Sandrine Pavoine

Carlo Ricotta

On the relationships between rarity, uniqueness, distinctiveness, originality and functional/phylogenetic diversity

Keywords: conservation priority ranks, diversity index, equivalent number, evolutionary history, functional traits, quadratic entropy

Rarity reflects the low abundance of a species while distinctiveness reflects its quality of being easy to recognize because it has unique functional characteristics and/or an isolated phylogenetic position. As such, the assemblage-level rarity of a species' functional and phylogenetic characteristics (that we name 'effective originality') results from both the rarity and the distinctiveness of this species. The functional and phylogenetic diversity of an assemblage then results from a compromise between the abundances and the effective originalities of the species it contains. Although the distinctiveness of a species itself depends on the abundance of the other species in the assemblage, distinctiveness indices that are available in the ecological literature scarcely consider abundance data. We develop a unifying framework that demonstrates the direct connections between measures of diversity, rarity, distinctiveness and effective originality. While developing our framework, we discovered a family of distinctiveness indices that permit a full control of the influence one wants to give to the strict uniqueness of a species (=its smallest functional or phylogenetic distance to another species in the assemblage). Illustrating our framework with bat phylogenetic diversity along a disturbance gradient in Mexico, we show how each component of rarity, distinctiveness and originality can be controlled to obtain efficient indicators for conservation. Overall our framework is aimed to improve conservation actions directed towards highly diverse areas and/or towards species whose loss would considerably decrease biodiversity by offering flexible quantitative tools where the influence of abundant versus rare, and ordinary versus original, species is understood and controlled.

Introduction

Generally speaking, biodiversity is the range of many different characteristics of biological systems (see the Convention on Biological Diversity, https://www.cbd.int/doc/legal/cbden.pdf [accessed in August 2021], for the primary definition). In species assemblages, biodiversity thus emerges because species are not equivalent in their abundance and biological characteristics, their functional traits and phylogeny. To simplify the writing, hereafter we use the acronym 'FP' to mean 'functional or phylogenetic'. In local assemblages, often many species are rare, having small population size, and only a few species dominate in abundance (e.g., [START_REF] Hughes | Theories and Models of Species Abundance[END_REF]. Another increasingly studied aspect of rarity [START_REF] Pavoine | From phylogenetic to functional originality: Guide through indices and new developments[END_REF][START_REF] Violle | Functional Rarity: The Ecology of Outliers[END_REF][START_REF] Chapman | Both rare and common species make unique contributions to functional diversity in an ecosystem unaffected by human activities[END_REF][START_REF] Kondratyeva | Reconciling the concepts and measures of diversity, rarity and originality in ecology and evolution[END_REF] is the rarity of a species' FP-characteristics. We consider two terms often used to designate the rarity of a species' biological characteristics: distinctiveness and originality. Distinctiveness is the quality of being easy to recognize because of being different from other things (Cambridge Dictionary, 2021). In our context, the distinctiveness of a species can thus be defined as its quality of being easy to recognize because it has some unique functional characteristics or as its quality of being easily found in a phylogenetic tree because it belongs to an old, speciespoor clade. Originality is the quality of being special and interesting and not the same as anything or anyone else (Cambridge Dictionary, 2021). This is why the originality of a species can be considered as the assemblage-level rarity of the FP-characteristics associated to this species [START_REF] Pavoine | From phylogenetic to functional originality: Guide through indices and new developments[END_REF]. This originality depends on whether abundance data are considered (Fig. 1).

In absence of abundance data, the concept of originality is equivalent to that of distinctiveness [START_REF] Pavoine | Is the originality of a species measurable?[END_REF]. This is because the species own rarity is discarded and the rarity of a species' biological characteristics is only linked to the proportion of functional traits or phylogenetic history that are unshared with other species (see, e.g., the distinct species 1 and 2 in Fig. 1a,b,c). In global conservation studies for example abundance data are rarely considered (e.g., [START_REF] Gaüzère | Rapid adjustment of bird community compositions to local climatic variations and its functional consequences[END_REF]. For more local ecological studies however, abundance data are often available and often reveal meaningful to analyze ecological systems (e.g. [START_REF] Enquist | The commonness of rarity: Global and future distribution of rarity across land plants[END_REF]. Consider that a focal species j is distinct from other species in a defined species pool. If in an assemblage, its few functionally sibling or close relatives "dominate" the assemblage by their high abundance or if species j is itself very abundant, then the assemblage-level rarity of the FP-characteristics associated to species j is actually low and the originality of this species is in fact low (e.g., species 1 and 2 in Fig. 1d,e). Inversely, consider that species j has low distinctiveness in the species pool (where abundance data is discarded).

Species j may still be effectively original in an assemblage where its abundance and those of its sibling, close species are all low (e.g. species 3 to 8 in Fig. 1d,e). Hereafter, we will refer to this abundance-based definition of FP-originality as the effective FP-originality.

Combining the different aspects of a species rarity (its low abundance; the distinctiveness of its traits; its isolated position in the phylogeny), for the scope of this paper, we will thus consider two intuitive conditions that a measure of effective FP-originality should respect.

Consider an assemblage composed of N species with relative abundances In our context, the effective originality of a species can thus be considered as its quality of being of ecological interest because it is special, all both rare, unique and distinct from all other species in an assemblage.

Our aim here is to develop a unifying framework that demonstrates the direct connections between community-level measures of diversity, and species-level measures of rarity, (hereafter more simply named quadratic diversity; [START_REF] Rao | Diversity and dissimilarity coefficients: a unified approach[END_REF] for the measure of Ricotta et al.

We thus extend below these three originality indices and unify them in a common framework on the link between community-level diversity indices, and species-level rarity, distinctiveness and originality indices.

Quadratic diversity as a mean of effective originalities

Several diversity indices can be viewed as functions of distinctiveness or effective originalities. For example, Isaac et al. (2017) developed the index ED of evolutionary distinctiveness which is a partitioning of [START_REF] Faith | Conservation evaluation and phylogenetic diversity[END_REF] phylogenetic diversity index into individual species contributions; [START_REF] Ricotta | Measuring the functional redundancy of biological communities: A quantitative guide[END_REF] showed that quadratic diversity can be viewed as mean species originality; [START_REF] Violle | Functional Rarity: The Ecology of Outliers[END_REF] underlined that [START_REF] Webb | Phylogenies and community ecology[END_REF] MPD and NNTD indices of phylogenetic diversity can be also viewed as mean species distinctiveness and easily translated to the context of functional diversity (see [START_REF] Kondratyeva | Reconciling the concepts and measures of diversity, rarity and originality in ecology and evolution[END_REF] for a review of such relationships between distinctiveness or effective originality and diversity). Our framework relies on several diversity indices that can be viewed as weighted means of effective originalities, in other words as expected effective originalities. Here, we start with one of such indices, the quadratic diversity, and, in the next section, consider parametric extensions of it. Let ij d be the FP-dissimilarity between species i and j such that

0 ij d  and 0 jj d  .
Dealing with FP-diversity, we can define the FP-distinctiveness of species j as the weighted mean FP-dissimilarity of j from all other species in the assemblage [START_REF] Ricotta | Measuring the functional redundancy of biological communities: A quantitative guide[END_REF]):

1

N i j ij ij j p D d p     (1)
where the quantity [START_REF] Medellín | Bat diversity and abundance as indicators of disturbance in Neotropical rainforest[END_REF] [START_REF] Grenié | funrar: An R package to characterize functional rarity[END_REF]. We chose to name it here D for "Distinctiveness". j D satisfies condition C1 of a measure of effective originality but not condition C2, as it does not depend on the abundance-based rarity of species j (only on the relative abundance of other species and on the FP-dissimilarity with other species). A community-level measure of expected FP-distinctiveness defined as

1 N j j j D p D    
would not behave as an index of diversity (see Appendix A for details).

An index of FP-diversity can however be obtained by combining a species abundance-based rarity with its FP-distinctiveness. For ij d in the range [0, 1], consider ij s , the similarity between species i and j calculated as

1 ij ij s d  . Let 1     N j i ij
i ps be the ordinariness of species j, i.e., the expected similarity between an individual of species j and an individual chosen at random in the assemblage, including the individuals of species j itself. According to [START_REF] Leinster | Measuring diversity: the importance of species similarity[END_REF], j  can be thus interpreted as the relative abundance of all species that are FP-similar to j. Since

1 N i ij j ij pd    
, we have:

1 1 j j j D p     (2)
The denominator of Eq. 2, 1 ji p   , is the rarity of species j according to the well-known

Simpson diversity [START_REF] Simpson | Measurement of diversity[END_REF][START_REF] Patil | Diversity as a concept and its measurement[END_REF]. The numerator of Eq. 2,

1 jj O  
is identical to the species-level originality index K of [START_REF] Ricotta | Measuring the functional redundancy of biological communities: A quantitative guide[END_REF] that we mentioned in the Introduction section. j O satisfies both conditions C1 and C2 of an index of effective originality. The corresponding community-level measure of expected effective originality equals the quadratic diversity:

1 1 1          N N N j j j i ij j j i Q p O p p d
(3) the abundance-weighted average FP-dissimilarity between any two species in the assemblage.

In contrast to FP-distinctiveness Dj, effective FP-originality Oj accounts for the abundance of species j itself: to evaluate the rarity of the biological characters of a focal species when abundance data are considered, then the abundance of the focal species itself has to be considered. For example, in Fig. 1d, according to index Dj, species #1 is distinct because, among the 9 other species, each represented by one individual, only one species share its morphological aspects. However according to Oj, the effective originality of species #1 is low because this species is represented in the assemblage by 30 individuals, which amounts to 77% of the size of the assemblage. Dj and Oj thus drive different points of view on a species: its distinctiveness and its effective originality, respectively.

Parametric generalizations

As regards the measurement of community-level diversity, we consider below two possible parametric extensions of quadratic diversity. The parameter α of the first one controls the importance given to ordinary species in opposition to effectively original species:

1 1 1 1 N j j j K p            (4)
Eq. 4 was first developed by [START_REF] Ricotta | Towards a unifying approach to diversity measures: Bridging the gap between the Shannon entropy and Rao's quadratic index[END_REF]. For α = 2, 2 KQ  and for α tending to 1, it is a generalization of the Shannon index [START_REF] Shannon | A mathematical theory of communication[END_REF][START_REF] Ricotta | Towards a unifying approach to diversity measures: Bridging the gap between the Shannon entropy and Rao's quadratic index[END_REF].

When
1       N j j i ij ij O p d
that conveys a single viewpoint on effective originality: the average FP-dissimilarity between a focal species j and all species from the assemblage (including the focal species itself).

Knowing this unique definition of effective originality, varying α in K  thus enables to control how much the effectively original species versus the ordinary ones drives the level of FP-diversity.

We develop here the following alternative to index K  , named K   , where parameter α controls species' abundance instead of ordinariness, i.e. α controls the importance given to abundant species in opposition to rare species:

  1 1 | | 1 1 1 1 N N c j c j c j j u p K p               (5)
where c|j indicates the cth closest species from species j;

| | , 1| , c j c j j c j j u d d  
for c > 1 with 0|j = j and thus 0| , 0 jj d  and 0| jj pp  ; and

1 || 0 c c j i j i pp     . Compared with K  , K   does not require that the dij vary in [0,1]; the only necessary condition is dij ≥ 0. For α = 2, 2 KQ   .
For α tending to 1, it is a generalization of the Shannon index different from that associated with index K  (Appendix A). We provide in Table 1 a decomposition of K   in terms of rarity, distinctiveness and originality (see details and proofs in Appendix A). With K   , the effective originality of a species is measured by a parametric index

    1 1 | | 1 1 / 1            N j c j c j c O u p
that conveys different viewpoints on effective originality depending on the value of parameter α: varying α in K   enables to control how much the abundant species versus the rare ones drive the levels of FP-distinctiveness, effective originality and diversity. In

  j O ,     1 | 1 / 1     cj p
represents the global rarity of the set of c species that have the smallest FP-dissimilarity with species j (including j itself) and | cj u represents the excess of FP-dissimilarity with species j brought by the cth closest species compared to the c-1th closest species. Overall, the effective originality of species j, as measured by   j O , can be viewed as the rate at which the rarity of species tends to decrease with the FP-dissimilarity to the focal species j (Fig. 2). Low rate as in Scenario a of Fig. 2 corresponds to high effective originality as it means that abundant species tend to be FPdistant from the focal species; inversely, high rate as in Scenario c of Fig. 2, corresponds to a low effective originality as it means that abundant species tend to be FP-close to the focal species.

For the specific case of species characterized by their rooted phylogenetic tree, the parametric index α I of phylogenetic diversity proposed by [START_REF] Pavoine | Hierarchical partitioning of evolutionary and ecological patterns in the organization of phylogenetically-structured species assemblages: application to rockfish (genus: Sebastes) in the Southern California Bight[END_REF] (Table 2) is equivalent to K   applied to dij = the sum of branch lengths on the path from tip j to its most recent common ancestor with species i. In addition, in this particular case the associated species effective originalities (Table 2) are equivalent to those introduced by Kondratyeva et al.

(2019) who expressed index α I of phylogenetic diversity as a mean of the species originalities.

For α = 0, the phylogenetic effective originality associated with and its associated indices of distinctiveness and originality, in the special case where species are tips of a phylogenetic tree and the phylogenetic distance between species j and i is calculated as the sum of branch lengths on the path from tip j to its most recent common ancestor with species i. All functions of rarity, distinctiveness and effective originality discussed here are thus connected in a global framework in Tables 1 and2, highlighting the strong links between different facets of rarity, distinctiveness, originality and diversity.

I  (
All these diversity indices can be easily transformed into equivalent numbers of species: the number of evenly and maximally dissimilar species needed to obtain the level of FP-diversity observed in an assemblage (Tables 1 and2). The functions that transform the diversity indices we discussed here in terms of equivalent numbers of species do not change the way species assemblages are ranked from the least to the most diverse (Appendix A).

The special case of abundance-free distinctiveness indices

Abundance-free distinctiveness indices, particularly phylogenetic distinctiveness indices, are often used in conservation biology [START_REF] Isaac | Mammals on the EDGE: Conservation priorities based on threat and phylogeny[END_REF][START_REF] Redding | Measuring evolutionary isolation conservation[END_REF]. Imposing equal relative abundances for all species in our framework provides a useful family of such distinctiveness indices where abundance data is discarded as outlined below (see a complete introduction of the family in Appendix B).

In the special case of equal abundance for all species (pj=1/N for all j), the phylogenetic distinctiveness index    associated with phylogenetic diversity index  I (Table 2) becomes:

  11 1 ,root eq 1 b jb b C j NN L N               (6)
where Nb stands for the number of species that descend from branch b and C(j, root) for the set of branches between species j, tip of the phylogenetic tree, and the root of the tree. eq   

thus provides a parametric alternative to the most widely used index of species distinctiveness named "evolutionary distinctiveness" (ED; [START_REF] Isaac | Mammals on the EDGE: Conservation priorities based on threat and phylogeny[END_REF] or "Fair-Proportion" [START_REF] Redding | Measuring evolutionary isolation conservation[END_REF] and whose formula is:

  ,root b j b C j b L ED N    (7)
Both ED and eq j    can be seen as the sum, on the path from a tip to root, of the product of a branch length times a decreasing function of the number of species descending from that branch. While by construction, the value of ED is dominated by the length of the terminal branch that connects species j to the tree [START_REF] Redding | Measuring evolutionary isolation conservation[END_REF], the parameter α in eq

  
allows controlling the influence of this terminal branch (see Appendix B for more details).

Although the diversity indices of Tables 1 and2 , from tip j to the root of the tree ( eq jj a H

      , Appendix B). Another notable case is obtained with α = 2 because   2 1 ,root eq 11              N ij bi jb b C j d NN L NN (8)
where ij d is the sum of branch lengths on the path from tip j to its most recent common ancestor with species i. As a consequence, by varying parameter α between -∞ and 2 in eq j    one can obtain a range of strongly connected indices of phylogenetic distinctiveness along a gradient that goes from a strong influence of the terminal branch to the average distance between a species and all others in phylogenetic tree. eq j    can be seen as a quantification of the extent to which the loss of species j would weaken/threaten phylogenetic diversity, with two extreme views: α tending to -∞, where the loss of species j would imply the loss of its unique phylogenetic history (terminal branch length); and α tending to +∞, where the loss of species j would imply that all its phylogenetic history is threatened (length of the path between j and the root of the tree). Varying α between these two extreme cases controls the extent to which a phylogenetic branch from which species j descends can be considered secure if one or more other species also descend from it. The parametric index of distinctiveness makes it possible to study and compare all points of view in a consistent way and to know according to which point of view, if any, the safeguarding of a species should be considered a priority. Index eq j    thus responds to the critical need in conservation science to justify and organize indices of evolutionary distinctiveness which currently represent different trade-offs between the unique evolutionary history represented by a species versus its average distance to all other species, trade-offs that may impact on-the-ground decision on conservation priority [START_REF] Redding | Measuring evolutionary isolation conservation[END_REF]. Similarly, using even abundance in the distinctiveness index j D   (Table 1) provides a measure of FP-distinctiveness eq D   whose values vary (by varying α) from the smallest FP-dissimilarity between j and any other species (α → -∞), through the average FP-dissimilarity to all other species (α = 2), to the largest FPdissimilarity between j and any other species (α → +∞) (Appendix B).

Worked example

Data

As a case study, we considered changes in bat phylogenetic diversity across a disturbance gradient in the Selva Lacandona of Chiapas, Mexico. [START_REF] Medellín | Bat diversity and abundance as indicators of disturbance in Neotropical rainforest[END_REF] collected the abundance of bat species in four habitats (rainforests, cacao plantations, old fields and cornfields). We obtained the phylogeny of 34 observed species using a consensus ultrametric tree (function consensus.edge, R package phytools; Revell, 2012) on 9,999 credible birthdeath, tip-dated completed trees downloaded from http://vertlife.org/phylosubsets on 2021/02/12 [START_REF] Upham | Inferring the mammal tree: Species-level sets of phylogenies for questions in ecology, evolution, and conservation[END_REF]. For species names, we followed [START_REF] Upham | Inferring the mammal tree: Species-level sets of phylogenies for questions in ecology, evolution, and conservation[END_REF].

Branch lengths on the consensus tree were obtained using mean edge length, ignoring credible trees in which the branch is absent [START_REF] Revell | phytools: An R package for phylogenetic comparative biology (and other things)[END_REF].

Analyses

Our objective in the analysis of this dataset, in addition to simply illustrate our theoretical developments, was to depict how disturbance changes the level of bat phylogenetic diversity by identifying which among the rarity, distinctiveness, and effective originality of bat species is/are affected. For that, we calculated, in each habitat, the phylogenetic diversity using Y  and α I, for α varying between 0 and 3 with a step of 0.1. We explored then in detail the patterns of phylogenetic diversity in terms of effective originality. Additionally, we analyzed species distinctiveness in the species pool (using indices ED and eq

  
that both discard abundance data).

Results

In the species pool, for values of α of 1, 2 and 3, according to eq

  
the two Vespertilionidae species (Bauerus dubiaquercus and Myotis keaysi) are the most phylogenetically distinct, being the sole members of their clade in the dataset (Fig. 3). The least distinctive species are the two Artibeus species as they are in the most species-rich part of the phylogenetic tree, where both terminal and internal branches are numerous and relatively short. For low values of α, the distinctiveness is strongly influenced by the length of the terminal branches. This has two main consequences: the species perceived as least distinct are the two Carollia species that diverged recently; and the two Vespertilionidae species are no longer perceived as the most distinct, as their terminal branches are shorter than those of Thyroptera tricolor, Pteronotus parnellii and Mormoops megalophylla. The case, α = 0 is the closest to ED (Pearson correlation = 1). For 0 eq   and ED, T. tricolor is the most distinctive followed by the two Vespertilionidae and then P. parnellii and M. megalophylla; the least distinctive are the Artibeus species followed by the Dermanura species and then the Carollia species. A more complete visualization of variation in eq

  
as a function of α can be found in video S1.

Phylogenetic diversity decreases along the disturbance gradient (from rainforest, through cacao plantation and old fields, to cornfields) whatever the value of α we considered (from 0 to 3 in Fig. 4) for

I 
and for high values of α (α >= 1.5 in Fig. 4) for

Y  . With  Y , for low
values of α (α < 1.5 in Fig. 4) the phylogenetic diversity of old fields exceeds that of cacao plantations; and as α approaches zero (α < 0.5 in Fig. 4) the phylogenetic diversity of old fields even exceeds that of the rainforest. This is because old fields contain both B.

dubiaquercus and M. keaysi, the two Vespertilionidae species with the highest average phylogenetic distance to all other species, while only one of them was observed in the rainforest and in the cacao plantation, and none in the cornfields (Fig. 5). Indeed index Y  , as

K 
, uses the effective originality associated with the quadratic diversity and a parameter α that controls the relative importance given to ordinary species compared to effectively original species. For low values of α, the influence of the most effectively original species in the measurement of phylogenetic diversity increases. In contrast as shown above, α in I  , as in K   , controls the importance given to abundant compared to rare species, influencing the way the effective originality of a species is perceived (with strong influence of the terminal branch for low values of α). Compared to other habitats, cornfields lacked effectively original species (Fig. 5; see also Video S2 for a more complete visualization of variation in species originality    as a function of α). Phylogenetic diversity in cornfields was thus always the smallest (whatever α; and even strongly lower than that of other habitats when α approaches zero; Fig. 4). Species with the least effective originality were often either the Carollia species or the Artibeus species depending on the habitat and the value of α considered (Fig. 5).

However the relatively high abundance of P. parnelii observed in the rainforest, made this species perceived as one of the least effectively original in this habitat when rare species were given high weights in the measurement of phylogenetic diversity (α = 0, index α I, Fig. 5), despite its high distinctiveness (Fig. 3 and5; see also Video S3 for a more complete visualization of variation in species distinctiveness    as a function of α).

Discussion

The contribution of a given species to the biodiversity of an assemblage thus depends on its rarity and on the rarity of its FP-characteristics. Starting from [START_REF] Ricotta | Measuring the functional redundancy of biological communities: A quantitative guide[END_REF] measure of species-level originality, we have shown that quadratic diversity can be expressed as a mean of effective FP-originality values over all species in an assemblage, and that [START_REF] Faith | Conservation evaluation and phylogenetic diversity[END_REF] phylogenetic diversity index and [START_REF] Cadotte | Phylogenetic diversity metrics for ecological communities: integrating species richness, abundance and evolutionary history[END_REF] and [START_REF] Kondratyeva | Reconciling the concepts and measures of diversity, rarity and originality in ecology and evolution[END_REF] measures of effective phylogenetic originality can be both related to parametric extensions of quadratic diversity. This led us to develop a unified framework (summarized in Tables 1 and2) where diversity, rarity, distinctiveness and originality measures are intrinsically linked. 2015). Increasing α may reveal how much phylogenetic diversity depends on these rare species.

In our case study, when effectively original species were given high weights in the measurement of diversity, the old fields had the highest measure of phylogenetic diversity.

This was due to the presence of the two species from the Vespertilionidae family. Although distinct in our study area, the Vespertilionidae species represent a large family of bats at a global scale. This illustrates how the measurement of a species' distinctiveness is dependent on the reference species assemblage, on the data used to characterize species (here phylogeny), and thus on the taxonomic, phylogenetic and spatial scales of a study. A species' rarity, when measured relatively to the rarity of all other species rather than as an absolute value, is also dependent on these scales. According to the International Union for the Conservation of Nature (IUCN, 2021), all species of our case study are least-concern (i.e., neither threatened nor near threatened) at a global scale with either stable or unknown population trends except M. megalophylla that is least-concern but with decreasing population trends, and B. dubiaquercus that is near threatened (+ Vampyressa pusilla classified as data deficient). Our results showed that M. megalophylla and B. dubiaquercus are among the most phylogenetically distinct species in our study area. Only one individuals of M. megalophylla and three individuals of B. dubiaquercus were observed in our case study (out of a total of 2405 bats). M. megalophylla was observed in old fields and B. dubiaquercus in rainforest and old fields. These scarce occurrences prevented us to evaluate the possible direct effects of agriculture on their population size. However, both species are aerial insectivous species (Rodríguez-Aguilar et al., 2017) and insect-eating bats are known to be affected when high pesticide inputs are used in plantations (e.g., [START_REF] Estrada | Bat diversity in montane rainforest and shaded coffee under different management regimes in southeastern Chiapas, Mexico[END_REF]. Other studies showed that, in addition to cave collapse and vandalism, threats on M. megalophylla and B. dubiaquercus also concern their sensitivity to disturbance and habitat loss (IUCN, 2021). In an urban context in the highlands of Chiapas, it was shown that abundance of B. dubiaquercus tends to diminish outside forest; and the activity of both M. megalophylla and B. dubiaquercus increases with tree density (Rodríguez-Aguilar et al., 2017).

The whole framework can thus be used in ecological studies to reveal the relative contributions of each species to biodiversity, and to depict these contributions in terms of abundance-based rarity, species-level FP-distinctiveness and effective FP-originality. It can be applied from local to global scales provided abundance data are available and the species assemblage is clearly delimited. In the context of the sixth species mass extinction [START_REF] Ceballos | Accelerated modern human-induced species losses: entering the sixth mass extinction[END_REF], our framework could be used to evaluate how changes in species contributions may impact ecosystem services, via changes in biodiversity levels. It would allow following changes in species contribution to biodiversity under novel environmental conditions.

Relative abundances may change under novel environmental conditions with rare species for example benefiting from the reduction of population size of other species. If these species are functionally rare, an increase in their abundance can considerably increase the functional diversity of an assemblage and modify ecosystem functioning and the connected ecosystem services. The bioprospecting or option value associated to a species notably advocates as a precaution principle to protect species with the aim to give option to discover new uses of these species in the future, especially in medicine. Here functionally and phylogenetically unique species may be considered as disproportionally contributing to bioprospecting value [START_REF] Dee | When do ecosystem services depend on rare species?[END_REF]. Where and when they have low abundance, they might be in need of urgent conservation actions (e.g., the Van Gelder's bat, B. dubiaquercus, of our case study). Indeed, if environmental changes inversely lead to the extinction of currently effectively original species with key role in the ecosystem, these changes could yield the biological system to collapse, with potential drastic loss of ecosystem services. Effectively original species may for example be directly threatened when they are increasingly targeted by economic activities because of their combined aspects of rarity (e.g., private collections of rare, distinct specimens; safaris spotting rare, distinct species; e.g., [START_REF] Holden | High prices for rare species can drive large populations extinct: the anthropogenic Allee effect revisited[END_REF].

Links between the functional distinctiveness of a species and its abundance-based rarity have for example been observed in European estuarine fish communities, with identified potential consequences on the stability of these communities [START_REF] Teichert | Functional redundancy and sensitivity of fish assemblages in European rivers, lakes and estuarine ecosystems[END_REF]. Links between the functional distinctiveness of a species and its risk of extinction have also sometimes been observed: e.g., among anurans in Ecuador [START_REF] Menéndes-Guerrero | Extinctions of threatened frogs may impact ecosystems in a global hotspot of anuran diversity[END_REF]; globally in mammals and birds [START_REF] Cooke | Ecological distinctiveness of birds and mammals at the global scale[END_REF]. Species which are rare both in terms of low abundance and phylogenetic distinctiveness have sometimes been found to be threatened. For example, [START_REF] Uchida | Non-random loss of phylogenetically distinct rare species degrades phylogenetic diversity in semi-natural grassland[END_REF] observed that, in semi-natural grasslands of south west Japan, low-abundance and phylogenetically distinct species were threatened by land-use intensification, resulting in plant phylogenetic diversity loss.

To make our mathematical framework readily available for data analysis, we have updated and added R functions in package adiv of R (Pavoine 2020, 2021; R Core Team, 2021) (see also Appendix C). adiv is dedicated to the analysis of diversity in ecology and now integrates a range of functions to measure both species-level originality and community-level diversity.

It constitutes thus a useful tool to implement our framework. Compared to previous studies that connected a single index of distinctiveness or of effective originality to a single diversity index (e.g. [START_REF] Isaac | Mammals on the EDGE: Conservation priorities based on threat and phylogeny[END_REF][START_REF] Ricotta | Measuring the functional redundancy of biological communities: A quantitative guide[END_REF][START_REF] Violle | Functional Rarity: The Ecology of Outliers[END_REF], our parametric framework unifies, in a mathematical scheme, various well-used measures of diversity, distinctiveness and effective originality, previously scattered in the literature and developed in different contexts. It also allows disentangling the effects of rarity, distinctiveness, and effective originality on biodiversity levels to better identify how species together contribute to biodiversity levels. As such, it has the potential to improve studies on the mechanisms by which global changes affect biodiversity levels, by identifying which aspect of biodiversity they impact, be it rarity, functional distinctiveness or the global effective originality of some species, which was not possible with previously published connections between diversity and either rarity, distinctiveness, or effective originality. By developing it, we revealed a parametric family of phylogenetic distinctiveness indices that could complement the most currently used "evolutionary distinctiveness" index (e.g. [START_REF] Isaac | Mammals on the EDGE: Conservation priorities based on threat and phylogeny[END_REF][START_REF] Ibáñez-Álamo | Global loss of avian evolutionary uniqueness in urban areas[END_REF][START_REF] Potter | Do United States protected areas effectively conserve forest tree rarity and evolutionary distinctiveness?[END_REF][START_REF] Cooke | Ecological distinctiveness of birds and mammals at the global scale[END_REF] whose values are strongly dominated by the independent evolutionary history of a species (length of terminal branch in a phylogenetic tree with the species as tip; [START_REF] Redding | Measuring evolutionary isolation conservation[END_REF]. The parametric family indeed allows controlling the degree of influence of this independent evolutionary history on the distinctiveness index.

We provide also an equivalent parametric family for functional distinctiveness. Overall, our framework helps to provide justification, explanation and order when applying a quantitative reasoning to biodiversity, contributing to the development of efficient biodiversity indicators for conservation strategies.

Table 1

Summary of the links between rarity, distinctiveness, originality, and diversity developed in the main text with diversity indices Q, K  and K   .

Quadratic diversity a

First parametric extension a Second parametric extension a Species-level FPdistinctiveness 1

N i j ij ij j p Dd p       1 1 1 1 1 N i ij i j j ps D p               1 1 | | 1 1 1 1 N c j c j c j j u p D p             abundance- based rarity   1 jj p   1 1 1 j j p         1 1 1 j j p         effective originality j j j OU  j j j OD      j j j OD       Community-level species diversity b 1 2 1 1 N jj j N i i Sp p          1 1 1 1 N j j j N j j S p p                1 1 1 1 N j j j N j j S p p              FP-diversity 1 11 N jj j NN i j ij ij Q p O p p d           1 1 1 1 1 1 N j j j N i ij N i j j K p O ps p                     1 1 1 | | 1 1 1 1 N j j j N N c j c j c j j K p O u p p                    Species equivalents c 11 1/ NN i j ij ij E p p s        1 1 1 1 1 N N j i ij j i E p p s                   1 1 1 | 1 | 1 1 max max max ( ) 1                             N N i ij c j j c j j c d u E p p d d a 22   Q K K ; FP-components are expressed in terms of dissimilarities (dij ≥ 0) or similarities (sij = 1 -dij / dmax, with dmax ≥ maxij(dij)) between
species; Notations are identical as in the main text; see Table 2 for the special case of phylogenetic dissimilarities. b S = Simpson index [START_REF] Simpson | Measurement of diversity[END_REF][START_REF] Patil | Diversity as a concept and its measurement[END_REF]; S  = HCDT index (see e.g., [START_REF] Kondratyeva | Reconciling the concepts and measures of diversity, rarity and originality in ecology and evolution[END_REF].

c See [START_REF] Leinster | Measuring diversity: the importance of species similarity[END_REF] who first developed a function similar to E  .

Table 2

Adaptation of Table 1 in the case where species are tips of a phylogenetic tree.

First parametric extension a Second parametric extension a Species-level distinctiveness We simulated a set of 20 species, with a standard lognormal distribution for their abundance and a uniform distribution (between 0 and 1) to define the FP-dissimilarities with a focal species j (one of the 20 species). Here we considered α = 2 in   j O formula but our reasoning is valid for any nonnegative value of α. In Scenario a, species rarity tends to increase with the FP-dissimilarity with species j, while in Scenario c, it tends to decrease. Scenario b is a random scenario (where rarity is independent from the FP-dissimilarity with species j). In each scenario, the top graph provides each species' relative abundance (pi) as a function of its FP-dissimilarity to species j and the curve in the bottom graph shows how fast the rarity of the set of species whose FP-dissimilarity to species j is lower than a nonnegative value d (as measured by    

    1 ,root 1 1 1 b L b H b C j j j p p               1 ,root 1 1 1 bb b C j j j Lp p            effective originality j j j         j j j          Community-level FP-diversity b       1 1 ,root 1 11 1 j b N j j j H L b H H b C j N j j Y p p p                        1 1 1 ,root 1 1 N jj j N b jb j b C j Ip p pL                 Species equivalents c   1 1 1 1 ,root 1                           N j b j b j b C j H L E p p H H   1 1 1 1 ,root 1 N j b j b j b C j H L E p p H H                         a Phylogenetic

  effective originality of a given species j should increase with its functional or phylogenetic distinctiveness with respect to the individuals of other species in the assemblage, and C2. the effective originality of a given species j should increase with its abundance-based rarity, a decreasing function of j p .

  distinctiveness and effective originality. Thanks to this framework, we offer flexible quantitative tools, as guides to conservation strategies, where the influence of abundant versus rare, and ordinary versus original, species is understood and controlled. Our framework relies on, and combines, three abundance-weighted measures of originality: Cadotte et al. (2010) AED index and Kondratyeva et al. (2019) O index for phylogenetic originality; and Ricotta et al. (2016) K index for functional originality. These measures of originality have the advantage to be linked with standard measures of diversity: Faith phylogenetic diversity (PD; Faith 1992) for the measure of Cadotte et al., an abundance-weighted generalization of Faith's PD (Pavoine et al., 2009) for the measure of Kondratyeva et al. and Rao's quadratic entropy

  are meaningful only for nonnegative values of α (as otherwise rarity function rapidly tends to infinity for low values of α),

  The parametric indices developed in this framework allow regulating the importance given to abundant and ordinary species in FP-diversity measures. They include generalizations of the Shannon index (for α → 1) and the Simpson index (for α = 2) to functional and phylogenetic data. By increasing the value of their parameter α, the weight given to abundant and ordinary species in the measurement of FP-diversity increases in comparison with the weight given to rare and effectively original species. Low values of α may thus indicate regions with high diversity but a diversity that may be threatened by the rarity of the most effectively original species. Low values of the parameter α could thus be particularly relevant to obtain biodiversity indicators directed to the preservation of rare and distinct species while maintaining a high level of global diversity (e.g., Hidasi-Neto, Loyola and Cianciaruso,

  ij d , the sum of branch lengths on the path from tip j to its most recent common ancestor with species i. I  was first developed by Pavoine et al. (2009) and expressed in terms of species originalities by Kondratyeva et al. (2019). c See Chao et al. (2010) for functions related to E   .

Fig. 1 .

 1 Fig.1. Theoretical illustration of the concept of effective originality. We considered 10 theoretical species that form the species pool (panel a). We considered the morphological aspects of the species, imposing (panel b) that none of the species is exactly similar to another in the pool but that the species represented by the same symbols share many morphological characteristics. Two species are black with white broken lines and have the most distinct morphology in the species pool (panel c). We considered an assemblage where species numbered 1 is 30 times more abundant than all other species (panel d). Due to the skewed distribution of abundance, in this assemblage the white species with black lines have the highest effective originality (panel e).

Figure

  Figure size: 1.5 column

Fig. 2 .

 2 Fig. 2. Illustration of the measurement of effective originality of a species j according to index   j O .

O

  amounts to the surface of the shaded area below the curve. As expected, the effective originality of species j thus decreases from Scenario a, through Scenario b, to Scenario c.

FigureFig. 3 .

 3 Figure size: Double-column

FigureFig. 4 .

 4 Figure size: Double-column

FigureFig. 5 .

 5 Figure size: 1.5 column

Table 2

 2

	) reduces to

j

H is the sum of branch lengths from the focal species j to the root of the tree. This links, in a unified framework,

Ricotta et

al. K, Kondratyeva et al. O and Cadotte et al. AED measures of abundance-weighted originality. I  provides thus a consistent alternative to K   in the special case of a phylogenetic tree and we depict its writing in terms of rarity, distinctiveness and effective originality in Table 2. Similarly, we introduce in Table 2 a rewriting (named Y  ) of index K 

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or non-for-profit sectors.

Data statement

R scripts and data have been placed in Appendixes C to E. We also integrated scripts in package 'adiv' [START_REF] Pavoine | adiv: An R package to analyse biodiversity in ecology[END_REF][START_REF] Pavoine | adiv: Analysis of Diversity. R package version 2[END_REF] into functions named 'distinctAb ', 'distinctDis', 'distinctTree', 'FPdivparam', 'plot.FPdivparam' as indicated in Appendix C, and data in the dataset named 'batcomm'.


Morgane Mokhtari

Christophe Le Bourlot
email: christophe.le-bourlot@insa-lyon.fr

Jérome Adrien

Anne Bonnin

Takeshi Wada

Jannick Duchet-Rumeau

Hidemi Kato

Eric Maire

Microstructure characterization by X-ray tomography and EBSD of porous FeCr produced by liquid metal dealloying

Keywords:

Liquid metal dealloying is a promising technique to get nanoporous materials which are attractive for their excellent functional properties. From a (FeCr) x -Ni 1-x precursor, it is possible to get porous FeCr with this technique.

While immersing precursors in a molten Mg bath, Ni atoms selectively migrate into the Mg bath. After cooling down to room temperature, the resulting microstructure is a bi-continuous structure of FeCr and Mg (with Ni in solid solution). A final etching step removes the Mg solid-state solution phase. Precursors with different compositions were dealloyed and imaged in 3D. From X-ray tomography images, porous FeCr characteristics as solid fraction, phase size, specific surface were extracted. EBSD maps were acquired on samples before and after dealloying.

We will show that precursor composition is a key parameter to control pores and grains size while dealloying parameters (time and temperature) are key parameters to control ligaments size and in fine specific surface. By controlling precursor composition and dealloying parameters it is now possible to get dealloyed nanoporous metals with desired grain and porous microstructure by liquid metal dealloying.

Introduction

Nanoporous metals have attracted considerable attention for their excellent functional properties including high catalytic activity, sensing capabilities and surface-enhanced Raman scattering [START_REF] Erlebacher | Hard materials with tunable porosity[END_REF][START_REF] Snyder | Oxygen reduction in nanoporous metal-ionic liquid composite electrocatalysts[END_REF][START_REF] Lang | Nanoporous metal/oxide hybrid electrodes for electrochemical supercapacitors[END_REF][START_REF] Chen | Nanoporous copper with tunable nanoporosity for SERS applications[END_REF]. Properties at the nanoscale differ significantly from those of their dense, bulky counterparts. Liquid metal dealloying is a promising technique to get nanoporous common materials. The liquid metal dealloying method is a selective dissolution phenomenon of a solid alloy precursor. One component (referred to as the soluble component) is soluble in the metallic melt while the other (referred to as the targeted component) is not. When the solid precursor is immersed in the metallic melt, only atoms of the soluble component dissolve into the melt inducing a spontaneously organized bi-continuous microstructure formed by the targeted and sacrificial phases (solidified metallic bath). After cooling down to room temperature, this sacrificial phase can be removed by selective chemical etching to obtain the final porous material with the targeted composition. It has enabled the preparation of micro/nanoporous structures of pure metals (Ti, Fe, Cr, Nb, Ta) [START_REF] Wada | Dealloying by metallic melt[END_REF][START_REF] Wada | Three-dimensional open-cell macroporous iron, chromium and ferritic stainless steel[END_REF][START_REF] Kim | Sub-micron porous niobium solid electrolytic capacitor prepared by dealloying in a metallic melt[END_REF][START_REF] Mccue | Kinetics and morphological evolution of liquid metal dealloying[END_REF], alloys (TiCrZr, TiZr, TiNb, TiHf, FeCr, NiCr) [START_REF] Wada | Three-dimensional open-cell macroporous iron, chromium and ferritic stainless steel[END_REF][START_REF] Okulov | Dealloying-based interpenetrating-phase nanocomposites matching the elastic behavior of human bone[END_REF][START_REF] Okulov | Dealloyingbased metal-polymer composites for biomedical applications[END_REF][START_REF] Mokhtari | Cold-rolling influence on microstructure and mechanical properties of NiCr-Ag composites and porous NiCr obtained by liquid metal dealloying[END_REF] and non-metals (Si, C) [START_REF] Wada | Bulk-nanoporoussilicon negative electrode with extremely high cyclability for lithium-ion batteries prepared using a top-down process[END_REF][START_REF] Yu | Three-dimensional bicontinuous porous graphite gerenated in low teperature metallic melt[END_REF].

The study of the microstructure of these porous materials is not easy because it requires 3D information. The solid fraction morphology of porous TiCrZr and NiCr was already 3D imaged [START_REF] Mokhtari | Cold-rolling influence on microstructure and mechanical properties of NiCr-Ag composites and porous NiCr obtained by liquid metal dealloying[END_REF][START_REF] Chen-Wiegart | 3D morphological evolution of porous titanium by x-ray micro-and nano-tomography[END_REF]. The dealloying mechanisms of Ta-Ti with different Ta/Ti ratio have been studied and a similar grain microstructure is observed before and after dealloying [START_REF] Mccue | Kinetics and morphological evolution of liquid metal dealloying[END_REF].

The mechanical, electrochemical properties of these porous materials are directly linked to their microstructure. The dealloying mechanism of FeCr-Ni was already studied in liquid and solid Mg bath [START_REF] Wada | Three-dimensional open-cell macroporous iron, chromium and ferritic stainless steel[END_REF][START_REF] Wada | Evolution of a bicontinuous nanostructure via a solidstate interfacial dealloying reaction[END_REF][START_REF] Wada | Preparation of hierarchical porous metals by twostep liquid metal dealloying[END_REF]. Morphology of liquid metal dealloying resulting porous structure was mainly analyzed in 2D so far in the literature. In order to extract more information on porous structures, 3D analysis is available today and has been shown to be preferable. After a preliminary study on Titanium alloys [START_REF] Chen-Wiegart | 3D morphological evolution of porous titanium by x-ray micro-and nano-tomography[END_REF], FeCr-Ni, the material of the present study, was observed by X-ray tomography in [START_REF] Chonghang | Three-dimensional morphological and chemical evolution of nanoporous stainless steel by liquid metal Dealloying[END_REF]. In this previous study however, the authors focused on short dealloying time with the evolution of pores and residual Ni at the early stage of dealloying. Contrarily to this previous work, in the present study, we focus on long dealloying time. This enables us to have a coarsened structure with no residual sacrificial element inside the ligaments. In the present paper we also aim at analyzing the microstructure of the solid phase. In this respect, grain microstructure is very important to explain dealloying mechanisms and the resulting mechanical behavior. To study the grain microstructure, one of the most appropriate tools is EBSD (Electron BackScatter Diffraction) analysis. EBSD analysis on porous materials made by liquid metal dealloying has not been reported yet in the literature.

In the present paper EBSD was used to analyze grain microstructure before and after dealloying and X-ray tomography to image porous microstructure in 3D. The analysis for different precursor compositions was conducted to be able to predict sample microstructure and in fine sample properties.

Materials and Methods

A Fe-Cr-Ni alloy (the precursor) and a Mg molten bath were used in this work. The strategy for selecting precursor and molten bath element is described in [START_REF] Wada | Dealloying by metallic melt[END_REF]. (Fe 80 Cr 20 ) 30 Ni 70 , (Fe 80 Cr 20 ) 50 Ni 50 and (Fe 80 Cr 20 ) 70 Ni 30 ingots were prepared using arc melting as described in [START_REF] Wada | Three-dimensional open-cell macroporous iron, chromium and ferritic stainless steel[END_REF]. The resulting precursors of around 1 cm thickness were cold-rolled until reaching the thickness of 1 mm. For EBSD analysis one part of precursors were heat-treated 5 h at 1093 K. For dealloying, cold-rolled precursors 1 mm thick sheets were directly cut without further heat treatment into 30 * 20 mm rectangular samples.

The dealloying step was carried out on these samples with a temperature of 1093 K for 1 h in a Mg melt bath under a high-purity He atmosphere to give FeCr-Mg bicontinuous structure.

For EBSD precursor samples, cross sections were mechanically polished (the final step was performed using a colloid alumina solution). After dealloying, the cross sections were polished using an Ar ion beam polisher (EM-3500 Hitachi, Japan). The EBSD acquisitions were performed on JEOL 7100F (JEOL, Japan) and were analyzed with OIM analysis software.

For X-ray tomography imaging, samples were cut with a precision micro-cutting machine into a rectangular rod with 0.5-1 mm in sides and 1 cm in length and then polished into a tip shape. The selective etching step was carried out using highly concentrated nitric acid to dissolve the Mg based solid-state solution. This etching resulted in microporous Fe 80 Cr 20 .

To validate the use of laboratory X-ray source, X-ray tomography measurement were conducted both using synchrotron X-ray source on Tomcat beamline (SLS, PSI, Swiss) and using laboratory X-ray source on an easytom tomograph (RX Solutions, France). Acquisitions in Tomcat were done with an monochromatic source of 25 keV and a voxel lateral isotropic size of 0.163 μm and they were reconstructed in 32 bit floating point representation from 2D projections using optimized software based on Fourier methods using a Simultaneous phase and amplitude extraction ("paganin") [START_REF] Hintermüller | Image processing pipeline for synchrotron-radiation-based tomographic microscopy[END_REF][START_REF] Marone | Fast reconstruction algorithm dealing with tomography artifacts[END_REF][START_REF] Marone | Regridding reconstruction algorithm for real time tomographic imaging[END_REF][START_REF] Paganin | Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object[END_REF]. Acquisition on an easytom tomograph were done using a polychromatic source operated at a voltage of 100 kV and a voxel lateral isotropic size of 0.3 μm and the reconstructions were obtained using the commercial software provided with the tomograph.

The reconstructions were analyzed with Fiji free software [START_REF] Schindelin | Fiji: an open-source platform for biological-image analysis[END_REF] on 90 μm side cubes. All compositions were imaged and analyzed on both devices and quantitative analysis gave similar results for both tomographs. For the sake of conciseness, only results provided by easytom tomograph from (Fe 80 Cr 20

Results and Discussion

Grain Microstructure Characterization

Cold-rolled Samples

In this part the EBSD maps on precursor samples were acquired directly after cold-rolling without recrystallization. Fig. 1 presents the grain microstructure before and after dealloying with Index Quality (IQ) + Inverse Pole Figure (IPF) maps. Before dealloying the microstructure is highly affected by cold-rolling: grains are elongated along the cold-rolling direction and the degree of indexation is low: most of the maps are black i.e. non-indexed. This result is similar to others coldrolled samples [START_REF] Barekar | Microstructural evaluation during melt conditioned twin roll casting (MC-TRC) of Al-Mg[END_REF][START_REF] Suwas | Crystallographic Texture of Materials[END_REF][START_REF] Zhao | Effects of intercritical annealing temperature on mechanical properties of Fe-7.9Mn-0.14Si-0.05Al-0.07C steel[END_REF]. The influence of the composition of precursor samples on EBSD map is not evident. Precursor samples present only FCC phase while after dealloying samples present only BCC phase [START_REF] Wada | Preparation of hierarchical porous metals by twostep liquid metal dealloying[END_REF]. After dealloying 30%FeCr and 50%FeCr present small randomly oriented grains composed of a small number of ligaments while 70%FeCr presents large grains, each grain including many ligaments of similar orientation.

From phase diagrams, 30%FeCr-Ni and 50%FeCr-Ni should be FCC phase at room and dealloying temperature without allotropic transformation between both. However 70%FeCr-Ni should be composed of both FCC (main) and BCC phase at room temperature [START_REF] Marsh | Limits of new phase in the Fe-Cr-Ni system[END_REF][START_REF] Hayes | Thermodynamics of stainless steels[END_REF]. At higher temperature (over 673 K) 70%FeCr-Ni is composed of only FCC phase [START_REF] Pugh | Iron-chromium-nickel ternary system[END_REF]. Because of low inter-diffusion coefficient between Fe and Ni, it is probably more difficult to reach equilibrium at low temperature. This is a good explanation why BCC phase could not be observed [START_REF] Cacciamani | The Fe-Ni system: thermodynamic modelling assisted by atomistic calculations[END_REF]. From phase diagrams again [START_REF] Marsh | Limits of new phase in the Fe-Cr-Ni system[END_REF][START_REF] Massalski | Fe-Cr phase diagram[END_REF], Fe 80 Cr 20 is composed of BCC phase, in agreement with our EBSD analysis.

Table 1 presents the average grain size measured by two methods (i) given by OIM analysis software where grain are assimilated to circle (ii) calculated from the average of six manual measurements on the EBSD map. One grain for each map after dealloying is circled in white in Fig. 1 to highlight what we considered as a grain in this manual measurement. We can see when the amount of solid increases in the porous sample, the difference between the two methods decreases. In fact, for 30%FeCr, the measurement given by the software corresponds to ligaments size and not grain size because the analysis is done in 2D and the connection between ligaments is not obvious while for 70%FeCr even in 2D the connection between ligaments is present: OIM analysis software could capture the full porous grain.

Before dealloying grains are highly affected by cold-rolling. Because dealloying is performed at high temperature, we can expect a recrystallization of the precursor sample prior or during dealloying. The difference between 30%/50%FeCr-Ni and 70%FeCr-Ni could be explained by different recrystallization rate or/and dealloying rate. This material (70%FeCr-Ni), could indeed have time to fully recrystallize before dealloying while for the two other ones (30%/50%FeCr-Ni), recrystallization could be slower.

After Recrystallization

In this part, cold rolled precursor samples were subsequently recrystallized 5 h at 1093 K under highly vacuumed atmosphere and then air quenched. Fig. 2 presents the grain microstructure before and after dealloying with Index Quality (IQ) + Inverse Pole Figure (IPF) maps. After recrystallization, grains exhibit a rather polygonal shape with an average diameter of around 25 μm which are typical for recrystallized austenite [START_REF] Abedi | Continuous dynamic recrystallization in low density steel[END_REF][START_REF] Qin | Electron backscattered diffraction analysis of the effect of deformation temperature on the microstructure evolution in a typical nickel-based superalloy during hot deformation[END_REF][START_REF] Shi | Twin boundary characters established during dynamic recrystallization in a nickel alloy[END_REF]. Before dealloying the effect of the composition on the microstructure is not strong. Samples also present only a FCC structure. After dealloying, only BCC phase is presented. 30%FeCr and 50%FeCr presents small grains composed of a couple of ligaments which look randomly oriented while 70%FeCr presents large grains including many ligaments. This result is different compared to previous studies where no microstructure change was observed between precursor and dealloyed samples [START_REF] Mccue | Kinetics and morphological evolution of liquid metal dealloying[END_REF][START_REF] Van Petegem | On the microstructure of nanoporous gold: an X-ray diffraction study[END_REF][START_REF] Gwak | Weakened flexural strength of nanocrystalline nanoporous gold by grain refinement[END_REF]. In our case however, dealloying occurs at high temperature and phase transformation occurs between precursor and dealloyed samples. Precursor shape was almost preserved during dealloying and shrinkage was almost not noticed after dealloying for all samples, with no influence of the recrystallization step. This makes a difference compared to the dealloying of cold-rolled Au-Pd-Ag sheets where a large shrinkage was observed [START_REF] Detsi | Enhanced strain in functional nanoporous gold with a dual microscopic length scale structure[END_REF]. This difference between our results and the ones from chemical dealloying can be explained by the very low targeted element amount (5 at.% Au in precursor). Moreover liquid metal dealloying is done at high temperature, which plays a similar role than heat treatment on the microstructure.

While comparing Figs. 1 and 2 after dealloying, no significant difference is observed. This means that the different behavior between 30%/50%FeCr and 70%FeCr could not be explained by the absence of recrystallization treatment after cold-rolling step. To explain this difference, another hypothesis can be postulated: if one considers the same interface precursor/dealloyed sample, when the amount of FeCr is low, Fe and Cr can be considered as spatially isolated. Moreover for low FeCr amount in precursor, dealloying front rate is high and this promotes diffusion and displacive transformation. As a consequence, nucleation of new grains is easy and grain size is rather small. On the contrary when the amount of FeCr is high and the dealloying rate low (controlled by the Ni diffusion rate in the solid phase), a diffusive phase transformation is observed (Fig. 2c) with large grains due to grain growth rather than nucleation of new grains [START_REF] Fisher | Fundamentals of Solidification[END_REF]. So the difference of behavior could be partly explained by this competition between crystal nucleation and crystal growth. For 30/50%FeCr-Ni samples, nucleation would be the dominant mechanism while for 70%FeCr-Ni sample, that would be grains growth.

Morphology Characterization

Qualitative Analysis

Fig. 3 shows one reconstructed slice for each sample extracted from the reconstruction. Two phases are visible. The lighter phase corresponds to the FeCr phase and the darker phase to the air. The amount of white phase is qualitatively coherent with the samples expected composition. Fig. 3a obtained using the synchrotron tomograph seems to have similar FeCr volume fraction and ligaments size to Fig. 3b ones (same type of sample but scanned using our laboratory tomograph). In the case of the synchrotron image, the contrast is better.

Table 1

Comparison between grain size given by the software and measured on the map. For 30 and 50% samples, the software actually measures the ligament size.

Quantitative Analysis

Volume was filtrated with a bilateral filter plugin added in Fiji. The intensity curve with gray level presents two peaks (one for each phase) and the threshold value was chosen between these two peaks.

Table 2 summarizes an inventory of all parameters measured from X-ray tomography images. Ligaments/pores size and connectivity were measured with Fiji function/plugin "local thickness" and "find connected area" [START_REF] Schindelin | Fiji: an open-source platform for biological-image analysis[END_REF][START_REF] Hildebrand | A new method for the model-independent assessment of thickness in three-dimensional images[END_REF][START_REF] Saito | New algorithms for Euclidean distance transformation on an n-dimensional digitized picture with applications[END_REF]. Volume fraction, specific surface and tortuosity were measured with homemade plugins described in [START_REF] Maire | X-ray tomography applied to the characterization of highly porous materials[END_REF]. Euler number was measured with Morpholibj plugin [START_REF] Legland | MorphoLibJ: integrated library and plugins for mathematical morphology with ImageJ[END_REF].

The value of porous FeCr volume fraction is close to the one expected from the FeCr/Ni atomic fraction of the precursor. During the dealloying reaction, Ni atoms are removed from the FeCr-Ni lattice and the remaining FeCr atoms reconstruct a FeCr lattice. According to the reported lattice parameter of FeCr-Ni and FeCr, the volume of the unit cell divided by the number of atoms, the theoretical FeCr volume fraction divided by precursor atomic fraction is 1.03. The experimental results are in good agreement with this predicted value [START_REF] Wada | Preparation of three-dimensional nanoporous Si using dealloying by metallic melt and application as a lithium-ion rechargeable battery negative electrode[END_REF][START_REF] Doroeolskii | Solid solutions in the V-Ti, V-Fe, Fe-Cr and Fe-Ni systems[END_REF][START_REF] Dyson | Effect of alloying additions on the lattice parameter of austenite[END_REF][START_REF] Mohammadzadeh | Correlation between stacking fault energy and lattice parameter in nanocrystalline Fe-Cr-Ni austenitic stainless steels by atomistic simulation study[END_REF].

Fig. 4 (left) presents the phase thickness distribution of FeCr and Air phases for all samples. As already mentioned in literature for a similar process, phases display an unimodal and somewhat scattered thickness distribution [START_REF] Mokhtari | Cold-rolling influence on microstructure and mechanical properties of NiCr-Ag composites and porous NiCr obtained by liquid metal dealloying[END_REF][START_REF] Chen-Wiegart | 3D morphological evolution of porous titanium by x-ray micro-and nano-tomography[END_REF][START_REF] Liu | A general dealloying strategy to nanoporous intermetallics, nanoporous metals with bimodal, and unimodal pore size distributions[END_REF]. In Table 2, the average phase thickness of both phases is reported. We can firstly notice that even if resolution and contrast were higher with SLS facilities, there is no incidence on our phase distribution. This means that even if the lab tomography shows a poorer contrast, segmenting our both phases correctly was still possible. Remarkably, FeCr phase thickness distributions are similar for all samples. This clearly shows that in our case, ligaments size are independent of precursor composition, it was also noticed in ref. [START_REF] Chonghang | Three-dimensional morphological and chemical evolution of nanoporous stainless steel by liquid metal Dealloying[END_REF]. Because phase thickness distributions are similar and materials densities are different, pore distributions must on the contrary be dependent on the composition of the precursor. If we assimilate our structure to a perfect alternation of ligaments and pores, the pore size divided by ligament size is equal to pore fraction divided by FeCr fraction. We calculated the theoretical pore/ligament size for various FeCr amount. Then, we plotted our experimental values and the theoretical ones in Fig. 4 (right). Our experimental values match fairly well the theoretical ones. This allows us to validate a rule for controlling pore size by simply tailoring the composition, time and temperature of dealloying.

The average ligament size is 4.8 ± 0.3 μm for 1 h dealloying at 1093 K. In ref. [START_REF] Wada | Three-dimensional open-cell macroporous iron, chromium and ferritic stainless steel[END_REF], Fe 30 Ni 70 precursors were dealloyed in similar conditions to our study so some of their result can be extrapolated. Fig. 3b in their paper shows the evolution of the ligament size (measured by SEM), for different dealloying times. By using Eq. ( 3) of this paper for the time and temperature used in our study, the expected ligament size is 5.08 μm [START_REF] Wada | Three-dimensional open-cell macroporous iron, chromium and ferritic stainless steel[END_REF]. This is very close to what we have measured in Table 2. Similar ligament size for different precursor compositions suggests similar activation energy. This series of results shows that the addition of Cr has a limited influence on the coarsening behavior.

One of the main interests of microporous materials is their large specific surface area. Specific surface in (m 2 /kg) and (m 2 /m 3 ) are displayed in Table 2. For different compositions, specific surface is different while ligaments size is constant. This means that the specific surface depends on sample density. These values of specific surface are comparable to porous NiCr obtained with the same methods [START_REF] Mokhtari | Cold-rolling influence on microstructure and mechanical properties of NiCr-Ag composites and porous NiCr obtained by liquid metal dealloying[END_REF] but very low compared to nanoporous gold obtained by dealloying in aqueous solution [START_REF] Lakshmanan | Surface area of nanoporous gold: effect on temperature[END_REF]. In the present study, our samples, were not particularly designed to get very small ligaments. However if necessary, ligament size can be easily decreased for example by using solid state inter-facial dealloying reaction (reported ligament size down to 400 nm in [START_REF] Wada | Three-dimensional open-cell macroporous iron, chromium and ferritic stainless steel[END_REF]) or even liquid metal dealloying at lower temperature and shorter dealloying time (reported ligament size down to 650 nm in [START_REF] Wada | Evolution of a bicontinuous nanostructure via a solidstate interfacial dealloying reaction[END_REF]).

Detsi et al. linked the specific surface area (S) with the porous sample's characteristics: ligament size (d), bulk density (ρ) and a constant C in Eq. ( 1) [START_REF] Detsi | On the specific surface area of nanoporous materials[END_REF].

= * S C ρ d (1)
Specific surface vs B is plotted in Fig. 5a. Graphically, the value of C constant can be extracted. We measure C = 3.7 ± 0.4. Values reported in the literature for nanoporous gold is C = 3.7 and C = 3.4 for the simulation of single gyroid nanostructure [START_REF] Lakshmanan | Surface area of nanoporous gold: effect on temperature[END_REF][START_REF] Detsi | On the specific surface area of nanoporous materials[END_REF][START_REF] Tan | Surface area and pore size characteristics of nanoporous gold subjected to thermal, mechanical, or surface modification studied using gas adsorption isotherms, cyclic voltammetry, thermogravimetric analysis, and scanning electron microscopy[END_REF]. This clearly shows that the structure of our materials is close to single gyroid structure. Finding a similar C for nanoporous gold and our material shows one more similarity in the structure of materials elaborated by different dealloying processes. Eq. ( 1) is therefore validated for our material and can thus be used to calculate the suitable ligament size for a given specific surface.

For various applications where the transport of electrical charges is a key property (such as capacitors), it is important to have a perfectly connected structure and a high specific surface [START_REF] Erlebacher | Hard materials with tunable porosity[END_REF][START_REF] Snyder | Oxygen reduction in nanoporous metal-ionic liquid composite electrocatalysts[END_REF][START_REF] Lang | Nanoporous metal/oxide hybrid electrodes for electrochemical supercapacitors[END_REF][START_REF] Chen | Nanoporous copper with tunable nanoporosity for SERS applications[END_REF]. Connectivity of both phases for all samples is equal to 1. This value validates the fact that our materials exhibit a bi-continuous structure. The connectivity value is a good way to measure if the structure is perfectly connected or not. Euler characteristic and genus are also often used for this purpose [START_REF] Michielsen | Aspects of mathematical morphology[END_REF][START_REF] Mangipudi | A FIB-nanotomography method for accurate 3D reconstruction of open nanoporous structures[END_REF][START_REF] Chen-Wiegart | Morphological and topological analysis of coarsened nanoporous gold by x-ray nanotomography[END_REF][START_REF] Lilleodden | On the topological, morphological, and microstructural characterization of nanoporous metals[END_REF]. Genus is calculated as (Euler number / 2 + 1) and corresponds to the number of connections inside the structure [START_REF] Mangipudi | A FIB-nanotomography method for accurate 3D reconstruction of open nanoporous structures[END_REF][START_REF] Chen-Wiegart | Morphological and topological analysis of coarsened nanoporous gold by x-ray nanotomography[END_REF][START_REF] Lilleodden | On the topological, morphological, and microstructural characterization of nanoporous metals[END_REF]. We measured respectively around 500, 1300 and 3000 connections for 30% FeCr, 50%FeCr and 70%FeCr. As expected, the number of connections increases with the amount of solid phase in the sample. However the genus of 70%FeCr is 6 times higher than the one of 30%FeCr whereas the ratio of the amount of solid phase is only 2.3. This means that 70% FeCr is better connected than 30%FeCr. Dividing the genus per unit volume with the cube of the specific area provides a dimensionless genus called scaled genus. Our scaled genus (0.02 to 0.06) is comparable with the one's found for nanoporous gold (from 0.0015 to 0.147) [START_REF] Mangipudi | A FIB-nanotomography method for accurate 3D reconstruction of open nanoporous structures[END_REF][START_REF] Chen-Wiegart | Morphological and topological analysis of coarsened nanoporous gold by x-ray nanotomography[END_REF][START_REF] Lilleodden | On the topological, morphological, and microstructural characterization of nanoporous metals[END_REF].

Transport properties, diffusivity and conductivity properties are depending on tortuosity. The effective conductivity (C eff ) and diffusivity (D eff ) are linked with the intrinsic conductivity (C i ) and diffusivity (D i ), the tortuosity (τ) and the phase fraction of the conductive phase (p) in respectively Eqs. ( 2) and ( 3

) [54] = * C C p τ eff i (2) = * D D p τ eff i ( 3
)
The tortuosity (τ) is a factor only depending on the porous sample's geometry. Tortuosity of both phases can be found in Table 2. Tortuosity is defined by the ratio of the length of the curve to the Euclidian distance between the ends of the curve. A tortuosity equal to one corresponds to pores parallel to the transport direction as for example honey comb cells. A tortuosity included between one and three means than the transports properties are lower than the intrinsic transport properties of the conductive phase and also relatively good connectivity and low meandering of the pores [START_REF] Xue | Switchable imbibition in nanoporous gold[END_REF]. A tortuosity equal to 3 corresponds to an isotropic spatial distribution of the pores [START_REF] Bear | Dynamics of Fluids in Porous Media[END_REF]. In our case, bicontinuous structure does not enable to have isotropic pores/matters so this is consistent with a tortuosity below 3. All the values that we measure for τ are rather small, between 1 and 1.3. The low tortuosity of our materials suggests from Eqs. ( 2) and (3) that the material should exhibit good diffusion and transport properties compared to materials with higher tortuosity. Our tortuosity is comparable to the one of microporous TiCrZr obtained by liquid metal dealloying [START_REF] Chen-Wiegart | 3D morphological evolution of porous titanium by x-ray micro-and nano-tomography[END_REF].

In 1935, the volume fraction (ρ) and the tortuosity was linked by [START_REF]Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen[END_REF] in Eq. ( 4) where γ and a are constant depending on the morphology, porosity, material, and particle-size distribution of the material. Usually people take γ = 1 and adjust a, or a = 1.5 and adjust γ. The simplest case corresponds to γ = 1 and a = 1.5 [14,54] Except some small areas (around 2 vol%) with a higher tortuosity, tortuosity is small.

= * - τ γ ρ a 2 1 (4)

Conclusion

From (FeCr) x -Ni 1-x with x = 30, 50 and 70 precursors were dealloyed 1 h at 1093 K in Mg melt. EBSD maps were acquired on samples before and after dealloying. A phase transformation happened during the dealloying process: precursors samples were FCC phase while dealloyed samples were BCC phase. Different behavior were observed depending on precursor composition: 30 and 50%FeCr present small grains while 70%FeCr presents large grains resulting in different grain formation mechanisms. Etched samples were 3D imaged by Lab and synchrotron X-ray tomography. Porous sample morphological parameters were extracted and similar results were given by both Lab and synchrotron tomography. From our analysis porous sample morphological characteristics can be extracted: volume fraction is directly linked to precursor composition. Dealloying time and temperature control the ligaments size while precursor composition controls the pore size. Specific surface is linked to ligament size by a well-known equation and suitable specific surface can be obtained calculating the dealloying time and temperature. Specific surface area of porous FeCr is very close to the one of single gyroid microstructure. Our tortuosity and connectivity are close to 1 and independent of the volume fraction. This proves the presence of a bicontinuous structure and shall lead to good transport properties.

Precursor composition is a key parameter to control pore and grain size while dealloying parameters (time and temperature) are key parameters to control ligament size and in fine specific surface. In conclusion, this study shows how to design porous FeCr with desired grain size and morphology.

  ) 30 Ni 70 , (Fe 80 Cr 20 ) 50 Ni 50 and (Fe 80 Cr 20 ) 70 Ni 30 precursors and as comparison from (Fe 80 Cr 20 ) 30 Ni 70 precursors imaged on Tomcat (referred to as 30%FeCr-Ni SLS and 30%FeCr SLS for precursor and porous samples respectively) will be shown in Section 3. (Fe 80 Cr 20 ) 30 Ni 70 , (Fe 80 Cr 20 ) 50 Ni 50 and (Fe 80 Cr 20 ) 70 Ni 30 precursors will be referred to as 30%FeCr-Ni, 50%FeCr-Ni, 70%FeCr-Ni respec-Cr 20 ) 30 Ni 70 , (Fe 80 Cr 20 ) 50 Ni 50 and (Fe 80 Cr 20 ) 70 Ni 30 precursors will be referred to as 30%FeCr, 50%FeCr and 70%FeCr respectively.

Fig. 1 .

 1 Fig. 1. IQ + IPF EBSD map before and after dealloying from precursors after cold-rolling without recrystallization from (a) 30%FeCr-Ni (b) 50%FeCr-Ni (c) 70% FeCr-Ni samples. Arrows correspond to cold-rolling direction (RD) and dealloying reaction direction (DD). White circle corresponds to one grain.

Fig. 2 .

 2 Fig. 2. IQ + IPF EBSD map before and after dealloying from precursors after cold-rolling with recrystallization from (a) 30%FeCr-Ni (b) 50%FeCr-Ni (c) 70%FeCr-Ni samples. Arrows correspond to cold-rolling direction (RD) and dealloying reaction direction (DD).

Fig. 3 .

 3 Fig. 3. Images extracted from X-ray tomography scan of (a) SLS 30%FeCr (b) 30%FeCr (c) 50%FeCr (d) 70%FeCr.

2 Fig. 4 .

 24 Fig.[START_REF] Chen | Nanoporous copper with tunable nanoporosity for SERS applications[END_REF]. Evolution of phase thickness for different volume fraction (left) and evolution of pores/ligaments size ration with volume fraction (right).

Fig. 5b corresponds

  Fig. 5b corresponds to the plot of Eq. (2). Our value shows almost a good agreement with Bruggeman law. Fig. 5c corresponds to the air phase tortuosity value of 70%FeCr. White points correspond to the highest tortuosity measure (around 2.2) and dark blue to the lowest (1).

Fig. 5 .

 5 Fig. 5. (a) Specific surface as a function of the ligament size. (b) Evolution of tortuosity factor of air phase with volume fraction. The line corresponds to Bruggeman relation with γ = 1 and a = 1.5. (c) Air phase tortuosity values map of 70%FeCr; white corresponds to highest tortuosity and dark blue to the lowest. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Table 2

 2 Morphological samples parameters.

		SLS 30%FeCr	30%FeCr	50%FeCr	70%FeCr
	Volume fraction (vol%)	0.31 ± 0.03	0.28 ± 0.03	0.46 ± 0.03	0.68 ± 0.03
	Average ligaments size (μm)	5.0 ± 1.5	4.5 ± 1.3	5.0 ± 1.2	4.8 ± 1.6
	Average pore size (μm)	6.2 ± 2.0	6.5 ± 2.2	4.1 ± 1.2	3.0 ± 0.6
	Connectivity FeCr/air ( ± 0.0002)	1.0000/1.0000	1.0000/1.0000	1.0000/1.0000	1.0000/1.0000
	Genus ( ± 100)	4.10 2	5.10 2	13.10 2	32.10 2
	Scaled genus ( ± 0.02)	0.02	0.02	0.03	0.06
	Specific surface (m 2 /kg)	121 ± 4	138 ± 4	109 ± 3	79 ± 2
	Specific surface (m 2

Acknowledgement

This work was supported by Région Rhone-Alpes [CMIRA 138249], Erasmus+ program and the French Ministry of Higher Education and Research.