solid solution strengthening of high and medium entropy alloys[END_REF], but also resistance to corrosion [START_REF] Lee | Effect of the aluminium content of AlxCrFe 1.5 MnNi 0.5 high-entropy alloys on the corrosion behaviour in aqueous environments[END_REF][START_REF] Wang | Study of the surface oxides and corrosion behaviour of an equiatomic CoCr-FeMnNi high entropy alloy by XPS and ToF-SIMS[END_REF], improved wear resistance [START_REF] Firstov | Wear resistance of high-entropy alloys[END_REF][START_REF] Hsu | Wear resistance and high-temperature compression strength of fcc CuCoNiCrAl 0.5 Fe alloy with boron addition[END_REF] and significant thermal stability [START_REF] Ahmad | Structural stability of high entropy alloys under pressure and temperature[END_REF].

The phase stability has been explained by the large configurational entropy, ∆ mix S = -R i x i ln x i expected to play an important role in the solid solution formation, in particular at the equiatomic composition [START_REF] Yeh | Nanostructured high-entropy alloys with multiple principal elements: Novel alloy design concepts and outcomes[END_REF]. However, this idea has been challenged by Otto [START_REF] Otto | Relative effects of enthalpy and entropy on the phase stability of equiatomic high-entropy alloys[END_REF] and several studies demonstrate that maximizing configurational entropy alone is not sufficient to form a multi-component solid solution [START_REF] Laurent-Brocq | From high entropy alloys to diluted multicomponent alloys: Range of existence of a solid-solution[END_REF][START_REF] Bracq | The fcc solid solution stability in the Co-Cr-Fe-Mn-Ni multi-component system[END_REF]. The phase stability in multi-component alloys is controlled by the Gibbs energy, which includes an enthalpy contribution as well, sometimes difficult to measure and separate from the contribution of the entropy (configurational and other ones [START_REF] Ma | Ab initio thermodynamics of the CoCrFeMnNi high entropy alloy: Importance of entropy contributions beyond the configurational one[END_REF]).

It is also interesting to deviate from the ideal equiatomic composition, as known as Canthor phase [START_REF] Cantor | Microstructural development in equiatomic multicomponent alloys[END_REF], but this makes the large composition range impossible to explore with an experimental approach only. In this respect, the development of predictive tools is necessary. Several empirical models tried to predict the formation of a solid solution phase [START_REF] Troparevsky | Criteria for predicting the formation of single-phase high-entropy alloys[END_REF][START_REF] Zhang | Solid-solution phase formation rules for multi-component alloys[END_REF], however they are not very accurate and are proven to be not predictive enough [START_REF] Laurent-Brocq | From high entropy alloys to diluted multicomponent alloys: Range of existence of a solid-solution[END_REF]. Another approach, which demonstrated good efficiency to predict the formation of a single solid solution, is based on the thermodynamical analysis of the system using the Calphad methodology [START_REF] Sundman | Computational Thermodynamics: The Calphad Method[END_REF][START_REF] Saunders | CALPHAD (Calculation of Phase Diagrams): A Comprehensive Guide[END_REF]. Based on the description of the Gibbs energy, it uses the binary and ternary descriptions to extrapolate higher-order systems. But, the capability to correctly predict the stable phases and the thermodynamic properties is dependent on the quality of the database in which the binary interactions represent the major contribution. Ternary interaction parameters are not always used to optimize the experimental ternary phase diagrams because ternary thermodynamic data are rarely available. Possible special features of HEA compared to conventional solutions open the question of their need and of the potential need of even higher order interaction parameters. This is why, it appears useful to have both an evaluation and a more physical description of the mixing enthalpy of high entropy alloys.

In the present work, the faced-centered cubic (f cc) phase of the Co-Cr-Fe-Mn-Ni system is analyzed as a case study. The aim of this paper is to investigate the contribution of the enthalpy of mixing to the stability of this quinary phase for a better understanding of the "high entropy alloys" concept. For this purpose, systematic density functional theory (DFT) calculations on special quasirandom structures (SQS) were performed in a comprehensive manner on the f cc phase in every 5 2 = 10 binary, 5 3 = 10 ternary, 5 4 = 5 quaternary and the quinary systems. The magnetic contributions have been considered carefully.

Methodologies

First-principles calculations

The mixing energy and cell volume of the f cc solid solutions has been investigated with the SQS methodology [START_REF] Zunger | Special quasirandom structures[END_REF]. This method allows to reproduce the random disorder of a solid solution at a given composition with a finite number of total atoms per unit cell. The generated SQS-cells are used as input for an additional DFT calculation [START_REF] Hohenberg | Inhomogeneous electron gas[END_REF][START_REF] Kohn | Self-consistent equations including exchange and correlation effects[END_REF] to estimate the formation or mixing enthalpy: enthalpy is calculated by subtracting the DFT total energy of a SQS-cell to the molar fraction weighted sum of the DFT energies of the pure elements in a f cc structure:

∆ mix H = E SQS (x i ; i = {A, B, C, D, E}) - A,B,C,D,E i x i E f cc (i) (1)
In the present work, the 10 binaries have been treated by considering a 16-atom cell in f cc-SQS structure for 3 different compositions (x A = 1 4 , 1 2 , 3 4 ). The SQS structures have been generated from Wolverton's work [START_REF] Wolverton | Crystal structure and stability of complex precipitate phases in Al-Cu-Mg(Si) and Al-Zn-Mg alloys[END_REF], with a pair correlation function equal to the random alloy for all pairs up to the 7th nearest-neighbor for x A = 1 2 and to the 4th nearest-neighbor for x A = 1 4 ; 3 4 . In addition, 2 simple dilute solutions (x A = 1 16 , 15 16 ) have also been considered by substituting a single atom in the same 16-atom supercell.

Furthermore, the f cc solution has been studied in the 5 3 = 10 ternaries. For this, a 24-atom cell and three 32-atom cell SQS structures taken from the literature [START_REF] Shin | Enthalpy of mixing for ternary fcc solid solutions from special quasirandom structures[END_REF] have been used for the equiatomic composition (x A = x B = x C = 1

3 ) and three compositions

(x B = x C = 1 4 and x A = 1 2 , x A = x C = 1 4 and x B = 1 2 , x A = x B = 1 4 and x C = 1
2 ), respectively. For the quaternaries, the 5 4 = 5 equiatomic compositions have been calculated with a 64-atom cell, whereas the quinary equiatomic composition has been calculated with a 125-atom cell also taken from the literature [START_REF] Gao | Applications of special quasi-random structures to high-entropy alloys[END_REF], where only pair correlation functions up to the third nearest shell are considered to evaluate the SQS generation.

All SQS structures have been calculated using the DFT within a pseudo-potential approach using the Vienna ab initio simulation package (VASP) and projector augmentedwave (PAW) method [START_REF] Kresse | Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set[END_REF][START_REF] Kresse | From ultrasoft pseudopotentials to the projector augmented-wave method[END_REF]. The generalized gradient approximation (GGA) described with the Perdew-Burke-Ernzerhof functional (PBE) [START_REF] Perdew | Generalized gradient approximation made simple[END_REF] has been used with a cut-off energy of 400 eV and a high k-mesh density (see details in Supplementary Materials).

The convergence criterion is only given on energy argument with difference of 0.01 meV, ie ∼1 J/mol. The residual force is reduced up to less or the same as the energy difference between two ionic steps.

The estimation of the energy error bars from SQS and DFT with an element permutation test on the different subsystems are given in Supplementary Materials. For equiatomic binairies, there is no effect of the A/B permutation since pair correlation functions up to the 7th neighbors are equal (null). For higher order systems, our sample test gives a standard deviation around 0.2 kJ/mol. While keeping the original crystal symmetry, the structures have been relaxed (volume and internal parameters relaxations) within 3 magnetic states:

• without spin polarization, or non-spin polarization (NSP)

• with collinear spin polarization (CSP)

• with non-collinear spin polarization (NCSP).

Whereas there is no difference for binaries between the calculations done with collinear or non-collinear spin polarization (CSP or NCSP), more stable magnetic states have been obtained with NCSP calculations for ternary, quaternary and quinary systems. Because of CPU time consumption, only equiatomic compositions have been investigated for NCSP calculations with one additional composition in the Co-Fe-Ni system.

Thermodynamic computations

The Calphad method relies on a self-consistent description of the Gibbs energy of each phase of a given system. This description is made thanks to semi-empirical functions, which depend on temperature and composition. The parameters are optimized on reliable experimental phase diagram and thermodynamic data, and also DFT results. The fitting parameters are stored in dedicated databases.

Thermo-Calc has developed a database dedicated to HEA [START_REF] Mao | TCHEA1: A thermodynamic database not limited for "high entropy" alloys[END_REF] shown to correctly predict the f cc solid solution in this system [START_REF] Bracq | The fcc solid solution stability in the Co-Cr-Fe-Mn-Ni multi-component system[END_REF]. It will be used in the present work to compare mixing enthalpies and volumes as a function of composition to our results. 1 K was chosen for the enthalpy of mixing in order to have a full account of the magnetic contribution. We have checked that this is not an issue for the Calphad calculation even if the database has not been evaluated at so low temperature. Excepted the magnetic contribution, no changes of the enthalpy of mixing is observed at so low temperature. 50 K was chosen for the volume calculation since, for several elements, the volume diverges at low temperature. 50 K is a good compromise between this effect and keeping a temperature low enough to allow comparison with DFT results.

In order to be able to compare SQS data of binary and higher order systems and to evaluate the contribution of the higher order interactions compared to the average of binary interactions, we treated our DFT data within a Calphad-like approach. The binary mixing energies were fitted to a first degree polynomial with the Redlich-Kister description [START_REF] Redlich | Algebraic representation of thermodynamic properties and the classification of solutions[END_REF]:

∆ mix H ij (x i ; x j ) = x i • x j ( 0 L + 1 L(x i -x j )) (2)
The predicted enthalpy of mixing for higher order systems is estimated by combining the binary descriptions according to the Muggianu extrapolation formula [START_REF] Muggianu | Enthalpy of formation of liquid Bi-Sn-Ga alloys at 723 k, choice of an analytical expression of integral and partial excess quantities of mixing[END_REF], such as in the ternary case:

H(x A , x B , x C ) = 4x A x B (2x A + x C )(2x B + x C ) ∆ mix H AB x A + x C 2 ; x B + x C 2 + 4x A x C (2x A + x B )(2x C + x B ) ∆ mix H AC x A + x B 2 ; x C + x B 2 + 4x B x C (2x B + x A )(2x C + x A ) ∆ mix H BC x B + x A 2 ; x C + x A 2 . ( 3
)
For higher degree system, the equation can be generalized as given by Hillert's reference handbook [START_REF] Hillert | Phase equilibria, phase diagrams and phase transformations: their thermodynamic basis[END_REF]:

H({x i }) = i j>i 4x i xj (1 + x i -x j )(1 + x j -x i ) ×H ij 1 + x i -x j 2 ; 1 + x j -x i 2 . ( 4
)

Results

The calculated enthalpies of mixing for equiatomic compositions are collected in Table 1. The Redlich-Kister parameters fitted from binary SQS-DFT calculations are presented in Table 2.

Binary compositions

The results for each binary system are collected in the grid of Fig 1 . The DFT calculations with spin polarization (CSP equal to NCSP) are represented with red circles with the Redlich-Kister fitted curve also in red. The stability range of the f cc phase is also represented. The mixing enthalpies computed using TCHEA4 database [START_REF] Mao | TCHEA1: A thermodynamic database not limited for "high entropy" alloys[END_REF][START_REF]Thermo-Calc Software, TCHEA4 -TCS High entropy alloy Database[END_REF] are indicated by a blue line. On each graph, is also indicated the volume calculated by DFT (compared with the Zen's law) [START_REF] Zen | Validity of "Vegard's Law[END_REF] and from the TCHEA4 database [START_REF]Thermo-Calc Software, TCHEA4 -TCS High entropy alloy Database[END_REF].

The mixing enthalpies are generally not found to be very large. They reach -10 kJ/mol only for Co-Cr, Cr-Fe and Cr-Ni. They are always negative except in the case of Co-Mn and Mn-Ni. The values are comparable with other SQS studies of similar systems [START_REF] Fedorov | Phase stability and magnetic properties in fcc Fe-Cr-Mn-Ni alloys from first-principles modeling[END_REF][START_REF] Tamm | Atomic-scale properties of Ni-based FCC ternary, and quaternary alloys[END_REF][START_REF] Gao | Thermodynamics of concentrated solid solution alloys[END_REF].

For the volume curves, the observed differences between SQS data and TCHEA4 are mainly caused by the differences obtained for the pure elements, in particular Mn. These differences may be explained by the difficulty to assess molar volumes in the database for metastable structures (among the five elements, only f cc-Ni is stable at 0 K). Otherwise, the deviation or the absence of deviation to the Zen's law is generally reproduced.

Ternary compositions

Analyzing and visualizing the enthalpy of mixing in a ternary system is not an easy task. An interesting way is to draw, in an A -B -C system, the three isopleth sec- On each graph, the SQS-DFT calculations without magnetism (NSP) and with the two states of magnetic ordering (CSP and NCSP) are indicated. The Muggianu extrapolation obtained from the fit of the binary SQS calculations (Eq. 3) is also shown together with the calculation obtained using TCHEA4 database.

tions
NCSP calculation results show that the ternary configurations are significantly more stable than those calculated with CSP (sometimes by 7 kJ/mol for Co-Fe-Ni). In rare cases (such as Cr-Fe-Mn), the most stable state was not found using NCSP because of the difficulty to find the magnetic non-collinear ground-state.

The sign of the mixing enthalpy is always negative from a SQS-DFT calculation, involving attractive interaction, with the only exception of the Co-Mn-Ni system. Additionally, in most systems, such as Co-Fe-Ni, where the magnetic contribution is the most significant, the NCSP energies reproduce the extrapolation made from the binary systems extremely well.

At this point, one may conclude several important facts for the ternary systems:

• the magnetic contribution is not negligible and the fundamental state could only be reached by non-collinear spin polarization, NCSP (not only collinear, CSP);

• the good agreement with the Muggianu extrapolation shows that the binary interactions are the main contributions. The additional ternary contribution is always positive except for Co-Fe-Ni where it is almost null;

• the TCHEA data are of same order of the magnetic set of results.

Quaternary and quinary compositions

Only the equiatomic compositions are investigated by SQS-DFT for the quaternary and quinary systems. Thus, 5+1 compositions have been calculated, and the enthalpy of mixing is always negative, as shown in Table 1. The Muggianu extrapolation of the binaries seems to predict correctly the enthalpy for the Co-Cr-Fe-Mn, Co-Cr-Mn-Ni and Co-Fe-Mn-Ni with a difference of less than 1.5 kJ/mol. However, for the Co-Cr-Fe-Ni and Cr-Fe-Mn-Ni, the value of mixing enthalpy is higher than that from the binary prediction. The difference reaches 3 kJ/mol for the quaternary system without manganese.

The difference between CSP and NCSP is very small for quaternary systems compared to the ternary case and null for the quinary. This suggests a maximum of the noncollinear contribution for the ternary case which may cancel out with the increase of complexity of the systems. The heat of mixing of the quinary Cantor alloy Co-Cr-Fe-Mn-Ni was calculated at -5.51 kJ/mol whereas the value from the binaries extrapolation is by 1.4 kJ/mol more stable.

Discussion

Zen's law

One of the most interesting points in analyzing the volumes obtained from the SQS calculation is the possibility to examine the deviation to the Zen's law in regard with the mixing enthalpies. A negative deviation could be anticipated to accompany negative heats of mixing. The results show that this is neither the case for our SQS nor in the TCHEA4 results. Most often absence of any deviation is noticed which may be related to the weak mixing energies that have been calculated. But the cases for which a larger deviation is observed (Co-Cr, Co-Fe) cannot be related to an energy deviation with larger amplitude.

Binary systems

The analysis of the binary systems shows that the energies of mixing are nearly always negative but are rather weak (in average -4 kJ/mol at the equiatomic composition) as expected from transition metals with comparable electronic structures. Good agreement is obtained between our calculations of the mixing enthalpy and the results from the TCHEA4 database for many systems (Co-Fe, Co-Mn, Co-Ni, Cr-Mn, Fe-Mn, Fe-Ni).

Less good agreement is obtained for other systems (Co-Cr, Cr-Fe, Cr-Ni and Mn-Ni).

Let us recall that TCHEA4 database is encrypted and that the origin of the thermodynamic parameters originating from the different thermodynamic optimizations that have been used are not known.

Different things should be considered to explain the apparent discrepancies. First, the stable homogeneity range of the f cc phase should be taken into account. It has been indicated in Fig. 1. Thermodynamic properties outside this range can be difficult to establish in a thermodynamic assessment since there are no experimental data that can be used to adjust the description for these compositions. It is interesting to see that the systems in which the f cc phase is stable over the complete composition range have a similar description with both approaches (with the exception of Mn-Ni system).

The second point concerns the evaluation of the so-called lattice stability of the f cc phase i.e. the energy difference between the f cc and the stable phase for elements that are not stable in the f cc structure. It is very well known that DFT calculations give different results compared to Calphad estimations [START_REF] Wang | Ab initio lattice stability in comparison with calphad lattice stability[END_REF][START_REF] Van De Walle | Invited paper: Reconciling SGTE and ab initio enthalpies of the elements[END_REF]. This has been attributed

to the instability of some elements in the f cc structure yielding inconsistent DFT estimations. This is particularly true for body-centered cubic (bcc) elements and, in our case, Cr. Within this framework, it is striking to see that the systems giving the worse agreement are those containing Cr and, in particular, in the Cr-rich region. Problems on the reported enthalpies of mixing may arise from the fact that the pure Cr reference is wrongly calculated, that SQS may be or may not be also mechanically unstable and that atomic relaxation of mechanically unstable structures may result in wrong estimation of the total energy (maximum estimated error of 2 kJ for Cr 0.5 Mn 0.5 ). We however choose to keep the raw data without a correction such as that proposed for pure components in

Ref. [START_REF] Van De Walle | Invited paper: Reconciling SGTE and ab initio enthalpies of the elements[END_REF] because the procedure is too complex for SQS structures and the detection of the instability would necessitate phonon calculations for all the 50 structures presented in this work. The error should affect the Cr-rich compositions only. We believe it is negligible (less than 1 kJ) in the range of composition investigated for the HEA in which Cr composition is not higher than 33 at.% and for which the binary compositions at 50 at.% Cr at most are used for Muggianu's extrapolation. Using mixing energies rather than total energies also diminishes partly the risk of errors. On the hand, in cases like the Cr--Mn system where the calculated mixing enthalpies tend not to converge to zero as we approach pure Cr, the error is also suppressed by using the SQS values for a fitting function which is forced to zero for the pure components.

Another feature that should be considered is the fact that the Calphad data have been evaluated at temperatures where the systems are no longer magnetic. Of course, magnetism is considered in the modeling and the reported Calphad are computed at low temperature. But, the overall results may differ from the ground state that is calculated by SQS and DFT approaches.

Ternary systems

We would like to emphasize the importance of magnetism for the calculation of ternary configurations. It appears that binary alloys can be fairly well calculated using CSP and no major difference was observed when calculated with NCSP. This is not the case for ternary configurations for which NCSP yields significantly more stable states than CSP.

Note that the NCSP calculation is much more complex and that one is never completely sure that convergence is obtained toward the most stable magnetic configuration. In few cases, more stable states are found for CSP meaning that the global minimum has not been reached for NCSP. The lower values will be considered for the rest of the discussion.

On the other hand, NCSP results are found much closer than CSP to the prediction of ternary mixing energies using Muggianu extrapolation and the fit of the SQS binary data. The difference is always positive corresponding to a positive ternary interaction (opposite to the binary interactions), except for Co-Fe-Ni where it is almost null. The maximum is obtained for Cr-Fe-Ni system in which the difference between calculated and the extrapolated mixing enthalpies reaches 4.7 kJ/mol.

The comparison with the extrapolation made with TCHEA4 shows relatively good agreement for the ternary systems for which the binary sub-systems were in agreement (in general without chromium).

Quaternary and quinary systems

For quaternary and quinary alloys, the difference between CSP and NCSP states vanishes. As shown in Fig. 3, we find again in both quaternaries and the quinary the positive interaction already shown for the ternary alloys, i.e. the calculated values are not as negative as expected from the only extrapolation of the calculated binary systems (lines in Fig. 3). We show that the mixing enthalpy of the quinary Cantor alloy could have been estimated from the only binary contributions to within 1.9 kJ/mol. The TCHEA4 results for the Cantor alloy deviate somewhat from our calculation.

Additionally, we present in Fig. 4, for each system order n, the difference between its enthalpies of mixing (Eq. 2) with circle symbols and those calculated from the extrapolation of n -1 systems order with empty triangle symbols. It shows that the contribution of the n -1 systems to the enthalpy of mixing decreases with n and is almost null for n = 5 (i.e. the quinary alloy can be perfectly calculated from the quaternary systems).

This is an indication that there is no special "cocktail" effect for this composition and that its thermodynamic properties can be well anticipated from the lower order systems, essentially binary and ternary systems, as usually observed in any system.

Note also that the stability of the quinary alloy is not very large because, if elements are no longer taken in the f cc state but rather in their reference state, the formation enthalpy becomes positive (+7.2 kJ/mol, in agreement with +7.6 kJ/mol in [START_REF] Gao | Applications of special quasi-random structures to high-entropy alloys[END_REF]) indicating that the solid solution is not stable at 0 K relative to its pure components. Entropy contribution at high temperature is therefore necessary to stabilize the phase.

Summary and conclusions

Using the SQS+DFT methods, the mixing enthalpy of the f cc phase has been investigated for the all sub-systems of the quinary Co-Cr-Fe-Mn-Ni, they are found almost always negative suggesting attractive interactions. It was shown that magnetism is not negligible. For ternary systems, consideration of non-collinear spin polarization is necessary, but is tricky to calculate. All the calculated energies were compared with the results obtained with TCHEA4 database. There is no obvious correlation between the sign of mixing enthalpy and the cell volume in the solid solution. The study of higher-order systems shows that the binary interactions are dominant and that higher-order interactions diminish progressively with the system order and cancel for the quinary system. In spite of a negative mixing energy, the quinary equiatomic alloy has a positive formation energy and needs an entropy contribution to be stable at high temperature.

  joining the pure elements with the equiatomic binary compositions, i.e. A -BC, B-AC, C-AB. The ternary f cc SQS cells of four compositions (A 33 B 33 C 33 , A 50 B 25 C 25 , A 25 B 50 C 25 , A 25 B 25 C 50 ) have been calculated. The three sections are illustrated in three individual plots compiled in Fig. 2 for the Co-Fe-Ni system as an example and in Supplementary Materials for the other systems.

Figure 1 :

 1 Figure 1: Enthalpy of mixing (J/mol) and volume ( Å3 /at) for the f cc phase in the 10 of Co-Cr-Fe-Mn-Ni binary sub-system.

Figure 2 :Figure 3 :

 23 Figure 2: Enthalpy of mixing (J/mol) for the f cc phase in three isopleth sections of Co-Fe-Ni system

Figure 4 :

 4 Figure 4: Enthalpy of mixing of each system order n (J/mol) calculated from the pure elements or from the extrapolation of the n -1 order systems in the f cc phase of Co-Cr-Fe-Mn-Ni system.

Acknowledgments

This work has benefited from a French government grant managed by ANR within the frame of the national program Investments for the Future ANR-11-LABX-022-01.

DFT calculations were performed using HPC resources from GENCI-CINES (Grant 2020-96175) and supercomputer at IMR, Tohoku University (No. 16S0403).