From varieties in contact to the selection of linguistic resources in multilingual settings

Isabelle Léglise (CNRS, SeDyL) & Santiago Sánchez Moreano (Université Paris 3 Sorbonne Nouvelle,SeDyL) This chapter focusses on various methodologies we can rely on to study heterogeneity and dialectal variation among multilingual speakers. We first focus on linguistic variation in contact settings and present a methodology to describe heterogeneous and multilingual corpora and show how languages sometimes overlap. A second part focusses on (dialectal) language boundaries and how speakers may sometimes use unmarked elements showing fuzziness or reorganization of language boundaries. The role of ideology is highlighted in discourse but also at play in language practices and in doing-being 'multilingual', 'urban and modern' or performing authenticity. The third part focusses on how speakers use dialectal and linguistic resources from their linguistic repertoire in their everyday life interactions as stances and acts of identity.

1) Variation in multilingual settings

Although we know societal multilingualism is the norm and monolingualism is the exception, a special case, linguistic description, from the 16th century onwards, focused on languages or dialects (in fact a language or a dialect at a time) by forgetting the multilingual environment of the speakers. The "persistent use of language as a synecdoche for community" (Gal & Irvine 1995, 968) relies on a pervasive ideology: to consider languages as spoken within monolingual and monocultural communities. This ideological construct is still very lively although it was criticized 50 years ago because it leaves on the side questions of variation, multilingualism, and social construction of language [START_REF] Hymes | Why Linguistics Needs the Sociologist[END_REF]). So we should conceptualize communities as imagined entities, or imagined communities [START_REF] Anderson | Imagined Communities: Reflections on the Origin And Spread of Nationalism[END_REF].

Sociolinguistic research on variation has also mostly focused on monolingual population even if variationists knew the speech communities under consideration were heterogeneous and socially and linguistically diverse. The pioneering studies on social stratification in New York [START_REF] Labov | The Social Stratification of English in New York City[END_REF](Labov , 1972) ) or social differentiation of English in Norwich [START_REF] Trudgill | The Social Differentiation of English in Norwich[END_REF] excluded for example non-native speakers and focused on intra-varietal change (Labov 1994: 20). But communication ordinarily takes place not in monolingual linguistic communities perceived as homogeneous, but in a multilingual "contact zone" [START_REF] Pratt | Arts of the Contact Zone[END_REF]. Research on multilingualism and code-switching (Fishman 1967;Gumperz 1982;[START_REF] Myers-Scotton | Social Motivations for Codeswitching: Evidence from Africa[END_REF][START_REF] Auer | Code-Switching in Conversation[END_REF]Muysken 2000 among others) took this multilingual contact zone seriously. But it has been difficult to integrate the results of these different lines of research, perhaps because it is widely assumed that the "variability found in biand multilingual speech communities is more extensive than that found in monolingual and majority-language communities" (Sankoff 2002: 640)? There is still a relative lack of published research on variation in multilingual settings (but see for eg. [START_REF] Meyerhoff | Social Lives in Language. Sociolinguistics and Multilingual Speech Communities[END_REF][START_REF] Léglise | The Interplay of Variation and Change in Contact Settings[END_REF], except in the sub-field of dialects in contact [START_REF] Gumperz | Dialect Differences and Social Stratification in a North Indian Village[END_REF]Trudgill 1986;Siegel 1987;[START_REF] Mesthrie | Koinéization in the Bhojpuri-Hindi Diaspora, with Special Reference to South Africa[END_REF][START_REF] Kerswill | Creating a New Town Koine[END_REF]Auer, Hinskens & Kerswill 2005, etc.).

Most studies in contact linguistics and code-switching nevertheless assumed also languages as discrete and bounded systems in contact, and terms as matrix language or code-alternation are good examples of this hypothesis [START_REF] Léglise | Pratiques Langagières Plurilingues et Frontières de Langues[END_REF]. A shift in focus can be noticed within the last decade, from how bounded communities come into contact and with what kind of linguistic outcomes, to how social actors produce and negotiate social meanings, social positions and relations through their language practices. Within the scope of multilingualism, lots of terms were coined the last 10 years to give a voice to this heterogeneity such as crossing (Rampton 2005), trans-idiomatic practices [START_REF] Jacquemet | Language & Communication, Multilingualism and diasporic populationsSpatializing practices, institutional processes, and social hierarchies[END_REF], translanguaging (Garcia 2009, Garcia & Li 2013[START_REF] Creese | Translanguaging in the Bilingual Classroom: A Pedagogy for Learning and Teaching?[END_REF] or (poly)languaging [START_REF] Jørgensen | Polylanguaging in Superdiversity[END_REF]. [START_REF] Hall | Code-Switching, Identity, and Globalization[END_REF], in their review on 40 years of research on codeswitching, show very clearly how contemporary research, much centered on linguistic repertoires, wonderfully illustrates disruption of presumed connections between language, community, identity and spaces. This has of course an impact on how to deal with linguistic variation in multilingual settings, and the three sociolinguistic waves [START_REF] Eckert | Three Waves of Variation Study: The Emergence of Meaning in the Study of Linguistic Variation[END_REF]) took also the same way in monolingual environments, from the analysis of varieties of global communities to communities of practices and to individuals' practices.

In this paper we will address language variation as a linguistic resource in multilingual, heterogeneous language practices and look at what kind of methodology we can rely on. In the first part, we will focus on language tagging in plurilingual heterogeneous corpora before getting to two case studies: language practices among Maroons in French Guiana, and with Kichwas in Cali, Colombia. We will consider how social actors make use of their linguistic repertoire to communicateto what purpose and with what kind of construction of social meaninginstead of looking at the consequences of their "dialects in contact"1 .

2) Methods for analyzing variation in heterogeneous corpora

Following the French tradition in sociology of language and sociolinguistics, we consider language practices as social practices. Pratiques langagières is a term coined 40 years ago to insist on the fact that language practices are determined and constraint by social order and at the same time construct social meaning, produce social effects and contribute to change it (Boutet, Fiala & Simonin 1976). Social practices, social formations and symbolic power are to be understood here through a theory of practice such as the one of [START_REF] Bourdieu | Outline of a Theory of Practice[END_REF]. In a multilingual environment, these language practices are heterogeneous [START_REF] Léglise | The Interplay of Variation and Change in Contact Settings[END_REF] in the sense that they are made of linguistic resources attributable to various sources and due to the diverse linguistic repertoires of the languagers. Languaging may constitute a good translation. As [START_REF] Jørgensen | Polylanguaging in Superdiversity[END_REF] puts it: "[a languaging perspective] emphasises that people do not primarily use 'a language', or 'some languages', but use language, linguistic resources."

To describe precisely the linguistic made-up of these language practices, a methodology has been developed through various research programs2 . We use the term "plurilingual corpora" to designate the heterogeneous corpora we created based on heterogeneous language practices (i.e made of resources from multiple languages within the same recordings). They illustrate not only instances of codeswitching and codemixing but also languaging through the use of linguistic bricolage. They are performed by plurilingual speakers with varied skills, competence and diverse repertoires [START_REF] Blommaert | Repertoires Revisited: 'Knowing Language' in Superdiversity[END_REF]. They also exhibit variation and non-standard forms, often neglected by monolingual corpora, or controlled for by general parameters (such as the types of speech). These plurilingual corpora are still few in number, not readily available to the linguists' community, and rarely "processed" by the available computer software. They pose formidable problems, not only of the identification of forms but also of their transcription and annotation.

In corpus linguistics, the Text Encoding Initiative (TEI) proposes a set of standards for annotating corpora. We refer the reader to [START_REF] Vaillant | A la croisée des langues : Annotation et fouille de corpus plurilingues[END_REF] for technical details on the annotation system established but would like to mention a major adaptation for heterogeneous corpora. The TEI recommends identifying the basic language of each sentence, and noting in angle brackets when an item from another language is introduced, for instance as <foreign item belonging to language x>. A first problem, when faced to heterogeneous practices, is that the systematic allocation of one basic language for each speech turn is not possible (see [START_REF] Léglise | Plurilingual corpora and polylanguaging, when corpus linguistics meets contact linguistics[END_REF] for a discussion). In most cases, we observe several linguistic resources in the same speech turn produced by the same speaker. For example, in (1), a client is addressing an employee of the national electricity company, in Cayenne, French Guiana ; the first line can be attribute to Créole guyanais (a French-based Creole), the second line seems to begin in Creole and to end in French, whereas the third line seems to begin in French and to end in Creole. Here, bold stands for Créole guyanais, roman for French and italics+bold are for Antillan Creole insertions.

(1) Corpus EDF Clapoty -Nelson / Léglise3 a. Rather than selectingoften arbitrarilyone language as the matrix language (Myers-Scotton 1993b, see Nunez & Léglise 2017 for a discussion), we propose to consider that the utterance is multilingual and we represent this visually here by a frame. We, then, identify elements assignable to a particular language within each multilingual turn.

A second problem, when faced to heterogeneous practices, is that many elements can be attributed to various languages. For example, in (1), i at the beginning of the third line can be both interpreted as third person singular pronoun in spoken French (classical pronunciation of i for il) or in Creole (being Créole guyanais or Créole antillais). These homophonous diamorphs or bivalent elements have been long identified in the literature on code-switching (Muysken 1990, Woolard 1998) but never treated as such in corpus linguistics. We follow here the proposition of [START_REF] Ledegen | Prédicats "flottants" entre le créole acrolectal et le français à La Réunion : exploration d'une zone ambiguë[END_REF] to adopt a double or floating transcription in order to show the various possible interpretations, so the third line would offer the following possibilities:

(2)

c. i problème i té problem i(il) m' a donné […] comme té ni problem 3SG 1SG have given as if.CONJ TE.PST have problem.N (

s)he gave me […] as if there was a problem

We see clearly here how languages sometimes overlap and imagine what it could imply for the study of linguistic variation.

3) (Un)marking dialectal language boundaries

As a first case study, we will focus now on the Maroon people in French Guiana who speak 3 related English-lexified Creoles that originally emerged on the plantations of Suriname, namely Aluku, Ndyuka and Pamaka. These languages are associated with three independent Maroon communities bearing the same names and residing in French Guiana, especially along the Maroni River, for more than two centuries or are recent migrants from the interior of Suriname. Based on sociohistorical and comparative linguistic data, linguists argue that they are dialects of a common language called Nenge(e), Businengetongo or Eastern Maroon Creole [START_REF] Migge | Creole Formation as Language Contact: The Case of the Surinamese Creoles[END_REF], Goury & Migge 2003).

They may also speak Sranan Tongo, the mother tongue of the descendants of slaves who did not flee the plantations of Suriname. Sranan Tongo also serves as a lingua franca in multiethnic Suriname and is the language of the urban population (Carlin & al 2015). Although all these 4 English-based Creoles descend from the same plantation varieties, they differ somewhat linguistically and are not fully mutually intelligible due to partially different linguistic developments. For example urban varieties of Sranan Tongo have been subject to relatively strong influence from Dutch.

Over the last 20 years, mobility to urban contexts has led to an expansion of the linguistic repertoires of the population which are now more diverse or heterogeneous. Maroons had to acquire some competence in the regional lingua franca Sranan Tongo or in Créole guyanais and French, the official language, to interact with the members of the other local populations. The expansion of the linguistic repertoire has also led to changes in individual and community language practices so we find a greater use of Sranan Tongo in in-group encounters. On the coast where the members of the different Maroon groups are now in regular contact with each other and with members from other local social groups (e.g. Haitians, Amerindians, Créole guyanais, metropolitain French), Maroons have develop a different sense of ethnic belonging. Unlike previous generations of Maroons who generally emphasized inter-Maroon differences, they now emphasize the similarities that exist between the different Maroon groups. It is linked to a new pan-Maroon identity that transcends the traditional ethnic divisions.

While interested in the emergence of language varieties (Migge & Léglise 2011), we showed that this process of identity formation was giving rise to processes associated with koineization (i.e. involving mixing of features from different related regional dialects, levelling of such features, formal reduction, and finally focusing of a new 'mixed' variety [START_REF] Siegel | Koines and Koineization[END_REF]. Dialect mixing involving Sranan Tongo and the Maroon varieties is now widespread even in in-group encounters. We also found processes of leveling in progress. For example, ethnically marked morphosyntactic features such as the verbal negation marker (á (Ndyuka), án (Pamaka/Aluku)) and the negative potential marker (man (Pamaka, Aluku), poi (Ndyuka)) were replaced with more ethnically neutral equivalents that usually come from Sranan Tongo, namely no and kan.

Using the method we developed in-between has made possible to show that in some corpora almost all the turns can be attributed to one or the other variety, to one or the other language, as in Example 3 where the discussion between three men in a bar could be viewed as either Nenge(e) or Sranan Tongo (9 elements could be both, and 4 from Sranan).

(3) Discussion between men in a bar in Saint-Laurent (corpus CLAPOTY_Migge) Our deliberate choice to present all the possibilities in the transcriptions transforms the way we look at the corpora. Rather than viewing extract (3) as spoken in Nenge(e) with some inserted items of Sranan Tongo (Migge and Léglise 2011), we can conclude that the speakers prefer to use items common to both languages to express themselves. In doing so, they use bivalent, unmarked elements showing fuzziness or a form of reorganization of language boundaries (Léglise 2017). They may signal the fuzziness also expressed by the name they can put on it, takitaki, a useful label (Migge & Léglise 2013) which avoids reference to ethnic identification (Pamaka, Ndyuka, Aluku), language names and boundaries such as Nenge(e) vs. Sranan Tongo. Because language denominations and categorizations are always situated, takitaki offers a useful fuzziness to express homogeneity or differentiate ways of speaking when necessary ("this is all the same language" vs. "there are different kinds of takitaki").

But, at times, people select, meaningfully a particular and iconic feature from their linguistic repertoire to signal differences. For example, in urban settings young people generally employ a form of code-switching with Sranan Tongo to construct themselves as sophisticated and urbanized Maroons and code-mixing to assert membership in the social group of young men whose salient properties are modern urban sophistication (Migge 2007, Migge & Léglise 2013). They may also include linguistic resources from other languages (such as Dutch, English or French) in a way of doing-being multilingual, urban and modern to differentiate from more traditional ways of speaking linked to rural area.

4) Linguistic resources as stances and acts of identity

Our second case study focusses on how speakers use dialectal and linguistic resources from their linguistic repertoire in their everyday life interactions as stances and acts of identity. Examples here are taken from features from several varieties of Spanish spoken by Quichuas in Cali, Colombia. Cali is the third largest city in Colombia with more than two million people including indigenous (0.55 %) and afrocolombian (26.95 %) minorities. A 71.56% of people are self -recognized as non-ethnic, i.e. metis and white people. Although this seems to be a clear and steady image for Colombian Administrative Department for Statistics (DANE, 2005) 4 , the heterogeneity of Cali's populations seems to be more complex and dynamic if we look at it through an ethnographic perspective. Indigenous populations, for example, in most of the cases, have been forced by social, economic, and security factors to leave their original territories to look for better life conditions in the cities. In the cities, they are socially organized and recognized by local and national authorities. They constitute Cabildos urbanos, a sort of institutional and administrative forms of organization for indigenous populations in urban settings. One of these groups is the Ecuadorian Kichwas who come from two different regions in Ecuador: Imbabura and Chimborazo (Map 1).

They live in Cali since the sixties and today, at least two generations have been born in the city. Most of them work in the downtown streets selling hand-made products and traditional clothes, as well as other manufactured products. Their labor informal activities lead them to interact constantly with the local retailers and with the other indigenous groups. The estimated number of Kichwas in Cali is 376 people, gathered in 90 families [START_REF] Anacona | Estudio de Caracterización de Pueblos Indígenas: Kofán, Misak -Guámbianos, Quichuas, Ingas, Yanaconas, Nasas habitantes de Santiago de Cali[END_REF]. The asymmetrical contact of populations in Cali results in a multi-ethnic and multicultural environment shaped by different dynamic social variables such as ethnicity, mobility, transnational migration, integration to the economic and social life, and language practices. This complex social situation has important sociolinguistic consequences which entails, for example, the use of specific linguistic forms drawn on speaker's linguistic repertoires.

Kichwas in Cali speak the Ecuadorian varieties of Andean Spanish (AS), a broader Spanish variety largely described in the literature as being influenced by Quichua and Aymara languages [START_REF] Haboud | Quichua y Castellano en los Andes Ecuatorianos[END_REF][START_REF] Molina | El contacto lingüístico en el español andino peruano. Estudios pragmático-cognitivos[END_REF]Palacios Alcaine 2005b;[START_REF] Muntendam | On the nature of cross-linguistic transfer: A case study of Andean Spanish[END_REF][START_REF] Adelaar | The Languages of the Andes[END_REF][START_REF] Muntendam | On the nature of cross-linguistic transfer: A case study of Andean Spanish[END_REF]Pfänder 2009). AS has its own characteristics which make it different from other Spanish varieties. One of these features is the high frequency of OV syntactic constructions, the vocalic system reduction, the simplification of pronoun system, and the particular use of gerunds, among others. Kichwas also adopt and use linguistic forms of Caleño Spanish (CS), the local Spanish variety, less known in the literature 5 but making part of the Colombian Spanish described by some studies in dialectology [START_REF] Giraldo | El español hablado en Colombia[END_REF][START_REF] Flórez | El Atlas lingüístico-etnografico de Colombia (ALEC): nota informativa[END_REF][START_REF] Patiño | Español, lenguas indígenas y lenguas criollas en Colombia[END_REF][START_REF] Monroy | Dialectos del español de Colombia. Caracterización léxica de los sub dialectos andino-sureño y caucano-valluno[END_REF]. Probably, CS's most salient feature is the well described phonetic reduction of /s/ [START_REF] Brown | Syllabe-final and Syllabe-initial /s/ Reduction in Cali, Colombia : One Variable or Two?[END_REF][START_REF] Espinoza | Variación del fonema /s/ en contextos de juntura de palabra en el español caleño: una mirada sociolingüística1[END_REF] 6 .

They also speak Imbabura Quichwa (IQ) and Chimborazo Quichua (CHQ), two varieties of Andean Highlands Quichua 7 . However, a language shift situation (Thomason and Kaufman 1988), favored by the local Spanish hegemony, is observed in this context. These Quichua varieties are relatively well-known in the literature, although they may be considered as non-standard varieties 8 . Interestingly, these two varieties may serve as a means for indicating difference between Kichwas from Imbabura and Chimborazo. 5 Varieties of Spanish spoken in Bogotá, Medellín, Barranquilla, Valledupar, for example, are being studied within variationist projects such as the Proyecto para el estudio sociolingüístico del español de España y de América (http://preseea.linguas.net/). 6 Phoneme /s/ reduction is a characteristic feature of several varieties of spoken Spanish in America and Spain (Brown and Brown 2012, 89). It is a wide-spread and well-known phonological phenomenon in Spanish linguistics [START_REF] Lipski | On the Weakening of /s/ in Latin American Spanish[END_REF]. 7 From the northern branch of the Quechua family 8 For there are no studies of these varieties in contact with Spanish in urban settings in Colombia,

The fact that AS and CS but also IQ and CHQ have come into contact due to the Kichwas' transnational migration to Cali has at least two consequences: 1) their linguistic repertoires have become heterogeneous; and 2) their language practices are heterogeneous and show evidence of phenomena that can be describe as codeswitching, dialect-switching, languaging and other (contact-induced) linguistic variations at various (phonetic, morphosyntactic, interactional) levels. We believe that the choice of one specific linguistic marked form within conversations is socially meaningful as we will show below. As for the annotation, we decided to annotate as unmarked Spanish the forms with no particular specificity. For example, in (4) the speaker uses buenas, a form of unmarked Spanish (A). Then he uses a linguistic form attributable both to Caleño Spanish (B), or to Andean Spanish (C). In fact, it may belong to C (Andean Spanish) as we observe the lengthening of the vowel /a/ and the Ecuadorian phonetic variant [ɻ]. These two elements are phonetic characteristics of Equadorian Andean Spanish. This element may also belong to B (Caleño Spanish) since we can observe the reduction of phoneme /s/. Whereas si in ( 5) is the only element unequivocally marked as Caleño Spanish [B] because of its pronunciation.

(4) For the purpose of the exposition, in the case of ABC, in (5), we may indicate below only A for unmarked. And finally, we may label a whole turn as an instance of a particular variety, as in ( 6) where the word order (OV) and the use of the discursive maker nomás9 are criteria to label the turn as Andean Spanish.

( Example (7) shows how a speaker, as a social actor, makes linguistic choices to express brief affiliation. M is a Kichwa old lady working down-town. She spends her worktime with other colleagues who may be indigenous, afrocolombian or non-ethnic. C is a nonethnic Caleño who asks M, in the first turn, to keep an eye on his stuff (001). He uses undeniably forms of Caleño Spanish. Interestingly, M answers affirmatively to his request using a Quichuan affirmative adverb ari in the second turn (002). C recognizes this element as Quichua and replies introducing in the third turn (003) a linguistic form of Quichua: shuhuas (thieves). Then M replies again using Quichua (004).

(7)

Yes

Following [START_REF] Auer | The pragmatics of code-switching: a sequential approach[END_REF] interactional annotation 10 , we observe the pattern B2 D1 BCD2 D1: it means C starts using Caleño Spanish, while M follows in Quichua. In 003, C uses not only forms of Caleño Spanish, but selects a specific form which may be construed as Quichua (D) or Andean Spanish (C), two language varieties spoken by M. He shows he is able to understand the forms of Quichua introduced by M and to use a term that was not supposed to belong to his linguistic repertoire (he is supposed to only speak Caleño Spanish).

This sequence also shows the interesting use of the term ['ʧuwah], shuhua in Quichua. First, ['ʧuwah] shows evidence of phonetic variation as a consequence of the appropriation C makes of it. In fact, C produces the original Quichua word ['ʃuwah], as a voiceless palato-alveolar affricate [ʧ], the common realization of the written letters "ch" in Spanish. Moreover, he produces a final /s/ reduction, i.e., /s/ aspiration. ['ʧuwah] means "thief" or "thieves" and it is usually used among Kichwas to alert the others about the presence of a thief nearby, without the thief himself realizing it. The fact that C uses it shows evidence of his strategy for convincing M to keep an eye on his stuff while he goes momentarily away. It may also be considered as a way of expressing a social position of brief affiliation to the Kichwas group in order to support his request, which can be considered as a search for solidarity among retail traders.

Example ( 8) also shows the use of a specific linguistic feature from one of the Spanish varieties in contact in Cali. M is a kichwan old lady speaking Andean Spanish and Quichua. T is a caleño speaker who is talking to L, her niece, a 10 year-old girl. She says to L that she should stay with M to learn Quichua. But, before L answers, M replies for her using an affirmative adverb clearly realized as Caleño Spanish. The phonetic transcription shows the reduction of /s/, a well-known feature of Caleño Spanish.

(8) This choice is socially meaningful. T uses first Caleño Spanish and is followed by M who uses a specific form of Caleño Spanish. By using it, M expresses an affiliation to the majority group to which T belongs. This affiliation is the consequence of the asymmetrical contact situation in which Kichwas evolve in Cali. Due to the linguistic ideology which says that Caleño Spanish (as a language spoken by the majority group) benefits of a positive view rather that the Andean Spanish variety spoken by a minority indigenous group, Kichwas tend to align with their interlocutors to express this affiliation.

However, the linguistic choice of an element may also express differentiation or exclusion. For example, in excerpt (9), the use of Andean Spanish shows that M switches from A to C after the evaluation that she makes of an element represented in the discourse situation. S asks M if L is her daughter. M answers negatively. Then S asks M if L speaks Quichua, but M does not answer this question directly. Rather, she uses a SOV syntactic structure and a restrictive adverb (nomás), both highly frequent in Andean Spanish (004).

(9) By using these forms, M positions herself as a speaker of Andean Spanish, therefore as belonging to the group of Kichwa from Cali, a group to which L does not belong to. By doing this, M excludes L from the groups of Kichwas.

As we may see, the selection of linguistic features is socially meaningful in theses interactions as it entails social implications for speakers within conversation. These choices may construct social positioning expressing brief affiliation and differentiation. However, if we look in depth to the interactional sequences, these social positionings are framed within a broad act of stance. Social identities are linguistically indexed through different related indexical processes that contribute directly or indirectly to the construction of social identities. According to [START_REF] Bucholtz | Identity and interaction: a sociocultural linguistic approach[END_REF]'s model for analyzing identity as it emerges in social interaction, these processes may include, among others, displayed evaluative and epistemic orientations to ongoing talk, as well as interactional footings and participant roles. The selection of specific linguistic forms from heterogeneous linguistic repertoires in Cali contributes to the expression of evaluative and epistemic orientations. That is what Ochs (1992) calls "stance", a concept developed by Du [START_REF] Bois | The Stance Triangle[END_REF] and [START_REF] Kiesling | Stance in context: Affect, alignment and investment in the analysis of stancetaking[END_REF]. Stance is an act of social positioning implying an evaluation of an element (stance focus) existing in the discursive context by a social actor (stancetaker). It implies also his positioning vis-à-vis the stance focus and his/her alignment or dis-alignment vis-à-vis another social actor.

The examples discussed above show how a speaker may align or dis-align with his/her interlocutor within conversation through the choice of a specific linguistic form belonging to his/her linguistic repertoire. These (dis-)alignments take part on a single act of stance which includes an evaluation of some element in the communicative environment. This element will determine a positioning by the speaker and then an alignment or dis-alignment shaped on the selection of a linguistic form. For example, in (7), the stance focus is the utterance in 002 produced in Kichwa by speaker 1. Speaker 2 evaluates this focus and then positions himself as someone being able to use Quichua words to support his requests. This positioning is followed by an alignment with his interlocutor. The whole act is a stance of brief affiliation by the speaker 2 to the group of Kichwas to accomplish his request.

In (8), the stance focus is the fact that T speaks the prestigious local variety of Spanish. M evaluates this focus and positions herself as belonging to the majority group by aligning herself with T. By doing so, she leaves behind, momentarily, her own ethnicity to accomplish an act of stance of brief affiliation.

On the contrary, we observed in (9) that the selection of specific linguistic features may also express differentiation and exclusion. In this interaction, the stance focus is the question asked by S in 003 about the fact that L speaks Quichua or not (speaking Quichua meaning that she may belong to the group of Kichwas). M evaluates this question and dis-aligns with S. This dis-alignement means that either S neither L belong to the group of Kichwas from Cali.

Conclusion

Language variation is a linguistic resource in multilingual, heterogeneous language practices. The annotation method we followed reveals the heterogeneity of language practices and at the same time show how languages or varieties sometimes overlap making it irrelevant to draw arbitrary lines and boundaries between linguistic resources. This has a great impact on the way weas linguistconsider the attribution of language (or variety) labels to linguistic forms.

Language practices among Maroons in French Guiana showed the strong tendency for languagers to use unmarked or bivalent elements that could belong to two or more languages (or language varieties). As language denominations and categorizations are always situated, it has no sense to ask people what they think about the belonging of a form to a language (or language variety): their response may vary as often as the context changes and in relation to the local or more global necessity of affiliation or differentiation. These language practices tell us that using bivalent elements as linguistic resources is a way of showing fuzziness and challenging language boundaries. In this case, it may be a way to assert a pan-Maroon identity (by challenging the varieties boundaries) or a way to perform urbanity and masculinity (by challenging language boundaries).

Language practices with Kichwas in Cali showed that the selection of linguistic features is socially meaningful as it constructs social positioning expressing brief affiliation and differentiation. However, if we look in depth to the interactional sequences, these social positionings are framed within a broad act of stance. These stances observed in social interactions may reveal larger mechanisms of identity construction towards the majority society, and as Kichwas in Cali. At a local level, they confirm that languagers sometimes affiliate with imagined communities (Kichwas or Caleños for example) at their convenience through the selection of specific linguistic forms within their heterogeneous linguistic repertoires, or sometimes show, in purpose, dis-affiliation as ways of differentiation or exclusion. All are illustrations of social meanings and positionnings constructed by language practices.

For such an analysis, we refer the reader to[START_REF] Léglise | The Interplay of Variation and Change in Contact Settings[END_REF] for the consequences of contact between French and Creole,Migge & Léglise (2013) for varieties of English-based Creoles in contact and SanchezMoreano (2015) for the consequences of contact of Quichua on Spanish.

The methodology was developed within the CLAPOTY project, funded from 2009 to 2014 by the ANR under the number 09-JCJC-0121-01 and further tested within the Language Contact program funded by the French Investissements d'Avenir -Labex EFL program (ANR-10-LABX-0083) whose PI was I. Léglise.

The recording and first transcript comes from[START_REF] Nelson | Le contact de langues au travail : l'étude de l'alternance codique entre les langues français-créole dans les situations de service à l'accueil direct d'EDF Guyane[END_REF]. Annotation was done through the Clapoty project.

Departamento Administrativo Nacional de Estadística, 2005.

Nomás in Andean Spanish would be influenced by the enclitic prefix -lla from Quichua (Cerro ́n-Palomino 2003, 193).


H Amara

C Bichara

F Ducastelle

Interaction of carbon clusters with Ni(100) : Application to the nucleation of carbon nanotubes

Keywords: numbers: 97.10.Gz, 97.30.Qt, 97.80.Gm

In order to understand the first stages of the nucleation of carbon nanotubes in catalytic processes, we present a tight-binding Monte Carlo study of the stability and cohesive mechanisms of different carbon structures deposited on nickel (100) surfaces. Depending on the geometry, we obtain contrasted results. On the one hand, the analysis of the local energy distributions of flat carbon sheets, demonstrate that dangling bonds remain unsaturated in spite of the presence of the metallic catalyst. Their adhesion results from the energy gain of the surface Ni atoms located below the carbon nanostructure. On the other hand, carbon caps are stabilized by the presence of carbon atoms occupying the hollow sites of the fcc nickel structure suggesting the saturation of the dangling bonds.

I. INTRODUCTION

During the past few years, there has been a tremendous increase in research activities devoted to single-wall carbon nanotubes (SWNTs) mainly because of their extraordinary mechanical, electrical, and chemical properties. Future nanoscale applications will however necessitate the production of pure SWNTs in significant quantities and also the selective formation of nanotubes with welldefined chirality or diameter. This obviously requires to understand their growth mechanisms. SWNTs are generally produced via techniques working at high temperatures (arc discharge [1], laser ablation [2]) or at medium temperature (chemical vapor deposition [3]). The presence of a transition metal catalyst (such as Fe, Co, Ni, . . . ) has been found necessary for the synthesis of SWNTs but the exact role played by the catalyst is still unclear. The similarities between the samples synthesized from different techniques suggest a common growth mechanism and the so-called vapor-liquid-solid (VLS) model is frequently mentioned [4,5]. In this scheme, the growth of SWNTs begins with the formation of a metallic particle supersaturated with carbon; then carbon segregates towards the surface and finally carbonaceous structures emerge from the catalyst surface. From the theoretical side, several modelizations at an atomic scale have been performed using either phenomenological potentials [6,7,8,9] or first principles methods [4,14,15,16].

To some extent these computer simulation works confirm the VLS model but with some differences between the high temperature and medium temperature routes. In chemical vapor deposition methods the size of the tubes is generally determined by the size of the particle. Actually several modelizations show that carbon atoms at the surface self-organize and form graphene-like sheets wrapping the particle. Once formed these sheets no longer interact with metallic atoms and can detach to form nanotubes. This accounts for a so-called tangential process. In other situations, principally within the high temperature route, bundles of nanotubes grow perpendicularly to the surface and it is still a challenge to understand why this can be more advantageous than a tangential growth. Some arguments based on classical nucleation and growth thermodynamic models have already been put forward [17,18]. Carbon atoms at the surface of the metallic catalyst are assumed to condense in the form of graphene flakes. Then the metallic substrate can help to saturate the dangling bonds and this is presumably favoured by the formation of a cap, the energy cost due to the curvature induced by the presence of pentagons, being more than compensated by the reduction of the number of dangling bonds [18]. This model has been substantiated by Fan et al. [14] who performed ab initio energy calculations of different arrangements of carbon atoms on a Ni(100) surface. Among the different carbon geometries tested, they found actually that the most stable one is a hemispherical cap forming a tube embryo.

These calculations are quite accurate but computationally very demanding and the structures cannot be fully relaxed. Furthermore these calculations do not provide an easy access to a local analysis of the bonding. Fan et al. have thus developed a simplified phenomenological analysis of the energies to characterize this effect of dangling bond saturation [19]. A more precise analysis of the chirality fixed by the embryos growing on a (111) surface has been made by Reich et al. [16] based on the use of the SIESTA ab initio code. They confirm the bond saturation effect and show that the possible chiral selectivity depends more on the carbon-metal bonds than on the energy of the different carbon caps. In this calculation however, the metallic (nickel) atoms remain unrelaxed, a strong approximation, considering the order of magnitude of the NiC interaction, and no local energy analysis can either be achieved.

In this paper, we study the adhesive properties of carbon clusters on a Ni(100) surface using a tight-binding (TB) total energy model. This model relies on local (atomic) energy calculations using the recursion method and is coupled with Monte Carlo (MC) simulations in order to relax the structures. Our objective is to establish if the nucleation of SWNTs is favoured by the elimination of dangling bonds, as suggested by the above mentioned authors and also to discuss the role of the displacements of the metallic atoms. We thus present an analysis of the local energy distributions in order to determine which atoms (carbon or nickel), and to what extent, are stabilized when various carbon clusters are put in contact with a metallic surface.

II. METHOD

In our study, the interaction between metal (here nickel) and carbon atoms is treated within the semiempirical tight-binding model described in Ref. [20]. Only s, p electrons of carbon and d electrons of nickel are taken into account. The total energy of an atom i is splitted in two parts, a band structure term that describes the formation of an energy band when atoms are put together and a repulsive term that empirically accounts for the ionic and electronic repulsions :

E i tot = E i band + E i rep . (1)
The total energy of the system, E tot , then writes :

E tot = i atoms E i tot . (2)
The band energy, E i band , is given by :

E i band = E f -∞ (E -ǫ 0 i )n i (E)dE, (3)
where E f denotes the Fermi level and ǫ 0 i is the atomic energy level. We use the recursion method to calculate the local electronic density of states n i (E) on each site [23]. Only the first four continued fraction coefficients, (a 1 , b 1 , a 2 , b 2 ) corresponding to the first four moments of the local density of states are calculated exactly. The continued fraction is then expanded to the N th using constant coefficient equal to a 2 and b 2 (here N = 40). The local density of states n i (E) is then expressed as a set of N poles (E j i , j = 1, N ) with weights (A j i , j = 1, N ) by diagonalizing a tridiagonal matrix of size N . Thus, the band energy of an atom i writes :

E i band = jmax j=1 A j i E j i (4)
In the above equation the highest occupied energy level j max depends on each site i. [25]). More details are given elsewhere [20]. This approach which correctly describes the chemical bonds of a mixed metal-carbon system, is able to deal with systems of a few hundreds of atoms over large time scales. As compared to the standard diagonalization technique, the recursion method is local and then gives direct access to the total energy of each atom i (see Eq.4). This possibility to easily analyze local energy distributions will be very useful in the following. This atomic interaction model is then implemented in a Monte Carlo code, based on the Metropolis algorithm, which allows to relax the structures [22]. Finally different tests were performed to check the reliability of our model. As an example, a simple relaxation of C atoms covering a Ni(100) surface induced the "clock" reconstruction observed experimentally by Klink et al. [12]. This reconstruction has also been studied theoretically [26,27] and corresponds to the rotation of the four Ni atoms surrounding a C atom in an Ni(100) surface, clockwise and counterclockwise alternatively (see Fig. 1)

In a previous study, we presented Grand Canonical Monte Carlo simulations of the dissolution and segregation of C in the presence of a Ni(111) surface [21] in order to understand the processes involved in the catalytic growth of carbon nanotubes. At finite temperature (1000 and 1500 K), we analyzed the self-organization of carbon atoms close to a nickel surface and identified a succession of four types of configurations, corresponding to an increase of the carbon chemical potential (and consequently an increase of the number of carbon atoms): single C atoms adsorbed on the surface or incorporated

III. RESULTS

Here, we use Monte Carlo simulations in the canonical ensemble (with a fixed number of Ni and C atoms) to relax the atomic structures and analyze the stability of specific carbon structures deposited on a nickel surface. We perform MC simulations on a Ni slab presenting a (100) surface. This is a four-layer slab of 288 atoms, with bond lengths of 2.48 Å, separated by a 20 Å thick vacuum region along the z axis and periodic boundary conditions. The slab size is 17.60 × 19.36 × 5.28 Å3 . We consider several configurations of carbon atoms, with bond lengths of 1.42 Å, deposited on the metallic surface: molecules (C 5 and C 6 rings), a finite linear chain of 12 atoms parallel to the surface, flat flakes of 24 and 54 atoms (see Fig. 2a-b) and curved structures such as a flake of 20 atoms containing a pentagon (see Fig. 2c) and the cap of a (6, 6) tube (see Fig. 2d).

First, we study the stability of these carbon clusters in the presence of the metallic substrate at 500 K. At this temperature, the C 5 and C 6 rings are dissociated to present C-C bonds of roughly 2.2 Å and C-Ni bonds of 1.9 Å. The twofold coordinated atoms belonging to the finite linear chain interact moderately with the metallic surface (in the 1.0 eV/atom range), while we observe that end of chain, one-fold coordinated, C atom tends to penetrate the metallic substrate, to occupy a semi-octahedral site of the Ni(100) surface. This process has also been observed experimentally by Fujii et al [28]. The stability of the clusters seems to depend on the number of of C-C bonds, with a large enough number of such bonds resulting in a stable cluster. For small number of C-C bonds, the competition between the formation of a carbide and the phase separation results in the formation of a Ni-C alloy. For the other structures, the number of C-C bonds is large enough to stabilize the carbon clusters despite the presence of the metallic surface. The flakes of 24 (Fig. 2a) and 54 (Fig. 2b) atoms remain in plane and we also do not observe any dissociation for the flake of 20 atoms containing a pentagon (Fig. 2b). The presence of this defect induces a curvature of the structure in agreement with Euler's theorem. Last, the interaction between the metallic particle and the open tube cap stabilizes it by preventing its closure (Fig. 2d).

We then calculate the adhesion energies per carbon atom, E a , given by :

E a = (N C + N N i )E C/N i -N C E C -N N i E N i N C , (5)
where N N i and N C are the number of Ni and C atoms. E C/N i , E N i and E C represent the total energies (per atom) of the carbon-covered Ni slab at different surface coverages, the clean Ni slab and the relevant isolated carbon structure, respectively. They are calculated at 0 K on fully relaxed configurations. In each case, the minimum energy structures are obtained by performing simulated annealing. In this process the initial atomic configuration, initially at high temperature (1000 K), is slowly cooled down so that the system is at any step at thermodynamic equilibrium. The major advantage of this method is its ability to avoid becoming trapped in local minima. Typical runs consist in 10 3 external Monte Carlo loops, each of them randomly performing 10 3 atomic displacements trials. For all the structures tested, the negative value of the adhesion energies, listed in Table I, means that the carbon cluster is stabilized by the surface. For the flakes deposited parallel to the surface, the results clearly show that it is favorable to incorporate pentagons during the initial stages of nucleation. It is therefore tempting to think that the structure containing defects can deform into a dome to better saturate the dangling bonds at their edges.

To investigate further, we analyze the local energies of Configurations Flake Flake Flake Linear chain Capped tube (Fig. 2a) (Fig. 2b) (Fig. 2c) (Fig. 2d carbon and nickel atoms in all the previous cases in order to determine which C and Ni atoms atoms gain energy.

As an example, we plot our results for the flake of 24 atoms in a histogram form in Fig. 3. After relaxation of the isolated carbon flake (away from the metal surface), we identify three populations of carbon atoms (full lines in Fig. 3a). First, the three-fold coordinated atoms denoted by A : 6 atoms in a local configuration equivalent to an infinite graphene sheet in which each atom is in a three-fold coordinated state with sp 2 hybrids forming strong covalent bonds. The energies lie between -7.5 and -7.4 eV/atom, close to the total energy of a perfect graphene sheet calculated with our TB model. Then, the 6 three-fold coordinated atoms located at the periphery of the structure and denoted B: their energies lie between -7.3 and -7.2 eV/atom. The difference between the two previous populations is simply due to the changes in their second neighbor shell. Finally, we have 12 two-fold coordinated atoms at the periphery of the flake, with energies between -5.8 and -5.7 eV/at. These atoms only have two neighbors, so that, in principle, one sp 2 hybrid is left unbonded, forming a dangling bond. With this analysis, we can estimate the energy of a dangling bond at about 1.80 eV: it is the energy difference, appearing on the histogram, between a C atom located at the surface and a three-fold coordinated A atom. This is close to the 1.78 eV [29] and the 2.26 eV [14] found in finite cluster by LDA studies and smaller than the 2.2 to 3.3 eV [30] found in previous LDA works in the case of a tube. Studying now the flake in contact with the metallic surface, we can see that the three populations are still present, but modified. In Fig. 3 the histogram still presents three clearly identified regions (dotted lines) equivalent to those described previously, with a larger distribution due to the accuracy of the relaxation and to the fact carbon atoms that are equivalent in an isolated flake may become no longer equivalent when the flake is deposited on a surface. The small differences observed however suggest that carbon atoms remains essentially unaffected when deposited on the nickel surface, implying that, in this sense, the interaction between the metallic surface and the carbon structure do not saturate the dangling bonds. This observation is somewhat different from the conclusions drawn by Fan et al., based on ab initio calculations, and therefore requires further analysis. In agreement with them, we nevertheless notice that the energy balance presented in Table I reveals a weak adhesion energy, meaning that the flake is more stable on the surface than free standing. Since this stabilization does not come from the carbon atoms, these results imply that nickel atoms should be stabilized by the presence of carbon. We confirm this with the analysis of the local energies of Ni atoms shown in Fig. 3b and Fig. 4. In the case of the bare relaxed Ni slab, we observe two different populations (full lines in Fig. 3b). The first corresponds to the bulk atoms which have 12 neighbors and energies between -4.4 and -4.3 eV/atom. The total energy of a face centered cubic bulk Ni is E i tot =-4.44 eV/atom in our TB model. The second population corresponds to the 8-fold coordinated surface atoms. These are less stable than atoms in bulk and have energies between -4.0 and -3.8 eV/atom. In the presence of the carbon structure, both populations are still present. It is important to note that 288 atoms of Ni have been reported on this histogram. Thus, the difference observed on Fig. 3b which seems weak when compared to the analysis of C atoms where only 24 atoms appear, corresponds to a larger contribution to the total energy. The quantitative analysis of the local energies shown in Fig. 4, demonstrates that the nickel atoms, located at the surface and in direct interaction with the carbon structure gain energy. On the one hand, the average total energy of nickel atoms at the surface and far away from carbon atoms is around -3.90 eV/atom. This value is similar to the one obtained without carbon atoms (full lines in Fig. 3b); on the other hand, the nickel atoms close to the carbon structure are stabilized (Fig. 4). The energy gain is approximatively equal to 0.4 eV/atom and is associated with a slight reconstruction of the Ni surface underlying the flake, as shown in Fig. 4. In the case of the other flat structure containing 54 carbon atoms, the local energy analysis leads to the same conclusions: the local energies of carbon atoms are not modified but surface nickel atoms below the flake are stabilized with roughly the same energy gain.

The same energy balance is obtained for the flake containing a pentagon. For the linear chain, the one-fold coordinated, end of chain, carbon atoms interact strongly with the surface and penetrate into the metallic substrate to occupy a semi-octahedral site of the (100) Ni structure (Fig. 5) with an energy gain of 0.5 eV. The same effect appears with the tube embryo where some carbon atoms are located in the fourfold hollow sites. These results show clearly that the anchoring of carbon atoms to the catalyst is energetically favorable during the nucleation process. They gain energy by occupying the semi-octahedral sites of the fcc nickel (100) surface.

Finally the adhesion process of carbon sheets on Ni (100) is slightly more complex than anticipated. The adhesion energy of flat sheets is mainly due to the energy gain of the nickel atoms below these sheets. When they curve to form caps, the energy gain becomes concentrated on the carbon and nickel atoms atoms close to their edge.

In this case one might argue that dangling bonds are saturated, but finally, this is a weak effect which does not play very much in favour of curved caps: the energy of adhesion of flat and curved sheets are similar and small compared to dangling bond energies, for small clusters at least.

At this point it is useful to stress that it can be difficult to choose between various decompositions and normalizations of energies of formation or adhesion. Depending on the point of view, it can be fruitful to count energies per carbon atom of the whole structure (flat sheet) or per edge atom (curved sheet). Our model has the advantage to provide us with unambiguous local (site) energies. Bond energy analyses could be performed as well but would be more complex to use in the present context.

IV. DISCUSSION

Our results allow us more generally to understand the mechanisms by which carbon atoms adsorb on the Ni(100) surface. Experimentally, investigations including low-energy electron diffraction [10], surface extended-xray-absorption fine structure [11], and scanning tunneling microscopy [12] have provided a large amount of information on the interaction of C with Ni(100). When carbon adsorbs on Ni(100) with a coverage smaller than one third of a monolayer (ML), it occupies hollow sites and does not change the symmetry of the outermost metal layers (Fig. 5). This situation results in strong interactions (see table II) and corresponds to the incorporation of isolated or one-fold coordinated C atoms in the hollow sites, observed during our calculations.

As the coverage exceeds 0.33 ML, the surface undergoes a reconstruction known as the "clock reconstruction" described previously (Fig. 1). This reconstruction, which has also been studied theoretically [26,27], is indeed observed during our MC calculations (Fig. 4). The adhesive properties of carbon clusters on Ni(100) surface then result from a competition between the formation of Ni-C and C-C bonds. For structures containing few C-C bonds, the competition between a carbide and a demixed system favors the formation of Ni-C bonds and the C atoms occupy the hollow sites of Ni(100) present at the surface. For planar flakes, C-C interactions predominate and, as described above, the weak energy of adhesion is principally related to the energy gain of the (reconstructed) metallic surface below the flakes. Tube embryos have an intermediate behaviour: C-C interactions predominate on the caps and Ni-C anchoring bonds predominate at their edges.

It is interesting to compare briefly our results with those obtained for the (111) surface. At low carbon concentration the situation is similar, with predominant Ni-C interactions, even if, due to the different topology of the surfaces, the preferred position for the (111) surface for carbon is a subsurface octahedral site. At higher concentration we expect the formation of a graphene sheet in quasi perfect epitaxial relationship with the Ni substrate and a vanishingly small energy of adhesion [21]. We have also calculated the energy of adhesion of a small flat cluster made of 24 atoms. As in the case of the (100) surface we find a weak energy of adhesion equal to -0.26 eV/C atom, and the local energy analyses show that the energy gain is concentrated on the edge atoms, without any important saturation of the dangling bonds. In their analysis of different caps on Ni (111) Reich et al. have also noticed that the relative energies involved are also fairly weak [16].

V. CONCLUSION

In summary, we have presented tight-binding Monte Carlo simulations of the interaction of carbon structures with Ni(100) surfaces Our calculations show that during the first stages of the nucleation the incorporation of defects and the anchoring of carbon atoms to the surface are energetically favorable. Even in the presence of a metal substrate, the carbon dangling bonds parallel to the surface still exist and the metal-carbon system gains energy by reconstructing the underlying metallic surface, close to the well-known clock reconstruction. For curved structures, the formation Ni-C bonds is more efficient, and C atoms prefer to be located in semi-octahedral sites of Ni present at the surface.

Part of this work has been supported by the Belgian

Program on Interuniversity Attraction Poles (PAI5/1/1) on "Quantum Size Effects in Nanostructured Materials".

FIG. 1 :

 1 FIG. 1: (a) Top view of a surface coverage equal to 0.5 ML. (b) Clock reconstruction leading to a (2 × 2)p4g symmetry.

FIG. 2 :

 2 FIG. 2: Stability at 500 K of carbon clusters (top and side views) deposited on Ni(100). (a) flake of 24 atoms containing 7 hexagons; (b) flake of 54 atoms containing 19 hexagons; (c) curvature of the flake of 20 atoms containing a pentagon; (d) embryo of (6,6) nanotube.

FIG. 3 :

 3 FIG. 3: Analysis of local energies for a flake of 24 carbon atoms in interaction with a Ni(100) surface. (a) Histogram of C atoms. (b) Histogram of Ni atoms.

FIG. 4 :

 4 FIG. 4: Histogram of Ni atoms close to the flake of graphene. Reconstruction of the Ni(100) surface when C atoms are present.

FIG. 5 :

 5 FIG. 5: Illustration of a C atom occupying a hollow site of Ni(100) surface.

  The set of C-C, Ni-Ni and Ni-C tight-binding interaction parameters used was shown to be transferable to different atomic configurations by comparison with ab initio calculations. It reproduces the expected phase separation tendency for the

Ni-C alloy, with an enthalpy of mixing of +1.0 eV/atom for NiC in the (metastable) rocksalt structure fitted to ab initio data (ABINIT code). The heat of solution of a C atom in a Ni matrix is then found equal to +0.4 eV/atom, in agreement with the experimental value of Shelton et al. (0.49 eV/atom [24]) and the first principles calculations of Siegel et al. (+0.2-0.35 eV/atom

TABLE I :

 I Adhesion energies per carbon atom of the carbon structures deposited on the surface of Ni(100).

TABLE II :

 II Binding energies (E b ) of a C atom on a Ni(100) slab. R is the distance between C and Ni atom.

		E b (eV)	R ( Å)
	Theory a	-7.43, -6.50, -8.10,	1.91, 1.85, 1.77
		-7.69, -8.43
	Experiment a	-7.35, -7.37, -7.55	1.80, 1.82, 1.89
	Present work	-8.21	1.90

a From Ref.

[31]

and references cited therein.


Anis Smara

Abderrezak Rezzoug

Jean Lévêque

Rachid Ibtiouen

New Type of Oversaturated Superconducting Machine

This paper presents the principle and design of an experimental bench of an inductor of an oversaturated superconducting machine with radial flux density. A designed experimental bench is under construction to validate the computation and the principle of the inductor. Also, this work describes the encountered problems and lays out the tools that permit to study a high power oversaturated superconducting machine.

INTRODUCTION

The properties of superconductors, such as high current density, diamagnetic behavior and magnetic flux trapping, opened the door to large possibilities in terms of electrical machines design [START_REF] Ailam | Design and testing of a superconducting rotating machine[END_REF][START_REF] Lin | A novel fully superconducting geared machine[END_REF][START_REF] Lin | Topologies for fully superconducting machines[END_REF][START_REF] Jeong | Design of a 10-mw-class HTS homopolar generator for wind turbines[END_REF][START_REF] Kawamura | Superconducting super motor and generator[END_REF][START_REF] Rezzoug | Non-conventional Electrical Machine[END_REF]. Their feasibility, however, is not trivial because of many constraints imposed by such materials.

To build a simple coil we can mention, without being exhaustive, the constraints (T < T c , J < J c , B < B c ) which must be respected in any point of the superconductor. For the mechanical constraints, we note the limited curvature radius > R c-min and the forces applied on the superconductor.

As a consequence, the calculation of any superconducting machine is more difficult than that of any conventional one which benefits from more than a hundred years of industrial experience. Taking into account the experience in the domain, we can affirm that superconducting machines can be competitive in the area of large power machines. Within this frame, earlier we proposed an original structure of an oversaturated machine with an axial magnetic flux spatially modulated in the angular direction, and we also proposed a scheme of a machine, based on the same principle, in which a radial flux density is used [START_REF] Rezzoug | Cryogenic supersaturated synchronous machine optimization and first experiment results[END_REF][START_REF] Manfe | Prototype 'Supersat 001 -Probative results[END_REF].

This paper presents a work developed to verify and illustrate the principle of a radial oversaturated machine by studying the air gap flux density distribution created by an oversaturated radial inductor. After the presentation of the principle and general scheme, design of the inductor is made. The design of the future experimental bench is analyzed.

BASIC PRINCIPLE OF AN OVERSATURATED MACHINE

The magnetic moments of a ferromagnetic piece, plunged in a high magnetic field, are oriented along the flux lines of this field. The magnetic flux inside the material can be expressed as:

B = μ 0 ( H + M ) ( 1 )
where B, μ 0 , H, and M represent the magnetic flux density (T ), magnetic vacuum permeability (H/m), magnetic field (A/m), and magnetization (A/m), respectively. Equation ( 1) is valid for all ferromagnetic materials and for all values of the magnetic field. However, increasing the external magnetic field will increase the value of magnetization M . If the external field is high enough, and as long as it is present, the ferromagnetic material will be saturated, and the value of M will be constant. The ferromagnetic material could be considered as a permanent magnet. The question is whether there is any interest in using oversaturated ferromagnetic pieces instead of permanent magnets. The response is "yes" if we consider the volume of rare earth, a strategic material necessary to manufacture permanent magnets and the price of which continually increases. We should also consider the process to fix them and their cost in the case of building a large power machine (typically several MW). Compared to classical superconducting machine structures, the cryogenic part of our machine is fixed. The only rotating part will be composed of massive ferromagnetic pieces. In addition, this type of machine is quasi insensitive to the armature reaction, and the recovery of the required magnetization is fast after the disappearance of any default. We also have to take into account that the evolution of new superconducting materials is in "daily" progress, which makes superconducting machines more promising.

However, the main problem is to obtain a source of magnetic field with a high value in a large volume. The solution can be inspired by the expertise found in the domain of static coils used for MRI or for large physics instruments. In any case, it seems to be obvious that superconducting coils constitute the best solution to build the source of magnetic field.

Taking into account all the above considerations, we propose to study the feasibility of an inductor of an oversaturated machine with radial flux density distribution.

FROM THE PRINCIPLE TOWARD THE DESIGN

The topology of a radial oversaturated inductor as well as the back iron is schematized in Figure 1. The inductor is composed of regularly distributed massive ferromagnetic pieces magnetized by superconducting coils. In this case, we expect to find a square wave flux density distribution around the inductor varying between B 0 and (B 0 + μ 0 M ).

At this point, we would like to draw the attention that in this machine, the torque is not due to the difference between the direct and quadrature inductances, as in a classical reluctance machine. This term, (Ld-Lq), is close to zero when the ferromagnetic pieces are oversaturated. The rotor must be considered as equivalent to a one polarity permanent magnets machine. The calculation of torque or EMF is done in a classical way as for any synchronous machine.

To obtain a sufficiently high value of the flux density, we use two superconducting coils. The armature can be made of copper or superconductors. An environmental confinement of the magnetic field needs a magnetic shield or supplementary coils.

Figure 1 shows that by using two coils, fed by opposite currents, we obtain a magnetic field with a radial component in the volume confined between them. It is important to note that a final design consists of superconducting coils with their individual cryostat and an armature composed of copper or superconductor winding with its own cryostat. The only rotating part is the one composed of magnetically oversaturated ferromagnetic pieces.

To conclude, one can highlight the potential advantages of the proposed axial and radial structures: the coils are static and consequently need a very simple cryogenic system compared to those used to cool down a rotating part, which is the case for many superconducting machines. The armature winding is also static and can be composed of copper or superconductors. Consequently, the only rotating part is the one composed of ferromagnetic pieces which is very robust as demonstrated with an axial structure studied and tested [START_REF] Rezzoug | Cryogenic supersaturated synchronous machine optimization and first experiment results[END_REF][START_REF] Manfe | Prototype 'Supersat 001 -Probative results[END_REF].

The study of the inductor, composed of superconducting coils and oversaturated ferromagnetic pieces, is presented in this paper. The magnetic field distribution, firstly determined in a theoretical study, will need experimental validation.

DESIGN OF AN INDUCTOR

Assumptions

The first problem to consider in this topology is to generate a high magnetic field in a relatively large volume to oversaturate ferromagnetic pieces. This can be assured using superconducting coils. Operating below their critical temperature, such coils can transport current densities high enough to generate the needed magnetic field. Such advantages are accompanied with cooling problems, but several solutions can be considered (liquid helium or nitrogen, cryo-cooler). Considering the prices of superconducting wires, NbTi superconductor is chosen and will be cooled by liquid helium.

To make the cryogenic process easy to implement, all the parts of the apparatus are static and will be put in an existing cryostat, thus its internal diameter constitutes a limit for the dimensions of the superconducting coils and the inductor.

The total length of the inductor is limited so that it has enough volume of liquid helium to cool down the system

Description of the Inductor

The system of coils and ferromagnetic pieces, arranged according to Figure 1, creates, at the level of the armature winding, an angular variation of the magnetic flux density. The oversaturation prevents, in addition, the demagnetization caused by the armature winding reaction.

The design of the experimental bench consists of calculating the superconducting coils and the size of the ferromagnetic pieces. We also add the external back iron because it is an important part for the design of the inductor. All the parts for this probative apparatus are static.

The coils are made of a low temperature superconducting wire (NbTi). The design of the coil must take into account the B(J) law of the superconducting wire at an operating temperature. For an air core coil system, the operating current density represents the point of intersection between the load line of the coils and the characteristics of the superconducting wire (see Figure 2). The load line B coil (J) is the maximum of the flux density inside the coils as a function of the current density. We should notice that adding ferromagnetic pieces changes the magnetic flux distribution inside the coils. As these ferromagnetic pieces are oversaturated, their behavior is linear, but the behavior of the back iron is not linear. This point constitutes a difficult problem to solve in the design. The characteristics of the used superconducting wire are mentioned in Table 1 and Figure 2.

In the upcoming sections, we present a first topology that represents the oversaturated superconducting radial machine inductor designed for the chosen cryostat. This topology, at those conditions, show a low flux density variation at the armature windings. A second topology is then presented after a coils shape analysis. It consists of a new shape of coils and ferromagnetic pieces inside and outside the cryostat to increase the flux density variation at the armature winding.

First Topology

Taking into account the constraints cited above. The dimensions, in reference to Figure 3, are mentioned in Table 2. The value of the used current density is chosen so that we reach, for the operating temperature, a maximum value using B(J) law of our superconducting wire.

The distance between the coils represents the active length of the armature winding. This distance should be a result of a compromise between the volume of the energy conversion medium, which must remain sufficient, and the value of the radial flux density, which must maintain the orientation of the magnetic moments. The dimensions must reflect the compromises and choices to maximize the ΔB × L product at the armature winding while taking into account the geometric constrains of this experimental bench (see Figure 4). Increasing the product ΔB × L increases the torque of the machine.

Considering the nonlinearities of the problem and taking into account the assumptions presented in the previous section, the dimensions of this first topology are the results of a non-exhaustive parametric study on the coils and ferromagnetic pieces (see Table 2).

The computations are performed by using finite element software [9], with a nonlinear B(H) curve for all ferromagnetic pieces. It leads to the results presented in Figure 4. In this case, the computed current density is 500 A/mm 2 .

In order to verify the oversaturation criterion, we visualize the magnetic flux inside the ferromagnetic piece along the r-axis, as illustrated in Figure 3. Now that the criterion of saturation is verified, we can see in Figure 4(b) the variation of the angular radial magnetic flux density at the surface of the inductor, r = rp2. This variation is about ΔB(rp2) = 1 T. These results validate the principle of the oversaturated machine.

The variation of Br, at r = rc1, 17 mm far away from the inductor, where the future armature windings will be located, is about ΔB(rc1) = 0.7 T. We note that the variation diminishes. This is due to the large air gap imposed by the cryostat of this probative bench. One can note that this problem is avoided in a complete machine, with a classical room temperature armature, because the air gap can be shortened to a few millimeters as in a classical machine. However, due to the large air gap the amplitude of the space harmonics decreases drastically, and this is an undeniable advantage. If the project deals with a machine with a superconducting armature, the large air gap is inevitable, and improvements must be brought to the design. Considering the B(H) curve of the used ferromagnetic material (see Figure 5(a)), we expect to have an angular variation of Br(θ), at the surface of the ferromagnetic pieces r = rp2, of around 1.8 T. Unfortunately, as shown in Figure 4(b), we only obtain a Br variation of 1 T. The difference is explained by the fact that the superconducting coils generate a magnetizing field oriented in the (r, z) plan as shown in Figure 5(b). To increase the ratio Br/Bz, we perform a coil shape analysis. This analysis is specifically made for this experimental bench with its geometric constraints. A shape analysis for a large machine will have the same purpose, but the approach will be adapted for other types of constraints.

Coils Shape Analysis

Taking into account the large air gap, we present the analysis that leads to finding an excitation system able to obtain a flux density with a value as high as possible at the armature. To overcome the problem caused by the large air gap, we can oversaturate the ferromagnetic pieces outside the cryostat. To remedy to the problem caused by the orientation of the magnetizing field, we perform a coil shape analysis to create a radial saturating magnetic field outside the cryostat, in the region where the external ferromagnetic pieces will be placed. We call this region the saturation zone (see Figure 6). We mention that there exist in the literature some studies on superconducting coils shapes in which criteria are mainly to increase the current density [START_REF] Jiang | Investigation of step-shaped field coil for large capacity superconducting motors[END_REF].

Till now, the excitation system is formed by superconducting coils of a rectangular cross sectional area. The maximum current density that can be carried by the coils is limited by the B(J) curve of the superconductor. For an operating temperature, the current density depends on the shape and position of each magnetic component (coils, ferromagnetic pieces and back iron).

Our goal is to generate the largest radial component of the magnetic field outside the cryostat in a region that happens to be situated away from the external radius of the coils.

The main idea is to play on the shape of the cross sectional area of the superconducting coil, knowing that any modification on the shape results in a change of the current density allowed in the superconductor as well as the orientation of the flux lines.

Figure 7 shows the basic cross sections we chosen for the study (shape "a", "b" and "c"). For each cross section shape, we compute the maximum current density allowed by such geometries and use it to compute the magnetic field along the external vessel of the cryostat, at r = rs1 (see Figure 6).

Our investigations to enhance the radial flux density component along the computation line in Figure 6 lead us to studying shapes that are combinations of the basic ones, "b" and "c".

The external radius of each shape is fixed by the cryostat radius r2. The distance between the two coils is the same for all configurations and is 70 mm.

The shapes are presented in Figure 7. Table 3 mentions the variable parameters of shapes "a", "b" and "c".

For each configuration presented in Figure 7, we compute the maximum current density allowed by the superconducting coil. Afterwards, we use the latter to compute the radial magnetic flux density and the orientation of B relatively to the radial axis at the line of computation in Figure 6.

Considering the symmetry of the structure, we perform the computations only on half of the machine, between 0 and -L/2 along the z axis. The results of computations are presented in Figure 8.

Shape "a" is a squared cross sectional area, and we notice that its current density is about Shapes "b" and "c" have the same surface as that of shape "a". We obtain them by changing the ratio between the length and width of shape "a".

(a) (b) (c) (d) (e)
Shape "c" is large along the radial direction, and we notice that its maximum current density is about Jc = 433 A/mm 2 and is higher than that of shape "a"; however, the amplitude of the radial component and the angle between the flux density vector B and the radial axis indicate that the flux lines created by the latter, in the region of magnetization, are mainly oriented along the z axis.

Shape "b" is large along the z-direction; we notice that its maximum current density is lower than that of shapes "a" and "c"; however, its radial component is the highest. This is explained by the fact that its deviation from the radial direction is the lowest. So far, shape "b" is the best candidate to meet our considerations; unfortunately, the amplitude of the radial component is too low to oversaturate a ferromagnetic material.

Although the coils have the same distance between them and the same cross sectional area surface, a change in the ratio between the length and width of the surface changes the maximum current density allowed in the coil. This is because the changes in shape change the distribution of the flux lines, so the maximum of B inside the coil can be lowered or increased.

After this brief analysis, we want to get the advantages of both shapes "b" and "c". The idea is to combine the two shapes to get a more radial flux density by superposing both of their flux lines. The two simple possible configurations are presented as shapes "d" and "e".

We notice that shape "d" has a higher current density and higher amplitude of the radial component than shape "e"; however, shape "e" has a more radial orientation (see Figure 8). Choosing shape "e", we notice that the amplitude of the radial flux density is ranging between 1.2 and 1.4 T. This is the limitations of this configuration with the geometric sizing constraints.

Second Topology

In the previous part, we have considered the coils alone in the space. A structure of an inductor is more complicated, and its final design becomes difficult.

To reach an over saturating value outside, in the region of magnetization, we can use ferromagnetic pieces between the coils inside the cryostat. Their role is to increase the amplitude of the radial component of configuration "e" to permit an oversaturating value in the predefined region of magnetization.

At this point, the topology is well defined: two superconducting coils with the shape of configuration "e". Between those coils we place ferromagnetic pieces. Outside the cryostat we place another set of ferromagnetic pieces. The latter have to be oversaturated with the magnetic field created by the coils and inner pieces (see Figure 9).

Parametric Study

To generate a saturating magnetic field for the ferromagnetic pieces while increasing the product ΔB ×L at the level of the armature, we perform a parametric study on the parameters of the machine (see Figure 9). The only fixed parameters are the outer radius of the inductor limited by the cryostat and the distance L between the superconducting coils. Table 4 mentions the parameters and precisely the ones that were varied and the ones that were fixed. Not forgetting that for each set of parameters we find a new maximum value of current density.

The parametric study yields a huge number of results that need analysis and treatment to get the desired results. The fruit of such a study is presented in Table 4 in reference to Figure 9. The results of computations are the subject of Figure 10. The obtained current density is 660 A/mm 2 . Figure 10(a) shows the results of computations performed on a radial line passing through the middle of the ferromagnetic piece at z = L/2; we notice that the inner pieces are oversaturated. The outer pieces are sufficiently magnetized.

Figure 10(b) represents the variation of the radial flux density at the surface of the inductor, r = rp2, and at the inner surface of the back iron, r = rc1, at (z = L/2).

We note that the variation of the flux density at the surface of the inductor, r = rp2, is higher than that obtained in the first topology and is about ΔB(rp2) = 1.3 T. This is due to the new shape of superconducting coils as well as the contribution of the outer ferromagnetic piece. We also notice that the variation of the angular flux density at the armature winding, r = rc1, has also increased and is about ΔB(rc1) = 1 T for the fundamental. This shows the benefits of using two sets of ferromagnetic pieces.

INDUCTOR MANUFACTURE

Thanks to the obtained results, an experimental part is now under development as shown in Figure 11. To achieve that, additional constraints must be taken into account. First of them deals with the acting magnetic forces between the components of the inductor, and the others concern the thermal losses due to Joule effect in the electric connection cables of the superconducting coils. High temperature superconducting current leads are planned to be used to reduce the above mentioned losses and consequently the liquid helium consumption.

CONCLUSION

This paper contributes to the design of a new topology of electrical superconducting machine of a radial flux density using oversaturated ferromagnetic pieces. This work presents the study of an inductor for an experimental bench. The aim of the work is to verify and illustrate the principle and compute the inductor air gap flux density.

The design of the experimental bench highlights the problems encountered to build such a machine and lays out the tools that can help to study the design of large power superconducting oversaturated radial machine.

Considering the constraints on the dimensions, such as the large air gap caused by the cryostat, variation of the angular magnetic flux density of the first topology is relatively small and is about 0.75 T.

To increase the inductor air gap flux density, a coils shape analysis is performed and allowed to conclude that the "L" shape constitutes a suitable solution, for this range of parameters, and enhances the radial component of the magnetic field, even away from the external radius of the coils themselves.

Ferromagnetic pieces are placed inside and outside the cryostat. This configuration allows us to place an armature winding with an air gap as small as the one of conventional machines, 2 mm in our case.

The parametric study allows us to find the dimensions that result in a mainly radial magnetic flux density in the region of magnetization.

Figure 1 .

 1 Figure 1. Scheme of the topology of the inductor of the oversaturated superconducting machine.

Figure 2 .

 2 Figure 2. NbTi B(J) Law.

  Figure 4(a) represents the variation of the flux density on radial line passing through the ferromagnetic piece at z = L/2. The two vertical

Figure 3 .

 3 Figure 3. Scheme of the first topology.

Table 2 .

 2 Dimensions (Figures2-3). space occupied by the ferromagnetic piece. The latter is clearly oversaturated because the value of the magnetic flux inside it largely exceeds 2 T.

2 ΔB

 2 norm of B along a radial line passing through a ferromagnetic piece at z=L/

Figure 4 .

 4 Figure 4. Results of computations on the first topology.

Figure 5 .

 5 Figure 5. Flux lines and B(H) curve of the used ferromagntic material.

Figure 6 .

 6 Figure 6. Scheme showing the saturation zone.

Figure 7 .Table 3 .Ja = 234 A/mm 2 .

 732 Figure 7. Shapes of the superconducting coils for a new topology.

Figure 8 .

 8 Figure 8. Results of computations for different coil shapes.

Figure 9 .

 9 Figure 9. Second topology scheme.

Figure 10 .

 10 Figure 10. Results of computations on the second topology.

Figure 11 .

 11 Figure 11. Pictures of the inductor under construction.

Table 1 .

 1 NbTi wire characteristics.

		supplier	SUPERCON INC
		Reference	56S53Multi filament
		Number of filaments	56
		Filament diameter	48 µm
		Bare wire diameter	0.50 mm
		Insolated wire diameter	0.54 mm
		Cu/Su ratio		0.9 : 1
	0	500	1000	1500	2000	2500
			J [A/mm²]

Table 4 .

 4 Dimensions of the second topology.

	Model's parameters