François Jullien

Jean François Billeter

Baptiste Mélès

Experience and subjectivity

Final remarks 17

Introduction1

The concept of subjectivity as we know it, especially since Descartes, does not appear as such in ancient Chinese thought. In Confucius' and his followers' work, it can not be found anywhere: there is no universal form of subjectivity, but always princes or subjects, parents or children, husbands or wives, elder or younger brothers. With the exception of friendship, no social relationship is symmetrical: there are as many kinds of subjectivity as kinds or social relationship (五倫 wǔlún). What comes closer to the Western notion of subjectivity appears much later, under the influence of Indian philosophy, especially Buddhism. But the fact that subjectivity is not explicitly expressed does not imply that one can do without it. Subjectivity may remain hidden, or implicit, awaiting to be revealed by commentators. Thus, what is in stake is not to know if subjectivity has been theorized by Chinese thinkers, but rather to know if sinology may use this concept as a principle when studying Chinese classics.

This question was recently raised in the controversy between two sinologists, the Swiss Jean François Billeter (born in 1939) and the French François Jullien (born in 1951). The general public heard of it for the first time when Billeter's book Contre François Jullien was published in 2006; but the controversy has begun much sooner, twenty years ago, in 1989, in the French journal Études chinoises.

Here will be briefly exposed the works of both sinologists, especially from a methodological point of view, and then the controversies that, as we will see, emerged quite naturally. We will finally examine if the existence of subjectivity in ancient Chinese thought is a merely sinological question, or more generally a philosophical one.

1 Antithetic methods

François Jullien's conceptual method

François Jullien frequently explained why he first decided to study Chinese thought.2 After passing the French agrégation of philosophy, intending to know the foundations of the Western way of thinking, he planned to discover some other ways of thinking, and wondered where he could find them. The scope statement was to find some written culture, which would have not been linked to our history for a long time. Biblical, Hebraic, Arabic cultures were thus excluded. Indian culture was too close as well, Sanskrit being an Indo-European language. And as Japanese culture was partly influenced by China, he decided to study the later one.

"Another ways of thinking": this idea may justifiably remind of Foucault's injunction to "think differently" (penser autrement). 3 And Jullien makes no secret of this influence: he regularly quotes the few lines Foucault wrote about Chinese heterotopia, at the beginning of Les Mots et les choses.4 Maybe Jullien's project was also influenced by Heidegger's questions about Eastern tradition. China would have been for him a mean for acquiring, by comparison, a better knowledge of Greece. When wanting to know our actual way of thinking, Foucault had been investigating into history, comparing for example biology with natural history. Wanting to become aware of the structural characteristics of European philosophy, Jullien compared it with Chinese thought. All in all, Jullien sought in space what Foucault sought in time.

By method, what interests Jullien is Chinese thought as a tradition,5 rather than individual thinkers as individuals. This often leads him to attribute to Chinese thought what he has shown for one thinker. The book Procès ou création is thus subtitled as An introduction to Chinese literate thought, even though its unique object is Wáng Fūzhī (王夫之, 1619-1692).6 Wáng Fūzhī does not seem to be described for himself, but as an example, as a typical case.

Intending to highlight what constitutes Chinese thought as a tradition, Jullien almost always reads Chinese classics with the help of commentators. As an example, the Yìj īng (易經) is seen through Wáng Fūzhī's eyes in Figures de l'immanence. 7 Jullien's goal is to show what kinds of fundamental concepts are used by Chinese classics as well as by their commentators.

The main theses of Jullien's work are quite well-known. The ancient Chinese thought would be a thought of immanence, whereas European philosophy has been for a long time -and maybe always is -a thought of transcendance, be it the transcendance of God or the transcendance of subjectivity. 8 The Chinese thinkers often show continuity between oppo-sites, whereas the Western thought often radically distinguishes concepts, as substance and accident, being and nothingness, time and space, etc. In particular, the ancient Chinese thought did not distinguish subject from object, and did not conceive the notion of an abstract and universal subject.

Nevertheless, what we would like to underline is not really the thesis to which Jullien arrives, but the method he uses. There is a few texts in which Jullien speaks from experience, an experience which may be considered as pre-discursive; for instance, when comparing Mencius (孟子 Mèngzǐ, 380-289 BC) with Rousseau. [START_REF] Jullien | Dialogue sur la morale[END_REF] But in general, Jullien avoids speaking about experience, as well as Foucault in his Archéologie du savoir. 10 Even though Jullien avoids to express this in such abstract words, he seems to think that in a way, discurse precedes experience. Experiences are built of words and traditions. Jullien claims for the inheritance of Nietzsche, and reads Chinese texts as a philologist. He avoids to use the notion of experience, which may let one think that there is something universal before the discurse; without excluding it, Jullien refuses to presuppose it. As a philologist, he highlights the role of words, of concepts. There is no innocent word: if Zhuāngzǐ (莊子, IV th century BC) or Wáng Fūzhī regularly use the word of dào (道), one can suppose it is a fundamental concept, and examine both what these occurrences have in common and what this notion may mean in an abstract sense.

Jullien's theory of translation is a natural consequence of his philological method. Translation should fulfill two demands. The first one is to keep the importance of a word: almost each time the same Chinese word is used, it should be translated by the same French word. The second one is to keep the relative independance of two philosophical lexicons; by using the words of God, being, subject or some other capital concepts of Western philosophy, we could nothing but misinterpret what makes Chinese thought special, what constitutes it in itself. Translations must keep something idiomatic. This is the reason why Jullien often uses French words which do not belong to the classical philosophical lexicon, but to biology or other realms: "régulation," 11 "procès," 12 etc.

Not only Jullien finally observes no real subjectivity in ancient Chinese thought, but the lack of subjectivity has been made possible by his method itself. By method, Jullien has chosen to analyze Chinese thought from the concept point of view, much more than starting from the structure of conscience.

Jean François Billeter's cousciousness point of view

Unlike Jullien, Billeter does not study Chinese thought through a tradition. He never uses commentators, but always reads directly the texts themselves. Whereas Jullien partly seeks the meaning of a text in its outsides, Billeter exclusively searches for it in its inside.

But it is not enough to say that Billeter does not study China through a tradition: more than that, he often studies it going against the tradition. He thus frequently severely criticizes the Chinese tradition of thought, especially among Confucianists: Mencius is not only a poor philosopher, but a humourless hypocrite,13 while Zhū Xī (朱熹, 1130-1200) is a conservative ideologist. 14 In this respect, Billeter's point of view already differs from Jullien's, who intends to take the tradition as it is, without choosing nor judging.

As a consequence, Billeter avoids to use classes of philosophical systems. The term of Confucianism must be avoided, because Confucius' thought is very different from Mencius', Mencius' from Zhū Xī's, and Zhū Xī's from Zhāng Zǎi's (張載, 1020-1077). Over more than 2000 years, the Confucean tradition was subjected to so many interpretations, so many external influences such as Daoism or Buddhism, that speaking of "one" Confucianism would be as deceptive as speaking of "one" platonism to involve the thoughts of Plato, Augustine, Leibniz, Frege and Gödel in one and only one system. The same is true concerning Daoism, even more than Confucianism.

And this seems to be a kind of bias in Billeter's method. Going against the Chinese tradition, which used such abstract entities as Daoism (道家 dàoji ā) or Confucianism (儒家 rúji ā), Billeter always speaks of mere individuals. Not Daoism, but Zhuāngzǐ. Not Confucianism, but Confucius.15 And a philosopher seems to be interesting to him as soon as he distances himself from a school. He became aware of Confucius' depth when he separated him from Confucianism. 16 He studied Zhuāngzǐ as a unique philosopher, very different from the "Daoist" which the later commentators would have constituted.17 And for the same reason, he could nothing but be fascinated by the personality of Lǐ Zhì (李贄, 1527-1602), dissident philosopher who refused the whole system of mandarinate, examinations and Confucianism.

Billeter's first book was precisely focused on Lǐ Zhì. 18 He described the life of this philosopher, who began as a mandarine and was recognized as very talented. However he was putting on an act. 19 Without believing in what he wrote in his essays, he just knew what was expected from him and came up to these expectations (a behaviour which may be common to many examinations, in all eras and places). Instead of making the very successful career he could have legitimately dreamt of, he chose to both leave and denounce the system in his books A Book to Burn (1590) and A Book to Conceal (1599). What interested Billeter in Lǐ Zhì's work and life was this claim for individuality. 20 He studied Lǐ Zhì using some concepts of Pierre Bourdieu's work, and especially the notion of "symbolic capital;"21 he had in return some influence on Bourdieu, who entitled "A Book to Burn" the first chapter to his own book Homo Academicus.22 Bourdieu's concepts allow Billeter to consider Lǐ Zhì, neither as a mere free subject, nor as totally determinated by some kind of structures, but as an agent, located in some social field, with which he can interact, accepting or refusing it; and also, by contrast, to explain the conservatism of Neo-Confucianism.

Much later, when Billeter studies Zhuāngzǐ, the same methodological principles may be observed. Billeter refuses to include Zhuāngzǐ in some abstract notion of Daoism, and studies him as an individual, separate from Lǎozǐ (老子) as well as from his own self-styled followers. Billeter explicitly intends to understand Zhuāngzǐ better than two millenia of Chinese commentaries and about three centuries of Western interpretations, going alone against them all.23 The least we can say is that Billeter is not very sensitive to the argument from authority.

But what is his secret? How is it that he can be so close to Zhuāngzǐ's private thoughts? Billeter's advice seems quite simple: use your imagination, use your experience. As an example, Billeter translates and comments a text from Zhuāngzǐ (III, ii), in which a cook who perfectly handles knives describes his art. Almost all translators make him say: "what interests me is the Way, not mere technique." But the very abstract notion of dào does not really help us to understand the meaning of his words. On the contrary, Billeter chooses to translate this sentence as "what interests me is how things work, not mere technique." 24 The translation is much more mandarinat chinois de la fin des Ming.

precise; what allowed such a precision is that Billeter used his imagination to figure out what the cook intended to say. This led him to remember how children gradually acquire gestural habits, such as filling glasses or cutting bread. With this method, Zhuāngzǐ's book looses every abstraction, becomes close to us. Billeter's thesis about Zhuāngzǐ is that his main theme is to constitute a phenomenology of activity: he describes different kinds and degrees of activity, from the usual, irreflexive and almost hypnotic 25 one to the most reflexive and conscious one. [START_REF] Billeter | Leçons sur Tchouang-tseu[END_REF] These degrees of activity are kinds of intentionality: they constitute the way our consciousness relates to the world. Zhuāngzǐ thus describes subjectivity. 27 This subjectivity does never cease, even when Zhuāngzǐ teaches how to free oneself from the intentionality which is commited in our everyday activity: 28 "this experience does not make us discover some upper reality, but the source of our own subjectivity." 29 As we can see, experience and subjectivity can not be separated in Billeter's method.

Trusting experience, Billeter avoids to highlight concepts: he does not trust words, claiming to follow Wittgenstein's critic of philosophy as being fooled by language. Even the term of dào should be almost each time translated by a different word. 30 Zhuāngzǐ does not only cease to be a daoist: he does not even speak of dào any more. His thought is a phenomenology much more than a daoism.

So, Billeter's method's originality is to widely use the notions of subjectivity and experience, each of them sustaining the other. The universality of subjectivity allows us to have, as an intuition, the experience of what Zhuāngzǐ thinks, because ancient Chinese and contemporary Swiss people share the same structures of consciousness. And the universality of experience makes us understand what special structures of subjectivity are implicitly described by Zhuāngzǐ. Zhuāngzǐ does not use the notion of subjectivity, but it does not prevent him, as a human being, to describe or comment it.

2 Is Wáng F ūzh ī a phenomenologist?

A phenomenological projection

One can easily see how far Jullien's method is from Billeter's. Jullien studies traditions, Billeter individuals. Jullien follows commentators, Billeter sidesteps them. Jullien avoids personal interpretation, Billeter claims for it. Jullien identifies concepts, Billeter avoids them; similarly, Billeter trusts experience, Jullien is wary of it. Jullien tries to keep an internal unity to his translations, even though the result may sound strange to us; Billeter prefers to have an immediately understandable translation, even though the original lexical field may be lost. So both sinologists could nothing but disagree.

But fortunate to us, they did not content themselves with desagreeing. They disputed. And this contest has been at least as instructive as their isolated works already were.

In 1989, Billeter publishes in the French sinological journal Études chinoises a review of Jullien's book Procès ou création. 31 At that time, Billeter still has some respect and admiration for Jullien. He recognizes the great value of this book from a sinological point of vue, 32 writing in particular that the chapter about the Yìj īng is, "as far as I know, the best thing ever written in Western sinology about the philosophical interpretation of the Yìj īng." 33 So when Billeter, in 2006, pretends to doubt about Jullien's sinological value, the polemic dimension of his judgement should not be forgotten.

After having spoken very highly of Jullien's book, "from a sinological point of view," Billeter comes to criticize it, "from a non-sinological point of view." This critic consists in two points. The first one concerns the use of Wáng Fūzhī, not as a singular and original philosopher, but as a typical Chinese thinker. We have earlier underlined how important it was in Billeter's method to isolate thinkers in order to see what was proper to each of them; this critic is thus not surprising from him. The second critic concerns Jullien's reserve about one particular point in Wáng Fūzhī's thought, which 31 Jean François Billeter, "Comment lire Wang Fuzhi ?", in: Études chinoises IX.1 (Mar. 1989), pp. 95-127, url: http://www.afec-en-ligne.org/IMG/pdf/9-1Billeter.pdf.

32 Ibid., pp. 96ff. 33 Ibid., pp. 101ff. The controversy of 1989 was not enough to change Billeter's mind about Jullien's competence as a sinologist: in 1993, he still uses superlatives to mention Jullien's article " "Fonder" la morale, ou comment légitimer la transcendance de la moralité sans le support du dogme ou de la foi -au travers du Mencius" (Extrême-Orient Extrême-Occident, 6, 1985, pp. 23-81), in the review of Jörg Schumacher's book Über den Begriff des Nützlichen bei Mengzi: "Il est [...] regrettable qu'il [Jörg Schumacher] ne cite pas l'étude de François Jullien sur Mencius, qui est à mon sens le meilleur essai de synthèse paru en langue occidentale, et ne nous dise pas en quoi les résultats de ses propres recherches complètent, corrigent ou remettent en cause la vision du Mencius proposée par François Jullien." See Jean François Billeter, "Jörg Schumacher, Über den Begriff des Nützlichen bei Mengzi, Berne, 1993", in: Études chinoises XII.2 (Sept. 1993), pp. 179-181, url: http: //www.afec-en-ligne.org/IMG/pdf/12-2CR.pdf, pp. 180-181.

will happen to highlight the very sticking point between Billeter and Jullien.

This particular point is some parallelism Jullien observes in Wáng Fūzhī's thought, about the relationship between visible and invisible. For Wáng Fūzhī, influenced in that by Zhāng Zǎi's Confucianism more than Zhū Xī's,34 there is no opposition between visible and invisible, but continuity. The universe is constituted by the qì (氣), the breath of energy, which sometimes disperses itself, letting things become invisible, and sometimes gathers itself, making things becoming visible. So there is no opposition between being and nothingness, but continuity between different kinds of "being" -although this word is inappropriate, since China neither developped any theory of "Being," nor even has an unique word for it. So this kind of transition is what Jullien calls "process," as opposed to the European concept of "creation," where something may appear ex nihilo. And Jullien contents himself to indicate a similar process for consciousness described by Wáng Fūzhī: some feelings may at first be very weak, then imperceptibly rise, and finally be prevailing. They do not magically appear, which would correspond to "creation," but gradually rise. And Jullien does not go further about this parallelism between both kinds of process: the cosmological one and the psychological one.

This "modesty" 35 is precisely what Billeter reproaches Jullien for. Jullien just shows the parallelism, without trying to justify it: he describes without explaining. Following Billeter, one can not be satisfied with a mere description. We want to understand, we want the truth. How is it that Wáng Fūzhī assimilates the cosmological and the psychological processes? How could we justify it? That is to say: how can we come to agree with it? No understanding without agreeing: this seems to be Billeter's principle of method.

To make Wáng Fūzhī's theory understandable and acceptable, Billeter develops a theory of projection. 36 This cosmologico-psychological process would first be a merely psychological one, which is then projected onto cosmology. This theory of projection leads Billeter to think that Wáng Fūzhī's thought is fundamentally a phenomenology. For instance, Wáng Fūzhī writes that the wise, to be effective, must expell all the thoughts while acting. Billeter does not explain it with Chinese commentaries, but with an example of common life: when trying to hold a stick upright in our hand, we must not focus on anything, but be aware of the whole activity. If we focus on the stick or on the hand, we will fail, the stick will fall on the ground. 37 This is characteristic of Billeter's method, explaining difficult texts with an ordinary example. This allows Billeter to interpret Wáng Fūzhī's philosophy as a phenomenology of action.

As in Husserl's phenomenology, Wáng Fūzhī's describles the structures of consciousness. But there is a fundamental difference: whereas Husserl's phenomenology is reflexive, describing consciousness as consciousness, Wáng Fūzhī's is not. 38 He does not use, nor even know, the lexicon of consciousness. This phenomenology of action is thus encrypted: that is why projection plays a role. Having no phenomenological lexicon at his disposal, Wáng Fūzhī projected his phenomenology onto the external world. He projected onto the sky (天 ti ān) the structures of effective consciousness he became aware of -although he probably was not aware of this projection itself. So, is there a real parallelism between consciousness and reality? Actually, what Wáng Fūzhī asserts about reality is an indirect way of describing phenomenological processes.

Some problem could nevertheless rise with this interpretation: it seems to invert what Wáng Fūzhī himself writes. He does not compare the reality with our cousciousness, but conversely compares our consciousness with the cosmological process. Consciousness is for him a part of the world, and in this regard is secondary. There would be no projection from consciousness to reality, but, on the contrary, influence of the reality on our consciousness. To escape this objection, Billeter appeals to a "cybernetic loop." 39 "Without being aware of that, the philosopher observes the projection of these activities [of common effective consciousness] in the working of the external world. And this is how the external world becomes for him a model of supreme activity, on which he judges that human beings are invited to model their own activity." 40 There is a kind of round-trip, or feedback, between consciousness and reality. Natural consciousness credits the world with a sort of activity, and tries to imitate it, without being aware that this type of activity already, and first, belongs to itself.

A duality appears in Billeter's analysis, of which he is fully aware: subjectivity plays two roles. It is first an object, since Billeter describes Wáng Fūzhī's thought as a phenomenology, an inquiry about the structure of consciousness, from a subjective point of view. But it is also a method: "we are led to conceive Wáng Fūzhī's philosophy through a real phenomenology of activity and to interpret in return his philosophy starting from this phenomenology of activity." 41 Should we fear some petitio principii? It is indeed by no mean surprising to find as a result what has been presupposed at first. But Billeter does not fear this circularity; as well as he speaks about cybernetic loop, or feedback, one can see here some kind of hermeneutic circle. Far from being a mistake, this circularity is claimed for. Thus, adopting a 38 Ibid., pp. 111 and 119. 39 Ibid., p. 120. 40 Ibid., p. 120. 41 Ibid., p. 115. phenomenological point of view, Billeter's thought is throughout consistent.

What is the foundation of Billeter's method in the analysis of Wáng Fūzhī? Billeter makes no secret of it: we have to postulate the universality of human experience. Starting from the "possible universality of Wáng Fūzhī's philosophy, that is to say the possible adequation to definite universal data of the reality itself, or of the experience we have of it," 42 Billeter ends up in asserting that experience is "universal in its principles," and that "the elementary data of experience are [...] the ground on which [all philosophers] necessarily founded their systems." 43 So Billeter rigorously applies to Wáng Fūzhī's thought the methodological principles of his sinology.

We can for this occasion criticize this following simplistic way of presenting the controversy between Jullien and Billeter; a simplification which Billeter recently tended to feed. [START_REF] François Billeter | see also Jullien, Chemin faisant. Connaître la Chine, relancer la philosophie[END_REF] It consists in presenting Jullien as a mere philosopher, Billeter as a mere sinologist. This opposition naturally hides a value judgement: Billeter would be a patient scientist, having no general theory about Chinese thought, whereas Jullien would be a philosopher, remaining in conceptual abstraction, looking from far away at Chinese texts and forcing their interpretation. The one would be longsighted, the other short-sighted. But both parts are false: Billeter's several times asserted admiration for Jullien may suffice to attest his competence as a sinologist. And contrary to what Jullien's often condescending posture towards Billeter may let us think, [START_REF] Billeter | Jullien sometimes mentions a "weak thought[END_REF] Billeter is not less a philosopher. Although more discrete than Jullien, he also has his general theory of China; this theory, as we can expect, is much different from Jullien's. There is a "specifically Chinese propensity to project onto the world the forms of counsciousness in motion rather than that of the motionless one." 46 Chinese thinkers, especially Neo-Confucianists, 47 but also philosophers like Zhuāngzǐ, are phenomenologists of activity. Whereas "most of Western philosophers sought beyond the sensible world a motionless reality, attempting to give an account of it by the mean of conceptual constructions," "the Chinese philosophers studied the motion and the transformations of experience." 48 Even more explicitly, Billeter concludes: "My thesis is that the phenomenology of activity I just gave an example of constitutes the rational kernel of the thought of many Chinese philosophers, therefore maybe one of the foundations of Chinese thought in general." 49 Billeter's perspective on Chinese thought is thus not less ambitious than Jullien's.

In this respect, how can be interpreted, in Billeter's system, Jullien's work? Not only "modesty," but more than that, deeper than that: Jullien's sinology is a too Western one. Understanding Chinese thought, following Billeter, implies trusting one's experience, in order to meet up with Chinese philosophers on the ground they started from. Thus, interpreting Chinese thought in a conceptual way, as Jullien does, uproots it, and imports it into typical Western conceptual thinking. Jullien would therefore not have left the way of thinking he intended to distance himself from.

Billeter's review of Jullien's book is thus much more than a classical technical review: it is a sinological and philosophical manifesto. Billeter expresses his methodological principles, and does not only apply them to a particular Chinese thinker, Wáng Fūzhī: but he also applies them to sinology itself, through Jullien's work. This reveals a highly systematic spirit, which exceeds mere sinology.

Conceptual environment of "projection"

Jullien's answer is as systematic as Billeter's critic was. 50 As the notion of projection was the heart of Billeter's critic of Procès ou création, it will be Jullien's main target. Faithful to his methodological bias, Jullien first refuses to use this concept, arguing that it can not be found anywhere in Wáng Fūzhī's writings: "even though Chinese thinkers, and especially Wáng Fūzhī, conceived a definite mode of projection, between "feeling" and "landscape" (情 qíng and 景 jǐng, under the name of 寄託 jìtu ō), it is obvious as well that, as it is used in Chinese aesthetics, it could not refer to the perspective, so familiar to us that it became natural, of the speaking subject." 51 Billeter blamed Jullien for interpreting Wáng Fūzhī in a Western way: Jullien turns the argument against Billeter. The most Western-styled interpretation is not the one using conceptualization, but the one using the notion of projection. To avoid intellectual ethnocentrism, we should content ourselves to say that the parallelism between cosmological and psychological processes is not due to any projection from consciousness onto the external world, but, literally following Wáng Fūzhī, that consciousness has to imitate the world's kind of activity. Everything else would be overinterpretation.

The notion of projection actually conceals a system of concepts and postulates. First of all, projection, as a phenomenological and psychanalytical 49 Ibid., p. 115. See also Billeter, L'Art chinois de l'écriture. Essai sur la calligraphie, VI, p. 153 and p. 154 n. 4. 50 François Jullien, "Lecture ou projection : Comment lire (autrement) Wang Fuzhi ?", in: Études chinoises IX.2 (Sept. 1990), pp. 131-149.

51 Ibid., p. 142.

concept, presupposes subjectivity. 52 In other words, what in Billeter's mind was a hermeneutic circle is now interpreted as a vicious circle. The notion of activity is also "signed" by Greek philosophy, and does not appear as such in Chinese tradition. 53 Thus, when using phenomenological notions as given data, "we do not compare two thoughts any more, but adapt the Chinese one to ours, we project this one onto that one. And Chinese thought is not any more perceived starting from itself." 54 Jullien's argument is fully symmetric with Billeter's. Both sinologists blame one another for not leaving his own philosophical tradition, and for projecting onto Chinese thought his own way of thinking. More than that: as well as Billeter uses the phenomenological method to criticize Jullien's abstract conceptual method, Jullien uses the structural method to criticize Billeter's. No one of both sinologists can be blamed for inconsistency: they use the same method to analyze Chinese texts and to criticize other methods. At this stage, the controversy can be considered as locked. There is, so to speak, nothing to add. Arguments have been expressed with so much symmetry that a state of equilibrium seems to have been reached. The controversy could only continue with details, or with human passions. Probably for this reason, it had been buried for long years, until Billeter published his book Contre François Jullien in 2006.

Power and subjectivity in China

Genealogy of an archeology

The new controversy that appeared in 2006 was once again initiated by Jean François Billeter and is still not closed, both sinologists criticizing one another, without having to spell things out, in their recent publications. [START_REF] François | François Jullien, sur le fond[END_REF] The book Contre François Jullien is a critic of the "myth of the alterity of China," 56 which Jullien would have defended. The blame is both right and wrong: Jullien has always cared not to exagerate the alterity, and to criticize the idea of a radical alterity between Chinese thought and European thought. [START_REF] Jullien | Des catégories originales de l'interprétation poétique dans la tradition chinoise (Contribution à une réflexion sur l'altérité interculturelle)[END_REF] But for sure, Jullien generally paies more attention to differences than to similarities, considering for practical purposes that the main danger is the temptation of assimilation more than the excess of distance.

What Billeter refuses is more than this myth: it is the very idea of a Chinese tradition. He does not contest that something exists like a Chinese tradition, but refuses the interpretation which is generally given of it, and Jullien would have relayed. In this respect, adopting a sociological point of view, Billeter links two aspects of his sinological work: the sociological method of his book about Lǐ Zhì, and the phenomenological method of his writings about Zhuāngzǐ or Wáng Fūzhī.

Billeter intends to expose the genealogy of this myth of alterity. This myth, which would have influenced Marcel Granet, Victor Segalen or Pierre Ryckmans, but also Voltaire, came from the Jesuits, who brought an ideology which would have been made up under the Hàn (漢) dynasty (202 BC). 58 At that time, Chinese mandarines would have "so much exploited the culture that they recreated it and made of it the ground of the new order." 59 This myth would thus not only be a mere mistake, but a two millenia old plot. The goal of this lie was to make the empire look as if it was founded on nature: "to make people forget the violence and the arbitrary from which the empire was born, and with the help of which it remained, it had to appear as complying with the order of things. [...] It was the most effective way of guaranteeing the durability of the imperial government, of its hierarchies, of the forms of domination it imposed to people, of the submission they required. From this general recreation arose what Chinese people considered since then, and what is still considered, in China and elsewhere, as Chinese civilization." 60 Highlighting the specificity of Chinese thought, Jullien would thus be nothing but one more victim of this millenary lie.

Following the young Chinese historian Lǐ Dōngjūn (李冬君), 61 Billeter writes that even the authority of Confucius was artificially built, in order to submit the self, the subject, to the domination of the political system. 62 The imperial Confucianism took root in this kind of political canonization. Lǐ Dōngjūn concludes that there would only be a real progress when the individual gets back to the first place. 63 We can see here the link with Billeter's theory of subjectivity: subjectivity is universal, although it may be recovered or partly masked by oppression. Even when and where it seems to be absent, the only reason is that it has been repressed. Behind what Billeter analyzed earlier as a projection, there is a repression.

When Jullien studies Chinese thought as an intellectual tradition, he actually focuses on the very result of a political process: the "relative uniformity" of Chinese thought does not come from a community of ideal structures, but belongs to political history. 64 Even the classical book of Zh ōngy ōng (中庸), which Jullien translated and commented, 65 has a political meaning. 66

The whole thought of immanence that Jullien recognizes in Chinese thought is "linked to the imperial order." 67 Nothing, in Jullien's method, appears to be innocent. The insistence on concepts itself belongs to political means. The use of the abstract word of dào in the eclectic classical book Huáinánzǐ (淮南子) may thus conceal the real goal of this book, which is to "found on nature the imperial power." 68 Going further than the earlier opposition between phenomenology and a kind of foucaldian archeology, Billeter intends to write the sociological genealogy of an archeology. Not only the ancestry of Jullien's work is suspicious, but also its descendants: the idea of an alterity of Chinese thought has political consequences, 69 and, in particular, prevents the democratisation of China. 70 Going still further, Billeter applies his sociological principles to Jullien himself: the fascination of French intellectuals for what is called "Chinese civilization" may be related to the French system of university and grandes écoles, whose elitism resembles the Chinese mandarinate. However this argument may legitimately seem to be ad hominem, it has to be linked to the sociological method of Billeter, and reminds us the way Bourdieu criticized the tutorial system he came from. So this argument is not only ad hominem, it is a natural consequence of Billeter's bias.

Archeology of a genealogy

In his answer, Jullien criticizes the very idea of an "imperial ideology," 71 inverting the relationship between political ideology and system of thought: power and lie would not have been enough to impose a way of thinking for such a long time and so much different eras. Even an ideology must be founded on a system of thought. [START_REF]defending archeology against sociology: "si l'existence à une groupe social peut toujours expliquer que tel ou tel ait choisi un système de pensée plutôt que l'autre, la condition pour que ce système ait été pensé ne réside jamais dans l'existence de ce groupe[END_REF] So, behind genealogical arguments, archeological ones prevail.

Speaking of archeology, which Jullien does not, allows to highlight some foucaldian influence. Jullien denounces Billeter's thought as a "humanism," implicitly claiming for the inheritance of French "structuralism:" Lévi-Strauss, Lacan, Barthes, Foucault and Althusser had little in common, but this little probably was the critic of what they called humanism. Foucault, in the last pages of Les Mots et les choses, asserted that the concept of man had just recently been invented in our culture, and may soon disappear; subjectivity would thus not be an universal concept, but an ephemeral one. 73 The very word of humanism, considered as a blame, is enough to see how far Jullien follows Foucault; but one more evidence is the expression of "soft humanism" (humanisme mou), [START_REF] Jullien | 15 and XII, pp. 131-141. See also Jullien[END_REF] which appears in 1966 in a well-known interview with Foucault. [START_REF] Foucault | L'Homme est-il mort ?[END_REF] In Foucault's book Les Mots et les choses, phenomenology and sociology belong to the same humanist system of thought (épistémè). [START_REF]About phenomenology, see Foucault[END_REF] Thus even Billeter's kind of genealogy is interpreted and criticized in an archeologic way.

Jullien also justifies his use of commentars to study Chinese classics. Without them, one can not even understand the texts, because of the ancient Chinese syntax; 77 this is the reason why, attacking Billeter's rearguard, Jullien doubts of the possibility to study Zhuāngzǐ without their help, without examinating the relationship between this author and Lǎozǐ, Chinese sophists, Mohists, etc. 78 Using commentars avoids us to "blithely project our fantasy." 79 Defending his conceptual method, Jullien appeals for the "right to concept:" 80 "concept definitely is a instrument and abstraction is its result." 81 When we are not aware of it, we may inadvertently project our concepts -such as reality, being, subjectivity, act -onto heterogenous conceptual universes. 82 Should we conclude of it that there is no universal experience, that we can not share anything with ancient Chinese thinkers? Absolutely not: the very existence of sinology shows that a community is possible, not only of experience, but the "community of the thinkable." 83 Thus Jullien and Billeter both intend to form a community with ancient Chinese thinkers: but whereas Billeter founds it on experience and subjectivity, Jullien founds it on concept and discursive thought.

Conclusion

May we presuppose the universality of subjectivity and experience when studying ancient Chinese texts?

In philosophy of mathematics, at the beginning of the twentieth century, happened a controversy about the foundations of mathematics, which first seemed do be a merely technical one, but much later turned out to require a philosophical decision. Although arguments of both sinologists are taken from Chinese tradition, one can wonder whether such documents really may give any neutral answer, or whether the philosophical debate remains untouched.

That any discipline may give up a question and expect from philosophy an answer is generally a bad sign. When a discipline needs a philosophical decision to answer a technical question, the technical question may be considered as buried.

From a philosophical point of vue, one can see that the structure of each thought is sufficient to make a decision impossible. Not only each of these sinological thoughts is very systematically structured, but both sinologists draw in an insightful way the conclusions implied by their theses. Not only each thought, but also the relationship between both thoughts seems to be systematic. So it is to fear that one can not answer the question of the implicit existence of experience and subjectivity in ancient Chinese thought without having already chosen.

We could maybe transpose into this controversy what Jules Vuillemin wrote about science and philosophy in general: "no scientific discovery is by itself able to force a philosophical decision." 84

Final remarks

This text was sent to both sinologists, and Jean François Billeter graced the author with a detailed answer. Concerning the last point of the text, Mr. Billeter maintains that a thinking system may legitimately be condemned, not only for inconsistency, but also for its intellectual, moral or political consequences. This point of view was developped in his book Contre François Jullien. 85 84 Jules Vuillemin, What Are Philosophical Systems?, Cambridge: Cambridge University Press, 2009 (1986), IV, 3, p. 132. 85 Billeter,op. cit.,pp. 7,

2

  Jullien's conceptual method . . . . . . . . . . . . . . 2 1.2 Jean François Billeter's cousciousness point of view . . . . . . 5 Is Wáng F ūzh ī a phenomenologist? 8 2.1 A phenomenological projection . . . . . . . . . . . . . . . . . 8 2.2 Conceptual environment of "projection" . . . . . . . . . . . . 12 3 Power and subjectivity in China 13 3.1 Genealogy of an archeology . . . . . . . . . . . . . . . . . . . 13 3.2 Archeology of a genealogy . . . . . . . . . . . . . . . . . . . . 15 Conclusion 17

The author would like to thank Jean François Billeter for his benevolent and critical proofreading, Marion Duquerroy and Xiè Jīng (谢晶) for their careful advice and patience.

See for example François Jullien, Conférence sur l'efficacité, Opus, reprinted in La Philosophie inquiétée par la pensée chinoise, Paris: Seuil, 2009 (2005), p. 11; and almost every interview with this author.

See, among others,Michel Foucault, "Le Philosophe masqué", in: Dits et écrits II, 1976-1988, Quarto, Paris: Gallimard, 2001 (1994), pp. 923-929, p. 929; and Michel Foucault, Histoire de la sexualité, II : L'Usage des plaisirs, Tel, Paris:Gallimard, 1984, Introduction, pp. 16 and 19.

Michel Foucault, Les Mots et les choses, Tel, Paris: Gallimard, 1966, Preface, p. 9. See for example Jullien, op. cit., p. 14; François Jullien, Chemin faisant. Connaître la Chine, relancer la philosophie. Réplique à ***, L'Ordre philosophique, Paris: Seuil, 2007, IV, p. 34; François Jullien, La Philosophie inquiétée par la pensée chinoise, Opus, Paris: Seuil, 2009, Preface, p. iii; and many interviews withFrançois Jullien.

François Jullien, "La Chine au miroir de l'Occident", in: Le Monde diplomatique(Oct. 2006), pp. 22-23, url: http://www.monde-diplomatique.fr/2006/10/JULLIEN/14010: "I still dare to use this word ["tradition"], against Michel Foucault's will, because a tradition is precisely perceived from outside." See also François Jullien, De l'Universel, de l'uniforme, du commun et du dialogue entre les cultures, Paris: Fayard, 2008, XIII, p. 263: this word is relevant to compare cultures, although it is not for describing one of them from inside. Jullien probably alludes to Foucault's critic of the notion of "tradition," in Michel Foucault, L'Archéologie du savoir, NRF, Paris:Gallimard, 1969, II, i, pp. 31-32.

François Jullien, Procès ou création. Une introduction à la pensée des lettrés chinois, Des Travaux, Paris: Seuil, 1989.

François Jullien, Figures de l'immanence. Pour une lecture philosophique du Yi King, le Classique du changement, Biblio Essais, Paris: Livre dePoche, 1995Poche, (1993)).

Especially see Jullien, Procès ou création. Une introduction à la pensée des lettrés chinois; Jullien, Figures de l'immanence. Pour une lecture philosophique du Yi King, le Classique du changement; François Jullien, Zhongyong. La Régulation à usage ordinaire, Paris: Imprimerie Nationale, 1993; and passim in Jullien's work.

Jean François Billeter, "Confucius : l'énigme", in: Études sur Tchouang-tseu, Paris:Allia, 2006, pp. 163-192, especially pp. 176-177.

Jean François Billeter, Li Zhi, philosophe maudit (1527-1602). Contribution à une sociologie du mandarinat chinois de la fin des Ming, Travaux de droit, d'économie, de sociologie et de sciences politiques, Genève:Droz, 1979, II, p. 53 n. 9 and pp. 88-89.

This separation between Confucius and Confucianism, and the claim for following the Master against the tradition, was maybe inspired by Lǐ Zhì: ibid., ch. x, p. 212.

Billeter, "Confucius : l'énigme",

Jean François Billeter, Leçons sur Tchouang-tseu, Paris: Allia, 2004[START_REF] Billeter | Leçons sur Tchouang-tseu[END_REF], IV, pp. 135ff.

Billeter, Li Zhi, philosophe maudit (1527-1602). Contribution à une sociologie du

Ibid., ch. II,

The existence of forms of subjectivity in Lǐ Zhì's work would not be surprising: this thinker was partly influenced by Buddhism and the "intuitionist" philosopher Wáng Yángmíng(王陽明, 1472-1529). See ibid., ch. II and X.

21 Ibid., pp.

82ff.22 Pierre Bourdieu, Homo Academicus, Le Sens commun, Paris:

Minuit, 1984. 23 Billeter, Leçons sur Tchouang-

tseu, I, pp. 9-13.24 Ibid., pp. 15ff.: "Ce qui intéresse votre serviteur, c'est le fonctionnement des choses, non la simple technique." A similar translation and interpretation may be found in Jean François Billeter, L'Art chinois de l'écriture. Essai sur la calligraphie, Paris: Skira/Seuil,

Jacques Gernet, La Raison des choses. Essai sur la philosophie de Wang Fuzhi (1619-1692), NRF Bibliothèque de philosophie, Paris: Gallimard, 2005.

 35 Billeter, "Comment lire Wang Fuzhi ?", p.

123. 36 It had also been exposed in Billeter, L'Art chinois de l'écriture. Essai sur la calligraphie, VI, pp.

145-153.37 Billeter, "Comment lire Wang Fuzhi ?", pp. 114-115.


J Seiwert

I Cantat

Highlight

• Deviation from the Frankel's law predicting the soap film thickness is computed.

• The film thickness varies non-monotonically with the surfactant bulk concentration.

• The elastic interface behavior is recovered at small solubility.

• The predictions are compatible with available experimental data.

The generation of soap films when extracted from a bath at a constant velocity is one of the staple problems featuring hydrodynamics in the presence of surfactants. The great and constant interest in this seemingly simple set up has several origins: it appears in numerous industrial processes, and it features By extending the work of Landau, Levich [START_REF] Landau | Dragging of a liquid by a moving plate[END_REF] and Derjaguin [START_REF] Derjaguin | Thickness of liquid layer adhering to walls of vessels on their emptying[END_REF], Frankel [START_REF] Mysels | Soap films: Study of their thinning and a bibliography[END_REF] has first predicted the steady state thickness of the film and its power law dependency with the capillary number Ca = ηU/γ, with U the pulling velocity, 10 η the solution viscosity and γ its surface tension. Numerous experimental studies have confirmed the great accuracy of this theory, over a large range of capillary numbers and for a large sample of surfactant solutions [START_REF] Krechetnikov | Surfactant effects in the Landau Levich 460 problem[END_REF][START_REF] Ramdane | Thickening factor in Marangoni coating[END_REF][START_REF] Shen | Fiber coating with surfactant solutions[END_REF].

However, at large enough capillary numbers, several surfactant solutions exhibit a clear deviation from Frankel's law [START_REF] Lal | Formation of soap films from insoluble surfactants[END_REF][START_REF] Saulnier | What is the mechanism of soap film entrainment?[END_REF][START_REF] Champougny | The role of surface elasticity in liquid film formation: unifying Frankel and Landau-Levich-Derjaguin configurations[END_REF]: the measured film thickness 15 is lower than predicted by Frankel, and exhibits a maximum at a given capillary number. In [START_REF] Lal | Formation of soap films from insoluble surfactants[END_REF][START_REF] Saulnier | What is the mechanism of soap film entrainment?[END_REF][START_REF] Seiwert | Theoretical study of the generation of soap 475 films: role of interfacial visco-elasticity[END_REF][START_REF] Champougny | The role of surface elasticity in liquid film formation: unifying Frankel and Landau-Levich-Derjaguin configurations[END_REF], these observations are explained by a finite elasticity of the interface. Scaling analysis shows that a deviation from the Frankel's law, which assumes incompressible interfaces, is expected for capillary numbers larger than (E/γ) 3/2 , with E = ∂γ/∂(ln(A)) the elasticity of the interface.

20

Depending on the type of surfactant used, this may or may not be in the range of capillary numbers accessible to experiments, which explains why only some solutions exhibit the aforementioned deviation. The main shortcoming of this approach is related to the fact the underlying physical mechanisms at the origin of the elasticity E are most often not explicited.

25

For insoluble surfactants with negligible surface diffusion, or when the exchange of surfactant between the interface and the bulk is slow enough that it can be excluded from the process, this elasticity arises directly from the variation of surface tension with the surface excess Γ. Indeed, in that case mass conservation of surfactants implies that AΓ be constant so d ln(Γ) = -d ln(A)

30
and thus E = -∂γ/∂ ln(Γ). This elasticity is known as the Marangoni elasticity E M [START_REF] Couder | On the hydrodynamics of soap films[END_REF].

However, for soluble surfactants, exchange with the bulk phase modifies the mass balance equation and the interface elasticity becomes an effective quantity which potentially depends on the complicated interplay between surfactant we consider here, however, have a very peculiar aspect ratio, as their thickness is generally several orders of magnitude smaller than their extent.

In such confined films, the physical origin of interfacial stresses, and thus of the effective interfacial elasticity, is generally assumed to lie in surfactant depletion within the film [START_REF] Van Den Tempel | Application of surface thermodynamics to Gibbs elasticity[END_REF][START_REF] Couder | On the hydrodynamics of soap films[END_REF][START_REF] Quéré | Le mouillage dynamique des fibres[END_REF]. As interfaces stretch, surfactant molecules 45 within the volume of the film are adsorbed, thereby lowering locally their concentration, increasing locally surface tension and generating Marangoni stresses.

This depletion effect becomes important if the thickness of the film becomes smaller than the typical length l q = Γ/c, where Γ and c are respectively the interface and volume concentration of surfactant [START_REF] Quéré | Le mouillage dynamique des fibres[END_REF]. For typical surfactants used 50 in film withdrawal experiments, this length varies between 1 µm and 100 µm, which is comparable or larger than film thicknesses (1 µm to 10 µm).

When the thin film deformation is a pure stretching deformation, a film element constitutes a closed system. The surfactant mass balance, involving interface and bulk contributions, can thus still be used to relate area variations 55 to surface excess variations. This leads to the definition of an effective surface elasticity of the film, namely the Gibbs elasticity E G [START_REF] Van Den Tempel | Application of surface thermodynamics to Gibbs elasticity[END_REF][START_REF] Couder | On the hydrodynamics of soap films[END_REF], that consistently takes into account the depletion effect. In contrast, in more complex dynamical situations, such as film extraction, the velocity of the fluid is not homogeneous across the width of the film, and the local mass balance at the origin of the 60 definition of the Gibbs modulus is not valid anymore. Surfactants are convected along the film and interface stretching at a given place of the film can therefore lead to a surface tension increase at another place of the film. An effective surface elasticity, coupling locally area variation and surface tension, can not be rigorously defined.

65

We present here a model that rationalizes the depletion effect involved in the film generation process. The model takes surfactants convection explicitly

Model

The problem that we consider is sketched on figure 1. A film is withdrawn at a constant velocity U from a meniscus acting as a reservoir of surfactant solution (viscosity η, surface tension γ m ). We neglect the effect of gravity, so that the meniscus features, at rest, a constant radius of curvature r m . We 80 further assume the problem to be bidimensionnal (the film has an infinite width in the z direction, perpendicular to the plane of the sketch), and we focus on steady state: far from the meniscus, the film has a constant thickness 2h f .

In typical experiments, h f is on the order of micrometers, while the meniscus has a millimetric size. This large difference in sizes allows for the classical 85 division of the system in three distinct regions [START_REF] Landau | Dragging of a liquid by a moving plate[END_REF]: 1) the flat film at the top, which is translated at a constant velocity U ; 2) the static meniscus, a the region of negligible flow where the meniscus is unperturbed and; 3) the dynamical meniscus of length ℓ and of typical thickness h f bridging the two, where viscous and pressure forces balance. 90

Scaling analysis of Frankel's law

In Frankel's theory, the interfaces are incompressible, and they move with the tangent velocity U imposed by the operator. In that case, the viscous force (per unit volume) in the dynamic meniscus scales like ηU/h 2 f . Moreover, the dynamic meniscus connects the flat film, where the pressure is p 0 (the atmospheric pressure), to the static meniscus where the pressure is p 0 -γ/r m . The balance between pressure gradient and viscous forces thus writes:

ηU h 2 f ∼ γ r m ℓ
The length ℓ of the dynamic meniscus is determined by imposing that its curvature (of order h f /ℓ 2 ) matches that of the static meniscus (1/r m ) to ensure the continuity of pressure, so that ℓ ∼ r m h f . The well known prediction for the thickness of the film follows [START_REF] Mysels | Soap films: Study of their thinning and a bibliography[END_REF]:

h f = 1.34r m Ca 2/3
with Ca = ηU/γ the capillary number. A consequence of this scaling is the length of the dynamic meniscus: ℓ ∼ r m Ca 1/3 .

Marangoni stresses and elastic interfaces

Another prediction deduced from Frankel's model is the stress arising at 95 the interface in order to pull the film out of the reservoir. It does not appear explicitly in the calculation, since the imposed constant velocity U is used as A c c e p t e d M a n u s c r i p t boundary condition instead. However, it can be computed from the velocity field, since the surface tension gradient must balance, at each position along the interface, the viscous stress in the bulk of the film. This gradient is confined to 100 the dynamical meniscus, and scales like ηU/h f [START_REF] Cantat | Liquid meniscus friction on a wet wall: bubbles, lamellae and foams[END_REF].

In other words, the surface tension γ f in the flat film region must be slightly larger than the surface tension γ m in the static meniscus, in order to balance viscous dissipation and extract the film. Their difference ∆γ = γ f -γ m is [START_REF] Seiwert | Theoretical study of the generation of soap 475 films: role of interfacial visco-elasticity[END_REF]

∆γ γ = γ f -γ m γ m = 3.84Ca 2/3 (1)
Frankel's theory assumes perfectly incompressible interfaces, where any surface tension difference may be generated with negligible interface deformation.

For elastic interfaces, on the other hand, this difference in surface tension ∆γ originates from a relative increase of interface area ∆A/A which depends on 105 the interface elastic modulus: ∆A/A = ∆γ/E ∼ Ca 2/3 γ m /E. As pointed out earlier [START_REF] Lal | Formation of soap films from insoluble surfactants[END_REF][START_REF] Champougny | The role of surface elasticity in liquid film formation: unifying Frankel and Landau-Levich-Derjaguin configurations[END_REF][START_REF] Seiwert | Theoretical study of the generation of soap 475 films: role of interfacial visco-elasticity[END_REF], when Ca (E/γ) 3/2 , interfacial deformation becomes non negligible, and deviations from Frankel's law are observed.

The elastic model will appear as a limit, for poorly soluble surfactants, of the more complex model presented below.

Surfactant transport

Our model for surfactant transport is based on two main approximations:

1. surfactant concentration c(x) is homogeneous in the direction of the film thickness.

2. surfactant adsorption is instantaneous, so that bulk concentration c(x) is 115 always at equilibrium with surface excess Γ(x).

Stresses and surface deformations only occur in the dynamical meniscus, so the validity of these approximations needs to be satisfied in this region.

Concentration gradients across the thickness of the film decay by diffusion with Γeq ceq depends strongly on the concentration, and on the surfactant used. At 140 the critical micellar concentration (cmc), l q varies for example between 1 µm for Sodium Dodecyl Sulfate, and 1 cm for Triton X-100 [15]. Above the cmc, since Γ remains approximately constant, the above estimations need to be multiplied by a factor c/c cmc , that may be of the order of 10. With these values, t ads varies between 10 -5 -10 -1 s. For the fastest surfactants, t ads is always much 145 shorter than t dyn (10 -2 -1 s), and approximation 2 is well satisfied. For other surfactants, however, this approximation is not correct anymore: the slowest surfactants behave like insoluble surfactants in this problem. These set of approximations has been used previously to study the static thickness of soap films [16] and the (slow) drainage of Plateau borders [17] in 150 the field of gravity. We showed here that their validity can be extended to our case.

A c c e p t e d M a n u s c r i p t

In the following, we will also neglect any diffusion in the direction along the film (both in the volume and at the interface). Indeed, comparing a diffusion time based on ℓ with t dyn shows that longitudinal diffusion is negligible as long 155 as Ca ≫ (ηD/γr m )

3/4 ≈ 10 -6 . This is a very well satisfied approximation, since in most experiments other phenomena (such as disjoining pressure, evaporation, etc.) would become dominant at such low capillary numbers.

Lastly, we use the same set of hydrodynamics approximation as Frankel, namely: we assume a gravity free lubrication flow in the dynamical meniscus 160 and match it asymptotically to both the static meniscus and the film.

Shape of the dynamic meniscus

As we mentioned earlier, we model the dilution of surfactants that generates the necessary Marangoni stresses in the dynamic meniscus. The dilution is driven by interface stretching: correspondingly, interfacial velocity increases 165 along the dynamical meniscus (from the static meniscus to the film). Five variables need to be tracked: the half with h(x) of the film, the bulk and interfacial surfactant concentrations c(x) and Γ(x), the surface tension γ(x) and the interfacial velocity v s (x).

The adsorption isotherm and the equation of state relate γ and Γ to c (see 170 section 2.5), so three additional equations are needed to close the problem.

These come from the conservation of the volume flux along the film, the balance of stresses at the interface and the conservation of surfactant molecules.

With the usual lubrication assumptions, symmetry around y = 0 and an interfacial velocity v s (x), the x-component of the fluid velocity within the film writes, at dominant order in h f /ℓ (subscripts are used to denote derivations) :

v(x, y) = - γh xxx 2η y 2 -h 2 + v s (x) (2)
At steady state, the volume flux q (in the half film) must be constant along the film, and may be evaluated in the flat film region where q = U h f (with

q = U h f = γh xxx 3η h 3 + v s h (3)
The second equation relates the distribution of γ to the underlying flow by expressing the balance between the bulk viscous stress and Marangoni forces at the interface:

∂γ ∂x = η ∂v ∂y h = -γh xxx h (4)
The last relation that we use expresses the conservation of surfactant molecules.

Since we neglect diffusion in the x direction, the flux of surfactants is only due to convection, and it has a bulk and a surface contribution:

q surf = cq + Γv s .
It must be constant with respect to x at steady state:

∂ ∂x (cq + Γv s ) = 0 (5)
Boundary conditions for h and v s are specified in the flat film region: for

x → +∞, they must asymptote to constant values h f , and U . c, γ and Γ must also reach constant values c f , γ f and Γ f . However, in practice the concentration,

180
and thus the value of γ and Γ, is imposed in the static meniscus, which acts as a reservoir for surfactants due to its large size. Thus, the physical boundary condition for those variables is imposed for x → -∞, where c = c m , γ = γ m and Γ = Γ m .

System of equations 185

Surfactants are assumed to equilibrate instantaneously between the interface and the bulk (assumption 2), so that the equilibrium equation of state and adsorption isotherm may be used to relate respectively γ to Γ and Γ to c.

We discuss the shape of these functions in the next section, but we take them formally into account here by treating γ and Γ as functions of respectively Γ 190 and c only.

Equation 4 is rewritten, using equation 3, as:

∂γ ∂Γ ∂Γ ∂c c x = -3η U h f -v s h h 2 (6)
A c c e p t e d M a n u s c r i p t

Equation 5 is integrated between x and the flat film region, where For the sake of simplicity we use simplified analytical expressions for the adsorption isotherm and the equation of state. Note that it is not necessary for our calculations, and arbitrary functions could be used instead. The adsorption isotherm Γ(c) has been chosen as a derivable function satisfying the two following important properties: a rapid increase of Γ from 0 to Γ cmc when c increases from 0 to the cmc, and a saturation of Γ above the cmc. Indeed, we found that the non linearity in the isotherm is crucial for the generation of films (see section 4.

Γ(x) = Γ f , v s (x) = U , c(x) = c f , γ(x) = γ f , h(x) = h f U h f (c -c f ) + v s Γ -U Γ f = 0 (7)

Adsorption isotherm and equation of state

Γ Γ cmc = 1 + K Γ 2 c c cmc + 1 -K Γ 2 1 -ξ log e c/ccmc -1 ξ + e -c/ccmc -1 ξ ( 8
)
whose graph is plotted on figure 2 Finally, we assume an affine relationship between γ and Γ. The equation of state is thus

γ γ cmc = γ(0) γ cmc - Γ Γ cmc γ(0) γ cmc -1 (9)
with γ(0) the surface tension of pure water. We define the dimensionless variables as follows, and separate them by uppercasing, or tilding. h and v s are rescaled by their value in the flat film:

Numerics

H = h/h f , V s = v s /U
H XXX = 3 γ 1 -V s H H 3 (10) ∂γ ∂ Γ ∂ Γ ∂C C X = -3Ca c 2/3 1 -V s H H 2 (11)
αβCa c 2/3 (C -C f ) + V s Γ -Γf = 0 (12)
where we introduce the coefficient β defined by h f = βr m Ca c 2/3 (which is the quantity that we are trying to determine) and the coefficient α = r m c cmc /Γ cmc .

The coefficient α is a crucial non dimensional parameter of our model, which 225 compares the concentration of surfactants in the bulk to that at the interface.

It compares r m to the length l cmc = Γ cmc /c cmc , which is the thickness of a liquid layer containing as many surfactants as the corresponding interface, at the cmc. Depending on the surfactants used, it can range from 1 µm to 1 cm, and accordingly α ranges from 1 to 1000 for millimetric menisci.

Boundary conditions and resolution

For numerical reasons, the boundary conditions must be imposed in the flat film region. As X → +∞:

H = 1, H X = 0, H XX = 0 C = C f V s = 1 ( 13
)
The integration is started for H = 1 + ǫ 0 and X = X 0 with ǫ 0 a small but finite quantity. The starting values for the other variables and the derivatives Marangoni stresses at the interface are related to the distribution of surfactant along the film.

In this example, we impose C(-∞) = Cm = 5, and the concentration in the film is much

lower (C f ≈ 1.27).
of H are found by linearizing the problem in the flat film region. The profile is then integrated to X = 0, towards the static meniscus.

235

Typical profiles are shown in figure 3. The important feature is that we choose X 0 to be large enough that both H XX , V s and C tend towards well converged constant values as X decreases. This ensures the validity of the asymptotic matching to the static meniscus.

Matching to the static meniscus 240

The matching to the meniscus imposes that the curvature tends to 1/r m for negative x. This condition can be written with non dimensional variables as

1 r m = h xx (-∞) = H -∞ XX Ca c 2/3 h f (14)
Given

h f = βr m Ca c 2/3 this imposes β = H -∞ XX ( 15
)
This numerical parameter β appears in equation 12, and it can thus not be freely chosen. Instead, the matching to the meniscus requires to solve the A c c e p t e d M a n u s c r i p t

implicit equation β = H -∞ XX (β).
With β * the solution of this equation, the film thickness is obtained as:

h f = β * r m Ca c 2/3 (16)
In Frankel's problem, the solution is h F r = 1.34r m Ca 2/3 with a capillary number Ca = ηV /γ m . To compare our predictions to Frankel's, we use the fact that h f /h F r = (γ cmc /γ m ) 2/3 β * /1.34. Note that, as the surface tension variation is very small above the cmc, Ca and Ca c differ significantly only below the cmc and the distinction between both is made only when necessary.

245

Lastly, since the concentration of surfactants is in practice imposed in the static meniscus, a shooting method is used to select the right value of C f (concentration in the film), given the value of C m (in the static meniscus) that is imposed.

This procedure allows us to solve the problem, given the following set of inde-250 pendent numerical parameters: C m , Ca c , α and the parameters of the equation of state and of the isotherm equation γ(0), K Γ and ξ. Frankel's prediction, which corresponds to a vanishing interface extension, is recovered within 5 %, for Ca < 10 -5 . Consistently, as seen on figure 4.b, 260 the velocity difference 1 -V m between the static meniscus and the film, hence interface stretching, is negligible at these low capillary numbers. For some critical capillary numbers interface deformation becomes relevant, as illustrated by the larger velocity difference. Accordingly, h f departs from Frankel's law.

Results and discussion

Influence of the capillary number

A higher capillary numbers, there is no direct relation between interface behavior of h f /h F r for intermediate Ca is not observed for V m . Note that the physical thickness of the film (h f = β * r m Ca 2/3 ) does increase monotonically with Ca.

Scaling law analysis for the velocity 270

Although there is no direct relation between the velocity difference δv s between the film and the static meniscus and the thickness of the film h f , deviation from Frankel's law only occur for significant interface stretching, that is values of δv s of order one. A simple law can be derived for this parameter, based on the mass conservation. In this scaling law analysis, we separate formally the 275 dynamic meniscus from the static part and the flat film region, as sketched on figure 5. We limit our analysis to small deviations from Frankel's law. Γ, c, γ and v s are assumed constant within the film (with respective values Γ f , c f ,

γ f , U ) and in the static meniscus (Γ m = Γ + δΓ, c m = c + δc, γ m = γ -δγ, v m = U -δv s ).
We focus on the regime of small interfacial stresses, and we 280 treat the problem at first order in δc and δv s .

At steady state, the net flux of surfactants going through the dynamic meniscus must be zero:

(c + δc)q + (Γ + δΓ)(U -δv s ) = cq + Γ f U
A c c e p t e d M a n u s c r i p t At first order, using q ∼ r m Ca 2/3 U , we get:

δv s = U Γ δΓ 1 + r m Ca 2/3 ∂c ∂Γ (17)
Lastly, the surface tension difference scales like δγ ∼ γCa 2/3 (see equation 1), so that our final prediction for interface stretching is:

δv s U = Ca 2/3 γ Γ ∂Γ ∂γ 1 + r m ∂Γ/∂c Ca 2/3 (18)
The surface dominated case corresponds to r m Ca 2/3 ∂c/∂Γ ≪ 1. In this limit, the predictions of the elastic interface model are expected and recovered, as verified in the section 4.3.

On the other hand, for a large meniscus radius, or above the cmc where 285 ∂Γ/∂c is very small, surfactants in the bulk are dominant (r m Ca 2/3 ∂c/∂Γ ≫ 1).

A new behavior is observed, which depends on the bulk concentration, as shown in section 4.4.

In non-dimensionnal variables, equation 18 becomes

δV s = 1 -V m = Ca c 2/3 (γ(0) -1) γ1/3 Γf 1 + α γ2/3 1 ∂ Γ/∂C Ca c 2/3 (19)
The variation of 1 -V m with the capillary number is plotted in figure 6. The two power laws are not well separated, but can still be observed. As Ca increases, δV goes from a surface dominated regime, where it increases as Cac 2/3 , to a bulk dominated regime with a Cac 4/3 dependence, as expected from equation 19 (dashed lines are guides of slope 2/3 and 4/3).

The limit of elastic interface

If the amount of surfactant in the bulk is much smaller than the amount of surfactant at the interface, then the exchange between interface and bulk becomes negligible and surfactant transport is governed by interfacial convection.

In that limit, our model coincides with a model of insoluble surfactant, leading 295 to a purely elastic interface, as studied in [START_REF] Seiwert | Theoretical study of the generation of soap 475 films: role of interfacial visco-elasticity[END_REF] and [START_REF] Saulnier | What is the mechanism of soap film entrainment?[END_REF]. This limit is recovered

when Γ f ≫ ch f = c f r m β * Ca 2/3
, or more precisely (using equation 18) when ∂Γ/∂c ≫ r m Ca 2/3 . The Ca dependence of the validity criterium simply stresses the fact that the relative contributions of surface and bulk transport vary with Ca, as seen in equation 12.

300

For simplicity we restrict the comparison to the case c m = c cmc . In that case, the previous condition becomes α = r m c cmc /Γ cmc ≪ Ca -2/3 and the Marangoni elasticity is E = -Γ∂γ/∂Γ = γ(γ(0) -1), as obtained from our equation of state 9.

We compare in figure 7 the results obtained with the elastic model discussed 305 in [START_REF] Seiwert | Theoretical study of the generation of soap 475 films: role of interfacial visco-elasticity[END_REF] and the results obtained with our calculations in different regimes. We

γ(0)-1 = 0.1, C f = 1 and α = kCa -2/3
, with k a constant. For α = 0.01Ca -2/3 310 or α = 0.1Ca -2/3 , both predictions for the thickness of the film agree, as expected. Frankel's law is observed at small capillary numbers and a thinner film

is obtained if E/γ ≫ Ca 2/3 .
For higher values of α (α = Ca -2/3 , α = 10Ca -2/3 or α = 100Ca -2/3 ), the solubility of the surfactant is increased and the bulk concentration becomes 315 non-negligible with respect to surface concentration. Surface extension can be cured by a reabsorption of surfactants. The surface tension variation for a given interfacial extension is thus reduced at high solubility, which corresponds qualitatively to a smaller effective elasticity. Consistently, the departure from Frankel's law occurs at lower capillary number for larger values of the solubility.

320

Note that, in figure 7, the bulk concentration is fixed: the variation of the ratio c/Γ thus corresponds to a variation of Γ, at constant c, that can be achieved by changing the surfactant used. The influence of the bulk concentration, for a given surfactant, is discussed below.

Effect of initial surfactant concentration c m 325

An important aspect of our model is the ability to predict film thickness variations as a function of the concentration in the meniscus. The most important feature is a strong film thinning observed when the concentration goes from one cmc to several cmc. 8, where The observed transition between a film close to the Frankel's film and a much thinner film can be understood on the basis of the velocity scaling given 340 by equation 19. The concentration decreases from the static meniscus to the film, and for 1 < C m < 6, the film is almost at the cmc, as seen in figure 9.a.

The variation of h

f /h F r with C m = c m /c cmc is shown in figure
For lower concentrations, the film is entirely below the cmc, whereas for higher concentrations, it is always above the cmc. It means that the non linearity of the isotherm (i.e., the fact that the concentration crosses the cmc, where ∂ Γ/∂C As visible on the two enlargements of figure 8, the rescaled thickness increases slightly with C m for C m < 1 and for C m > 6: surprisingly, for a given pulling velocity, the film thickness thus varies in a non-monotonic way with the concentration. For C m < 1 or C m > 6, the film is either always below the cmc or always above, as shown in figure 8. The variation of the surface concentration with the bulk concentration is thus affine: ∂ Γ/∂C is either 1 (below the cmc)

or K Γ (above the cmc). Equation 19 can then be simplified into

δV s = 1 -V m = Ca c 2/3 (γ(0) -1) γ1/3 Γ 1 + α γ2/3 1 K Γ Ca c 2/3 (20)
for the case C m > 1 (the case C m < 1 is obtained by setting K Γ = 1). In (by up to 50 %). We believe that measuring this quantity would provide the most direct way to confirm or infirm the relevance of our work to experimental situations. 410

Conclusion

In the present study, we predict the steady state thickness of soap films pulled from a bath of surfactants at constant speed. The originality of our work lies in the fact that we do not assume a particular effective interfacial rheology (i.e., incompressible or elastic or viscous interface). Instead, we take 415 explicitly into account the transport of surfactant molecules, and we deduce the Marangoni stresses at the interface, which serve as boundary condition for the hydrodynamics within the film, from their repartition.

In our model, the apparent rigidification of the interfaces comes explicitly from depletion/confinement effects: interface stretching lowers locally the con-420 centration of surfactants, because molecules in the bulk adsorb on the newly created interface. The peculiar thinness of the films make this mechanism particularly efficient to create interfacial stresses. We use two key approximations to track surfactants: their concentration is supposed homogeneous within the thickness of the film, and bulk and surface adsorbed surfactants are supposed 425 at equilibrium.

We show that this mechanism suffices to create the necessary stresses to pull a film out of a reservoir. For low enough capillary numbers, Frankel's law is recovered. Much like what is observed in experiments, deviations from this law occur at higher pulling velocities. We identify a regime of "insoluble" sur-430 factants, where the elastic interface model is recovered. Outside of this regime, bulk molecules have to be taken into account and lead to a less "rigid" interfaces.

A c c e p t e d M a n u s c r i p t

Most importantly, with our approximations, the entire problem is computed from equilibrium properties of the surfactant molecules. In particular, the resulting Marangoni stresses, hence the thickness of the film, depend strongly on 435 the form of the adsorption isotherm and the equation of state. We use here a simplified form for these two functions, but more realistic ones could be as easily incorporated. In fact, we found that the non linearity of the adsorption isotherm (namely the fact that its slope change abruptly around the cmc) is crucial to the generation of films.

440

Lastly, the most obvious prediction of our model is the thickness of the film, because it can be easily measured. We show that its results are compatible with available data by Saulnier and coworkers [START_REF] Saulnier | What is the mechanism of soap film entrainment?[END_REF]. Our calculations additionally compute the repartition of surfactants, and the interfacial velocity. Although these quantities are much harder to measure in experiments, they are much more 445 closely related to the mechanisms at work: the lower concentration in the film is a direct effect of depletion, while the lower interfacial velocity in the static meniscus comes from interface stretching.

Ackowledgements

We would like to thank E. Rio and B. Dollet for fruitful discussions. J. S.

450

acknowledges financial support from Région Bretagne (CREATE MOUSPORE)

and Agence Nationale de la Recherche (ANR-13-PDOC-0014-01-HYDROSURFDYN).

  A c c e p t e d M a n u s c r i p t *Graphical Abstract (for review) A c c e p t e d M a n u s c r i p t Generation of soap films with instantaneously adsorbed surfactants: concentration-dependent film thinning 1. Introduction

5 A

 5 c c e p t e d M a n u s c r i p t the same key ingredients as the central mechanisms for foam evolution and rheology, namely the coupling between hydrodynamics and surfactant dynamics.

35 2 A

 2 dynamics and hydrodynamics, through advection, diffusion and adsorption pro-c c e p t e d M a n u s c r i p t cesses. As a general picture, fast adsorbing surfactants may repopulate instantly any interface that is stretched, and should not lead to any interfacial gradients and Marangoni stresses: the effective elasticity or viscosity associated with exchanges with the bulk should vanish for such surfactants. The film systems that 40

A c c e p t e d M a n u s c r i p t into account by

  tracking surfactant concentrations in the volume and at the interface. We consider the case where surfactant adsorption is instantaneous: our model thus predicts the thickness of soap films for a given surfactant solution 70 from its equilibrium isotherm and equation of state. Both the incompressible interface behavior and the elastic interface behavior are recovered, as limiting cases in the parameter space. This model allows a better understanding of surfactant transport phenomena, and, despite its approximations, it does compare favorably with experimental data.

  75

Figure 1 :

 1 Figure 1: Sketch of the problem: only half of the film is represented, the dashed line represents an axe of symmetry of the problem. The film is extracted at a constant velocity U from a meniscus of constant radius of curvature rm. We compute its half thickness profile h(x), the interfacial velocity vs(x) and the concentration of surfactant c(x). Γ and γ are respectively the interfacial concentration of surfactants, and the surface tension.

  110

a typical time t ⊥ D = h 2 f /D ∼ r 2 m

 22 Ca 4/3 /D, where D is the diffusion coefficient 120 for the surfactant. On the other hand, changes in subsurface concentration 6 A c c e p t e d M a n u s c r i p t occur on the time scale of the transit in the dynamical meniscus t dyn = ℓ/U ∼ r m Ca 1/3 /U = Ca -2/3 ηr m /γ. Approximation 1 is valid as long as t ⊥ D ≪ t dyn , that is Ca ≪ Ca D = ηD/γr m . With typical aqueous surfactant solutions (D ≈ 10 -10 m 2 /s, η ≈ 125 1 mPa • s, γ ≈ 30 mN/m and r m = 1 mm), Ca D ≈ 10 -4 is close to the upper limit of the range of capillary numbers tested in experiments (Ca = 10 -6 -10 -3 ), and approximation 1 is well satisfied at low velocities. Evaluating approximation 2 requires the typical adsorption time t ads of the surfactant. To estimate it, we assume a linear kinetics adsorption (Henry kinet-130 ics), where the flux of adsorbed surfactant molecules is proportional to the bulk concentration, and to the deviation from the equilibrium interfacial concentration: j = kc (1 -Γ/Γ eq ) [15]. Assuming small deviations of c and Γ around an equilibrium value, the equation for ∆Γ = Γ -Γ eq becomes, at first order in deviations from equilibrium, d∆Γ dt = -kc eq ∆Γ Γeq . The interface thus repopulate 135 on a timescale t ads = Γeq kceq = lq k . The value of k depends on the surfactant used, and in the simplest cases it may be evaluated as a diffusion speed on a molecular length scale a [13]: k = D/a. With a = 1 nm and D = 10 -10 m 2 /s, k ≈ 0.1 m/s. The ratio l q =

175U

  the imposed velocity), giving the differential equation for h in the dynamic A c c e p t e d M a n u s c r i p t meniscus:

Figure 2 :

 2 Figure 2: Blue solid line: non dimensional adsorption isotherm Γ = Γ/Γcmc as a function of C = c/ccmc, which is affine far enough below and above the critical micellar concentration (C = 1), with slopes equal to 1 and K Γ . The transition region has a thickness ξ (throughout this study ξ = 0.15). Red dashed line: non dimensional surface tension γ = γ/γcmc as a function of C. The equation of state γ( Γ) is affine on the entire range, its only free parameter is the surface tension of the pure liquid γ(0) (throughout this study γ(0) = 72/35 ≈ 2.1).

  . The parameter ξ is the size of the transition region at the cmc. It does not affect the result of the model as long as it is small enough.195Below the cmc, the above expression asymptotes to the linear relation Γ/Γ cmc = c/c cmc . Due to the transition region, Γ is close, but slightly below, Γ cmc for c = c cmc . Above the transition, the adsorption isotherm reduces to Γ/Γ cmc = 1 + K Γ (c/c cmc -1). We chose an affine relation with a very small slope K Γ above the cmc instead of a strict saturation. A non vanishing value for K Γ is 200 indeed required for numerical stability. We additionally believe that this parameter K Γ is a physical parameter, even if difficult to measure. Stubenrauch and collaborators [18] measured for example a decrease of the surface tension of a solution of C 12 E 6 close to 2 mN/m as the concentration goes from c = c cmc to c = 10 c cmc . Even such a small variation of γ above the cmc may not be 205 neglected a priori: as we already stressed, in these films, minute surface tension variations (on the order of 0.01 % to 1%, that is 0.01 to 1 mN/m) have a strong effect.

  . The concentrations and the surface tension are rescaled by their value at the cmc: C = c/c cmc , Γ = Γ/Γ cmc , γ = γ/γ cmc . Consistently, 215 A c c e p t e d M a n u s c r i p t the space variable becomes X = xCa c 1/3 /h f with Ca c = ηU/γ cmc . This capillary number Ca is based on the value of surface tension at the cmc. It is slightly different than the capillary number used, for example, to describe experimental studies, based on the surface tension of the solution (at whatever surfactant concentration is used), which we denote by Ca. 220 Equations 3, 6 and 7 become, with Γ and γ explicit functions of C given by equations 9 and 8:

  230

Figure 3 :

 3 Figure 3: (a): profile H(X) of the dynamical meniscus (dashed line), for Cac = 10 -4 , α = 60, Cm = 5, K Γ = 0.001. We only show a portion of the solution, which is computed from X = 150 (flat film) to X = 0 (static menisus). The solid line is the curvature of the profile H XX : it reaches a constant value H -∞ XX ≈ 1.19 towards the static meniscus. (b): a direct illustration of interface stretching is given by the interfacial velocity Vs, which is much lower towards the static meniscus (Vm ≈ 0.42) than the imposed velocity in the flat film (Vs(+∞) = 1). (c):

Figure 4 .

 4 Figure 4.a shows the predicted value of h f /h F r as a function of Ca (in the

Figure 4 .

 4 Figure 4.b shows the value of the velocity in the static meniscus V m for the same parameters.

265Figure 4 :

 4 Figure 4: (a): Thickness of the film h f /h F r as a function of Ca. (b): Interfacial velocity in the meniscus Vm, as a function of Ca.

Figure 5 :

 5 Figure 5: Sketch of half a dynamical meniscus.

Figure 6 :

 6 Figure 6: Velocity decrease in the meniscus, as a function of the capillary number Ca, for Cm = 5, α = 1000 and K Γ = 0.1. δV = 1 -Vm, i.e. interface stretching, increases with Ca.

Figure 7 :

 7 Figure7: Thickness of the film h f /h F r as a function of Ca (Cm = 1, so Ca = Cac). Symbols are for the present model with α = 0.01Ca -2/3 (▽), α = 0.1Ca -2/3 (•), α = Ca -2/3 ( ), α = 10Ca -2/3 (⋄), α = 100Ca -2/3 (△). The other parameters of the calculations are chosen to allow for an easy direct visual comparison with the elastic case and thus test the robustness of the model, and may thus not correspond to real experimental values:K Γ = 1, γ(0) = 1.1.The solid line is the prediction of the model with elastic interfaces, with an elasticity E/γ = 0.1 = γ(0) -1[START_REF] Seiwert | Theoretical study of the generation of soap 475 films: role of interfacial visco-elasticity[END_REF]. It follows Frankel's law (dashed line) for Ca < 10 -4 , then decreases with Ca.

330

  we plot the result of our calculations for a constant pulling velocity, constant surfactant physicochemical properties, constant meniscus radius and increasing values of bulk concentration. For 1 < C m < 6, an important thickness decrease is obtained. Note that Frankel's film thickness h f r = 1.34(ηU/γ) 2/3 , used as a thickness reference, varies slightly with C m , as the surface tension varies (see 335 the surface tension in the film and in the meniscus in figure 9.b). However, this A c c e p t e d M a n u s c r i p t

Figure 8 :Figure 9 :

 89 Figure 8: Thickness of the film, rescaled by h F r , as a function of the imposed concentration of surfactant in the meniscus Cm = cm/ccmc, for Cac = 10 -4 (i.e. for a constant pulling velocity), α = rmccmc/Γcmc = 60 and K Γ = 0.001. h f /h F r increases slowly with Cm for Cm < 1, and for Cm > 6 (see the zooms on the relevant regions in insets). Between these values, it decreases much more rapidly with Cm. For these values of the parameters, h f /h F r always remains significantly smaller than 1.

A c c e p t e d M a n u s c r i p t effect is only

  sensible much below the cmc and does not play any relevant role here.

345 1 . 1 to 1 .

 111 changes abruptly) has to be taken into account in the intermediate concentration regime only. With the parameters of figure 8, we have K Γ = 10 -3 ≪ αCa c 2/3 = 0.13 ≪ The second term in equation 19, associated to bulk concentration effects, is thus negligible below the cmc, but non negligible above, as the coefficient ∂ Γ/∂C 350 suddenly decreases from 1 to K Γ . The velocity difference thus jumps from a small value below the cmc to a much higher value above the cmc, thus explaining the sudden thickness decrease for meniscus concentrations slightly above the cmc. Let us simplify the problem one final time by considering the limit where the 355 surface tension variation above the cmc is negligible and the tension variation below the cmc very large (which corresponds to incompressible interfaces, as in Frankel's work). Starting from a concentration in the static meniscus C m > 1, the concentration must first decrease to the cmc, which generates no interfacial stresses, but does stretches the interface. Below the cmc, the necessary stresses 360 may now be created with negligible concentration difference, hence negligible interface deformation. In other words, the entire interfacial elongation happens above the cmc, and corresponds to a decrease in concentrations from C m > Interface stretching thus increases with C m , which results in the thickness decreasing. The real curve is smoothened out by the small surface tension 365 variation above the cmc, and by the finite size of the transition region between the affine regimes of the isotherm. The physical origin of the thickness decrease A c c e p t e d M a n u s c r i p t still holds: it is a signature of the non linear isotherm.

figure 8 , 4 . 5 .Figure 10 :

 84510 figure 8, Ca c (i.e. the pulling velocity) is maintained constant, as well as the 370