Alan Turing

It is with great joy that I have accepted the invitation to write you a personal letter. Even in your scientifc writings your presence as a person is very strong, which is unusual for a mathematician. The traces of your personal life and your dramas extend beyond the limits of your own personal circumstances to concern us all -primarily because of the work that you did in World War II, but also because of the sufferings of a free young man, a homosexual, in a context that was totally hostile. I shall try to offer my own readings of some of your fundamental texts, placing them in relation to others that were equally revolutionary; and then I shall discuss with you some of the problems that we face, more than half a century later, including some that follow in the wake of your inventions. Science today is very different from the science that you knew, and it presents other challenges. Alongside the many possibilities that are offered to usalongside the potentials (and actualities) for a global interaction of networks of what we could call modern Turing machines that connect us all, and for a debate of everyone with everyone on the face of this planet, and for a memory of humanity made available to humanity -I shall also tell you how new forms of techno-science are deforming and emptying out the "meaning'" of the object of analysis, making it diffcult to arrive at an inventiveness of scientifc thought of the kind that you were so well able to express.

Knowing how to be within phenomena

First, you knew how to immerse yourself in phenomena, in the scientifc object, to give it 'meaning': initially, to make yourself as a machine (1936) 1 , then to live your dramatic question about being man-woman-machine (1950) 2 , fnally enabling us to see and almost to touch the continuous reshaping and the genesis of forms in a "hardware" without "software" (1952) 3 .

-The Machine

Your frst article reminded me of Archimedes, when he sees himself as a body in water and thus 1

comes to give us the principle of symmetrical thrusts. Or Einstein when he sees himself as photon "surfng" on a light wave to grasp the amazing invariance of the speed of light -just as, when we surf on a wave, the surrounding waves seem to be relatively unmoving. In the same way, you see yourself as "a man in the process of computing a real number": the "Logical Computing Machine" that you invent is effectively a "human computer". "Computing is normally done by writing certain symbols on paper. We may suppose this paper is divided into squares like a child's arithmetic book." "The behaviour of the computer at any moment is determined by the symbols which he is observing. and his 'state of mind' at that moment "(my emphasis). So its action is perfectly "stepwise", machinic, devoid of meaning, but ... human: you read 0 or 1 on the notebook / tape, you write 1 or 0 by moving to right or left, according to the internal state (of "mind") among a possible number of states q0, ..., qn. This is not a machine that is becoming human, but a man who is making himself into a machine. Did Hilbert want an absolute certainty of deduction in the "potential mechanisability of mathematics" 4 ; did he claim the deductive completeness of systems of formal calculus? Alright, so here is a man who reduces himself to a machine to effect logical-formal deduction: you, as mathematician, turn yourself into a formal deductive machine, a "computer [who] works by such a desultory manner that he never does more than one step at a sitting". And then you defne a function that is not computable, which thus escapes deductive mechanics: like Gödel, you destroy the formalist programme from within. Then, as the stated purpose of your article, you give the frst defnition of computable real number, thus also of non-computable ones. I shall attempt to reconstruct more closely the most signifcant ideas in your writings.

To show the limits of the formal system, which computes without meaning, you do not resort to the necessity of understanding what is being done; you do, however, show that the machine is incapable of calculating a function defned by a simple, purely formal diagonal procedure. In this you are imitating Gödel, but you do not follow the technique invented by Gödel -rather, you propose your formalism, which is much simpler and more human, too human to be machine, and yet machinic. And you accompany the invention with long explanatory discourses -you guide the reader step by step to your intuition, you are present in a human sense in the construction of your Logical Computing Machine, albeit so formal. This is the opposite of Gödel, whose 1931 5 article is a masterpiece of perfect formal rigour and closure, also in the writing, which is totally formal and "self-contained", almost unhuman, without any attempt at explanation, or an evocation of meaning, of the mathematical gesture proper to the construction carried out, apart from a few lines in the introduction. Gödel's article is an untouchable formal diamond: in no way can its proof be simplifed, or take shortcuts, except a possible weakening in the assumption of consistency and by rewriting the heavy notation -it is essential and perfect. Your article, on the other hand, is a pleasant chat; you take the reader by the hand, and discuss with us, and reason with us, all together. You are present with your humanity, as you will always continue to be later. Today, we are able to describe your formal machine in just a few lines, but in doing this we lose the originality of your inventive journey, the daringness of your turning yourself into a machine. That's why I'm so glad to be writing to you: in reading your mathematical articles, we get to know you as a person. Perhaps it's a bit like reading "Récoltes et Sémailles" by Alexander Grothendieck 6 , the great mathematician with no nationality, who was educated and lived in France, and was always out of place -as you were yourself, by reason of your character and your homosexuality. Alexander began his activity at the time of your death and, like you, was to do mathematics for only 20 years before then choosing scientifc, if not physical, suicide. I shall 4 See David Hilbert, Mathematical Problems, trans. Maby Winton Newson Bulletin of the American Mathematical Society 8 (1902), 437-479. Online at: https://mathcs.clarku.edu/~djoyce/hilbert/problems.html/ 5 Gödel, K. Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I", Monatshefte für Mathematik und Physik, 38: 173-198. Reprinted in Gödel 1986, pp. 144-195. 6 Récoltes et Sémailles, Réflexions et témoignage sur un passé de mathématicien, is an unpublished manuscript. The introduction is available in English at: https://www.babelio.com/livres/Grothendieck-Recoltes-et-semailles/529740/ return to this parallel.

Your result is a negative result, as were those of Poincaré (1892) 7 and Gödel (1931) 8 , who opened new ways by saying "no, that is not how it is": this or that programme and way of seeing things, which was dominant at the time, does not work. Poincaré demonstrates the possible predictive incompleteness of systems of non-linear equations: it is perfectly possible to determine a physical process with a system of equations, but its dynamics can be non-predictable ("et nous avons des phénomènes aléatoires" [and we have random phenomena], as Poincaré put it in 1902 9 ). Gödel, within any system that contains arithmetic, builds a formal assertion that is undecidable (the frst theorem of incompleteness) and which, moreover, implies the formal nondemonstrability of coherence (of the non-contradictoriness of arithmetic, second theorem). Your proof does not allow this latter virtuousity, the very fne gameplay of Gödel's second theorem, but, in compensation, it is much simpler and, as in the cases of Poincaré and Gödel, it paves the way for a new scientifc construction 10 . Following Kleene 11 , you then go on to prove that your Logic Machine is equally as expressive (it defnes the same class of functions) as the formal systems of Herbrand, Gödel, Kleene and the lambda calculus of Church 12 , all of which are of the 1930s. All very different logical-mathematical systems: these proofs of equivalence, between 1936 and 1940, will thus show that you all have invented a fundamental mathematical invariant, the class of formally computable functions. An invariant of Hilbertian writing and rewriting systems, i.e. of systems of transformation of signs into signs: any formal system in the style of Hilbert defnes (at most) such a class. The mystics, as always, will take such an invariant for an absolute.

Lambda calculus, a very elegant paradigm for systems of (re-)writing, will be the "means" for your (diffcult) proofs of equivalence. It is the system that is richest in "proper theorems"; it will provide the basis for logics with and without Types, from [START_REF] Church | A Set of Postulates for the Foundation of Logic[END_REF]Church ( 13 , 1940 14 14 ) and Gödel (1958) 15 to Girard (1971) 16 and Martin-Löf (1980) 17 18 . It is a means of great mathematical expressivity, at the heart of Mathematical Logic and of many of the programming styles that created modern computer science. In the 1970s to 1990s, the results of many people will make it possible to immerse lambda calculus and its Types into mathematical structures that also derive from a central notion in Grothendieck, that of "topos" 19 (categories of great structural and logical richness). These constructions gave a "geometric" meaning, possibly in continua, to your alphanumeric Discrete-State machine (what an honour for me to see my photo in the 1984 reprint of Henk Barendregt's classic book on lambda calculus, a few chapters after yours). Thus, again in 2008, new categories, inspired by Church's lambda calculus and the work of some of us between 1984 and 1990 20 , will be soundly described as "Turing categories" 21 . In summary, on the one hand, your Logic Computing Machine, for its simplicity and the separation between instructions and physical structure, inspired, after the war, the practical distinction between software and hardware, as well as the construction of operating systems and compilers -which are an implementation of your "Universal Machine". On the other hand, your work also opened the way to so much mathematics, of 30 and even 70 years later, thanks to both the bridging results inaugurated by yourself, with lambda calculus, and subsequently between this calculus and the mathematical theory of categories (and Gorthendieck's topoi).

I stress the point, however: all your results of the 1930s were born as "negative results". And so it was that your demonstrations, in placing the scaffolding for, and the limits to, an approach to knowledge, laid the foundations of new scientifc universes: Poincaré had done this as for the geometry of dynamic systems, Gödel, yourself, and the others I mentioned as for computability. To say "no" you had to refne an existing approach to the point of inventing new notions, which then showed themselves to be very fertile: Poincaré -bifurcations, homocline trajectories, the geometry of dynamical systems; Gödel and you -computable functions and machines. Who would dare today to propose a multi-million-dollar research project, the only way to have jobs, postgraduate researchers, collaborations, and activities guaranteed for three to fve years, to show that "no, it cannot be done ... this or that process is unpredictable, this or that system is incomplete, this or that function is not computable"?

-Man / woman / machine / Universe

The war interrupts your scientifc work and, as always in the twentieth century, it puts a freeze on all science. At best, dozens of great scientists like yourself will be set to work to fnd technical solutions to urgent problems in already established theoretical frameworks, from the chemistry of poison gases, and aviation, during World War I, and nuclear fssion for military purposes, or your computer with mechanical gears for decoding German codes, in World War II. Of course, in order to build it, you had to use your "arithmetical" talent, your ability to work on discrete combinations of integers and alphabetical letters, in the interplay between the two; but you will be obliged to build a machine that is traditional, a device of wheels and gears, such as had been in existence for decades, in order to fght the German machine with its cogs that continually generated different codes. The urgency of war did not allow you to develop your scientifc idea, the Logical Computing Machine with a software that was separate from the hardware, and your article of 1936 remained ignored for more than ten years -even by yourself -until, eventually, in peacetime, you and von Neumann could think about what would later become the "von Neumann architecture" 22 for modern electronic computers.

After the war, however, the resumption of scientifc thought will make you change your point of view on the Machine. You would frst grasp the practical importance of your mathematical invention, building a physical prototype of it, as I recalled, and then you understood that it could constitute a turning point in the representation of the world. Moreover, in your 1950 23 article you were to rename your Logical Computing Machine the Discrete-State Machine, thus highlighting the physical structure of its hardware, with discrete states, and the discrete-writable, alphabetical nature of its software.

I have to admit that I hesitated to read your 1950 article on "The Imitation Game". Is it possible, I asked myself, that a great man such as yourself would take up so uninspiringly a discovery of the 1920s, when the electrical discharges of neurons, the "spikes", were measured as beeps on a galvanometer? And the electrostatic and material deformations of the neurons, of enormous complexity, sort of "critical transitions" in a continuous dynamic -the electric-chemical-physical activity of a neuron -were then interpreted in simplistic terms as steps from a 0 (inactivity) to a 1 (discharge), like in the drawings in your children's machine notebook. I was amazed that your article could contain such a commonplace, that you might still think that an animal (or human) brain could be a Discrete-State Machine, 0 and 1 on a tape, written or deleted according to predetermined rules .... But no, actually you say exactly the opposite: "The nervous system is certainly not a discrete-state machine. A small error in the information about the size of a nervous impulse impinging on a neuron, may make a large difference to the size of the outgoing impulse." (p. 451)

In fact, what you are proposing is only an "imitation game", as you say on several occasions: the project is to fool a person who, by asking questions via a teleprinter, seeks to establish whether the respondent is a woman or a machine. In no way are you trying to fnd out how a human brain works, or to make a mathematical model of it; rather you are living the drama of a possible imitation. I say drama, because, in my opinion, you already know that the police might, at any moment, ask you the same question: but, are you a man or a woman? And so you make your machine answer this dull and insolent question, opening the way to a very rich symbolic game, but always a game -you, who knew from very young how to turn yourself into a machine, in the logical game of 1936, and maybe you were continually asking yourself that question: man / machine / woman? On the one hand, I must admit that I am not particularly interested in the psychological considerations that you adduce to convince the reader of the plausibility of the game of imitation and the possibility of tricking the reader: you do not go much beyond banalities such as "Do you have long hair? Can you write a poem? '' ... "Add 34957 to 70764, (Pause about 30 seconds and then give as answer) 105621" That's wrong: the machine has to imitate a woman and, as we well know, women, in mathematics.... Yet, Jean Lassègue claims that even in this case you lightly but deeply get into a profound semeiotic issue, concerning gender differences … maybe he is right 24 . Then, with great modesty, you dare to suggest that by 2000, in a game of not more than fve minutes, machines will have a 30% chance of passing themselves off as a woman, in a dialogue via teleprinter (p. 442). This was very far from the fantasies of perfect humanoid robots that were being promised to us a thousand times over by Classical AI. Today, the networks of artifcial neurons of the new AI, based on continuous variations of connectivity and capable of learningan idea to which you refer -are able gradually to establish invariants of images, and, after a lot 23Turing, A. 1950, 'Computing machinery and intelligence', Mind, 50: 433-460; also in Boden 1990,Collected Works (Volume 1) online at: https://www.csee.umbc.edu/courses/471/papers/turing.pdf 24Lassègue, J. 1996. "What Kind of Turing Test did Turing have in Mind ?", Tekhnema; Journal of Philosophy and Technology (ISBN : 2-9509944-0-7) (3) : 37-58. of computing, are able to distinguish a cat from a fat-iron (which, however, does not seek to imitate the cat) -no mean undertaking for a machine. These formally continuous neural nets are mathematically very different from your Discrete-State Machine, yet, so far, they still need to be implemented on the latter.

On the other hand, I was excited by the many observations you make of a physical-mathematical nature. They open the way to the other major article you were writing, the one about morphogenesis, which appeared in 1952 25 . Here, in contrast to your discrete-state machine, you grasp the role of the continuum, of the interplay between non-linear dynamics and physical measurement, which is always an interval, always approximate. Thus, on page 440 you say: "The system of the 'universe as a whole' is such that quite small errors in the initial conditions can have an overwhelming effect at a later time. The displacement of a single electron by a billionth of a centimetre at one moment could make the difference between a man being killed by an avalanche a year later or escaping." In other words, a disturbance, a fuctuation below the best measurement possible for a measuring device, at the appropriate scale -the man killed vs. the movement of an electron -can be amplifed over time into a phenomenon that is both observable and unpredictable. This is the crux of the deterministic unpredictability of phenomena that we represent with non-linear systems, sensitive to the initial conditions, those that Poincaré 26 had thoroughly analysed 60 years earlier and that very few had developed further: in them the role of measurement as an interval in the continuum (the in-principle completeness of all converging series, or of all convergent measurements) is crucial. Classical unpredictability, or randomness, arises at the interface between measurement and non-linear determination (equations, evolution function). Poincaré, Hadamard and some Russian mathematicians had worked on such systems in celestial mechanics, or in the classical physics of large systems. Perhaps these latter mathematicians, such as Pontryagin, were not even known to you (Kolmogorov 27 would only write in 1953 the frst version of what would then become, in 1963, the Kolmogorov-Arnold-Moser theorem, a pillar of those systems). Alone, therefore, or at most following on Poincaré, you grasp the importance of non-linearity, of its non-Laplacian character (determination does not imply predictability) far beyond the rational mechanics that was still being taught through to the 1960s and 1970s (including to myself, unfortunately, a very bad mathematical physics, alongside extraordinary courses in mathematics).

And so, in contrast to the sensitivity to initial conditions, or to the unpredictability of nonlinear dynamics in the continuum, you explain: "It is an essential property of the mechanical systems which we have called 'discrete-state machines' that this phenomenon does not occur. Even when we consider the actual physical machines instead of the idealised machines, reasonably accurate knowledge of the state at one moment yields reasonably accurate knowledge any number of steps later." 28 And you stress: "It will seem that given the initial state of the machine and the input signals it is always possible to predict all future states. This is reminiscent of Laplace's view that from the complete state of the universe at one moment of time, as described by the positions and velocities of all particles, it should be possible to predict all future states." 29 That is, maybe it can imitate a woman, but you stress -and still not many people understand it today -that your discrete-state machine is Laplacian, since its access to data is exact, its dynamic develops in a discrete space, it is a system of alphabetical writing and rewriting, as you explain clearly in a brief observation in which you compare alphabetical writing to ideogrammatic writing 30 . This is the key invariant property of the "knowledge of the state", in digital measurement, which is exact and develops exactly along the calculation process, something which does not happen in the physical Universe to which we "access" only by approximate or indeterminate measurement. Your "electron effect" (its displacement by a billionth of a centimetre with later macroscopic effects) anticipates by ten years the well-known "butterfy effect" of Lorentz (1962) 31 , which only came to be so called in 1972. As I have already written elsewhere, we should rather have been speaking in terms of Turing's "electron effect", a much earlier example of unpredictable (chaotic) dynamic, sensitive to initial conditions. But your 1950 article had been read principally in relation to Artifcial Intelligence; this did not allow people to grasp the tragic and symbolic theatre that you play, between a man, a woman and a machine, and even less your insight as a great mathematician who was working on morphogenesis, as a physical dynamic in the continuum.

So, decades after your very clear exposition, people are still saying (with Wolfram for example), that the Universe "as a whole" is a (very large!) Turing Machine, of which obviously living organisms, and the brains contained in them, are seen as "emergent" computational aspects, inasmuch as they are "computationally irreducible". And this, without even being able to give a rigorous mathematical defnition of computational irreducibility for a discrete process in two dimensions, see Wolfram, Cellular Automata, or A New Kind of Science, analysed in the margins of the doctoral thesis of Alastair Abbott, supervised by Calude and Longo, 2015. 32 The technical point is that computational irreducibility is not an invariant of "coarse graining" (changes of graining or scale), as shown since long 33 , but ignored by the ideologues of reducing nature to your machine, against your views. In addition, I have been vainly asking these computationalists of the Universe to tell us whether the fundamental constants of physics, which appear in all the relevant equations, namely G, c, h and the non-dimensional α, are computable real numbers, in the sense defned by yourself: two may be set equal to 1 … and the others? Theory and measurement will always give an interval or indeterminate values, producing the unpredictability of both classical and quantum dynamics -far away from Discrete-States Cellular Automata. In biology, these computationalist views of organisms have severely distorting research 34 , far away from your work on continuous morphogenesis, I will recall next.

There is something of the common to be understood in this computational folly that surrounds us -everything becomes discrete computation, numerics, digital programmes, from the Universe to the brain, from DNA to economics. Indeed, the whole of knowledge can be replaced by correlations evidenced computationally on huge discrete data bases -Big Data, which supposedly is going to make science obsolete. I am raising this with you in this letter because you are the right man to understand all of this -you who, with Gödel and Church, invented the theory of computability and you, the Discrete-State Machine by showing their limits w.r.to 30Lassègue, J., Longo, G. 2012 knowledge construction -producing an undecidable sentence, an incomputable function …. But you were also able to invent other scientifc paths, you had many strings to your bow, and you were always open-minded and curious about the world.

-The genesis of forms

About ffteen years ago, a biologist colleague told me that he was working on various issues of embryonic morphogenesis by following the work of an English mathematician, Alan Turing, and he asked if I knew him .... He was unaware of your work in Logic and Calculability and I knew nothing about your 1952 work, apart from the title! In fact, your work moves in a completely other universe, that is conceptual and mathematical. As a great thinker, you address another phenomenality by inventing other original tools for dealing with it. Except that then, in the end, you refect on correlations, and on a possible critical use of the pre-existing tools, the ones you had invented.

From the outset you stress: "The purpose of this paper is to discuss a possible mechanism by which the genes of a zygote may determine the anatomical structure of the resulting organism. The theory does not make any new hypotheses; it merely suggests that certain well-known physical laws are suffcient to account for many of the facts." 35 You then write a classical nonlinear system in the continuum: the model, as you call it, does not need "new hypotheses", especially not for the role of the " genes", to which I shall return. It is based on a system of equations, and is very simple. As usual, your genius way of working: frst, the almost childlike simplicity of the Logic Computing Machine of 1936, with which, however, one can reconstruct everything that is computable in the discrete; now, the elementary nature of this original idea of a chemical action that causes a reaction, and therefore a diffusion with waves that are both regular and irregular, that propagate in a continuum and that generate forms. A model that seeks to understand a process, which could be wrong ... but which, as you say, can "falsify" -an unusual term. So, this is not an imitation whose purpose is solely to "deceive a questioner", and which does not help to understand, let alone to falsify, but ... what could be falsifed by your model of morphogenesis? I shall try to understand this... The idea, very simple, but which nobody had posed in these terms before you, is therefore that a chemical action / reaction / diffusion can generate forms. In precisely those years Belusov, in Russia, had observed the phenomenon empirically, in a chemical reaction whose description, not believed at the time, was to be published only after a long delay 36 , and would only be understood after Zhabotinsky's 37 experiments in the 1960s. Thus, you intuit mathematically something that had not yet been observed experimentally, and had not been reported by others, but something that was possible: an equilibrium that is macroscopically homogeneous, but unstable, that a fuctuation, below the level of the observable, transforms into a dynamic of forms. Breakings of symmetry, catastrophic instability... these are the terms that you use. Here again, Poincaré and a few others had opened the way to the analysis of non-linear systems at equilibrium, but nobody had applied that vision of physical continuous dynamics to such areas, which you analyse in order to understand forms in biology.

35 Turing, A. 1952 Obviously, you work at a linear approach to the solution of the system of equations that you propose, but you go into a long discussion about the properties of non-linearity (the presence of second-order terms, as you put it). You do not propose a mathematics that is original or diffcult, as you say several times in your affectionate dialogue with the reader: you assure him that the equations and the calculations are elementary. But you invent an extremely original way of dealing with the problem of generating organic forms: the idea is, so to say, "very pure". As with your Machine, which computes like a child, on a squared notepad, now, you identify the terms of a physical dynamics that is minimal, but extremely expressive. As usual, you immerse yourself in the phenomenon, without pre-judgement, completely setting aside your previous experience, your own invention, the Logical Discrete-State Machine. You move beyond what Thom defnes as the fundamental aporia of mathematics: the discrete vs. the continuum. And your 1950 article, on imitation, with your observations on deterministic unpredictability and the electron effect, had already brought the aporia to light, the step that you were preparing yourself to take. Thus, you assign a central role to measurement, to access to phenomena: fuctuations below the best possible measurement are at the core of your analysis of continuous "symmetry breaking" and "catastrophic dynamics" (a wording by which you seem to refer to critical transitions, whose theory was yet to be invented). These notions make little sense in the discrete, or may at most be "imitated". But, at the end of the article, you pose the problem of implementing these dynamics on your Machines: you conclude that it will only enable the treatment of special cases and you promise yourself to do more work on this. The trial for homosexuality that will begin later that year will tear you away from that, and will lead to your suicide. So, a model, invented in order to understand, and not an imitation. And to falsify ... what? In conversations reported by Gandy 38 , you say that you didn't like Huxley's approach to Darwinian evolution. This was all centred on chromosomes and was to open the way to a new molecular biology that would see in DNA the complete programme of ontogenesis. For you, as you write in the article, genes are at most the producers of enzymes which are involved in the reactions that interest you, and it is the speed of this production, you say, that contributes to a process that is global, interactive, and based on physical continua, not "computational", even less "programmed". The idea of the descriptive completeness of the chromosomes, a fnite sequence of code-letters, did not sit well with you, who had shown, in your own way, the incompleteness of the axioms of arithmetic, also a fnite sequence of signs.

Schrödinger's little book of 1944, 39 the frst part of which proposed the idea of a "code" of biological forms inscribed in chromosomes, was already well-known. And von Neumann had already published, in 1951, an article on cellular automata 40 and had attributed to chromosomes the role of "programming of reproduction and ontogenesis". You do not cite them. What you propose is, however, compatible with the alternative identifed by Schrödinger in the second part of the book: ontogenesis as a dynamic based on the absorption of negative entropy, described in Gibbs's free energy terms 41 (together with Francis Bailly I have written about this, calling it antientropy, but the shadow of your work is in there too) 42 41 See, for the fullest explication of this, Gibbs, J. W. 1876. "On the Equilibrium of Heterogeneous Substances". Arts and Sciences. vol. 3, 1874-1878, pp.108-248. 42Bailly F., Longo G. Biological Organization and Anti-Entropy, in J. of Biological Systems, Vol. 17, n. 1, 2009.

Transactions of the Connecticut Academy of

of them very original: Child, D'Arcy Thompson, Waddington. [START_REF] Turing | The chemical basis of morphogenesis[END_REF] All of them were outside of the growing fxation on the idea of the completeness of the analysis of the chromosome as a way of understanding phylogeny and ontogeny (the latter was an embryologist-geneticist, but he too interpreted the action of chromosomes always within the interactions between the organism and the ecosystem). You, as the inventor of the notion of the "computer programme", of software that came to be defned independently of hardware, were against its use in biology. That's what your model demonstrates as false, as you hint at the beginning of your paper. In order to create forms (also biological) there is no need for a "predefned design", for a programme (you also say that your theory "does not make any new hypotheses; it merely suggests that certain well-known physical laws are suffcient to account for many of the facts." [START_REF] Turing | The chemical basis of morphogenesis[END_REF] ). The homunculus coded in DNA, a programme of ontogenesis and even of behaviour, which will become defnitively fashionable precisely in those years, is the opposite of the purely physical dynamics that you describe and that falsifies the need for a programme code for morphogenesis.

So, you, who distinguished software from hardware for us, and who thus invented the science of programming on discrete data types, and of software, describe now a dynamic that is purely hardware, without software, physical-chemical shaping and reshapings of forms in the continuum. This is the style of a great mind in addressing problems, always knowing how to renew ways of thinking, to enrich the gaze with new perspectives, and to invent or use a variety of tools, driven only by the desire to understand: the opposite of the fat transfer of just one conceptual tool, so much in fashion today -everything is information and digital computation. And in this way you opened a new pathway in science: after a delay of some decades, your analysis of morphogenesis was widely taken up and developed.

Intermezzo: Alexander Grothendieck (1928-2014)

I would like to tell you about another great mathematician who shares with you the drama of an early and deliberately chosen cessation of all activity, you with death, and him by extracting himself into the solitude of an agrarian life away from the world. But above all, I would like to emphasise your shared pursuit of "purity" of method, of an "ingenuousness" of the gaze, always innovative, which both of you devote to the object of your interest. You set about looking in a way that is different, original, "naive" (the ingenuousness of the child and his squared notepad, in your article of 1936), at areas that had already been studied by others, in computability, and so you open up new pathways; then, with just as much simplicity and originality, you invent a relatively simple, mathematical approach to morphogenesis. Alexander, like you, re-invents, with a new and very original and synthesising way of looking, huge and deep areas of mathematics. His mathematical notions are very "pure": they focus on invariance and maximal conceptual stability; they are very general without ever being empty. In this way, Grothendieck brought together previously distant constructions, offering surprising invariants, "bridging" notions and structures, shared, for example, by groups, topological spaces, manifolds of various kinds (differential, geometric, etc). Thus, he was able to bring together different areas of mathematics, transferring methods and correlating techniques [START_REF]Pour trouver un cadre commun permettant d'englober ces théories et d'autres » p. 119 in Grothendieck, A[END_REF] . For example, the "sheaves on a site", a notion of his that is diffcult and profound and motivated Grothendieck's notion of topos 46 , make it possible to move between discrete and continuous structures, beyond the foundamental aporia of mathematics -the interplay between the discrete and the continuum, which divides your activity into two distinct parts. Grothendieck's topoi produced a very interesting bridging between logic and geometry, also thanks to the works of Lawvere, since the '60s 47 , to the point of giving a ''geometrical'' meaning even to "impredicative defnitions", that terrible circular bogey for the logicians, from Russell to the present. Impredicativity becomes instead a very interesting property of logical closure, in the lambda calculus of Girard [START_REF] Girard | Une Extension de l'Interpretation de Gödel à l'Analyse, et son Application à l'Élimination des Coupures dans l'Analyse et la Théorie des Types[END_REF] , and a structural closure, in the categories between sets and topoi in the articles published by Hyland, Moggi and myself [START_REF] Hyland | A small complete category[END_REF] . The basis for this understanding was set by Dana Scott, in the late '60s and 70s, when he proposed the most successful approach linking computability, programming and categories50 . The ideas of Grothendieck, together with those of Serre and various others in France, in some sense touched on all felds of mathematics and were the main theoretical revolution of the post-war period, perhaps only comparable to the four or fve of the seventy years that preceded World War II (including the geometry of dynamic systems, relativistic physics, quantum physics ... and mathematical logic, to which you yourself contributed), but with even greater merits of unifying various and diverse felds.

After 20 years of activity, in the early 1970s, the scientifc purity of Alexander revealed its true nature, which was also ethical -as it was for you: you could not accept that you were at fault for being homosexual; you believed in the integrity of your person, as also of your way of thinking, you were perfectly ingenuous, but lucid, in the rigour with which you presented yourself, almost denouncing yourself, to the police, on the occasion of a petty theft suffered by an occasional companion. An ethically even more radical attitude was to lead Grothendieck to sever all relations with the academic world and the world of research: he felt that they were compromised by military funding, and not attentive to problems of the ecosystem. Pierre Lochack 51 , in a recent book, shows the continuity between his very diffcult life as a stateless person immigrating as a child into France, and the "asceticism" of his approach to mathematics and social life. The "purity" of Alexander's scientifc and moral vision joins the uncompromising political commitment of his parents, who were anarcho-syndicalists. His father in particular had been present in all the European revolutions, starting from 1905 in Russia. Jailed for ten years by the Tsar, released as a hero in 1917, he became hostile to Lenin, was persecuted by the Bolsheviks and ended up going into exile in Germany in 1921. As a result of his commitment against the rise of Nazism, when the Nazis came to power he took refuge in France, in 1933, and then, together with Alexander's mother, he went to fght in Spain in 1936. He died in 1942 in a German concentration camp. Alexander was to continue, in his own way, their uncompromising attitudeeven in mathematics, with the absolute rigour of a conceptual and ethical purity that gave no quarter, until the dramatic moment when he decided to drop everything and resign his position at 47Lawvere F.W. ( 1966) "The Category of categories as a Foundation for Mathematics", in Proc. Conf. on Categorical the IHES in Paris, fve years after having received the Fields Medal, at the height of his scientifc glory, and at the same age as yourself when you committed suicide. This way of working, this conceptual intransigence is often found in mathematics, and is perhaps one of its properties: in mathematical invention, there is always a revolutionary radicality and purity -a profound concept or a proof cannot be "patched", they cannot be the outcome of a compromise, even with the real. I have had teachers and friends, since my student years, and then collaborators and colleagues, who share these characteristics, people in whom "innocence" and a purity of ways of looking combine with a permanent re-invention and a depth of vision. Perhaps, at the best of times, every mathematician knows how to be equally uncompromising with concepts. They expect the maximum from them, they propose the purity of the simplest notions, like yourself, or the most general, like Grothendieck, or limit notions, as in Euclid's geometry, where "the line has no thickness" (defnition β) -the core invention of Western geometry, measures and organises the world with fgures made of pure edges, solely of lines without thickness.

-Networks and Big Data

In computing, over the last 30 years, there has been a major development, which you had not foreseen: the creation of networks. Networks of computers, all of which are individually extremely powerful, to an extent inconceivable to you (and thanks also to physics), are changing science and the world. A "symbolic" turning point, the third great writing revolution, says Clarisse Herrenschmidt, 52 after the invention of writing, and of alphabetic writing in particular, which she has studied as an archaeologist, and that of minted money, as a writing, as a symbolon, of value. Like every deep revolution in symbolism, and thus in human interaction, the present situation offers original challenges, that we still do not understand well, let alone control. These networks bring us all closer together, and offer us unprecedented opportunities to appreciate human diversity, and thus to enrich the experience of all, to provide the inspiration for new inventions, as a result of hybridisations and new syntheses.

However, the fact of having so many neighbours, as is suggested by the physics of "mean feld", can also imply that we all become "average", all the same, or nearly so. The challenge is open. The management of science is its a frst victim: bibliometrics, on which I have written a little article that is downloadable 53 from my web page (how lovely it is to have a web page accessible by anyone, where you can publish your own writings!), obliges us to work in dominant strands, where even a small advance can be quoted by many. Inventions like yours took ten, twenty, thirty years to be appreciated: the impact factor of journals is now computed by machines based on the numbers of citations of articles in the two years following publication. In mathematics and in physics ... it takes ten years just to understand a diffcult result on an original track, which is then ignored for a long time, unless it is a diffcult answer to problems that have been open for decades. Networked machines that compute the numbers of citations kill from the outset any attempt to venture, as you did, onto paths that are entirely new.

To this we have to add the madness, as I mentioned, of the "everything is computational", starting from the analysis of the living, the opposite of what you were able to offer, and creating the myth of the Universe as a Turing Machine, against your very precise observations. These colleagues, who are using the only technique that they know, and applying it to every possible 52Herrenschmit C. ( 2007) Les Trois Écritures, Langue, nombre, code. Paris: Gallimard 53Longo, G. (2014) Science, Problem solving and bibliometrics, in "Use and Abuse of Bibliometrics", Stockholm, phenomenon, fattening it onto a universe that is without meaning and made only of formal calculations, act as if yours is the last machine that man will be capable of inventing: it is coterminous with the world! I am convinced that we shall invent others, but these prophesies risk becoming self-affrming: piling computational techniques onto computational techniques, always in the same theoretical universe, to grasp the complexity of the Universe (of the brain, of DNA ...), in increasingly and abstrusively diffcult ways without the very simple purity and depth that mathematical invention requires -all of this prevents us from being able to think about the... next machine, which humankind will certainly fnd ways of inventing.

Discrete mathematical structures play a central role in all of this. Discrete databases are exact, and you can access them with precision. The great challenge of physical measurement is forgotten. The early 1900s, as I said, had brought this challenge to the limelight. Poincaré had grasped the role of the interface between the non-linearity of mathematical dynamics and physical processes, given by measurement; you yourself addressed this very well: fuctuations below the level of the measurable are amplifed into observable phenomena that turn out to be unpredictable. Quantum physics also begins, in 1900, from the question of the intrinsic indeterminacy of measurement, and the surprising discrete measure of the energy spectrum, in the continuum of space-time. All of this is set aside by computational dynamics, as alphanumeric systems of re-writing, as they are defned in generality. Starting with exact values, they always iterate in the same way: this is the correctness of the programmes. Then computer networks (Internet, typically) introduced the randomness proper to fuctuations in the space-time continuum, to perturbations in the operations of a node .... But the colleagues who work in networks and on concurrency call this form of randomness "do not care": everything possible is done to render it negligible. And they succeed -the networks operate with exactitude, with rare exceptions, thanks to the exactness of discrete data bases, with no nuances and without uncertainties in access to data. If discrete database networks become one with the world, if this is done without understanding the method which is thus imposed (the implicit grid of reading), one loses the meaning of the variation, which is "averaged out" by the average behaviours of networks, of the nuance, the approximation and the perturbation that contribute to novelty production, like in your analysis of morphogenesis. In particular, one loses the sense of the interface between our mathematical/theoretical description and the world: namely measurement. This is precisely what you were able to achieve, attributing a key role to fuctuations, from your "electron effect", below the level of measurability, that leads to a man being killed by an avalanche a year later, to the ones that, as you say, "trigger" morphogenesis. This helps us to understand the point at which your change of gaze, from the discrete to the continuum, enables us to talk about the world in another way: I shall return to this. The writing of equations, or of a function of evolution, of a dynamic, from Newton to Schrödinger, is not the "same thing" as the process of which they claim to be the "model", in the sense of your model of morphogenesis. Some out of this world Platonists still say that "a planet integrates a differential equation", forgetting, frst and foremost, that just two are suffcient, around the sun, for the system not to be integrable (and yet the planets move...). The equations, the function, propose or derive from a proposal of a causal structure, as in your analysis of morphogenesis, and are instruments of intelligibility and, in some rare cases, forecasting, mainly qualitative (here an attractor, there a singularity ... a certain type of forms ...). In more general terms, the equations can be derived from laws of conservation (energy, momentum...), and thus from symmetries, which structure them (the equations of motion, typically). Then man, or the machine if we are able to programme it properly, can apply algorithms to arrive at a solution, if they exist, or produce an approximate computation that "follows" the dynamic. We know that just a minimum of non-linearity is suffcient, namely the description of interactions (more bodies or agents), for the computation to diverge quickly from the physical dynamics. And this is easily shown, even without comparing the mathematical calculation with the physical process. In other words, it is not necessary to measure the process at the initial instant and at a later instant: it is suffcient to observe that a very small difference in the preferred (decimal) approximation (the 15 th decimal, say, for the very simple logistic function) gives rise to radical differences after a few iterations of the calculation (50 in that case, and it occupies the entire space of the phases). Since physical measurement (classical) is always an interval, this difference shows that a fuctuation below the best measurement possible makes the physical process unpredictable by a mathematical calculation. In quantum physics, measurement produces probability values that are real numbers, while calculation (the Schrödinger equation) is performed on complex numbers. In short, in either case, physical measurement is a fundamental and complex interface between our theoretical attempts, as far as possible mathematical, and phenomena; it shows the gap between them, and make human science possible, in the interplay between us and the world. The mathematical model (equations, typically) and the algorithmic calculations based on it are quite different from the physical process: measurement both connects and separates them, radically.

What, on the other hand, is done by these "computationalists" of the physical and biological world? They identify the Universe as a discrete structure, or rather as a discrete, alpha-numeric writing and they say, along with Wolfram 54 : "We can certainly imagine a universe that operates like some behaviour of a Turing machine." The systems for computability, as for instance Church's lambda calculus, and also your machine, are systems of re-writing -fnite strings of signs are transformed into other strings, following rules / instructions. In this way, without the problem of measurement, of the interface, as I said, the "physical" process, simulated and implemented in the machine, takes place exactly like the dynamics of re-writing of signs, in the discrete, and is identifed / identifable to symbolic dynamics: once the interface disappears, i.e. without the problem (which is huge, in physics) of measurement, the correspondence between mathematical writing and the world is exact. The world is a discrete-state machine, a machine for re-writing: the transformation of alpha-numeric strings into alpha-numeric strings. This is how Chaitin describes also biological dynamics, in Proving Darwin: "Life as randomly evolving software, software that describes a random walk of Increasing ftness in program space." In the discrete, without measure, DNA is made equivalent to a software and its physical materiality is not important: "we shall ignore bodies and metabolism and energy and consider purely software organisms". 55 In this approach, in physics, in biology, phenomena are "alienated" into a alphabetic formalism. They are no longer observed, because they are no longer measured. The computational world goes its own way, outside of the world, far from its physical and biological materiality, because in the actual world there are no written or numerical characters out-there. It is we who associate numbers (or letters) to phenomena and processes, through the diffcult challenge of measurement. The discrete replaces measurement and enumeration of acts of measurement, proper to continuous manifolds, with solely enumeration -a beautiful remark by Riemann 56 : in the discrete, one can only count.

Chaitin and Wolfram developed their theses, in both physics and biology, in many articles, and summarised them in two articles that appeared in a book published in your honour, where they seem to suggest: it is a pity that Turing did not understand this, but his machine is like the Universe as a whole, is like the dynamics of biology able to "immerse" yourself so deeply in phenomena, playing in the interface, getting a sense of the interplay between the discrete and the continuum, the role of measurement.

Perhaps the greatest catastrophe of anti-scientifc computationalism can be seen in the recent theory of "The End of Theories". In a series of widely quoted articles, informaticians or managers of very large databases explain that: "Correlation supersedes causation, and science can advance even without coherent models, unifed theories" 58 . In short, networked computers, bringing to light very extended correlations in huge databases, make it possible to predict and act, without the need to "understand": scientifc intelligibility is an uncertain luxury that is subjective and outdated, and theories are fallible proposals. Data, especially in large quantitiestera-terabytes, Big Data -is objective, is a new form of the absolute, is individually exact, expressed in digits. Thus, they argue, the larger that databases become, the more that statistical regularities, brought to light by computers, can govern us without the need to understand the meaning of the correlations, to interpret them, and without the need for theories about them, for interpretations.

Fortunately, mathematics allows us to demonstrate the absurdity of these claims: Cristian Calude and I have written an article about this 59 . Precisely the immensity of data involved has allowed us to apply the theorems of Ramsey and Van der Waerden. These make it possible to show that, given any "regularity", or any correlation between sets of numbers, you can fnd a number p large enough, such that every set with at least p elements contains a regularity (or a correlation between numbers) with the same structure. Now, since this applies to every suffciently large set (with at least p elements), this also applies when it is generated … by a random process. Indeed, we observe, almost all sets of fairly large numbers are algorithmically random (one can give a mathematical defnition of them, in terms of incompressibility), i.e., the percentage of nonrandom tends to 0 as p goes to infnity. So, if you observe regularities in increasingly large databases, it is increasingly likely that the inserted data are due to chance, in other words are perfectly without meaning and do not allow prediction nor action.

Which brings us to Frank Ramsey. You were not able to have known Ramsey personally. He too was a precocious mathematician at Cambridge. He died in 1930, at the age of 27. He was a translator and a friend of Wittgenstein, with whom you too were later to have an intensive exchange. 60 John Maynard Keynes was the strong, stable bond between you, an amazing group of attenders of each others lectures and friends (but, in your opinion, did Wittgenstein actually have friends ...?). They will certainly have talked to you about Ramsey, and I am sure that you will have liked his very fne result of fnite combinatorics 61 . Perhaps you might also have liked our simple application that demolishes the "Theory of the End of Theories", you who were always proposing increasingly original theories and mathematical frameworks, and putting forward different points of view, and turning yourself into a discrete-state machine, and inventing the software for it, and immersing yourself in continuous material shaping of forms, without software to programme them, and interpreting the real and your own invention of the real. And in the process profoundly changing our reality.

  . Instead you cite only three biologists, all 38Gandy, R. O. and C. E. M. Yates (eds.), 2001, The Collected Works of A M. Turing: Mathematical Logic, vol.4, Amsterdam: North-Holland. 39 Schrödinger, E. 1944, What is Life? The Physical Aspect of the Living Cell. Based on Lectures delivered under the auspices of the Institute at Trinity College, Dublin, in February 1943, Cambridge: University Press. 1944. 40Von Neumann, J., 1951, "The General and Logical Theory of Automata", in Cerebral Mechanisms in Behavior: The Hixon Symposium, New York: John Wiley & Sons.

  57 . A real insult to yourself, you who were 54Wolfram, S (2013) "The importance of Universal Computation", in A. Turing, his work and impact, Cooper ed., Elsevier. 55Chaitin, G. (2012), Proving Darwin. Making biology mathematical. New York: Vintage. 56Riemann B. "On the hypothesis which lie at the basis of Geometry", 1854 (Engl. by W. Clifford, Nature, 1973) 57Cooper, S. B. & Van Leuween, J. (eds.) (2013) Alan Turing. His work and impact. London: Elsevier

Translation:

  Ed Emery -4.xi.2017 58Anderson, C. (2008) "The end of theory: The data deluge makes the scientific method obsolete", WIRED. 59Calude, C. & Longo, G. (2017) "The deluge of Spurious Correlations in Big Data" in Foundations of Science Vol. 22, Issue 3, pp 595-612. Online at: http://www.di.ens.fr/users/longo/files/BigData-Calude-LongoAug21.pdf 60 See, for remarks on the discussions between Wittgenstein and Turing, Monk, R., 1991, Wittgenstein, the duty of genius, London: Vintage, pp.417-422

  "What is Turing's Comparison between Mechanism and Writing Worth?" Invited lecture, The Turing Centenary Conference (CiE 2012), Cambridge, June 18 -23, 2012; in Computability in Europe, LNCS 7318 (S.B. Cooper, A. Dawar, and B. Löwe, Eds.), pp. 451-462, Springer.31Lorentz, G. 1963 Deterministic nonperiodic flow. Journal of Atmospheric Sciences. Vol.20 : 130-141 32Abbott A. 2015. Value Indefiniteness, Randomness and Unpredictability in Quantum Foundations, University of Auckland and Ecole Politechnique, Paris,.33Israeli, N , Goldenfeld, N 2004 Computational irreducibility and the predictability of complex physical systems,

	Phys Rev Lett., Feb 20;92(7):074105.

34Longo, G. 2018 Information and causality: Mathematical Reflections on Cancer Biology. Organisms. A journal in Biological Sciences (to appear).

  , 'The chemical basis of morphogenesis', 36Belousov B. P. 1959 "Периодически действующая реакция и ее механизм"Сборник рефератов по радиационной медицине. 147: 145. "Periodically acting reaction and its mechanism", Collected Reports on Radiation Medicine, 147: 145. 37Zhabotinsky M. 1964 Периодический процесс окисления малоновой кислоты растворе (исследование кинетики реакции Белоусова). [Periodic processes of malonic acid oxidation in a liquid phase.] Биофизика [Biofizika], 9:306-311,.

Scott D. (1970) "Outline of a mathematical theory of computation" 4 th Ann. Princeton Conf. on Info. Syst. Sci.(pp. 169-176). Scott D. (1980) "Lambda-calculus, some models, some philosophy," The Kleene Symposium (Barwise et al. eds.) North-Holland.

May 2013. Proceedings, Wim Blockmans et al. (eds), Portland Press.

Acknowledgements I would like to thank warmly the translator and, with particular emphasis, Paolo Motta who made a wonderful work by completing the references, by a scholarly search and a perfect understanding of what I wanted to refer to.


Maël Pontoreau
email: pontoreau@protonmail.com

Emmanuel Le Guen

Christine Bourda

Jean-François Silvain

Optimization of highly concentrated dispersions of multi-walled carbon nanotubes with emphasis on surfactant content and carbon nanotubes quality

come

Introduction

Over the past three decades, carbon nanotubes (CNT) have drawn more and more attention of researchers all over the world. Among CNT, single-walled carbon nanotubes (SWNT) are described as a single layer of graphite rolled into a cylinder whereas multi-walled carbon nanotubes (MWNT) consist of sets of concentric cylindrical graphite shell. Typically, CNT are known as a carbon nanomaterial with a nanometric diameter and a microscopic length.

Due to their outstanding mechanical, thermal, electrical and optical properties [START_REF] Berber | Unusually High Thermal Conductivity of Carbon Nanotubes[END_REF]- [START_REF] Falvo | Bending and buckling of carbon nanotubes under large strain[END_REF], the development of CNT-based composites may find wide range of applications in polymer industry [START_REF] Jin | Characterization and nonlinear optical properties of a poly(acrylic acid)-surfactant-multi-walled carbon nanotube complex[END_REF], electronic components [START_REF] Pietrzak | Effect of Carbon Forms on Properties of Ag-C Composites Contact Materials[END_REF] or field emission devices [START_REF] Koohsorkhi | Fabrication of self-defined gated field emission devices on silicon substrates using PECVD-grown carbon nano-tubes[END_REF]. More recently, the use of this carbon nanomaterial in the preparation of nanofluids with improved thermophysical properties is under investigation for applications in heat exchangers [START_REF] Borode | Surfactant-aided dispersion of carbon nanomaterials in aqueous solution[END_REF]. However, as-produced CNT mainly exist as bundles arising from both chemical and physical entanglement of CNT due to van der Waals interactions between the tubes [START_REF] Thess | Crystalline Ropes of Metallic Carbon Nanotubes[END_REF] and the high aspect ratio of CNT respectively [START_REF] Hilding | Dispersion of Carbon Nanotubes in Liquids[END_REF]. In order to obtain homogeneous dispersion of CNT in nanotube-based composite materials and nanofluids, one of the key challenges is to deagglomerate the CNT.

Nowadays, solution based processing is the main route available which results in stable suspensions of CNT in which CNT are individualized [START_REF] Cheng | Improvement in properties of multiwalled carbon nanotube/polypropylene nanocomposites through homogeneous dispersion with the aid of surfactants[END_REF]. More precisely, most of the suspensions of CNT are obtained in aqueous solution by the joint use of ultrasonic treatment and organic species (polymers, surfactants …). On one hand, cavitation induced by the ultrasonic treatment [START_REF] Suslick | Sonochemistry[END_REF] de-agglomerates the bundles of CNT [START_REF] Strano | The Role of Surfactant Adsoprtion during Ultrasonication in the Dispersion of Single-Walled Carbon Nanotubes[END_REF]. On the other hand, adsorption of surfactant or polymers on the surface of CNT introduces repulsive interactions between the CNT surpassing the van der Waals interactions and preventing their re-agglomeration in solution. The adsorption of organic species on the surface of CNT is known as non-covalent functionalization and modifies the interfacial properties of CNT without compromising their structures unlike covalent functionalization, which consists in linking functional groups onto the CNT' surface [START_REF] Zakharychev | A study of the stability of aqueous suspensions of functionalized carbon nanotubes[END_REF], [START_REF] Bai | Aqueous stabilization of carbon nanotubes: effects of surface oxidization and solution chemistry[END_REF].

Even if the adsorption mechanisms of the surfactants on the CNT surface are under investigations [START_REF] Zueva | Structure and properties of aqueous dispersions of sodium dodecyl sulfate with carbon nanotubes[END_REF], the ability of surfactants to stabilize CNT in suspension has been studied multiple times. Different surfactants have been used such as sodium dodecyl sulphate (SDS) [START_REF] Zueva | Structure and properties of aqueous dispersions of sodium dodecyl sulfate with carbon nanotubes[END_REF]- [START_REF] Jiang | Production of aqueous colloidal dispersions of carbon nanotubes[END_REF], cetrimonium bromide (CTAB) [START_REF] Rausch | Surfactant assisted dispersion of functionalized multi-walled carbon nanotubes in aqueous media[END_REF] and Triton X-100 [START_REF] Richard | Supramolecular Self-Assembly of Lipid Derivatives on Carbon Nanotubes[END_REF]- [START_REF] Ramos | Dependence of carbon nanotubes dispersion kinetics on surfactants[END_REF]. Some studies show significant differences in CNT' stabilization depending on the surfactant's nature and concentration [START_REF] Cheng | Improvement in properties of multiwalled carbon nanotube/polypropylene nanocomposites through homogeneous dispersion with the aid of surfactants[END_REF], [START_REF] Ramos | Dependence of carbon nanotubes dispersion kinetics on surfactants[END_REF]- [START_REF] Javadian | Dispersion stability of multi-walled carbon nanotubes in catanionic surfactant mixtures[END_REF]. Measurement of zeta potential depending on the pH value of the aqueous solution is commonly used in these studies [START_REF] Jiang | Production of aqueous colloidal dispersions of carbon nanotubes[END_REF], [START_REF] Teng | Evaluating Stability of Aqueous Multiwalled Carbon Nanotube Nanofluids by Using Different Stabilizers[END_REF], [START_REF] White | Zeta-Potential Measurements of Surfactant-Wrapped Individual Single-Walled Carbon Nanotubes[END_REF].

Moreover, efforts have been devoted optimizing the time and the power of the ultrasonic treatment. Actually, transmission electron microscopy (TEM) and Raman spectroscopy analysis showed that « sonication induced CNT cutting » occurs [START_REF] Shelimov | Purification of single-wall carbon nanotubes by ultrasonically assisted filtration[END_REF] and the structure of CNT is damaged during the ultrasonic treatment [START_REF] Lu | Mechanical damage of carbon nanotubes by ultrasound[END_REF], [START_REF] Li | Grafting modification and structural degradation of multi-walled carbon nanotubes under the effect of ultrasonics sonochemistry[END_REF]. By using the results obtained by UV-Vis spectroscopy, the duration of the ultrasonic treatment has been reduced as much as possible in order to preserve the integrity of CNT while obtaining a high concentrated CNT suspension [START_REF] Grossiord | Time-Dependent Study of the Exfoliation Process of Carbon Nanotubes in Aqueous Dispersions by Using UV-Visible Spectroscopy[END_REF], [START_REF] Yu | Controlling the dispersion of multi-wall carbon nanotubes in aqueous surfactant solution[END_REF]. UV-Vis spectroscopy is usually used to compute the concentration of the suspensions based on previous experiments carried out on MWNT [START_REF] Bahr | Dissolution of small diameter single-wall carbon nanotubes in organic solvents?[END_REF]. At that time, few studies have been conducted on the influence of the CNT characteristics on CNT suspension. Contradictory reports exist on the effect of the length of the CNT [START_REF] Ntim | Size dependent aqueous dispersibility of carboxylated multiwall carbon nanotubes[END_REF], [START_REF] Wu | Length reduction of multi-walled carbon nanotubes via high energy ultrasonication and its effect on their dispersibility[END_REF].

Therefore, the objectives of our study are to provide details on the dispersion of MWNT in aqueous surfactant solutions using 3 surfactants and 3 MWNT of different morphologies. An emphasis is placed on the mass ratio between surfactant and MWNT prior to ultrasonic treatment as well as the determination of the concentration of the surfactant and MWNT stable in suspension.

Experimental

Materials

Three types of purified MWNT have been studied, namely, NC7000 Triton X-100 C8H17C6H4(OC2H4)9-10OH Non-ionic 0.12-0.56 13.5

Table 2 : Characteristics of the used surfactant (CMC: critical micelle concentration, HLB: hydrophilic-lipophilic balance)

Dispersion of MWNT in various surfactants solution

In order to compare the suspensions of MWNT depending on the surfactant's concentration, dispersions of MWNT were prepared by adding 750 mg of MWNT to 75 mL of an aqueous surfactant solution. The initial concentration of MWNT is thus kept constant for all experiments. This aqueous solution contains the surfactant (either SDS or CTAB or Triton X-100) at concentrations spanning from 0 to 10 mg/mL. All three surfactants were used with MWNT2 whereas MWNT1 and MWNT3 were used only with SDS.

All resulting aqueous solutions were sonicated for 1 h with an ultrasonic homogenizer (Bandelin Sonopuls® UW2200) with active intervals of 0.1 s, passive intervals of 0.9 s, amplitude fixed at 50% and a power equals to 200 W. Subsequently, the solutions were centrifuged for 5 min at 2680 g0 in order to bring bundles as well as residual catalyst particles to the bottom of the centrifuge tubes [START_REF] Zueva | Structure and properties of aqueous dispersions of sodium dodecyl sulfate with carbon nanotubes[END_REF], [START_REF] O'connell | Band Gap Fluorescence from Individual Single-Walled Carbon Nanotubes[END_REF], [START_REF] Ramos | Dependence of carbon nanotubes dispersion kinetics on surfactants[END_REF], [START_REF] Javadian | Dispersion stability of multi-walled carbon nanotubes in catanionic surfactant mixtures[END_REF], [START_REF] Krause | Dispersability and particle size distribution of CNT in an aqueous surfactant dispersion as a function of ultrasonic treatment time[END_REF]. The supernatants were selected and further characterized. For some characterizations, the supernatants were dried at 60 °C overnight. Each suspension will be referenced in function of the MWNT batch and surfactant used as well as the initial mass ratio surfactant/MWNT. For example MWNT2-SDS(0.5) refers to a MWNT2 suspension with SDS concentration equals to 5 mg/mL.

Investigation methods

Characterization of debundling

To confirm debundling of the MWNT in the supernatant, scanning electron microscopy (SEM) and TEM were performed using Tescan® Vega II and Jeol® JEM 1400+ respectively.

For each analysis, a droplet of the suspension was dried on the appropriate substrates (TEM grid cover with carbon film).

Characterization of suspension's content

Similarly to previous studies, the debundling of MWNT was characterized using UV-Vis spectrophotometer (Varian® Cary 5000) operating between 250-800 nm. In order to establish a relationship between the amount of suspended MWNT and the intensity of the corresponding absorption spectrum, the samples were diluted by a certain factor. Same spectroscopy cells measuring 10 mm have been used for all measurements. From 250 to 800 nm, the absorption of aqueous solutions of each surfactant was measured to be zero relative to the water used for baseline correction.

By drying the supernatant at 60 °C, solid particles consisting of a mixing of MWNT and surfactant can be collected. For each suspension, total concentration of MWNT and surfactant in the supernatant was computed by drying a specific volume of the supernatant and weighing the collected solid particles [START_REF] Zakharychev | A study of the stability of aqueous suspensions of functionalized carbon nanotubes[END_REF]. Moreover, quantification of carbon and hydrogenusing a CHNS elemental analyzer (Thermo Fisher Scientific®)was systematically performed to determine the relative amount of MWNT and surfactant in the collected solid particles. These measurements were repeated at least 3 times with about 3 mg of particles collected each time.

Pristine MWNT and surfactants were measured as references after being dried at 60 °C overnight. Standard deviations of the different measurements are used to compute absolute uncertainty.

A thermal gravimetric analyzer (Setaram® TAG 24) was used as second characterization technique to confirm the ratio between MWNT and surfactants. The thermogravimetric analysis (TGA) were performed between 25 to 600 °C at a heating rate of 5 °C/min under argon atmosphere.

Characterization of CNT's integrity

MWNT morphology was investigated by Raman spectroscopy (Horiba® LabRAM HR).

The excitation source was a He-Ne laser (632.8 nm) and the power was kept below 1 mW to avoid local heating of the sample. The band fitting of the spectra was carried out using Lorentzian and Gaussian functions as reported elsewhere [START_REF] Vallerot | Quantitative structural and textural assessment of laminar pyrocarbons through Raman spectroscopy, electron diffraction and few other techniques[END_REF].

Results and discussion

Debundling of MWNT

Firstly, it had be verified experimentally that the debundling of MWNT can only occur by the joint use of an ultrasonic treatment and a surfactant. Figure 1 shows observations of different MWNT2 aqueous solutions, some of which have undergone US treatment with or without a SDS concentration of 10 mg/mL. Instead of US treatment, aqueous solutions have been magnetically stirred at 500 rpm for 1 h. After being vertically stored for 3 days, it appears that stable suspension is only obtained by the combination of physical and chemical method.

Without surfactant (cf. Figure 1 b), MWNT2 are exfoliated but agglomerate in solution.

Agglomeration of MWNT2 is prevented by the surfactant adsorbed on MWNT's surface (cf.

UV-Vis spectroscopy analysis

As reported elsewhere, debundled MWNT dispersed in solution are active in the UV-Vis region and exhibit characteristic bands corresponding to additional absorption due to 1D van Hove singularities [START_REF] Yu | Controlling the dispersion of multi-wall carbon nanotubes in aqueous surfactant solution[END_REF], [START_REF] Kataura | Optical properties of single-wall carbon nanotubes[END_REF]. Therefore, UV-Vis spectroscopy can be used to detect and quantify MWNT in suspension. Assuming that the extinction coefficient is the same for each MWNT batch, taking account of the dilute factor and using the mean absorbance of the suspensions for a mass ratio above 0.5, maximum MWNT concentrations of all compared to MWNT2-SDS suspensions can be computed following the following equation :

C 𝑀𝑊𝑁𝑇 𝑚𝑎𝑥 (𝑆𝑢𝑠𝑝𝑒𝑛𝑠𝑖𝑜𝑛) C 𝑀𝑊𝑁𝑇 𝑚𝑎𝑥 (𝑀𝑊𝑁𝑇2-𝑆𝐷𝑆) =
A 500 𝑛𝑚 (𝑆𝑢𝑠𝑝𝑒𝑛𝑠𝑖𝑜𝑛) * Dilute factor (𝑆𝑢𝑠𝑝𝑒𝑛𝑠𝑖𝑜𝑛) A 500 𝑛𝑚 (𝑀𝑊𝑁𝑇2-𝑆𝐷𝑆) * Dilute factor (𝑀𝑊𝑁𝑇2-𝑆𝐷𝑆)

Table 3 shows the mean absorbance at 500 nm of each suspension for a mass ratio above 0.5 as well as the ratio of the maximum concentration of MWNT in each suspension to the maximum concentration of MWNT in MWNT2-SDS suspensions. discussions of the maximum concentration measured by each method will be done.

Weighing method

By drying and weighing precise volumes of suspensions of MWNT, evolutions of the total concentration of surfactant and MWNT in function of initial surfactant/MWNT mass ratio have been computed for each suspension (cf. Figure 7 and Figure 8). For each suspension, these evolutions can be divided into two parts. Firstly, from 0 to a certain surfactant/MWNT mass ratio, the concentration of collected particles soars.

Experimentally, this increase can be seen as supernatant becomes darker and darker which means that the concentration of MWNT being suspended increases. Secondly, when the mass ratio further increases, the increase of concentration becomes slighter. In accordance with the UV-Vis spectroscopy measurements, the shape of the graphs demonstrates that the concentration of MWNT remains stable in the supernatant above a threshold surfactant/MWNT mass ratio. Indeed, the threshold mass ratios are similar to those determined by UV-Vis spectroscopy measurements. The slight rise of concentration computed is only due to a further increase of the concentration of the surfactant in the supernatant.

For the suspensions MWNT1-SDS, the threshold mass ratio is unclear because of the low concentration of MWNT. However, it can be noticed that total concentration of SDS and MWNT in the supernatant is systematically lower than the initial surfactant concentration. It means that some SDS molecules are adsorbed on the surface of the bundles of MWNT at the bottom of the centrifugation tubes. In the case of MWNT1-SDS, the amount of surfactant adsorbed on the surface of MWNT would not be sufficient to prevent major bundling of MWNT in suspension and, thus, MWNT sedimentation upon centrifugation.

Relative proportion of SDS and MWNT in the suspension

Further investigations have been carried out to compute the actual concentrations of MWNT and of surfactant for each suspension of MWNT.

The hydrogen content of collected particles after drying has been measured using a CHNS elemental analyzer. Indeed, all three surfactants contain a significant amount of hydrogen whereas the amount of hydrogen in the MWNT is extremely low. Consequently, the measurement of the hydrogen content of the collected particles after drying leads to the computation of relative amount of MWNT and of surfactant. Due to low MWNT concentration in the MWNT1-SDS suspensions, it is not possible to compute the mass fraction of SDS thanks to the elemental analyzer. Actually, the amount of SDS is so high that the amount of hydrogen -For elemental composition measurement :

% 𝑤𝑡 𝑆𝑢𝑟𝑓𝑎𝑐𝑡𝑎𝑛𝑡 𝑠𝑎𝑚𝑝𝑙𝑒 = (% 𝑤𝑡 𝐻 𝑠𝑎𝑚𝑝𝑙𝑒 -% 𝑤𝑡 𝐻 𝑀𝑊𝑁𝑇 ) (% 𝑤𝑡 𝐻 𝑆𝑢𝑟𝑓𝑎𝑐𝑡𝑎𝑛𝑡 -% 𝑤𝑡 𝐻 𝑀𝑊𝑁𝑇 ) * 100
-For TGA measurement (computation for the mass loss at 400, 500 and 600 °C) :

% 𝑤𝑡 𝑆𝑢𝑟𝑓𝑎𝑐𝑡𝑎𝑛𝑡 𝑠𝑎𝑚𝑝𝑙𝑒 = (∆𝑚 𝑠𝑎𝑚𝑝𝑙𝑒 -∆𝑚 𝑀𝑊𝑁𝑇 ) (∆𝑚 𝑆𝑢𝑟𝑓𝑎𝑐𝑡𝑎𝑛𝑡 -∆𝑚 𝑀𝑊𝑁𝑇 )
* 100 The graphs show that surfactants' mass fractions are systematically superior to initial mass fractions. Generally, the mass fraction of surfactant decreases in a first time before increasing significantly from an initial mass ratio corresponding approximately to those determined by the two previous characterization techniques. This observation is once again consistent with the view that, above a certain initial surfactant/MWNT mass ratio, only the concentration of surfactant increases in the supernatant.

Evolution of the concentration of MWNT and surfactant in the supernatant

Combining the total concentration of surfactant and MWNT deduced by weighing with the mass fraction of surfactant, the evolution of MWNT and surfactant concentrations can be computed as shown in Figure 12 and Figure 13. At the threshold value, the surface of the MWNT are fully covered by surfactant molecules and the repulsive interactions between MWNT permit to maintain a certain amount of MWNT in suspension. Considering that CNT are nanomaterials exhibiting huge surfaces, the concentration of surfactant molecules needed is logically higher than the CMC of a given surfactant. Once the MWNT are fully covered by surfactant, the MWNT concentration is not influenced by a further increase of the concentration of surfactant. Consequently, optimized suspensions highly concentrated in MWNT and weakly concentrated in surfactant can be obtained following the procedure described in this paper. For example, MWNT2-SDS(0.3) and MWNT3-SDS(0.4) are considered as optimized suspensions.

Linear fitting, with slope equals to 0, have been performed in order to determine the MWNT' maximum concentration for each suspension. and MWNT3, at 500 nm, are equal to 48.7 ± 4.7 cm²/mg (mean of MWNT2-SDS, MWNT2-CTAB and MWNT2-TRIT suspensions) and 67.9 ± 4.6 cm²/mg, respectively. These values are consistent with those commonly used in the literature for low concentrated MWNT suspensions [START_REF] Rastogi | Comparative study of carbon nanotube dispersion using surfactants[END_REF]- [START_REF] Baskaran | Noncovalent and Nonspecific Molecular Interactions of Polymers with Multiwalled Carbon Nanotubes[END_REF]. Using these two extinction coefficients, the maximum MWNT concentration of MWNT1-SDS is equal to 0.62 ± 0.12 mg/mL or 0.44 ± 0.07 mg/mL, respectively. For MWNT1-SDS suspensions, the optimized initial mass ratio equals to 0.5.

Therefore, in the frame of our study, the maximum achievable concentration of MWNT does not depend on the surfactant used but mainly on the batch of MWNT used. Indeed, all suspensions with MWNT2 have approximately the same maximum MWNT concentration.

Moreover, using SDS as surfactant, the maximum MWNT concentration is significantly different for MWNT1, MWNT2 and MWNT3 despite the fact that MWNT2 and MWNT3 have the same initial length and diameter. However, the MWNT are not only defined by their initial length and diameter. Therefore, quality of the arrangement of the carbon atoms of the different batches of MWNT has been analyzed by Raman spectroscopy. It has to be mentioned that XPS analyses of the surface chemistry of the 3 different MWNT show almost identical results.

Raman spectroscopy analysis

Figure 14 shows the Raman spectra of pristine MWNT3. For each first order Raman spectra, five Raman bands of graphitic materials are identified and fitted by the procedure described elsewhere [START_REF] Vallerot | Quantitative structural and textural assessment of laminar pyrocarbons through Raman spectroscopy, electron diffraction and few other techniques[END_REF]. On the one hand, the G band (1580 cm -1 ) corresponds to sp²-C network. On the other hand, the D band (1350 cm -1 ) and D' band (1620 cm -1 ) are attributed to a double resonance Raman scattering mechanism and represent the defect in the C-lattice [START_REF] Vallerot | Quantitative structural and textural assessment of laminar pyrocarbons through Raman spectroscopy, electron diffraction and few other techniques[END_REF].

I (1170 cm -1 ) and D" (1500 cm -1 ) bands are attributed to disordered graphitic lattice and amorphous carbon respectively [START_REF] Düngen | Investigation of Different Pre-Treated Multi-Walled Carbon Nanotubes by Raman Spectroscopy[END_REF], [START_REF] Sadezky | Raman microspectroscopy of soot and related carbonaceous materials: Spectral analysis and structural information[END_REF]. The ratio ID/IG is commonly used to evaluate the amount of defects in the structure of CNT; the lower the ratio, the better the structure of the MWNT. Nevertheless, it should be stressed out that further investigations using a larger number of different MWNT batches are necessary to make a statistically relevant conclusion on the strength of the correlation between MWNT quality and the maximum achievable MWNT concentration. For example, adding defects by covalent functionalization on CNT surface and reproducing the experimental protocol presented in this article could be interesting.

Nevertheless, it will have to be taken into account that misarrangement of carbon atoms on CNT surface and functional groups are different types of defects which could have different impact on surfactant adsorption. It has been proven that the maximum MWNT concentration depends mainly on the quality of the structure of the MWNT rather than their initial length, the surface composition and the nature of the surfactant. Adsorption of surfactant may be disrupted due to the presence of structural defects. The maximum MWNT concentration achievable varies from 0.50 to 7.5 g/L depending mainly on quality of the MWNT.

The high concentrated MWNT suspensions may find many applications as nanofluids or as part of nanocomposites processes. Future studies could focus on demonstrating the strength of the correlation between MWNT quality and the maximum achievable MWNT concentration or on the optimization of the initial CNT nanotube concentration, the time and the power of ultrasonic treatment.

Figure 1 dFigure 1 :Figure 2 :Figure 3 :

 1123 Figure1 d). Even after the centrifugation step, the supernatant remains opaque which means that a significant amount of MWNT2 stays in suspension in the aqueous solvent.

  Figure 4 illustrates UV-Vis spectra of MWNT2-SDS suspensions made with different SDS/MWNT initial mass ratios. Similarly to other studies, a maximum around 250 nm can be noticed which means that the suspensions are composed of individual MWNT.However, the maximum absorbance of MWNT suspensions varies significantly depending on the initial surfactant's concentration. Firstly, the absorbance increases gradually with the surfactant/MWNT mass ratio. In a second time, the absorbance remains interestingly constant. This result means that a maximum concentration of CNT in solution is reached when the mass ratio is above 0.3. UV-Vis spectra of all other MWNT suspensions exhibit similar behavior. Figure5and Figure6plot the absorbance measured at 500 nm for MWNT-SDS suspensions and MWNT2-surfactant suspensions respectively. Depending on initial surfactant/MWNT mass ratio, a certain mass ratio exists above which the absorbanceand consequently the concentration of MWNT in suspensionreaches a maximum. Consequently, these results show that, above a certain mass ratio MWNT/surfactant, the surface of all the MWNT, deagglomerated by the US treatment, is saturated with surfactant molecules. A further increase of the concentration of surfactant in solution does not affect the MWNT. These threshold mass ratios depend on both the MWNT and the surfactant used. They are typically ranging from 0.2 to 0.5. It can be noticed that these threshold values do not correlate with the CMC of the surfactants.

Figure 4 :Figure 5 :Figure 6 :

 456 Figure 4 : Evolution of the absorbance of MWNT2-SDS suspension in function of the initial mass ratio MWNT2/SDS

Figure 7 :Figure 8 :

 78 Figure 7 : Evolution of the mass of the collected particles in function of the initial mass ratio SDS/MWNT for MWNT-SDS suspensions

  measured is too close to the amount of hydrogen in the SDS. Moreover, TGA analysis have been also used to quantify the proportion of MWNT2 and SDS in the dried MWNT2-SDS suspensions. Under inert atmosphere, MWNT are very stable until 600 °C whereas SDS decomposes significantly after 200 °C. Consequently, the height of mass loss above 200 °C can be related to the mass fraction of SDS. All the TGA curves obtained for MWNT2-SDS suspensions are shown in Figure 9.

Figure 9 :

 9 Figure 9 : TGA analysis of MWNT2-SDS suspensions (5 °C/min, Ar)

Figure 10 :Figure 11 :

 1011 Figure 10 : Evolution of the mass fraction of SDS in function of the initial mass ratio SDS/MWNT for MWNT-SDS suspensions

Figure 12 :Figure 13 :

 1213 Figure 12 : Evolution of the suspended MWNT concentration in function of the initial mass ratio Surfactant/MWNT

Figure 14 :Table 5 :

 145 Figure 14 : First order Raman spectra of pristine MWNT3

Table 1 .

 1 TM provided by Nanocyl®, MWNT provided by US Research Nanomaterials® and MWNT GCM331 provided by Carbon Nanotubes Plus®. The manufacturer's specifications of the MWNT are grouped in For ease of understanding, MWNT will be referred MWNT1, MWNT2 and MWNT3 respectively.

	Suppliers	Synthesis method	Purity (wt.%)	Length (μm)	Outside diameter (nm)	Surface Area (m²/g)
	MWNT1 (Nanocyl®)	CCVD	> 90%	1 -2	9 -10	250 -300
	MWNT2
	(US Research	CCVD	> 95%	10 -30	10 -20	> 200
	Nanomaterials®)
	MWNT3
	(Carbon Nanotubes	CCVD	> 98%	10 -30	10 -20	> 150
	Plus®)

Table 1 :

 1

	X-

Manufacturer's specifications of the MWNT The 3 surfactants used are SDS (Fisher Scientific ®), CTAB (Serva®) and Triton

Table 3 :

 3

		0.46 ± 0.05	0.30 ± 0.03	0.51 ± 0.02	0.40 ± 0.02	0.43 ± 0.02
	C 𝑀𝑊𝑁𝑇 𝑚𝑎𝑥 (𝑠𝑢𝑠𝑝𝑒𝑛𝑠𝑖𝑜𝑛) C 𝑀𝑊𝑁𝑇 𝑚𝑎𝑥 (𝑀𝑊𝑁𝑇2-𝑆𝐷𝑆)	1 ± 0.11	0.13 ± 0.01	2.22 ± 0.09	0.87 ± 0.04	0.93 ± 0.04

Ratio of maximum concentrations of MWNT for the different suspensions according to the Beer Lambert law

In this Table

3

, only the ratio of maximum MWNT concentrations is presented. Actual concentrations of MWNT are determined by the following experimental method. Further

Table 4

 4

	resumes the maximum MWNT

Table 4 :

 4 Maximum MWNT concentrations of the different MWNT suspensionsThe evolutions of the computed MWNT concentration are therefore consistent with the absorbance of the suspension measured by UV-Vis spectroscopy. MWNT3-SDS suspensions are the most concentrated suspensions. The ratio of the maximum MWNT concentration of MWNT3-SDS suspension to MWNT2-SDS equals to 1.64 ± 0.11, which is relatively close to ratio computed by UV-Vis spectroscopy (2.22 ± 0.09). Moreover, the suspensions elaborated with MWNT2 have approximately the same maximum concentration despite the fact that anionic, cationic and non-ionic surfactants were used. Therefore, for such highly concentrated MWNT suspension, this result indicates effectively that the stabilization effect would be mainly a steric one. According to the concentrations computed, the extinction coefficients of MWNT2

  Table 5 resumes the ID/IG ratios (average of five acquisitions) of as received MWNT and MWNT after US treatment with optimized MWNT/SDS mass ratio. Firstly, it can be noticed that the ratio ID/IG only slightly increases after the US treatment, which is typical of non-covalent functionalization.Regarding the number of defects from one batch to another, a correlation can be established between the maximum achievable MWNT concentration and the quality of the structure of the MWNT. Whether on pure or suspended MWNT, MWNT3 presents the lowest ratio ID/IG and the highest maximum suspended MWNT concentration while MWNT1 presents the highest ID/IG and the lowest maximum suspended MWNT concentration. Actually, defects may disrupt the adsorption of SDS molecules on MWNT surface. Indeed, according to some reports, SDS molecules are adsorbed on the surface of the CNT by following the graphite network of CNT and forming supramolecular structures[START_REF] Richard | Supramolecular Self-Assembly of Lipid Derivatives on Carbon Nanotubes[END_REF]. As defects detected by Raman spectroscopy are misarrangement of carbon atoms in the graphite network, one may think that defects disrupt the supramolecular organization of SDS around CNT. More generally, defects may disrupt organization of all kind of surfactants on CNT surface. Consequently, presence of defects lower the amount of surfactant that may adsorb on MWNT. With less surfactant, the repulsive interactions between MWNT decrease and thus the maximum concentration of MWNT in suspension.