Marie Perrier

Translating imaginary territories in fantasy short fiction

Translating a text set in an imaginary world poses specific problems to the translator, since the original often indirectly builds on cultural bearings to create a setting both related to reality and endowed with its own peculiarities and exotic features. The translator will need to find the same thin balance between familiarity and strangeness so as to allow its reader to enter the imaginary world. This paper article will focus on a precursor of the genre of fantasy, the Irish writer Lord Dunsany. Instead of a tapestry whose threads become invisible, his imaginary world is built like a tessellated literary body. Coherence and identity are no to be found in a wealth of details and lengthy descriptions, but in powerful, striking nodes of meaning. A few examples from the various translations of his early tales into French will help us show that this in turn becomes a challenge to the translator, who has to identify these nodes and translate them, finding a new balance in fragmentation which still allows the reader to grasp the ethereal unity of Dunsany's imaginary territory.

Translating imaginary territories in fantasy short fiction

Marie Perrier (University of Lille -France) Since it was popularized by Tolkien's The Lord of the Rings trilogy in the second half of the twentieth century, the genre of fantasy has been championed by very long narratives which strive to create extremely detailed imaginary settings, both lands of the impossible and credible coherent backgrounds. However, fantasy is not irreconcilable with short fiction. This article will focus on a precursor of the genre, the Irish author Lord Dunsany who began his career by writing collections of short-stories set in mythical lands. Instead of a tapestry whose threads become invisible, his imaginary world is built like a fragmented and tessellated literary body. Coherence and identity are no to be found in a wealth of details and lengthy descriptions, but in powerful, striking nodes of meaning, some of them as obvious as proper names or invented words, some as subtle as recurring rhythms or sound games.

Translating a text set in such an imaginary world poses specific problems to the translator, since the original often indirectly builds on cultural bearings to create a setting both related to reality and endowed with its own peculiarities and exotic features. The translator will need to find the same thin balance between familiarity and strangeness so as to allow its reader to enter the imaginary world. A few examples from the various translations of Dunsany's early tales into French will help us show that this in turn becomes a challenge to the translator, who has to identify these nodes and translate them, finding a new balance in fragmentation which still allows the reader to grasp the ethereal unity of Dunsany's imaginary territory.

I. Fantasy worlds of earth and sea

Lord Dunsany began his literary career in 1905 with the publication of The Gods of Pegāna, a collection of short stories describing his own original pantheon of deities. It seemed in keeping with the Irish tendency of the Literary Revival which entailed the collecting and rewriting of Irish myths, legends and folk-tales according to a political agenda of cultural independence. The Gods of Pegāna's title was reminiscent of Lady Gregory's Gods and Fighting Men, a personal version of the legends of the Tuatha de Danaan, published just one year before in 1904. But the strange diacritic on Pegāna reminded one of Greco-Roman metrics, not of specifically Gaelic or Celtic roots, and the name itself evoked either an indistinct primitive time (by its paronymic proximity to the word 'pagan') or a prehistoric remote past of universal myth (if we read it as an anagram of 'Pangea').

Dunsany kept elaborating his own fantasy universe in the following collections of tales he published until the First World War, and he is now mostly known as a forefather of modern fantasy and a notable influence on J.R.R Tolkien, who is widely considered as the creator of the genre itself. Tolkien's success made his work a reference of the characteristics of worldmaking in fantasy, which he defined in an essay entitled 'On Fairy-Stories' where he introduced the notion of the Secondary World, the world of fiction, and its relationship to the Primary World:

Enchantment produces a Secondary World into which both designer and spectator can enter, to the satisfaction of their senses while they are inside; but in its purity it is artistic in desire and purpose. Magic produces, or pretends to produce, an alteration in the Primary World. (Tolkien 1947: 8) Enchantment, which stands for magic, is considered as the essence of the fantasy world, but also needs to be related to our everyday reality. As an historian and linguist, Tolkien achieved this by designing an extremely rich, structured and detailed fictional world, complete with cosmogony, history, religions, languages, and accompanied with a scholarly apparatus of footnotes, maps, glossaries, appendices, tables, etc. In other words, the enchantment of Middle-Earth comes from abundance, from discovering that there is always more to it, and in perfect coherence with what one already knew. Most fantasy novels now are based on this model, frequently extending over trilogies or even more voluminous cycles. By Tolkien's principles, fantasy is a literature of impossibility where the power of invention may be unleashed, but also needs consistency as it unfolds. In the recent study Building Imaginary Worlds, Mark Wolf states that 'if a secondary world is to be believable and interesting, it will need to have a high degree of invention, completeness, and consistency.' But each of these criteria both encourages and limits the use of the other two, which is why 'all three properties must be considered simultaneously as the world takes shape and develops' (Wolf 2013: 34). In Dunsany's case, there was no lack in invention, as made obvious by the numerous exotic names, be they places, objects or characters, and their colourful descriptions. Though his plots may sound familiar to a modern reader, Dunsany was a pioneer in writing such 'stories of gods and heroes, magicians, and exotic cities of the "East" or "at the edge of the world"', stories that 'have had an immense influence on subsequent fantasy and are the immediate ancestors of much contemporary sword and sorcery or heroic fantasy' (Schweitzer 2013: 55). However, in terms of completeness and consistency, Dunsany breaks the rules: his fantasy world is unstable and changing like myths from oral tradition, with various versions, sometimes even redundant or contradictory. No matter how rich the invention, one can never know if his world is becoming more complete, because its boundaries are open, and can therefore never be entirely explored. To Tolkien's solid Middle-Earth, a territory with fixed boundaries and a solid timeline, we can therefore oppose Dunsany's Central Sea, which is the starting point of The Gods of Pegāna's prologue: 'There be islands in the Central Sea, whose waters are bounded by no shore and where no ships come --this is the faith of their people.' (Dunsany 2000: 534) Both names are vague and generic, but contrary to Tolkien's, Dunsany's locus is multiple, in the form of an archipelago where each story is like an island, and inbetween is a liquid desert, unknown and untraveled, with all the dangers and fears that entail the powerful image of the sea. In short, Tolkien creates territories, whereas Dunsany depicts places, in the sense Patrick Chamoiseau has given to these terms:

A place is open and thrives on this openness; a Territory erects borders. The place can be moved, it acknowledges and activates its multiple sources; the unblinking Territory braces against its unique root. The Place pertains to Diversity; the Territory imposes Universality. The Place can only be perceived as a thousand stories entangled; the Territory gains strength from one History. (Chamoiseau 1997: 205-206. My translation.) Dunsany relies on the short format to depict not a coherent and stable background but a world that remains loose and changing. The format of the short-story cycle here enhances the theme of isolation and fragmentation, creating a kaleidoscopic effect of refraction. How then does he allow the reader to connect with his unstable imaginary world? First of all, the titles and prefaces of his collections create a unity based on a unity of place and logic of dreams that prevents the reader from seeing each story as independent but leads him to expect the stories to be connected to a certain degree, creating cycles. It has been pointed out that 'central to the dynamics of the short story cycle is the tension between the one and the many' (Forrest Ingram, quoted in Mann 1989: 18); indeed, multitude is a challenge to the reader of Dunsany's tales, for there is a risk that the tremendous amount of exotic names -gods, kingdoms, cities, mountains, rivers -will represent too much information to process. In a Tolkien-like narrative, one would need a glossary, because each name would fit in the overall narrative. But there would be no point in doing so with Dunsany, since he seldom makes the names recurrent from one tale to the other. He keeps inventing new ones, ephemeral and transient, displaying the joy of naming things with sheer creativity, unbound by previous occurrences of names, and this is where the essence of his world and style lie. The reader therefore has to accept being carried away by the flow of the narration, from one name to the other like old Irish heroes sailed from one wondrous island to the next in the traditional sea-voyage tales, or Imrammas. Dunsany progressively creates a sense of familiarity through other devices than narrative coherence, founded on orality and repetition with variation: the names are different but sound patterns and rythms create a sense of unity (Assarnees / Poltarnees / Plegathanees, the Yann river / the desert of Azrakhan / Istahn / Zahn / Ozhan / the prophet Syrahn).

It thus appears that the short format is not incompatible with the creation of a powerful Secondary World, albeit one very different from those that the genre of fantasy has made us familiar with today. Most readers turn to Dunsany because they are already infatuated with the genre, and are bound to feel unsettled by the Irish lord's versatility and his multifaceted world. His work is like a mosaic or a tessellated work of art. He has bent compared to an impressionist whose tales blur the edges of reality: 'a typical Dunsany story is like a painting which breaks down into dots and globs of paint if one examines it closely' (Schweitzer 1989: 15). This kaleidoscopic effect is one of Dunsany's tool for creating a sense of wonder, which is the essence of fantasy and a challenge for the translator to recreate for a different readership in a different literary field

II. Dunsanian worlds in translation

The preceding considerations take relevance when considered within the context of the survival of the literary work, which depends on its diffusion and availability to new readerships, including in other countries, entailing linguistic translation. What and how to translate are pivotal questions, translation being the most usual mode of reading foreign fiction, and most readers will not turn to the original to draw comparisons. Therefore, the translated text is likely to stand as a strictly faithful representation of the original in their mind.

In the case of short fiction, representation begins with the editorial choice of format, title, and the selection of the texts, which give the stories various frames that whet the reader's appetite for some aspects of the original, on a metonymic mode at the macroscopic level.

Such metonymies are to be found in the way that translation is always a partial process, whereby some but not all of the ST is transposed, and in the way that translations represent STs by highlighting specific segments or parts, or by allowing specific attributes of the STs to dominate and, hence, to represent the entirety of the work. (Tymoczko 1999: 282) Moreover, the translator has to make microscopic choices as well as he works his way through the source-text, by choosing which aspects of the original will be retained. Each of these individual choices is like an island of meaning within the mosaic of Lord Dunsany's world, considering its fragmented nature, and each translation will therefore present a slightly different facet of this universe, as we shall see thanks to two examples.

Sprinkling reality within fantasy

With Dunsany, the Primary and Secondary World's relationship is fluctuating; they can be geographically distinct, and sometimes the otherworld is more of a symbolic dreamland, either personal or shared collectively. This makes it important for the translator to distinguish sprinkled references to real places within the dream world as important nodes of meaning.

My first example comes from the story 'In the Twilight'. The narrator has fallen off a boat and, drowning, sees visions of his past taking him back to the world of childhood. The Secondary world is here metaphorical, built up on a lyrical crescendo of emotions and the slow evocation of Ireland. The title itself is strongly connoted, since the Irish Revival was also known as the Celtic Twilight, after a book by Yeats which ended on a poem called 'Into the Twilight'. Throughout the story, Dunsany sprinkles his text with other words where one can read connotations of Ireland. Here is for instance the first sentence of the story:

The lock was quite crowded with boats when we capsized. (Dunsany 2000: 181) Two of the three existing French translations have the word lock translated quite unproblematically as l'écluse, but Patrick Reumaux chose to render it as le loch. We can assume he favoured the rendering of the Gaelic connotation enclosed in the sound proximity between the English lock and the Irish lough even at the cost of slightly distorting the denotated meaning. However, the French reader will most likely think of Scotland and its famous Loch Ness, and might not be aware of the common etymology that would allow him to relate this term to Ireland.

Later on the narrator has visions of the Bog of Allen, where he reconnects to his childhood and all his senses are gradually overwhelmed with feelings of Ireland, via the colour of the mosses, the smell of the burning turf, and finally the sounds that he ends up associating with his native country. Let us compare the three French translations of this passage:

And far away there arose and came nearer the weird cry of wild and happy voices, and a flock of geese appeared that was coming from the northward. Then their cries blended into one great voice of exultation, the voice of freedom, the voice of Ireland, the voice of the Waste; and the voice said 'Goodbye to you. Goodbye!' and passed away into the distance; and as it passed, the tame geese on the farms cried out to their brothers up above them that they were free. Then the hills went away, and the bog and the sky went with them, and I was alone again, as lost souls are alone. [START_REF] Dunsany | Time and the Gods[END_REF] In the source-text, several stylistic effects contribute to the powerful lyricism of the passage: the multiplicity of voices merging into unity via the incantatory repetition of the word 'voice', no less than six times, and each time having another qualifier ('of exultation', 'of freedom', 'of Ireland', 'of the Waste'), which makes all these words part of the same web of meaning associated with one reality. There is a final echo of characteristic Irishness at the end, just before the fantasy world fades away; with the word 'bog'.

The capital letter to 'the Waste' is significant, especially since the term is a key-word of Dunsany's literary idiolect. It thus provides a link between this individual story and others; the careful reader may then realize that in the Dunsanian language, the word 'Waste' carries undertones of Ireland, and this may apply even in stories with no reference to the Primary World. This is but one of Dunsany's 'pet-words', a word that 'finds its way again and again into the work and provides a key to its author's temperament.' [START_REF] Miller | Minor Magus, the fantastical writings of Lord Dunsany[END_REF]) Laura Miller identifies 'little" as the most significant of Dunsany's pet-words but Max Duperray has pointed out a number of other recurrent terms or phrases: 'exiled', 'the fields we know', 'shadow', 'prophecy', 'as they had prophesied', 'dream', 'stream', 'home', 'beyond' (Duperray 1979: 161, 176, 191, 291). Being part of the Dunsanian idiolect, these words represent a challenge to the translator who needs to spot them not only within a story, but also throughout collections, and even from one book to the other. Then he may choose one strict equivalent in French or resort to variation, hence risking losing this incantatory aspect of the original prose. Julien Green's translation plays down the Irish connotations of this passage: the bog becomes a more generic 'swamp' ('marais'), and, more importantly, Green erases the segment 'the voice of Ireland' from his translation, which destroys the building up of a connotation at the macroscopic scale. There is something of the Celtic cliché in the choice of 'la lande sauvage' (the moorland) for 'the Waste', and it connotes a romantic feeling that minimizes the sense of utter desolation of the source-word. 'The Waste' is actually very tricky to translate into French since it connotes both a natural expanse -a desert -and the decay of civilization and industry, garbage, ruins, a fall from grace, etc. The Wasteland is also a strong motif of Celtic literature, referring to a land cursed with barrenness and famine, which is of course of particular resonance with nineteenth-century Irish history (see [START_REF] Nitze | The Waste Land: A Celtic Arthurian Theme[END_REF]. Both P. Reumaux and A.-S. Homassel tried to find a one-word target equivalent to maintain the dramatic effect of the capital letter. The former chose 'la voix du Désert' (the voice of the Desert), which reduces the richness of connotations but at the same time, does resonate with Dunsany's other tales, especially his Arabian Nights-flavoured stories. A.-S. Homassel translates it as 'la voix de la Solitude' (the voice of Loneliness), favouring a more abstract feeling that sums up the atmosphere of the tale, the narrator being gradually abandoned by all his memories as he is drowning and dying. In French, in a somewhat dated literary language, the word solitude can also metonymically refer to a desert place, which may account for this choice.

To conclude, depending on the translation one reads, the fantasy Ireland that is created through stylistic devices and imagery will be more or less in keeping with the lyricism of a Romantic tradition, more or less linked to the rest of the Dunsanian world or to one's own preconception of Ireland and Irishness.

The underlying principles of the mythopeic fantasy world

My second example stems from the mythopoeic character of Dunsany's prose itself on the one hand, and from what may be considered an irreductible difference between the languages on the other. In a number of tales, when mimicking the mythic mode of classical literature, Dunsany often personifies the forces of nature to use them as godlike characters. This is conveyed in the text by the use of a capital letter which turns the phenomenon into a proper name and by using gendered pronouns instead of the impersonal it. This may pose two problems in French, where all nouns are assigned a given gender by default. First, even if Dunsany's choice of gender happens to be the right one in French, the visual marker of the personal pronoun where one would expect the impersonnal will pass unnoticed. Second, the translator may have to change the character's gender, not to offend the logic of the French.

In all of Dunsany's French translations, no translator has dared to break this grammatical rule. In 'Night and Morning', Dunsany has the masculine Night tell stories to the feminine Morning, which is consistent with an earlier story where the Dawn was a little girl. A.-S.

Homassel inverts the two genders: 'our Lord the Night' becomes 'Notre maîtresse la nuit' ('Our Mistress the night'). There is no incidence on the narrative itself, but each occurrence of this stylistic process is a way for Dunsany to consolidate his pseudo-mythological world and breathing life into divinized natural forces. This is a subtle way of creating unity and consistency as well as enchantment, easily lost in translation.

For instance, let us focus on the Sea, since we have seen it is one of the most pregnant motifs of Dunsany's fantasy world. In the story 'Poltarnees, beholder of the Ocean', the young men of a kingdom surrounded by mountains keep leaving their homes, crossing the mountains to behold the sea, and none ever come back. The hero Athelvok has promised his fiancée, the princess, that she is more beautiful than the sea and that he will come back after he has understood why. He eventually gets across the mountain and there discovers the sea: And there below him was the old wrinkled Sea, smiling and murmuring song. And he nursed little ships with gleaming sails, and in his hands were old regretted wrecks, and mast all studded over with golden nails that he had rent in anger out of beautiful galleons. (…)At that moment the Sea sang a dirge at sunset for all the harm that he had done in anger and all the ruin wrought on adventurous ships; and there were tears in the voice of the tyrannous Sea, for he had loved the galleons that he had overwhelmed, and he called all men to him and all living things that he might make amends, because he had loved the bones that he had strewn afar. And Athelvok turned and set one foot upon the crumbled slope, and then another, and walked a little way to be nearer to the Sea, and then a dream came upon him and he felt that men had wronged the lovely Sea because he had been angry a little, because he had been sometimes cruel; he felt that there was trouble among the tides of the Sea because he had loved the galleons who were dead. (Dunsany 2000: 241-242) Throughout this passage, the ambivalence of the Sea as a sublime masculine figure is striking: he is characterized both by his violence, reminiscent of when he was a young conqueror god, and by new emotions such as love and regret. Both Julien Green and A.-S.

Homassel translated this passage using the feminine, making the Sea appear as an old woman. This affects our reading on three levels.

First, it disrupts echoes of significance linked to previous tales from Dunsany's body of work, in which the Sea is almost systematically presented as endowed with masculine features. 'The Coming of the Sea' (The Gods of Pegāna) tells of the arrival of the god Slid 'whose Soul is in the Sea' as a foreign conqueror god who comes to brutally invade the first gods' lands. In Time and the Gods, he has become fully integrated into the pantheon ('A Story of the Dawn'). And even if he is not named in 'Poltarnees', we can feel the presence of this figure, although older now, at the end of the tale. Such a connexion cannot be made if the reader is unaware of the gender shift that has been operated by the translator.

Second, feminizing the Sea alters the economy of the story itself; as it turns it into a seductive and whimsical siren ('la charmante Mer') and presents the princess and the Sea as rivals for the hero's love. The violence brings to mind the figure of a monstrous Charybdis, and the image of the old woman evokes a witch prone to fits of anger and hysterical destruction, even though she can also prove a benevolent and loving old mother. This distortion unravels the perception of the Sea as a symbol in the passage per se. Beholding the sea is here described as an experience not of physical or emotional love, but of the sublime, entailing both the fear of an overwhelming power and the fascination for a paradoxically frail and evanescent presence.

Finally, feminizing the Sea's violence alters our perception of the novelty of Dunsany's imagery. The Sea's destructive power appears weaker compared to the expressions of feminine love and the desire to atone, which are unexpected from a raging seagod, while it seems rather conventional for an old woman looking back at her life. Where the source-text challenged the traditional vengeful Poseidon figure through the subtle gender orientation, the French presents us with a more conventional figure. According to Maria Tymoczko 'human beings are not very good at hearing new stories: we have a tendency to reinterpret them, to reshape them so that they become versions of stories we already know' (Tymoczko 1999: 48). Even if the ambiguity of extreme violence and love juxtaposed is preserved, the originality of the image may be overlooked by the French reader who will associate this tale to patterns he knows, to representations of the Sea as a mother figure. From the subjectivity of the translator to the subjectivity of the target-reader, a detail such as gendered pronouns may therefore produce a significant change. The French version of the tale creates a different mythology than Dunsany's, hence a different fictional world.

Conclusion

Fantasy as a genre is considered as difficult to translate because of the specific discrepancy between its combined reliance on sheer imagination and realist techniques to ensure the reader's suspension of disbelief. Dunsany's fantasy short fiction takes us one step further into difficulty, since his imaginary worlds are plural and depends on stylistic devices and individual passages whose importance may be overlooked. The representation of the fictional universe will not be the same depending on whether the reader has read only one of Dunsany's tales, either isolated or within an anthology, one full collection or several of them.

Only by becoming more and more familiar with his universe may the reader start seeing the meshes of echoes and fluctuant patterns of his prose. Contrary to Tolkien, Dunsany relies on repetition more than expansion; the elements that keep coming back constitute nodes of meaning, keys to the fantasy world. What makes him so special is that you don't need all the keys, or to find them in a specific order.

References to Ireland and mythopoeic personifications are but two examples of such nodes of meaning, and there are much more in Dunsany to which the translator is confronted: archaic language, metaphors, the building up of rhythm, the clues of oral storytelling, are all tessels of this great mosaic. Such a fragmented fictional world is challenging but also admirably fit for translation, and even multiple translations. The metonymic aspect of translation entails that, as one reads more than one version, the chances to identify new elements as key nodes of meaning increase. Reading fantasy short fiction in translation may help us forget about the age-old debate on faithfulness and loss unavoidable in translation and become more aware of the gain and richness that this rewriting act makes possible, giving us keys to open a virtually infinite number of doors on the Central Sea of mankind's imagination.


J Mainka

G Maranzana

J Dillet

S Didierjean

O Lottin

A simple alternative to the expression of finite Warburg diffusion impedance in porous electrodes by considering oxygen consumption along the air channel

The classical expression of the finite Warburg element used for analysing electrochemical impedance spectroscopy (EIS) spectra is obtained assuming that the reaction occurs at the electrode/membrane interface and that the oxygen concentration at the gas channel/gas diffusion layer interface is constant. An alternative to this expression is proposed: it takes into account the variation in oxygen concentration along the gas channel due to the oxygen-reduction reaction (ORR). This pseudo-2D approach is well suited for investigating mass transfer limitations in the fuel cell membrane-electrode assemblies (MEA). Simple numerical applications starting from data already published by different authors show that with this new expression, conclusions about the mass transfer limiting layer and its main characteristics can be significantly modified. This improvement to the usual Warburg impedance could contribute to a better understanding of mass transfer limitation in fuel cells and eventually, to the design of MEAs in which diffusion losses are minimal.

Introduction

Fuel cell development has made considerable progress in recent years, so that their incorporation in mobile and stationary power applications becomes a realistic aim for the near future. Among the different categories of fuel cells (depending on the fuel and on the type of electrolyte), proton exchange membrane fuel cells (PEMFC) are very promising because of their moderate operating conditions (temperatures < 100 °C, gas pressures from 1 to 3 atm (1)). Fed by pure hydrogen and air, they do not emit any greenhouse gases.

Fuel cells must be able to operate at high current densities with a satisfying efficiency and reasonable gas consumption. Under these conditions, the slow oxygen diffusion to the cathode reaction sites is a significant impediment, which has a direct influence on the oxygen reduction reaction (ORR) kinetics: for instance, various studies highlight the role of the gas diffusion layer in oxygen mass transport limitations in PEMFC (2,3). Hence, there is a need for mass transfer studies pointing out clearly the origin of the oxygen diffusion limitations, for which EIS can be of great interest besides of being a powerful technique for investigating electrochemical reaction kinetics in the porous electrodes.

The first one-dimensional steady-state diffusion model proposed by Springer et al. (4) made it possible to analyse fuel cells performance dependence on water management. Since, many improvements have been brought to the theoretical models. Most of them focus on charge and mass transfer dynamics in the cathode active layer (5)(6)(7). Eikerling and Kornyshev (6) even developed a model describing potential and concentration gradients through the active layer in the direction perpendicular to the electrode surface. In parallel, EIS allows to investigate the different processes occurring in PEMFCs (8) and to analyse in situ fuel cells behaviour (9,10). One of the most common expressions of diffusion impedance used in fuel cell equivalent circuits is the finite Warburg element (11). It is based on simple assumptions: Fick's diffusion of oxygen, surface reaction, and a constant concentration at the gas channel boundary of the diffusion medium. The resulting expression of the diffusion impedance yields surprisingly good results. However, in order to improve this model, a decreasing oxygen concentration profile along the gas channel can be used as a boundary condition. This does not change radically the expression of the diffusion impedance, but it is a first step for obtaining a more accurate mass transfer model.

Theoretical Model

Before accessing the active reaction sites, air and hydrogen have to flow through different media: the bipolar plates ensuring the gas supply via the flow-field channels, the gas diffusion layers (GDL) homogenising the gas propagation over the electrodes, and the catalyst layers consisting of a porous mixture of carbon powder and catalyst particles embedded in the electrolyte. Gases have to flow through their pores and the electrochemical reactions occur on (or in) the solid phase. Mass transfer models should take into account as well as possible the geometrical characteristics of these successive media but actually, the finite Warburg element describes only gas diffusion through a homogeneous medium.

Mass transfer model

According to the majority of the authors (5,6,12), the oxygen reduction reaction is the limiting factor for the energy conversion process in PEMFC and on top of that, the access of oxygen to the reaction sites remains problematic, due to the (possible) presence of liquid water in the MEA and to the low oxygen partial pressure when the fuel cell is fed by air (1). On the other hand, since the cathode is the place of water production, the ionic resistance of the well-hydrated polymer can be neglected. As a consequence, the main hypotheses of the one-dimensional cathode model, where oxygen transport occurs only by diffusion are:

• The fuel cell is isotherm and isobar.

• Ohmic drops in the active layer and in the GDL are neglected due to their high ionic and electronic conductivity. • Mass transfer of O2 and H2O is assumed to occur only by diffusion in the pores of the active layer and the GDL. • There is no H2O diffusion through the agglomerates. These are usual assumptions for one-dimensional mass transfer modelling in fuel cells. In order to obtain a pseudo-2D model (Figure 1), which takes into account oxygen concentration variation along the gas channel, three more hypotheses are made:

• Mass transfer limitations in the gas channel are neglected.

• Time variations of the oxygen concentration in the channel are neglected.

• The Tafel slope b is constant between the inlet and the outlet of the air channel.

The electrode surface is given by L × d; the gases flow along the channel length L.

Usually, it is preferable to use a Butler-Volmer formalism to describe oxygen reduction kinetics (13) but for sufficiently high current densities and in steady state, the activation overpotential follows a Tafel law of the form:

 = = [1]
In equation [1], the faradaic current density jf (y) and the oxygen concentration cO2 (x,y) vary along the gas channel, i.e. as functions of the y coordinate (Figure 1). The cell average current density J is given by:

0 1 L f J j dy L =  [2]
It is worth mentioning that by convention, the activation overpotential as well as the faradaic current density at the cathode side are usually negative. However, for convenience their absolute values are considered in the following: ( )

,0 f j y t  .
Under the conditions mentioned above, the molar fluxes of oxygen and water in the channel depend also on their y-position. Thus, the molar flux of oxygen 2 () O Ny corresponds to that at the inlet of the gas channel minus the flux consumed at the cathode:

( ) 0 02 02 4 y f in d j dy N y N F =-  [3]
Supposing that all water produced by the fuel cell remains in the air channel, it comes:

( )

0 20 20 2 y f in HH d j dy N y N F =+  [4]
The fuel cell being fed by air, the oxygen molar ratio at the GDL/gas channel interface as a function of y is given by: Neglecting the integral in the denominator of [6], it can finally be approached by: ( ) ( )

( ) ( ) ( ) ( ) 02
0 02 0 02 02 02 , 1 y f d j dy S I c x y c SH  - == +  . [7] With ( ) 0 , x c c 2 O 0 2 O  = =
. According to the previous hypotheses, mass transport of oxygen through the pores of the GDL and those of the electrode is described by Stefan-Maxwell equations (1,14,15). However, since the binary diffusion coefficients of O2/H2O and O2/N2 are close to each other, it seems reasonable to use Fick's 1 st law. Assuming that the electrochemical reaction takes place only at the active layer/membrane interface (x = 0), the 1 st Fick's law can be written:

( ) 2 2 0 O O eff x y c N y D x =  =-  . [8]
The gas diffusion and active layers being porous, it is necessary to use an effective diffusion coefficient eff D (16,17) taking into account their porosity ε and possibly, the presence of Knudsen diffusion (1,17). The most common expressions obey Archie's law (18,[START_REF] Torquato | Random Heterogeneous Materials: Microstructure and Macroscopic Properties[END_REF]:

m eff D D  = [9]
Where m is an exponent varying between 1.5 and 4. In the case of a 3D medium, there is a wide consensus for using m = 3/2, which corresponds to the differential effective medium approximation introduced by Bruggeman [START_REF] Torquato | Random Heterogeneous Materials: Microstructure and Macroscopic Properties[END_REF][START_REF] Bruggeman | Berechnung verschiedener Physikalischer Konstanten von heterogenen Substanzen[END_REF]. m = 3/2 is probably appropriate for the active layers, in which the orientation of the solid phase does not follow a privileged direction (although this is not true in the agglomerate models using a cylindrical description). In the gas diffusion layers however, the solid phase can be considered as two-dimensional since the carbon fibres are mainly parallel to the electrodes. In this case, the result of the differential effective medium theory is m = 2 [START_REF] Torquato | Random Heterogeneous Materials: Microstructure and Macroscopic Properties[END_REF]. In a liquid, the diffusion is governed by shocks between particles and D corresponds to the molecular diffusion coefficient Dmol. In a gas, two cases have to be distinguished depending on the values of the pore size and of the molecules mean free path  (14):

2 A PN 2 RT    = [10]
Where NA is the Avogadro number and  the mean molecule diameter. If the mean free path is much smaller than the pore size inter-molecule collisions govern the diffusion process and D = Dmol. If the mean free path is of the same range as the pore size ( ≈ dpore) molecule collisions with the pore wall become significant and the Knudsen diffusion must be considered. This is not the case in the GDL where the typical pore size is between 20 and 50 m (21). On the other hand, Kong et al. (22) showed that in most of the active layers, the pore size is between 30 and 60 nm. Therefore, the Knudsen diffusion has to be taken into account. Since the mean free path of the gas molecules is of the same order of magnitude as the pore size, there exist various more or less sophisticated expressions combining D K and Dmol for estimating the diffusion coefficient D. One of the most simple is the Bosanquet formula

K mol D 1 D 1 D 1 + = the Knudsen diffusion coefficient of species i K i D can be expressed by i pore K i M RT 8 d 3 1 D  = .

DC solution

In steady-state conditions, cO2 and Ni (with i = O2, N2, H2O), as well as jf and ηact are constant in time. As the different species are neither consumed nor produced in the diffusion layers, their molar fluxes depend only on the y position. Nitrogen being inert, its flux is null while oxygen and water fluxes can be expressed by jf (y): Solving the two Fick's diffusion equations in steady-state, knowing the oxygen concentration at the channel/diffusion layer interface [7] yields an expression for cO2 (x,y). Fick's 2 nd law in steady-state conditions is given by:

2 02 2 0 eff c D x  =  . [12]
Using equation [11] relating jf and the molar oxygen flux, the 1 st Fick's law [8] gives:

( )

2 0 4 f O eff x y jy c D xF =  =  . [13]
This leads to the following expression for oxygen concentration in steady-state:

( ) ( )

02 0 0 02 02 02
, 1 14

y f f eff IS d j dy j x c x y c IS H FD   -  = - -  +   . [14]
For comparison, the oxygen concentration in the classical 1D model is expressed by:

( ) 0 02 02 1 4 eff Jx c x c FD    = - -   . [15]
The difference between [14] and [15] lies in the oxygen concentration at the gas channel boundary. In the one-dimensional model, it is constant over the electrode surface, whereas in the two-dimensional model it varies between the gas inlet and the outlet.

AC solution

When a small sinusoidal perturbation Δjf (y,t) is added to the mean (DC) current density ( )

f t
jy, the oxygen concentration cO2 fluctuates around its steady-state value with the same frequency. Solutions for the current density and the concentration must obey the following conditions:

( ) ( ) ( ) ( ) 02 02 02 02 , , , , ( , ) , exp t
c x y t c x y t c x y c x y iwt  = - =  [16] ( ) ( ) ( ) ( ) , , ( ) exp
f f f f t j y t j y t j y j y iwt  = - =  . [17]
And the time-averaged values of the oxygen concentration jy correspond to the steady-state solution [14]:

( ) ( )

02 0 0 02 02 02 () () ,1 14
y f t f t t eff IS d j y dy jy x c x y c IS H FD   -  = - -  +   . [18]
Thus, in boundary conditions [7] and [13], the current density is replaced by its timeaveraged expression ( )

f t
jy. Fick's 2 nd law in AC conditions is given by:

2 02 02 2 eff cc D tx  =  [19]
Resolving this system of partial differential equations yields the following expression for cO2(x,y,t):

( ) ( ) ( ) ( ) 2
        -     -            = - + - -     +                    [20]
At x = 0, the variation in oxygen concentration ΔcO2(x = 0,y,t) is given by: ( ) ( )

2 02 0, , tanh 4 f iwt eff eff jy jw c x y t e D F jwD     = = -     , [ 21
]
which makes it possible to express the faradaic impedance, defined as:

2 2 0 act act act O f f f O f x c Z j j c j    =     = = +     . [22]
The first term

act f j   
is the charge transfer resistance Rct, which can be obtained by deriving the Tafel law [1]:

( ) () tc f t b Ry jy = , in Acm 2 . [23]
The second term . Therefore, the mass transfer impedance can be calculated with [21] and [17], which yields:

( )

2 2 02 tanh 4 ( 0, ) eff cO eff t jw D Z y b F c x y iwD      = =
.

[24]

The oxygen concentration at x = 0 can be obtained with [18]. The expression of the diffusion impedance [24] can also be written:

( ) ( ) 2 2 tanh d D co d d iw Z y R iw   = , [ 25]
with the low frequency limit Rd 2D [27]

At the first sight the expression of the pseudo-2D diffusion impedance is identical to a classical finite Warburg element (11):

( )

1 tanh d D Wd d iw ZR iw   = , [28] with 1 02 4 ( 0) D d eff t b R FD c x  = = .
[29]

The characteristic diffusion time τd is also the same as in the pseudo 2D model [27]. Actually, the difference lies in the expression of the low frequency limit 1 /2 DD d

R

: in the pseudo 2D model [START_REF] Fischer | [END_REF] the oxygen concentration 02 ( 0, ) t c x y = depends on the y position, which should improve the precision of parameter estimation starting from the global diffusion impedance as well as from its local values along the cell.

Results and discussion

Applied to actual fuel cell systems, the pseudo-2D diffusion model gives access to the global and local mass transfer characteristics under specified working conditions. It is used to analyse data taken from the literature (15,23) in order to open a discussion about its accuracy and interest.

of local mass transfer

It can be useful to assess the local gas diffusion conditions in a given MEA in order to determine the optimal operating parameters. From [START_REF] Fischer | [END_REF] and [18], an explicit expression for the diffusion resistance ( )

2D d
Ry can be obtained. Calculating the 2 D d R profile between the gas inlet and the gas outlet reflects the local mass transfer dynamics, which affects in turn the oxygen reduction kinetics:

( )

2 2 0 0 2 2 () 4 ( ) 1 D d y Of t eff O f t O b R IS d j y dy FD c j y IS H   = - - +  .
[30]

The results illustrated below are deduced from data published by Bultel et al. (15). They correspond to a H2/air PEMFC operating at 60 °C and with an inlet relative humidity of about 90%. The characteristics of the gas diffusion layer and the main operating conditions are summarized in table I.

Table I:

Values of the parameters used for mass transfer kinetic analysis (15).

Tcell [°C] 60 H [-] 0.9 p [atm] 1 SO2 1 [-] 2 (if not mentioned otherwise) Aelectrode [cm 2 ] 0.5 L 1 [cm] 1 d 1 [cm] 0.5 δGDL [µm] 500 (if not mentionend otherwise) εGDL 0.2 2, 2 O H O GDL eff D [m 2 s -1 ]
3. ( )

2D d
Ry profiles between the air inlet and the air outlet for different air stoichiometries. The curves tend toward infinity for SO2 < 1.93 according to eq. [31] and to the values of the parameters given in table I. Ry for each position y. Fig. 2 shows the modelled profiles of ( )

2D d
Ry under the operating conditions of table 1 and for air stoichiometries varying between 1.5 an 8. The increase in ( )

2D d
Ry from the air inlet to the air outlet depends on the air stoichiometry. There is a critical value of the air stoichiometry ( )

2 crit O
S  -depending on the diffusion medium thickness δ-for which no oxygen can access to the active sites beyond a limiting length ylim. As a consequence, for y approaching ylim, 2 D d R tends toward infinity.

( )

2 crit O
S  is defined as the stoichiometry factor for which the oxygen concentration at the electrode/membrane interface at the gas channel outlet

( ) 2 0, O c x y L == is null.
In steady-state conditions and from [14], it comes:

( )

0 02 2 0 02 1 4 crit O eff c S J cH FD  = -+ [31]
In the example of Figure 2, ( )

2 crit O
S  is about 1.93 and for SO2 = 1.5, there is no oxygen reaching the electrode/membrane interface beyond ylim/L = 0.78. The analysis of ( )

2D d
Ry profiles makes it possible to determine in which conditions the diffusion impedance can be described by a one-dimensional model (when, for instance, the maximum relative variation over the electrode length

( ) ( ) ( ) ( ) 22 2 2 0 0 DD dd D d D d R L R RL R - = is below 10%).
For this example ( 15), a one-dimensional behaviour would be achieved only for air stoichiometries SO2 above 11. Ry between the air inlet and the air outlet for different values of the diffusion layer thickness and with an air stoichiometry SO2 = 2 (starting from data of Bultel et al. (15), table I).

The curves in Figure 3 show that the thickness of the diffusion layer can have a great impact on mass transfer kinetics: when SO2 = 2, reducing its initial value (500 µm With relatively simple hypotheses, this alternative expression of the Warburg diffusion impedance shows that in certain conditions, mass transfer limitations due to oxygen consumption along the channel can be quite significant: the usual 1D expression remains satisfying only for large values of the air stoichiometry which do not correspond to usual operating conditions. Further investigations should focus on the time-variations of the oxygen concentration resulting from upstream consumption and convection, which should entail -in some cases-a time-shift between oscillation in current density and in oxygen concentration, as a function of the flow velocity (24).

Identification of the main characteristics of diffusion EIS could also be used to estimate the geometrical characteristics of diffusion layers. As 1 /2

DD d R
and τd depend both on the diffusion medium thickness δ, on the effective diffusivity Deff, and indirectly on the porosity ε [9], the experimental impedance spectra can be used to identify the value of these parameters. Starting from the expression of the characteristic diffusion time τd [27], that of ( )

2 02 0 02 02 1 1 2 4 1 d tc D d JR R S Fc SH    =+   - + .
[34]

The limiting diffusion layer can be determined by comparing the value of δ identified from the experimental spectra to the actual thickness of the gas diffusion layer and of the active layer. Furthermore, comparing the results of equations [32] and [33], shows whether oxygen transport can be considered as one dimensional or 2-dimensional. Examples of results are presented in the following, using data published by Rubio et al. (23), which are obtained with a single PEM cell of S = 100 cm 2 operating at 12.5 A. The fuel cell is fed by pure H2 and air (at 1 atm) with a stoichiometry SO2 = 2.3 (23).

According to Rubio et al., the charge transfer resistance Rtc = 13.2 mΩ and the diffusion resistance Rd = 7.97 mΩ. Since the cell temperature was not controlled, it has to be estimated, as well as the air inlet relative humidity. Knowing that the fuel cell was not humidified, we chose Tcell = 60 °C and RH = 0.1. The identified values for δ and Deff are given in table II. The differences between values of the diffusion layer thickness δ and of the effective diffusion coefficient Deff obtained with the usual Warburg expression and with the one resulting from the pseudo-2D approach shows clearly that gas consumption along the air channel has a significant influence, at least in these operating conditions. The classical one-dimensional Warburg expression yields a value of the effective diffusion coefficient typical of active layers [table II]; the corresponding value of the porosity (starting from an equivalent -molecular and Knudsen-diffusion coefficient equal to 6.9 10 -6 × εAL 3/2 [9]) is also within the range generally considered for active layers. However, using the finite Warburg impedance, the estimated thickness of the diffusion layer is almost 2 orders of magnitude larger than that of an active layer (≈ 10 µm). As a consequence, the 1D model does not allow conclude about the limiting layer for mass transfer. Using the pseudo-2D approach [34], both the porosity and the equivalent diffusion coefficient are within ranges typical of gas diffusion layers. The estimated thickness is about 3 times larger than actual GDL thickness, which could be explained by gas diffusion in the y direction below the channel rib (increasing the total diffusion length) and/or by a possible mass transfer resistance in the gas channel. Thus, the pseudo 2D approach returns results pointing out the gas diffusion layer as the limiting mass transfer layer, in accordance with some previous works (15).

Conclusions and perspectives

EIS is frequently used to investigate the origin of the mass transfer impedance in cathode gas diffusion and active layers. However, the classical expression of a finite Warburg element used for analysing impedance spectra is obtained assuming that the oxygen concentration at the gas channel/gas diffusion layer interface is constant. This hypothesis is much constraining when the gas stoichiometry is low, and it could lead to wrong estimates of mass transfer parameters characterising the diffusion media. An alternative to this expression is proposed: considering gas consumption along the gas channel entails minor changes to the classical Warburg expression but the first numerical applications show that conclusions about the mass transfer limiting layer and its main characteristics can be significantly modified. The estimated value of the GDL thickness is significantly higher than the actual one, which could be linked to a possible mass transfer resistance in the gas channel and/or to constriction effects in the GDL, below the channel rib. Considering oxygen consumption also makes it possible to investigate variations in local impedance spectra and in the diffusion kinetics along the air channel. These first results are only of qualitative interest and should be shortly confirmed by specific experimental investigations. Other alternatives to the expression of Warburg impedance will also be evaluated, by considering for instance mass transfer resistance in the gas channel or at the interface between the channel and the GDL.

Figure 1 .

 1 Figure 1. Schematic representation of the pseudo 2D diffusion model. Oxygen concentration variation in the channel is taken into account for mass transfer modelling.

=

  at the air inlet. Thus, the mean oxygen concentration at the GDL/channel interface is assumed equal to its value in steady-state (time variations are neglected):

  mass transfer impedance Zco2.Electrochemical impedance spectroscopy analyses small perturbations around the steady-

  Figure 2.

Figure 3

 3 Figure 3. Profiles of

  (15)) by one half decreases significantly the mass transfer limitations ( ( ) 2D d RL  is reduced by a factor of about 14) and the ( ) 2D d Ry profile becomes almost flat. When SO2 = 4, ( ) 2D d RL  depends less on the diffusion layer thickness but ( ) 2D d RL  is still divided by about two when dividing the diffusion layer thickness also by two.

  [23], one can obtain relations giving the thickness of the diffusion medium. It depends on the operating conditions and on the impedance parameters τd and

						1 /2 DD R	[29, 30] and that from the charge
						d
	transfer resistance
	1 /2 DD R	, which must be determined experimentally:
	d
					01 02 tc D d JR Fc d R  4  1  =+ 	.	[32]
			( 1 fd 0 02 IS + () t y f jy 02 d j  - =+  ) 2 02 Fc H 4 0 1 () tc D d t R R IS y dy  	.	[33]
	Actually, equation [33] expresses δ as a function of y and it is necessary to consider
	its mean value over the electrode length L. In the absence of other information, assuming
	a homogeneous current density,	j	=	/ I dL	, it comes:
				f
					y

tc

R

Table II :

 II Diffusion layer thickness δ and effective diffusion coefficient identified starting from experimental results of Rubio et al.(23) and comparison with typical values from the literature. Oxygen is assumed to diffuse in gas phase in the GDL and the active layer: the possible presence of liquid is accounted through the actual value of the porosity ε.

	Model	δ [m]	Deff [m 2 s -1 ]	εGDL	εAL
	1D	0.834 10 -3	9.94 10 -7	0.18	0.27
	pseudo 2D	1.2 10 -3	1.96 10 -6	0.25	0.43
	Literature	GDL: 0.42 10 -3 [27] AL: 10 10 -3 (est.)	GDL: 3.2 10 -5 x εGDL 2 AL: 6.9 10 -6 x εAL 3/2 (25)	0.2 -0.6 (5)	0.2 -0.35 (26)