Carlo Rovelli

A critical look at strings

Following the invitation of the editors of Foundations of Physics, I give here a personal assessment of string theory, from the point of view of an outsider, and I compare it with the theory, methods, and expectations of my own field.

I. INTRODUCTION

"Perché i discorsi nostri hanno a essere intorno al mondo sensibile, e non sopra un mondo di carta." Galileo Galilei, Dialogo sopra i massimi sistemi I am not an expert on strings. I follow the results announced as main string achievements, but I have not worked in the field. I have therefore much hesitated before accepting the invitation by Foundations of Physics to express a view on the theory. I have eventually decided to accept, in the hope of giving a contribution to the overall debate on the theory, because a central problem addresses by string theory is also addressed by the research direction in which I work.

For a considerable number of years, strings have represented a huge intellectual investment, aiming at a complete theory capable to describe the world at the elementary level, including quantum gravity. Today, the problem is obviously not yet solved. String theory is incomplete, far from describing precisely our real world, and its foundation is poorly understood. But the difficulties of a similar task are arduous and advances are necessarily slow. Strings provide tantalizing hints, partial answers, intriguing mathematical tools, and the tentative architecture of a grand overall picture to solve the problem.

In such a situation it is hard to evaluate string theory in isolation. An evaluation can only be made by comparing the theory with alternative research directions. Following the indications of the editors of Foundations of Physics, I try here to asses the results of string theory by comparing them with results and methods of my own field of research. I hope that this can contribute to put the results of string theory in perspective, and seeking a sober evaluation of the relative merits and the relative potential of different research directions, in a field where the final answer is not yet known.

String theory has recently been evolving into a toolbox, with tentative applications to fields such as QCD, strongly interacting fluids, or pure mathematics. I will not comment on the interest of string theory for these fields. This should be evaluated by QCD theorists, condensed matter physicists, or mathematicians. I focus on the motivating claim of string theory, which is to describe the real world beyond what is well accounted-for by the particle-physics standard-model and classical general relativity, and in particular to concretely describe the regimes where the quantum property of gravity cannot be neglected. This last problem is my own specific field of interest.

II. WHAT WE KNOW, WHAT WE DO NOT KNOW, AND WHAT IS THE PROBLEM

Before addressing merits and shortcomings of the tentative solution provided by string theory, let me briefly recall the problem on the table, which string theory means to solve. The Standard Model and classical general relativity are spectacular theories that have enjoyed an empirical success with few -if any-equal in the history of science. Today, these theories (with neutrino mass and cosmological constant) seem able to account for virtually anything we can measure, with the notable exception of dark-matter phenomenology. These are the currently established fundamental theories that summarize what we know about the physical world at the most elementary level we can access. But this set of theories does not allow us to compute what happens in all physical regimes, is patchy, and manifestly incomplete.

Specifically, if we want to compute the scattering amplitude of two point-particles interacting gravitationally, as a function of their center of mass energy E and impact parameter b, we can do so using (non-renormalizable) perturbative quantum general relativity as an effective theory, but predictivity breaks down when E becomes of the order of b in Planck units. Therefore current established theory gives no prediction whatsoever on what happens to particles that scatter at that energy. Such lack of predictivity is particularly relevant in important physical situations, such as early cosmology, some aspects of black hole physics, and our understanding of the shortscale structure of physical space. We need new theory to understand this physics.

Furthermore, there are major theoretical and conceptual shortcoming of the current theory. Ultraviolet divergences appear to indicate that there is something important we miss at short scale. General relativity is a beautiful theory with a tight formal structure and a minimal number of free parameters, but the Standard Model is a patchwork with a number of free parameters that calls for an explanation. The conceptual structure of the Standard Model includes aspects (fixed spacetime, global Poincaré invariance, local energy conservation...) which play a major role in dealing with its quantum properties, but are profoundly different from that of general relativ-ity (dynamical spacetime, no global Poincaré invariance, no local energy conservation, general covariance...): if we want a coherent picture, we need a way to combine the two. In particular, if the central physical tenet of general relativity is correct, namely if the geometry of physical space is a physical field, then the quantum character of this field imply that physical geometry is "quantum geometry". What is quantum geometry? Can we find a complete and consistent theoretical picture where these issues are resolved?

There are two problems raised by this situation. The first is to complete the picture and make it consistent. This is the problem of quantum gravity, since what is clearly missing is the understanding of quantum gravitational physics. A second, distinct, problem, is unification, namely the hope of reducing the full phenomenology to the manifestation of a single entity. (QCD completes the standard model and is consistent with electroweak theory, but is not unified with it.) String theory is an attempt to solve the two problems at once, namely to provide a quantum theory of gravity within a unified picture (a "final theory").

Below, I discuss the extent to which these problems are solved by the current state of string theory, in comparison with other approaches. I focus on the approach to quantum gravity in which I work, loop quantum gravity. (Recent reviews of string theory to which I am particularly indebted are [START_REF] Mukhi | String theory: The first 25 years[END_REF], and [START_REF] Blau | String theory as a theory of quantum gravity: A status report[END_REF] which focus on strings as a theory of quantum gravity. A recent overview of loop gravity, with relevant references is [START_REF] Rovelli | Loop quantum gravity: The first 25 years[END_REF], a technical introduction is in [START_REF] Rovelli | Zakopane lectures on loop gravity[END_REF].) It is important to stress upfront, however, that the problems addressed by strings and loops do not coincide. Both theories aim at a quantum theory of gravity in order to complete the current theoretical description of the world and and make it coherent, but string theory assumes the working hypothesis that this can only be achieved in the context of a unified theory, capable of addressing also questions that are outside the scope of loop gravity (such as: why these matter couplings and not others? why four dimensions? what is the final theory of nature? and so on.)

III. ULTRAVIOLET FINITENESS

A major achievement of string theory is the control of the ultraviolet divergences of conventional quantum field theory. An actual proof of ultraviolet finiteness is still lacking. At least, I have searched and much asked around, but I have not yet been able to find a reference with such a proof. But string theorists appear to be convinced of finiteness, and I believe them.

Indeed string theory provides intuitive ways for seeing how singularities are resolved. When point-particles scatter at very high energy, the stringy degrees of freedom "open-up", effectively spreading over a finite spacetime region, smoothing out the interaction region. From a different perspective, in the perturbation expansion the Riemann surfaces corresponding to high momentum are the "thin" ones, but a modular transformation relates such surfaces to non-degenerate ones, effectively avoiding the ultraviolet regime. The reason for the stringy ultraviolet finiteness can therefore be traced to the very hypothesis at the basis of string theory, namely the existence of an infinite number of degrees of freedom besides the ones we see, defining extended elementary objects. These can smooth out the standard quantum field theory divergences.

These result can be compared with the ultraviolet finiteness of loop quantum gravity. Here the proof of ultraviolet finiteness is straightforward. 1 Unlike string theory, the physical ground for the ultraviolet finiteness of loop quantum gravity does not require any additional input beside general relativity and quantum mechanics, and it can be understood as follows. Since the geometry of space is a quantum field, it has quantum properties. In particular, the spectrum of the geometrical quantities can be computed and turns out to be discrete. At short scale, spacetime geometry is therefore effectively discrete, in very much the same manner in which the energy of a harmonic oscillator is discrete. Therefore there is literally "no room" for ultraviolet divergences in the theory: there is no short-distance beyond the Planck scale. This becomes manifest in computing transition amplitudes, where one sees explicitly that high momenta are cut off at the Planck scale. 2Ultraviolet finiteness is therefore a main achievement both for string theory and for loop gravity. But it is realized differently in the two theories. In both cases, it reflects an intrinsic physical limitation in measuring distances shorter than the Plank scale. But in the first, it follows from a novel hypothesis about Nature; in the second, it is a direct consequence of quantum theory and general relativity.

IV. QUANTUM GEOMETRY

The peculiar features of general relativity, and in particular its large gauge symmetry, have followed a curious fate along the evolution of string theory. To start with, general relativity was treated as a conventional field theory in the string context. It was considered the effective low-energy manifestation of something else, like Fermi weak-interactions theory, and dealt with using basic quantum field theoretical "non-general-relativistic" tools: expanding around a fixed background, relying heavily on global Poincaré invariance and so on. That is, relying on notions that are at odds with the symmetry of general relativity. In the hands of theoreticians mostly coming from the particle-physics tradition, general relativity was treated in a way that appeared to general relativists to betray its central physical ideas. But the peculiar features of general relativity and of the quantum aspects of spacetime have eventually resurfaced and are playing an increasingly important role in string theory today.

The main effect of the large symmetry group of gravity, and the main teaching of general relativity, indeed, is that the world is not a given spacetime over which dynamical degrees of freedom evolve. Rather, spacetime itself is a dynamical entity. In a quantum theory, spacetime itself is a quantum entity, whose structure cannot be assigned a priori. A number of developments of string theory aim at coming to terms with this deeply unconventional and novel aspects of the world that is directly implied by the physics of general relativity and quantum theory. For instance, with states that have no natural continuum spacetime description (say, a vacuum which is a tensor product of conformal field theories). Similarly, the difficulties of defining local bulk observables in a general relativist context which have long been discussed in the quantum gravity literature are now being increasingly discussed in the string literature.

Of course, much physics can be derived by choosing a background and computing around it. A state of a background-independent theory constitutes a background, and physics around that state will be -obviouslybackground dependent. There is no conflict between this background dependence about a chosen ("vacuum") state, and fundamental background independence, any more than there would be between quantum electrodynamics and its expansion about the field of an atomic nucleus. The difficult problem is another one: whether the full definition of the theory, and in particular the characterization of its degrees of freedom, requires a background to start with or not. For instance, if we reinterpret general relativity as the theory of small fluctuations around a fixed space-time, we lose most of the interesting phenomena it predicts, such as the Schwarzschild solution, the dynamics of the universe, black holes's horizons, and so on. In other words, the problem is whether or not we have a quantum theory with a clear definition of a state space capable of listing all possible background states.

There have been numerous beautiful attempts to find this fully background independent formulation of string theory, such as string field theory, matrix theory, holography... But full background independence of string theory is not yet properly understood.

The way this fundamental issue is addressed in string theory is often indirect. For instance, attempts are made to describe the bulk quantum geometry of spacetime by using the ADS-CFT conjecture, thus trying to describe what we do not know (quantum gravity) in terms of conceptual tools that we control (flat-space quantum field theory on the boundary). Analogously, the string theory calculations of black hole entropy exploit the relation between the strong-coupling genuinely-gravitational regime of interest, and the weak-coupling regime where conventional flat-space tools can be used, and states can be counted. Again, string cosmology often addresses the highly non-Minkowskian geometry of early cosmology by an hypothesis, that sounds bizarre to relativists: an overall larger Minkowski space where everything happens. In all these cases, instead of addressing the real problem, which is to learn how to do physics where background spacetime plays no role, the strategy is to try to circumvent the problem, bringing back the calculations to the familiar pre-general-relativistic conceptual framework. The reason of this, of course, is not lack of imagination or courage from string theorists. String theory gives glimpses and hints of how a genuine theory of quantum geometry could be, with general states having no Riemannian spacetime interpretation at all -like a general state of a quantum particle is not necessarily similar to a classical localized particle-but for the moment it is far from providing a complete coherent picture of quantum geometry.

This must be compared with the picture of quantum geometry offered by loop gravity. Contrary to the string case, loop gravity addressed upfront the problem of describing the fundamental degrees of freedom of a theory without a fixed background spacetime. The result is that everything is conceptually clear, fully general relativistic, and well defined. There is a Hilbert space, whose states have a clean interpretation as quantum states of the geometry. These do not live over a background, but themselves build-up spacetime. The quanta of the theory are "quanta of space", quantum bricks that build up spacetime. The mathematics of quantum geometry is clear at the level of mathematical physics, as well as at the conceptual physical level. A formalism for computing well defined background-independent observables, as well as perturbing around a given background, is known. These techniques are perhaps unfamiliar to many, and might look strange at first sight, but so did string theory for many years, before becoming fashionable.

It seems to me that the clarity of the picture of quantum geometry is definitely a plus for loop quantum gravity that lacks in string theory.

V. OVERALL PICTURE

The beauty of string theory, on the other hand, is that it offers a tentative overall picture capable of bringing together in a natural and compelling way so many aspects of the world. It provides an ultraviolet consistent the-ory of gravity and at the same time has natural room for gauge symmetry, unification, holography, all fused in an interrelated net that suggest the existence of a compelling overall architecture. Even not working in the field, one cannot fail to appreciate the tantalizing aspects of the relations unraveled by the string research. It is very tempting to believe that beyond all these relations there should be a remarkable coherent edifice.

The difficulty is that for the moment we see only bits and pieces of the hypothetical complete edifice. In particular, we do not see the foundations: the basic degrees of freedom and the basic equations. The sentiment that this beautiful underlying theory should exist is strong among the people immersed in string theory, and is reinforced by the discovery of the beautiful relations -dualities-relating the different bits. It is difficult for an outsider to fully appreciate the support of this sentiment, but string theorists appear to be convinced of the existence of the underlying theory. They might be right, but the fundamental theory, if it exists, is still outside our control. Until we see it, its beauty and its physical consistency are hypothetical.

This can be compared with loop quantum gravity. The scope of the theory is much narrower, because the theory does not pretend to be a unified theory, does not select the matter couplings, and does not aim at being the final theory of the world. But the elementary degrees of freedom, which are the quanta of space, or, equivalently, the quanta of gravitational field, are clearly defined. The basic operators are well defined. The dynamics can be compactly presented with three equations. The overall structure of the theory is complete and simple.

Loops and strings differ in another key respect. Strings are based on a definite physical hypothesis: elementary constituents of the world are extended objects. The hypothesis might be right. Or wrong. The world might not be supersymmetric and 10 dimensional.

Loop gravity, on the other hand, is grounded in quantum theory and in the symmetry underlying general relativity, a symmetry today generally expected to survive at high energy. Loop gravity is just a general covariant quantum field theory, with degrees of freedom reducing to Riemannian geometry at low energy. Loop gravity can very well turn out to be wrong as well, of course. But if the theory is wrong, it must be so for some more subtle reason, which, in any case, would still teach us something about the quantum world at the general covariant quantum level.

Of course, assuming that the basic physical tenets of general relativity and quantum theory remains valid at the Planck scale is an extrapolation. But extrapolation has always been the most spectacularly effective tool in science. Maxwell equations, found in a lab, work from the atomic to the galactic scale. Up to contrary empirical indications, always possible, a good bet is that what we have learned may continue to hold.

VI. DESCRIBING THIS WORLD

Let me now come to what I see as a serious shortcoming of string theory. The interest in the theory exploded around 1985, when E8 × E8 and the heterotic string appeared to be the unique viable option, and the low-energy field theory of such a string, compactified to 4 dimensions on a suitable class of 6-dimensional manifolds, was shown in a classic paper of Candelas, Horowitz, Strominger and Witten [START_REF] Candelas | Vacuum Configurations for Superstrings[END_REF] to yield qualitatively correct phenomenological properties, including parity violation. The central promise of that paper and the hope it raised was that a realistic string theory incorporating and generalizing the Standard Model, plus gravity, was round the corner. I think that one can safely say today, in hindsight and despite the defining historical role played by the paper, that the hope grounding that paper, which sparked all that interest, was misplaced.

String theory would be in a stronger position, if it could exhibit a mechanics yielding the SU (3) × SU (2) × U (1) gauge group, the particle content of our world, the three generations, no supersymmetry at our scale and so on. Understanding this was its original aim. So far, string theory fails to describe our world as see it. It describes, instead, lots of worlds, in all sort of higher dimensions, generally with cosmological constant having the wrong sign, with "microscopical" internal spaces of cosmological size, and so on. This is a beautiful theoretical world, with marvels and surprises, but where is our world in it? Until the description of our world is found in this immense paper edifice, it seems to me that caution should be maintained.

This can be compared with the situation in loop gravity, or with other approaches to this problem, that might shed light on some of these issues, like for instance Alain Connes's non commutative geometrization of the Standard Model [START_REF] Chamseddine | Universal formula for noncommutative geometry actions: Unification of gravity and the standard model[END_REF]. Again, loop gravity does not pretend to provide a unified picture of nature, to tell us what is the matter content of the universe, or to determine the number of dimensions of spacetime. But the theory is compatible with a description of the world as we see around us: four dimensions, no supersymmetry, fermions and a certain Yang-Mills gauge group. Like all successful physical theories developed so far (QED, QCD, or general relativity) it is compatible with unphysical couplings. The ambition of loop gravity is not to solve all problems of physics and provide the final theory. Its ambition is to provide a consistent theory of quantum gravitational phenomena, coupled with the matter that we find (with experiments) in the world.

On the other hand, scattering calculation around the Minkowski background in loop quantum gravity are being developed, but they are in a far more primitive stage than the scattering calculations one can do with string theory. Also, in the last couple of years, loop gravity has seen the development of an explicit formulation of the theory that includes fermions and Yang-Mills fields, and of a technique to compute scattering amplitudes around the Minkowski background. But these developments are recent, and the results are preliminary.

VII. UNIFICATION

There is one issue in which string theory appears to be in a definitely better position than quantum gravity. This is the issue of unification, where loop gravity has nothing to offer.

I think that it is important to emphasize the fact that the unification of the forces and the quantization of gravity are two conceptually distinct problems. The first is the old dream of having a single theory explaining everything. The second refers specifically to the present inconsistency between general relativity and the standard formulation of quantum field theory, and is a problem that has to be solved in order to have a coherent theory of the world. Solving the second does not necessarily imply solving the first: the quantum theory of the gravitational field can in principle be found without addressing the unification problem, like the quantum theory of the strong interactions has been understood and found without solving the problem of unifying them with the electroweak forces.

The idea is often put forward that the problem of quantum gravity can only be solved together with the unification problem. There are hints that this might be the case. The running of the Standard Model coupling constants appears to converge not too far from Planck scale, fermions and boson divergences tend to cancel in supersymmetric theories, and so on. In the history of physics, often two major problems have been solved at once, and the temptation to do the trick again is reasonable.

But even more often, however, hopes to solve two problems at once have been disappointed. When I was a student, the idea that the theory for the strong interactions could only be found by getting rid of renormalization theory at the same time, was an unquestioned mantra, repeated by everybody. It turned out to be wrong. There are standard arguments against the possibility of finding a consistent quantum theory of general relativity alone. But these arguments hold in the context of standard local field theory, where fields operators are defined on a spacetime metric manifold. They are all circumvented by loop gravity by moving up to the proper context of a general-covariant quantum field theory.

Loop quantum gravity is a theory of quantum gravity that does not address the unification problem. It is like QED, or, more precisely, QCD: a quantum field theory for a certain interaction, which can be coupled to other interactions (affecting them), but is consistent by itself. The philosophy underlying loop gravity is that we are not near the end of physics, we better not dream of a final theory of everything, and we better solve one problem at the time, which is hard enough.

Back to the unification problem, does string theory actually solve it? Closed and open strings describe gravity and gauge theory. More than that, they can even be shown as two sides of the same physics, under certain conditions. This is very compelling. String theory definitely provides a unified picture in which gauge theory and gravity live together, and the nineteen or so parameters of the Standard Model are replaced by a single fundamental parameter. This is a strong plus.

But the initial objective of unification was far more ambitious. It was to understand what is beyond the Standard Model in order to be able to compute the value of the free parameters of the theory, in the same manner in which the Shrödinger equation allows us to compute chemical or condensed-matter parameters from fundamental constants. There is no computation of the Standard Model parameters from string theory. Nor a solution to the other puzzles in the theory: why is the cosmological constant so small? What is the origin of the three families? Can we give a better account of symmetry breaking? Little concrete physics has emerged from the theory so far. The results expected from a true unification do not seem to me to be there.

VIII. APPLICATIONS

Black holes thermodynamics is definitely a success of string theory, and in my opinion, the strongest evidence for its physical relevance. A similar success can be claimed by loop gravity. Both successes are partial in my opinion. The string derivation is still confined to, or around, extreme situations, as far as I know, and since it is based on mapping the physical black-hole solution into a different solution, it fails to give us a direct-hand concrete understanding of the relevant black hole degrees of freedom, as far as I can see. The loop derivation of black hole entropy gives a clear and compelling physical picture of the relevant degrees of freedom contributing to the entropy, but it is based on tuning a free parameter to get the correct Bekenstein-Hawking entropy coefficients, and this does not sound satisfactory to me either.

The crucial application to both strings and loops will probably turn out to be cosmology. This is the most likely domain where a window of opportunity for testing the theories might open. Loop cosmology is the most spectacular success of loop quantum gravity. The theory elegantly resolves the big bang singularity and predicts a sort of "bounce" from a previously contracting phase. When a collapsing universe reaches Planck-scale density its wave function opens up into a genuinely quantum state where classical space and time are ill defined. The quantum equation of the theory continue to hold, and the evolution can be studied across this non-classical region into a new expanding universe. This is similar of the picture of an electron falling straight into a Coulomb potential: the classical trajectory falls into the singularity. But the classical trajectory becomes ill-defined in the quantum evolution of the corresponding wave packet. Spacetime is ill-defined around the big bang like the classical trajectory of the electron around the center of the Coulomb potential.

The full quantum gravity effects are nicely summarized into an effective Planck-scale term that modifies the Friedmann equation, and an effort is under way to explore eventual testable consequences. Furthermore, inflation appear to be generic in this picture. The picture is simple, physically compelling and based only on standard general relativity and quantum mechanics, empirically well established physical inputs.

String cosmology is much developing as well, in a number of variants. The ability of the string to effectively resolve singularities and the possibility of topology change potentially provides important inputs to cosmology. I might be wrong, and this is vague, but for my general relativist formation, however, many concrete scenarios proposed by string theory to describe the big bang, in particular some brane cosmologies with configurations of branes in a background space-time, do not sound physically very plausible to me, compared to the clean simplicity of the loop-cosmology scenario.

Finally, string theory techniques may have potential applications to other domains of physics. These are very interesting, but in no way they testify in favor of the relevance of string theory for the fundamental interactions. Enormous intellectual investments have gone into string theory in the last decades and it would be strange if all the theoretical technology developed did not turn out to be good for something. Theoretical physics is pretty coherent and techniques developed in one field often turn out to be helpful elsewhere, irrespective of their success in the first place. After all, if string theory turned out to be useful for QCD, it would, in a sense, finally fulfill the aim for which its ideas were conceived at its very early initial stage, when Gabriele Veneziano wrote the dual amplitudes to describe strong interactions. To some approximation, there certainly are strings in the real world: the flux tubes of a confining gauge interaction.

IX. PREDICTIONS

Finally, although this should have probably been the first section, the main shortcoming of string theory is definitely its failure, so far, to produce any concretely verifiable physical prediction. To be sure, string theory has provided numerous "predictions", like short scale modifications of the gravitational force, black holes at CERN, dielectron resonances, or the existence of supersymmetric particles at low energy, but so far all these "predictions" have been falsified by observation. The theory has survived these failed predictions, because they were not solid predictions, but only hints of possibilities, effects compatible with the theory, but not necessary consequences of the theory. The real problem is that the theory does not appear, so far, to have any verifiable necessary consequence at accessible scales.

A burning difficulties is of course the landscape prob-lem. If there is an accurate string description of the real world, then there are probably so many of them to make the discovery of the right one virtually impossible and in any case devoid of predictive power.

In my opinion, this is serious. A physical theory that does not give predictions is not a good theory. We need definite predictions, like those that all good physical theories of the past have been able to produce.

Sometimes the strategy of saying "so is the world, we have to live with this", is put forward. I find this strategy unconvincing. Such a strategy would be questionable even if string theory had already proved itself as a physically correct theory of the world. But concluding that fundamental physics cannot anymore make definite predictions, just because a hypothetical theory turns out to be too weak to be predictive, is mistaking hypotheses for consequences.

As far as clear verifiable predictions are concerned, loop quantum gravity is in no better shape either. There are no experiments supporting loops, nor any other quantum theory of gravity. The simple question I have emphasized in the introduction -what is the scattering amplitude for two particles interacting gravitationally with a center of mass energy of the order of the impact parameter in Planck units?-does not have a clear answer yet, neither from strings nor from loops. Therefore the above condemnation of string theory applies equally to all other approaches to the problem of quantum gravity.

The closest to a verified prediction in the domain, as far as I know, comes from the poset approach to quantum gravity, which indicated the correct order of magnitude of the cosmological constant before its measurement [START_REF] Ahmed | Everpresent lambda[END_REF]. In this particular regard, string theory features particularly badly: not only it failed to predict a positive cosmological constant, but the very introduction of a positive cosmological constant appears to be at least problematic for the theory.

X. "IT DOES NOT WORK, THEREFORE LET'S DEVELOP IT FURTHER"

I think that the problem of describing our physical world at the elementary level beyond current established theories is open. String theory is one of the research directions among others aiming at solving this problem, with points of strength and weakness. Its main strength is its mathematical construction where gauge fields, fermions and the gravitational field can be seen as parts of an overall coherent construct. The theory has not delivered what it seemed to be almost there twenty years ago: a finite theory where the fundamental degrees of freedom are clearly identified, capable of describing our own world, with three fermion families, the SU (3) × SU (2) × U (1) gauge groups, the values of parameters of the standard model computable, and (we should add today) a small positive cosmological constant. The tentative predictions of the theory have so far been falsified. The development of the theory has constructed a toolbox that can perhaps be used in other contexts, but for the moment does not appear very effective for producing concrete results for high-energy physics. The picture of quantum geometry offered by the theory is still very unclear.

There has been a tremendous theoretical investment on strings, by far unmatched by alternative research directions, there have been successes, string revolutions and excitement. Seven years ago, I wrote a playful "dialog" to point out what I saw as the theory's shortcomings at the time [START_REF] Rovelli | A dialog on quantum gravity[END_REF]; reading the dialog today, it seems to me that those same difficulties are still open. Contrary to this, a theory like loop gravity has developed because the key problems open seven years ago have since been solved.

There is a compelling logical evolution that has lead from the particle-theory successes to strings. This path has been characterized by a sequence of spectacularly successful predictions (antiparticles, neutral currents, W and Z, various quarks, just to mention some) which at some point has turned into a sequence of spectacularly failed predictions (grand unified theories predicted proton-decay at 10 31 years, Kaluza-Klein theory predicted an observable scalar field, strings suggested effects of extra dimensions, supersymmetry has been "on the verge of being seen" year after year, . . . ) I think that we should keep in mind the possibility that a wrong turn might have been taken at some point along this path.

In recent years, various theories have developed following the logic: "it does not work, therefore let's develop it further". Perseverance may pay (it worked with Yang-Mills theories), but at a risk: a theory can grow on its own failures, enriching its structures to cover previous insuccesses.

There is certainly much beauty in strings. But beautiful ideas have turned out to be wrong in science, even ideas developed by large groups of scientists. (In the words of a quote attributed to Thomas Henry Huxley: "Science is organized common sense where many a beau-tiful theory was killed by an ugly fact".) The history of quantum gravity is particularly sprinkled with great hopes disappointed. I remember as a young student sitting in a major conference where a world-renewed physicist announced that the definitive theory of quantumgravity-and-everything, had finally been found. There were a few skeptics in the audience, regarded as zombies by the majority. Today most of us do not even remember the name of that "final theory". Or worse, we can think of more than one possibility . . . It is obviously not my intention to suggest that research in string theory should not be vigorously pursued. String theory is a spectacular intellectual achievement and it might well turn out to be the right track. It is a rich and elaborate theory, that deserves to be studied further, with the resolute aim of arriving at assessing its physical viability. If all the hopes of the string community are realized, it is a triumph.

But I think it would be a mistake to consider string theory as an established result about nature and therefore concentrate the attention solely on it. Also if the hopes of other research directions are realized, it would be a triumph. String theory appears of unmatched beauty to string theorists, but other ideas appear of unmatched beauty to others. What I think is important is to keep in mind that these theories are provisional.

I am not pessimistic. Major problems like the ones we are facing have sometimes resisted for a while in the history of physics, but a solution has generally been found eventually. The issue is open. I think that different path must be pursued. Completeness, internal consistency, full agreement with known low-energy physics, simplicity, and, ultimately, experience, will tell.

Infrared finiteness can be proven as well in an appropriate version of theory. See the review articles quoted above for specific references.

2 Geometry discreteness does not break Lorentz invariance in loop gravity because it is quantum mechanical. Eigenvalues do not transform continuously under a continuous symmetry. (The /2 eigenvalue of the angular-momentum component Lz does not transform continuously under rotation.) A boosted observer does not measure Lorentz-contracted discrete lengths: he measures a continuously-deformed probability distribution for the same spectrum of lengths. The hard-to-die idea that the existence of a minimal length at the Planck scale necessarily breaks Lorentz invariance is plainly wrong.

-I thank two anonymous referees for very useful criticisms and Matthias Blau for a detailed reading of the paper and a useful conversation. Thanks to Sal, a bit grown up and still without a job, for comments and suggestions.


Randy Mujica

Anusree Augustine

Matthias Pauly

Vincent Le Houerou

Gero Decher

Yann Battie
email: yann.battie@univ-lorraine.fr

Olivier Felix
email: olivier.felix@ics-cnrs.unistra.fr.

Macroscopic mapping of the linear in-plane anisotropy of nanocellulosic thin films by Mueller matrix polarimetry

Keywords: Layered structures (A), Nanocomposites (A), Anisotropy (C), Non-destructive testing (D), Birefringent thin films

The present work focuses on the characterization of the macroscopic optical anisotropy of cellulose nano-composite multilayer thin films prepared by grazing incidence spraying. The polarization properties of such films composed of in-plane aligned cellulose nanofibrils embedded in a polyelectrolyte multilayer are investigated by Mueller matrix polarimetry (MMP). Linear birefringence is the predominant polarization property resulting from the unidirectional organization of the cellulose nanofibrils, thus, the differential decomposition of the Mueller matrix measured on the films allows the determination of their actual structural birefringence, 〈Δn〉 = 0.032. Moreover, modeling the sample as an effective medium provides a parametrized expression that describes its linear birefringence as a function of the optical constants of the components, the volume fraction of nanofibrils, the distribution of orientation, and the film thickness. Therefore, MMP provides a fast, simple, and non-destructive method to characterize both the structural characteristics and the optical birefringence/dichroism of the films over large areas, which is of great interest for various optoelectronic applications.

Introduction

Over the past decades, there has been a great interest in developing materials possessing highly anisotropicoptical, electronic, and mechanical properties for various potential applications [START_REF] Xia | One-Dimensional nanostructures: synthesis, characterization, and application[END_REF][START_REF] Kim | Anisotropic optical film embedded with cellulose nanowhisker[END_REF][START_REF] Nan | Dissipative self-assembly of anisotropic nanoparticle chains with combined electrodynamic and electrostatic interactions[END_REF][START_REF] Guo | Quasi-1D ZrS3 as an anisotropic nanoreflector for manipulating light-matter interactions[END_REF]. The direction-dependent physical properties of these materials are based on the anisotropic nature of their constituents and their in-plane alignment within interfacial matrices. For example, the performance of optical components such as wave plates, polarizers, or multilayer mirrors mostly depends on splitting an incident light ray into two refracted rays (birefringence phenomena), which is caused by the optical anisotropy of the constituting material. Several methods have been adopted for analyzing the optical anisotropy of materials either in bulk or in thin films, including polarization-resolved reflectance and transmission spectroscopy [START_REF] Amani | Solution-synthesized high-mobility tellurium nanoflakes for short-wave infrared photodetectors[END_REF], polarization imaging [START_REF] Tong | Stable mid-infrared polarization imaging based on quasi-2D tellurium at room temperature[END_REF], or angle-resolved polarized Raman spectroscopy [START_REF] Wang | Field-effect transistors made from solution-grown two-dimensional tellurene[END_REF]. Recently, Mueller Matrix Polarimetry (MMP) has also been exploited for the highly accurate determination of the optical constants of systems exhibiting giant optical anisotropy such as transition-metal dichalcogenides (TMDCs) [START_REF] Hou | Birefringence and dichroism in quasi-1D transition metal trichalcogenides: direct experimental investigation[END_REF] and tellurene [START_REF] Guo | Giant in-plane optical and electronic anisotropy of tellurene: a quantitative exploration[END_REF], both promising materials to replace the inorganic solids and liquid crystals currently used in optical components. Moreover, this technique has been also used to explore the optical birefringence of artificial designs in cellulose nanocrystalline nanocomposites [START_REF] Mendoza-Galván | Linear birefringent films of cellulose nanocrystals produced by dip-coating[END_REF][START_REF] Chae | Anisotropic optical and frictional properties of Langmuir-Blodgett film consisting of uniaxially-aligned rod-shaped cellulose nanocrystals[END_REF][START_REF] Li | Preparation of optical waveplates from cellulose nanocrystal nematics on patterned polydimethylsiloxane substrates[END_REF]. Indeed, MMP is a powerful method for investigating both the structural and optical properties of nanostructured films and the correlation between them [START_REF] Battie | Generalized effective medium theory to extract the optical properties of two-dimensional nonspherical metallic nanoparticle layers[END_REF][START_REF] Keita | Determination of the optical properties and size dispersion of Si nanoparticles within a dielectric matrix by spectroscopic ellipsometry[END_REF]. On one hand, the polarization states of the scattered light are sensitive to small subwavelength scatterers [START_REF] He | A possible quantitative Mueller matrix transformation technique for anisotropic scattering media[END_REF] such as nano-objects, thus, MMP can be conveniently used for the macroscopic investigation of the sample's internal microstructure [START_REF] Chashchina | Mueller polarimetric imaging for fast macroscopic mapping of microscopic collagen matrix remodeling by smooth muscle cells[END_REF][START_REF] Ushenko | Mueller-matrix mapping of biological tissues in differential diagnosis of optical anisotropy mechanisms of protein networks[END_REF]. On the other hand, the off-diagonal Mueller matrix elements of optically anisotropic materials vary with the in-plane rotation of the optical axis [START_REF] He | A possible quantitative Mueller matrix transformation technique for anisotropic scattering media[END_REF][START_REF] Guo | Complete dielectric tensor and giant optical anisotropy in quasi-one-dimensional ZrTe5[END_REF][START_REF] Yang | Ultraviolet-visible plasmonic properties of gallium nanoparticles investigated by variable-angle spectroscopic and Mueller matrix ellipsometry[END_REF], which can be exploited for the mapping of the local orientations present in a sample [START_REF] He | Mapping local orientation of aligned fibrous scatterers for cancerous tissues using backscattering Mueller matrix imaging[END_REF]. Cellulose nanofibers (CNFs) have been growingly investigated for the improvement of the mechanical, thermal, and optical properties of nanocomposite materials [START_REF] Kim | Anisotropic optical film embedded with cellulose nanowhisker[END_REF][START_REF] Chowdhury | Improved order parameter (alignment) determination in cellulose nanocrystal (CNC) films by a simple optical birefringence method[END_REF][START_REF] Diaz | Thermal expansion of self-organized and shear-oriented cellulose nanocrystal films[END_REF][START_REF] Riester | Nanoindentation with a knoop indenter[END_REF][START_REF] Wicklein | Thermally insulating and fire-retardant lightweight anisotropic foams based on nanocellulose and graphene oxide[END_REF][START_REF] Reising | Effect of particle alignment on mechanical properties of neat cellulose nanocrystal films[END_REF][START_REF] Merindol | Bio-inspired multiproperty materials: strong, self-healing, and transparent artificial wood nanostructures[END_REF][START_REF] Merindol | Assembly of anisotropic nanocellulose films stronger than the original tree[END_REF] owing to the combination of desirable properties such as low density, high strength, chirality, biodegradability, non-toxicity, broad chemical modifying capacity, and availability. Optoelectronic applications of cellulose derivatives are based on the positive uniaxial birefringence exhibited by the crystalline domains of the long polysaccharide chain of cellulose. This intrinsic birefringence is maximized by the parallel structural arrangement of the anisotropic cellulose nanoparticles into solid films, which have been most commonly achieved by the application of external magnetic, electrical, or mechanical fields [START_REF] Chowdhury | Improved order parameter (alignment) determination in cellulose nanocrystal (CNC) films by a simple optical birefringence method[END_REF][START_REF] Diaz | Thermal expansion of self-organized and shear-oriented cellulose nanocrystal films[END_REF][START_REF] Reising | Effect of particle alignment on mechanical properties of neat cellulose nanocrystal films[END_REF][START_REF] Cranston | Formation of cellulose-based electrostatic layer-by-layer films in a magnetic field[END_REF][START_REF] Bordel | Orientation of native cellulose in an electric field[END_REF][START_REF] Gindl | Cellulose in never-dried gel oriented by an AC electric field[END_REF][START_REF] Ebeling | Shear-induced orientation phenomena in suspensions of cellulose microcrystals, revealed by small angle X-ray scattering[END_REF][START_REF] Orts | Enhanced ordering of liquid crystalline suspensions of cellulose microfibrils: a small angle neutron scattering study[END_REF][START_REF] Hu | In-plane aligned assemblies of 1Dnanoobjects: recent approaches and applications[END_REF]. Recently, we introduced an alternative simple and scalable methodology for imparting unidirectional alignment to flexible CNF within monolayers, the so-called Grazing Incidence Spraying (GIS) technique: by spraying a CNF suspension at shallow spraying angles (<20• ) to the receiving surface during the deposition of Layer-by-Layer (LbL) films [START_REF] Decher | Fuzzy nanoassemblies: toward layered polymeric multicomposites[END_REF], the shearinduced in-plane alignment of the nanofibrils in multilayer systems is achieved, which yields birefringent films as confirmed by optical microscopy between crossed polarizers as well as by the in-plane distribution of the apparent refractive index measured with a standard spectroscopic ellipsometer [START_REF] Blell | Generating in-plane orientational order in multilayer films prepared by spray-assisted layer-by-layer Assembly[END_REF]. Both of these characterization techniques are sufficient to demonstrate the effective birefringent nature of the anisotropic samples. However, they do not provide an accurate determination of the actual polarization properties of the films. Moreover, the homogeneity of the optical properties is dependent on the fabrication method. In the case of the GIS-assisted preparation of the multilayer films, the diverging flowlines of the CNF suspension at the outlet of the spraying nozzle produce different shear flow conditions along the receiving surface that translate into a characteristic cone-like shape of the 2D alignment field of the nanofibrils [START_REF] Blell | Generating in-plane orientational order in multilayer films prepared by spray-assisted layer-by-layer Assembly[END_REF]. The macroscopic mapping of the resulting birefringence is, therefore, more meaningful for the development of thin films with anisotropic properties than any localized characterization. Herein, we use Mueller matrix spectroscopic ellipsometry for a complete characterization of the polarization properties (including the macroscopic mapping of the optical anisotropy) of a cellulosic model film prepared by the combination of GIS and LbL. An effective medium theory is also introduced to describe the linear birefringence as a function of the internal characteristic of the nanocomposite film.

Experimental

Materials

Carboxymethylated nanofibrillated cellulose (degree of substitution of 0.087) was provided as a pulp containing 2.26 wt % concentration in water by Innventia AB (Stockholm, Sweden). The pulp was mechanically dispersed in Milli-Q water to obtain a suspension at 2 g L -1 and sonicated with a sonic dismembrator (model 505, Fischer Scientific, USA) for 20 min at 30% amplitude. The suspension was then centrifuged for 1 h at 9500 rpm in a ROTINA 420R centrifuge (Hettich Zentifugen, Ger-many). The resulting supernatant was first filtered on cotton wool and then with 5 μm PVDF syringe filters Millex-SV (Merck, Germany). The final concentration was typically 1.4 g L -1 as determined from dry mass measurements. Poly(vinyl amine) (PVAm, Lupamin 9095, 20% in water, Mw ≈ 340 000 g mol -1 ) provided by BASF (Ludwigshafen, Germany) was used to prepare solutions at a concentration of 1 g L -1 and pH adjusted at 8 with a solution of 0.5 M NaOH. Poly(ethylene imine) (PEI, 50 wt % in water, Mw ≈ 750 000 g mol-1) purchased from Sigma-Aldrich (Lyon, France), was dissolved in Milli-Q water under bath son-ication to a final concentration of 2.5 g L -1 . (100)-silicon wafers used as substrates were obtained from WaferNet, Inc. (San Jose, CA, USA). Quartz slides (25 mm × 25 mm x 1 mm) were acquired from Agar Scientific Ltd. (Stansted, UK). The silicon wafers and quartz slides were cleaned in a 50:50 solution (v/v) of Milli-Q water:ethanol and sonicated for 15 min in an ultrasound bath, after which they were blown dry with compressed air and further cleaned/ activated for 3 min in a plasma cleaner (Harrick Plasma PDC-002, USA) at a high RF power (~30 W).

LbL deposition

All films were deposited on a substrate (silicon or quartz) coated with a precursor layer of PEI. Alternated layers of CNF and PVAm were sprayed using intermediate rinsing steps with Milli-Q water. GIS deposition was performed semi-automatically with the help of a spraying system made in-house. The colloidal suspension of CNFs was pumped by a liquid handling pump into a stainlesssteel air atomizing nozzle (1/4J by Spraying Systems Co., Glendale Heights, IL, USA), which was like wise supplied with compressed air. The sample was kept in a vertical position using a rubber stopper and a pulling vacuum. The spraying direction was always pointing downward and the angle of the nozzle to the substrate surface was kept fixed at 10°. The liquid flow rate was set to 1 mL min -1 , the airflow rate to 40 L min -1 , and the spraying time at 60 s. The PVAm solution and rinsing water were sprayed manually (orthogonal to the substrate) for 5 s and 10 s, respectively, with AIR-BOY spray bottles (Carl Roth GmbH, Karlsruhe, Germany). A control sample in which the nanofibrils were deposited without any preferential orientation was prepared by dip-assisted LbL. For this, the substrate was alternately immersed in beakers containing either the polycation or the polyanion solution for 10 min using three intermediate rinsing steps of 1 min each in Milli-Q water and a drying step with compressed air.

Sample characterization

Tapping mode atomic force microscopy (AFM) was performed on a Bruker AFM Dimension Icon (Bruker, USA) with a Nanoscope V controller and silicon cantilevers (resonance frequency 300 kHz, nominal spring constant of 42 N m -1 , and nominal tip radius of 7 nm). Phase and height imaging channels were recorded simultaneously using a scan rate of 1.0 Hz with a resolution of 512 × 512 pixels2. The thickness of the samples deposited on silicon wafers was measured on a spectroscopic ellipsometry SENpro (SENTECH Instruments GmbH, Germany) in the spectral range between 400 and 800 nm using an incident angle of 70°. An effective medium with a homogeneous refractive index (n = 1.559) was used to fit the experimental data. This methodology has been demonstrated to be sufficiently precise for CNF-based LbL films [START_REF] Merindol | Bio-inspired multiproperty materials: strong, self-healing, and transparent artificial wood nanostructures[END_REF][START_REF] Merindol | Assembly of anisotropic nanocellulose films stronger than the original tree[END_REF][START_REF] Blell | Generating in-plane orientational order in multilayer films prepared by spray-assisted layer-by-layer Assembly[END_REF]. Transmission Mueller matrix (MM) measurements were performed with a spectroscopic ellipsometer UVISEL (HORIBA, Japan; beam diameter about 0.8 mm) at normal incidence in the spectral range 270-800 nm (limits of the equipment). This ellipsometer only allows determining the first three columns of the Mueller matrix, the last one being obtained by symmetry provided that the depolarization of the sample is negligible [START_REF] Arteaga | Complete Mueller matrix from a partial polarimetry experiment: the 12-element case[END_REF]. This was verified in a dual-rotating compensator ellipsometer (beam diameter of 1 mm). This ellipsometer has accuracy on the elements of the Mueller matrix lower than that of the UVISEL ellipsometer by one order of magnitude. However, unlike the UVISEL ellipsometer, the dual rotating compensator ellipsometer measures the complete Mueller matrix of the sample. The accuracy of the depolarization parameter is estimated at 0.008. The mapping was performed in transmission at normal incidence with the UVISEL ellipsometer. For this, the sample stage was moved manually using a grid size containing 135 points separated by 1 mm each. The contour plots presented in this publication were smoothed by an XYZ interpolation based on a Thin Plate Spline (TPS) algorithm [START_REF] Donato | Approximate thin plate spline mappings[END_REF]. Transmittance spectra of LbL films deposited on quartz slides were measured on a Cary 5000 UV-Vis-NIR spectrometer (Agilent Technologies, Inc., USA) in the spectral range between 200 and 800 nm.

Results and discussion

Preparation of unidirectionally nanostructured CNF films

Anisotropic CNF-based monolayers and multilayers were prepared by combining GIS and LbL assembly. The preferential 2D alignment of the CNFs parallel to the direction of the grazing incidence spraying was determined by image analysis of the AFM images obtained for a CNF monolayer deposited on a Poly(ethylene imine) (PEI)-coated substrate (Fig. S1), following a methodology reported in detail in our previous works [START_REF] Merindol | Assembly of anisotropic nanocellulose films stronger than the original tree[END_REF][START_REF] Blell | Generating in-plane orientational order in multilayer films prepared by spray-assisted layer-by-layer Assembly[END_REF][START_REF] Sekar | Anisotropic optical and conductive properties of oriented 1D-nanoparticle thin films made by spray-assisted self-assembly[END_REF][START_REF] Hu | Spray-assisted alignment of Layer-by-Layer assembled silver nanowires: a general approach for the preparation of highly anisotropic nano-composite films[END_REF]. Briefly, the distribution of the angle 𝜃 between the principal axis of the nanofibrils and the spraying direction was extracted using the plugin OrientationJ [START_REF] Rezakhaniha | Experimental investigation of collagen waviness and orientation in the arterial adventitia using confocal laser scanning microscopy[END_REF] of the software ImageJ [START_REF] Schneider | NIH Image to ImageJ: 25 years of image analysis[END_REF], which is based on the analysis of the structure tensor in the local neighborhood of each pixel. An average 2D nematic order parameter (S2D), weighted over the distribution of 𝜃, was calculated as:

S2D = 〈2 cos 2 𝜃 -1〉 (1)
This parameter takes values between 0 and 1, where the extreme values correspond to a random distribution (isotropic sample) and a perfectly parallel alignment of the imaged nanofibrils (anisotropic sample) respectively. Fig. 1a shows the effective alignment of the CNFs parallel to the spraying direction on a monolayer. and the dashed line corresponds to a linear fit (R 2 = 0.9984) of the data excluding the initial (precursor) layers whose thickness growth is influenced by the substrate and is not representative of that of the bulk film [START_REF] Mcaloney | Atomic force microscopy studies of salt effects on polyelectrolyte multilayer film morphology[END_REF].

The Layer-by-Layer assembly of thicker films was performed by alternating the grazing incidence spraying of the anionic nanofibrils with the conventional low-shear and orthogonal-to-the-surface spraying of Poly(vinyl amine) (PVAm) used as polycation. Verification of the CNFs alignment within thicker films by the aforementioned AFM-based methodology was not possible due to the increased roughness of the multilayer system (root mean square roughness of about 63 nm for a film composed of 97-layer pairs) and the formation of bundles with the increasing number of CNF layers (See Fig. S2). However, the structural birefringence of the sample was verified by optical microscopy between cross-polarizers (Fig. S3).

The thickness of the cellulose-based films prepared on silicon wafers was estimated by standard spectroscopic ellipsometry measurements. We adopted the approach of fixing an apparent refractive index of cellulose in between those reported parallel (1.576-1.595) and perpendicular to the crystal axis (1.527-1.534) [START_REF] Klemm | Comprehensive Cellulose Chemistry. Volume L: Fundamentals and Analytical Methods[END_REF], which has been previously found sufficiently accurate to characterize the thickness of CNF-based LbL films despite their anisotropy [START_REF] Merindol | Bio-inspired multiproperty materials: strong, self-healing, and transparent artificial wood nanostructures[END_REF][START_REF] Merindol | Assembly of anisotropic nanocellulose films stronger than the original tree[END_REF][START_REF] Blell | Generating in-plane orientational order in multilayer films prepared by spray-assisted layer-by-layer Assembly[END_REF]. It was found that the GIS films grew linearly with the number of layer pairs, at a growth increment of 9.0 ± 1.7 nm per layer pair (Fig. 1b). The uncertainty of this estimation was determined by the root-meansquare error (RMSE) of the linear regression. When the films reached about 60 layer pairs, it became too rough for the ellipsometric data to fit the optical model. Therefore, the thickness of thicker films was predicted using the extrapolation of the linear regression of the data shown in Fig. 1b. The unidirectionally aligned (CNF/PVAm)97 was thus estimated to be 875 ± 16 nm thick, with such uncertainty found by the propagation of error.

Modeling the optical behavior of the unidirectional CNF-based films

In the following, the (CNF/PVAm)x film is modeled as a homogeneous effective medium composed of straight cylindrical nanofibers embedded in a matrix. The average alignment of these nanofibers is along the 𝑒 ! &&&⃗ direction, while the axes of all cylinders are perpendicular to the 𝑒 " &&&⃗ direction. The laboratory frame of reference (𝑒 ! &&&⃗, 𝑒 # &&&&⃗, 𝑒 " &&&⃗) is related to the local frame of each cylinder (𝑢 ! &&&&⃗, 𝑢 # &&&&⃗, 𝑢 " &&&&⃗) by a rotation matrix R(𝜃) along the z-axis defined as:

𝑅(𝜃) = . cos 𝜃 sin 𝜃 0 -sin 𝜃 cos 𝜃 0 0 0 1 7 (2)
where 𝜃 describes the relative orientation of the cellulose nanofibers to the x-axis of the laboratory frame. By assuming that the normalized distribution of the orientation angle P(𝜃) of the nanofibrils is symmetric to the x-axis the effective dielectric tensor can be expressed as follows:

𝜀 $%% = .

𝜀 && 0 0 0 𝜀 '' 0 0 0 𝜀 (( 7

with:

𝜀 )) = (&+%)-! .%-"# / $,&& (&+%).%/ $,&& , (𝑖 = 1, 2, 3) (4)
𝜀 0% and 𝜀 1 are the dielectric function of the cellulose nanofibrils and the matrix, respectively, f is the volume fraction of cellulose nanofibrils within the film, and 𝛽 $,)) depends on three geometrical parameters L1, L2, L3 called depolarization factors (see the derivation of this model in the supplementary information).

According to this model, the medium is biaxial for 0 < S2D < 1 with neutral axes oriented along 𝑒 ! &&&⃗, 𝑒 # &&&&⃗ and 𝑒 " &&&⃗. On the contrary, the medium is uniaxial for S2D = 0 or S2D = 1. In the case of S2D = 0, cellulose nanofibrils have a random orientation in the (𝑒 ! &&&⃗, 𝑒 # &&&&⃗) plane and the optical axis of this medium is oriented along 𝑒 " &&&⃗. Thus, 𝜀 && = 𝜀 '' are the ordinary effective dielectric function while 𝜀 (( is the extraordinary effective dielectric function. In the case of S2D = 1, cellulose nanofibrils are perfectly aligned along 𝑒 # &&&&⃗ which corresponds to the direction of the optical axis. 𝜀 '' = 𝜀 (( are the ordinary effective dielectric function while 𝜀 && is the extraordinary effective dielectric function. Equation ( 4) allows calculating the diagonal elements 𝑛 )) of the effective refractive tensor. Then, by applying a first-order approximation, we can linearize the expression of 𝑛 )) as following:

𝑛 )) = 𝑛 1 + (-"# +-! )/ $,&& '0 ! 𝑓 ( 5
)
with 𝑛 1 the refractive index of the matrix.

The main polarization property expected for such a unidirectionally aligned film of thickness 𝑙 is linear birefringence (LB) obtained with light propagating along 𝑒 " &&&⃗. According to our model (equation ( 5)), the LB is given by:

𝐿𝐵 = '3 4 (𝑛 && -𝑛 '' )𝑙 = 3 40 ! (-"# +-! )² (-"# .-! ) 𝑓𝑙𝑆 '6 (6)
According to equation ( 6), the linear birefringence of the unidirectionally sprayed film is positive and increases with the difference between the dielectric functions of the nanofibrils and the matrix. Moreover, this property is also proportional to the volume fraction of nanofibrils, its distribution of orientation (S2D), and the film thickness.

Polarization properties of the unidirectionally aligned films

The polarization properties of the samples (diattenuation, retardance, depolarization, and their form, either linear, circular, or elliptical) were extracted from the differential (logarithmic) decomposition [START_REF] Azzam | Propagation of partially polarized light through anisotropic media with or without depolarization: a differential 4 x 4 matrix calculus[END_REF][START_REF] Ossikovski | Differential matrix formalism for depolarizing anisotropic media[END_REF][START_REF] Arteaga | Characterization of homogenous depolarizing media based on Mueller matrix differential decomposition[END_REF] of the measured Mueller matrix (the elements of the MM and its analysis can be found in Supporting Information). Fig. 2 shows the spectra of the four polarization properties extracted at the azimuth angles 0° and 90° for both the unidirectional and the control "random" film. We verified that circular dichroism (CD), circular birefringence (CB), and linear dichroism (LD) were negligible for both samples, while linear birefringence (LB) was non-zero in the unidirectional film. Consequently, the sample prepared by GIS-LbL presented an anisotropy of the refractive index. From these data, we calculated the effective birefringence 〈Δ𝑛〉 = 〈𝑛 # -𝑛 ! 〉 at the center of the unidirectional sample given by |𝐿𝐵| = 2𝜋|Δ𝑛|𝑙/𝜆, where 𝑙 is the film thickness (875 ± 16 nm). Note that the absolute error on the LB determined from our ellipsometer is smaller than 5 × 10-4. The resulting effective birefringence averaged over the wavelength range was 〈Δn〉 = 0.032 (see a plot of Δn = f (λ) in Fig. S5). This value is smaller than those reported for the birefringence of cellulose, which ranges from 0.045 to 0.062 depending on the source [START_REF] Klemm | Comprehensive Cellulose Chemistry. Volume L: Fundamentals and Analytical Methods[END_REF], but it lies between those reported for birefringent films prepared with CNC by dip-coating [START_REF] Mendoza-Galván | Linear birefringent films of cellulose nanocrystals produced by dip-coating[END_REF] and spin coating [START_REF] Cranston | Birefringence in spin-coated films containing cellulose nanocrystals[END_REF]. Knowing that the film thickness grows linearly with the number of layer pairs (Fig. 1b), we can estimate that the deposition of approximately 250 layer pairs might be suitable to use the film as a quarter waveplate (LB = 𝜋 2 ⁄ rad) for the UV spectral range. Moreover, according to our effective medium theory, the film birefringence can be increased by changing the polymer, improving the alignment, or increasing the volume fraction of CNF. Homogeneity of the optical properties over large areas is crucial for optoelectronic applications, therefore, two-dimensional analysis of the retardation is more meaningful for the development of birefringent films than the single-point approach. Consequently, the Mueller matrix of the unidirectional film was evaluated at different positions throughout the sample at a fixed wavelength of 300 nm (where this property was maximum). In the ideal case of perfect alignment and full packing of the CNFs, effective birefringence is 〈Δ𝑛〉 = 〈𝑛 # -𝑛 ! 〉. In the present case, though, the nanofibrils are aligned following the diverging flowlines of the cone-like deposition pattern (Fig. S1) [START_REF] Blell | Generating in-plane orientational order in multilayer films prepared by spray-assisted layer-by-layer Assembly[END_REF], the mapping of LB was thus calculated in a local coordinate system parallel to the mean axis of the nanofibrils to account for the gradient in the direction of orientations (see the detailed procedure in the supporting information). This correction allows mapping the angle of the local orientation of the internal CNFs throughout the film (Fig. 3a). We verified that at the centerline of the sample the orientation of the CNFs was effectively parallel to the spraying direction, but it was around ±30° at the border of the deposition pattern. The macroscopic mapping of the local orientation was consistent with the one obtained by AFM on a monolayer of CNFs as shown in Fig. 3b. Hence, we highlight the convenience of using the non-invasive Mueller matrix polarimetry technique for the verification of the unidirectional alignment of the reinforcing nanofibrils throughout the whole multilayer film. The ellipsometry approach probes simultaneously all the nanofibrils under the beam (diameter ~ 0.8 mm) and each spectrum is recorded in 5 s so that the 135-points maps shown in Fig. 3a andc can be obtained in approximately 12 min without requiring further image analysis. This time scale could be drastically reduced by using Mueller matrix imaging ellipsometry. For example, Arteaga et al. have proposed an imaging set-up with a micrometric resolution that can record a Mueller matrix image in 3s [START_REF] Arteaga | Characterization of homogenous depolarizing media based on Mueller matrix differential decomposition[END_REF]. Conversely, an AFM analysis carried out on a similar surface area would take several tens of hours to collect and analyze the images while probing only the surface of the sample. The macroscopic spatial variation of the linear birefringence, measured in the local frame of reference, is shown in Fig. 3c. This property, representing the effective optical anisotropy of the film, is maximum at the centerline of the deposition pattern and decreases symmetrically towards the edges. According to our effective medium theory (equation ( 6)), the spatial variation of LB gives the gradient of the products between the volume fraction, the 2D nematic order parameter, and the film thickness. The variation of this product agrees with the value of the 2D nematic order parameter in a monolayer (see Fig. 3d). Thus, the use of accurate local values of these variables (not available at present for our sample) will allow a simple, fast, non-destructive, and large-scale characterization method of the volume fraction of the reinforcing phase within the nanocomposite film.

Conclusions

Highly anisotropic films made of cellulose nanofibrils and poly(vinyl amine) were prepared using a Layer-by-Layer approach combined with the grazing incidence spraying of the reinforcing nanofibrils. The films are non-depolarizing and linearly birefringent as demonstrated by analyzing the spectral Mueller matrix measured by ellipsometry. Circular polarization properties were not observed despite cellulose being a chiral macromolecule. This is likely because the structural anisotropic organization of the cellulose nanofibrils maximizes the linear effects. The macroscopic mapping of the local orientation of the nanofibrils within the bulk of the film was easily achieved in a few minutes by Mueller matrix polarimetry. The orientation dependence of the Mueller matrix elements can be suppressed by applying rotation matrices on the Mueller matrix. Thus, Muller matrix measurements favor an accurate characterization of the optical anisotropy of the whole sample. Moreover, modeling the film as an effective medium provided a parametrized expression that describes the linear birefringence as a function of the local degree of alignment, the film thickness, and the volume fraction of the cellulose nanofibrils. Therefore, MMP provides information on both the film morphology (orientation and product between the volume fraction, thickness, and the nematic order parameter) and the optical birefringence/dichroism of the film. This MMP approach investigating the alignment in nanofibrillar composite films is much less localized than AFM, is less time-consuming, and is easily scalable to large samples. It can also be performed in situ and/or continuously during sample preparation and the results obtained represent the properties of the bulk film rather than only that of the surface. Thus, the presented methodology facilitates the characterization of thin films with in-plane anisotropy and is particularly promising for the development of nanocomposite thin films for optical applications such as polarizers, waveguides, or waveplates.

Fig. 1 .

 1 Fig. 1. (a) AFM height image of the monolayer of CNF (S2D = 0.85). Image size 2 × 2 μm2. The spraying direction is indicated by the white arrow. (b) Variation of the film thickness as a function of the number of layer pairs. The error bars represent the standard deviation over several measurements

Fig. 2 .

 2 Fig.2. LB, LD, CB, and CD spectra extracted from the Mueller matrix of the unidirectional sample at the azimuth angles 0° (solid black lines) and 90° (dashed red lines). For comparison, the spectra of a control sample prepared by dip-assisted LbL (i.e., a "random" film) at the azimuth angles 0° (solid blue lines) and 90° (solid green lines) are also plotted. Not visible curves are covered by others. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

Fig. 3 .

 3 Fig. 3. In-plane macroscopic MMP mappings of (a) the local orientation of the CNFs and (c) the corrected linear birefringence (LB) of a unidirectional multilayer sample prepared on a quartz slide with a size of 14 mm × 8 mm. The corrected LB corresponds to the linear birefringence measured in the local frame of CNFs (𝜃 = 0). The black arrows indicate the direction of the spraying jet. (b) and (d) are the macroscopic AFM mappings of the mean angle 𝜃 of preferential alignment of the CNFs and the nematic order parameter, respectively, determined on a monolayer deposited on PEIcoated silicon wafer with a size of approx. 60 mm × 32 mm. The dashed frames represent the comparative size of the quartz samples analyzed by MMP. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

Acknowledgments

The authors acknowledge the ministère de l'Enseignement Supérieur, de la Recherche et de l'Innovation (MESRI, France), the Centre National de la Recherche Scientifique (CNRS, France), and the International Center for Frontier Research in Chemistry (IcFRC, France) for their support. We also express our gratitude to Christophe Contal (ICS, France) and Mircea Rastei (IPCMS, France) for AFM tutorial, to the characterization platform of ICS for the UV-Vis measurements, to Innventia AB (Sweden) for supplying CNFs, and to BASF (Germany) for supplying PVAm. We thank the Ellipsometry core facility of LCP-A2MC (Université de Lorraine, http://lcp-a2mc.univ-lorraine.fr).

Appendix A. Supplementary data

Supplementary data to this article can be found online at: https://doi. org/10.1016/j.compscitech.2022.109889.


Dr Rongyu Sun

Dr Rajesh Nomula

Jean-Valère Naubron

Dr Nicolas Daugey

Prof Thierry F Buffeteau

Frédéric Castet

Patrick Y Toullec

Stéphane Quideau

Philippe A Peixoto
email: philippe.peixoto@u-bordeaux.fr

Prof. P Y Castet

Prof S Toullec

Dr P A Quideau

Peixoto

Dr J.-V Naubron

Asymmetric allenylation of alkynes mediated by chiral organoselenated reagents under oxidative conditions

Keywords: Allenes, Chiral Diselenides, Asymmetric Synthesis, Selenoxides

The reactivity of novel chiral lactamide-substituted diselenide-based reagents under oxidative conditions was exploited to develop a metal-free method for the preparation of enantioenriched allenylamides from simple alkynes in good yields, and with enantiomeric excesses up to 99%. The key of the success in this method is attributed to the hydrogen-bonded lactamide appendages that ensure configurational stability of chiral vinylselenoxide intermediates for an optimal enantiotopic β-synelimination step.

Due to their chemical structure made of two consecutive and orthogonal double bonds, allenes are unique molecular motifs possessing a wide range of synthetic applications in organic synthesis. In addition to their highly diverse reactivity features, another interesting aspect lies in their inherent axial chirality when both their sp 2 -carbon termini are properly substituted. While several methodologies exist to generate allenes, 1 notably relying on either prototropic or sigmatropic rearrangements, 2 nucleophilic substitutions on activated alkynes, 3 additions on enynes, 4 [1,2]elimination (or abstraction), 5 Wittig olefination, 6 or even from cyclopropanes (Doering-Moore-Skattebol reaction), 7 to date, still very few approaches are reported without the mediation of heavymetal-based reagents and/or starting from non-functionalized alkynes. In this context, the development of a general procedure based on a chiral organo-selenium-based reagent to generate enantioenriched allenes would represent an interesting progress in the field. β-syn-Elimination of alkylselenoxides 8 is probably amongst the most

used reaction in the selenium organic chemistry tool box, and is often applied to oxidize ketone into enones. 9 This occurs through a two step procedure consisting of a selenylation, followed by a β-synelimination of the alkylselenide moiety under oxidative conditions (i.e. after oxidation to the corresponding selenoxide -Scheme 1). In contrast to alkylselenoxides, the β-syn-elimination of vinylselenoxides to generate allenes is far less reported. This implies the formation of a transient vinylselenide intermediate, which can be oxidized to the corresponding vinylselenoxide before its β-synelimination under thermal activation (Scheme 1). Scheme 1. β-syn-Eliminations of alkyl-or vinylselenoxides / Access to potentially chiral allenes.

While described as a side reaction by Reich in 1980 through thermal activation, 10a the first detailed studies of such an allenylation were reported by Back and co-workers in 1983 and later in 1989 (Scheme 2). 10b,c After a radical mediated sulfoselenylation reaction performed on simple alkynes, followed by the oxidation of the diorganoselenium intermediate using m-CPBA, these authors showed that the β-elimination of the selenium motif could be performed at chloroform reflux to yield the corresponding racemic sulfonic allenes in good yields. In 2016, the Zhang group reported the simple formation of amidovinylselenides from alkynes in presence of a diselenide under oxidative conditions. 11 In this reaction, the oxidant (Nfluorobenzesulfonimide -NFSI) played both the role of an oxidant and a nucleophile as the sulfonamide moiety was added on the activated alkynes. In 2021, inspired by this report and in continuation of their own work on selenium-mediated allylic oxidations using NFSI, 12 the Breder group reported the first (racemic) organo-seleno-catalyzed oxidative allenylation of nonactivated alkynes 13 based on a β-syn-elimination from an amidovinylselenium(IV) intermediate (Scheme 2). The first asymmetric version of a vinylselenoxide β-elimination to generate allenes was reported by the group of Uemura in 1992. 14 This was performed starting from a vinylselenide that, upon asymmetric oxidation (using Sharpless or Davis chiral oxidation conditions), generated enantioenriched

R 2 Se R 4 H R 1 R 2 Se H R 1 X R 3 Ar O R 1 R 2 R 4 R 3 R 3 O Ar -ArSeOH -ArSeOH . R 1 R 2 R 3 X β-syn-elimination Axial chirality if R 1 ≠ R 2 Alkenes Alkyl selenoxides Vinylselenoxides R 3 R 1 R 2 H 1-(Ar-Se) 2 2-Oxidation R 2 R 4 H R 1 R 3 1-(Ar-Se) 2

2-Oxidation

Simple alkynes alkyl carbanions

(ArSe) 2 + ArSe(O)OH Dismutation (ArSe) 2 + ArSe(O)OH

Dismutation

vinylselenoxides, which were themselves able to evolved in situ at room temperature in 10 to 20 days to the corresponding allenes with 42% e.e. In 1995, the same group improved this approach using a chiral ferrocenyl selenium motif. The diastereoselective oxidation of the selenium atom at low temperature delivered enantioenriched vinylselenoxides intermediates (3 examples), which were able to evolve after 3 days at -20°C to the corresponding allenes with e.e. up to 89% and low yields (Scheme 2). 15 In contrast to previous reports, the allenylation step could here take place at low temperature; the reason for such kinetic differences being the nature of the starting electron-poor vinylselenides, a molecular feature easing the β-elimination step.

Scheme 2. Allenylations of vinylic-Se(IV) substrates by (stereocontrolled) β-Hsyn-eliminations.

Concomitantly to the efforts by Breder and co-workers, our group has initiated in 2019 a research program focusing on the enantioselective synthesis of allenes from non-activated alkynes using chiral diselenide reagents. [START_REF] Wirth | For reviews on Chiral Se Reagents[END_REF] The main challenge was thus to be able to reach high levels of enantioselectivities using chiral vinylselenoxides that can β-eliminate only at elevated temperature (Scheme 2). Indeed, as mentioned above, 10b,c,13 βelimination from vinylselenoxides bearing non-electron withdrawing substituents requires thermal activation; a feature incompatible with a perfect conservation of the chirality at a selenoxide center. 17 Our goal was thus to design chiral selenium-based reagents that could: 1-stabilize the absolute configuration of the selenium (IV) atom (and so, even at elevated temperatures) in order to, 2-promote a general reagent-controlled diastereoselective vinylselenoxyde β-synelimination towards highly enantioenriched allenylamides (Scheme 2).

Herein, we wish to disclose our own efforts, which ultimately resulted with the production of allenylamides from alkynes in good yields and with e.e. up to 99%.

Optimization of the reaction conditions using achiral diselenide reagents

Our work started with the optimization of the reaction conditions for the efficient preparation of vinylselenoxides from simple alkynes. This was done on 4-octyne, and initially required two steps consisting of: 1-a Zhang amidoselenation 11 performed with 68% yield in the presence of NFSI and o-anisyldiselenide, [START_REF]racemic model in view of the later implementation of stereodiscriminating motifs on the phenol[END_REF] and 2-a 82% yield m-CPBA mediated-oxidation of the transient vinylselenide 1. Interestingly, our investigations led us to consider the possibility of performing these two steps in a onepot procedure employing an excess of NFSI during the amidoselenylation step (See SI for further information on the optimization). Accordingly, this excess of NFSI allows the generation of a transient vinylic Se(IV) intermediate 2 that could be hydrolyzed by a 1M aqueous solution of sodium bicarbonate to furnish the desired vinyselenoxide 3 in 89% yield. With this one-pot alkyne amidoselenoxidation optimized, we next studied the β-syn-elimination of the arylselenoxide motif under thermal activation (Table 1). The first attempt was done according to literature precedents: 10a heating of 3 at 95°C (sealed tube) for 2h in benzene (0.1M) delivered the allenylamide 4 in 20% yield (Entry 1). The vinylselenoxide being moderately soluble in benzene, the reaction was run in CHCl3 and yielded 34% of allene 4 (Entry 2). While the starting material went to full conversion in these first two attempts, it should be noted that the allenylamide 4 was systematically isolated along with the reduced vinylselenide 1 as the sole side-product (isolated in 51% and 47%). To our delight, working under more diluted conditions (0.01 M) resulted in the formation of allenylamide 4 with 68% yield, and strongly decreased the undesired reductive reaction leading to 1 (Entry 3). As reported by Reich, 10a the addition of base (1 equiv. of DABCO) was finally evaluated, and delivered an optimal isolated 98% yield of the allene 4 with no detectable amounts of 1 (Entry 4). Remarkably, in this last experiment, the o-anisyldiselenide (See Scheme 3) was recovered in 65% yield. This corresponds to a nearly maximal recovery, as this reagent can theoretically be only reformed in 66% through the dismutation [START_REF] Gancarz | [END_REF] of the released phenylselenenic acid (ArSeOH converts readily in solution into Back (1983 & 1989) Racemic allenylations using vinylic Se(IV) intermediates equimolar amounts of ArSeSeAr and arylseleninic acid). Ultimately, a last attempt was perfomed under these last best conditions, but at room temperature instead of 95°C (Entry 5). No traces of allenylamide 4 were then observed, even after 7 days. This again confirms that thermal activation is mandatory for the β-syn-elimination to occur on such electron-rich substrates. Yield

R 2 R 1 H H . H R 1 N(SO 2 Ph) 2 R 2 (ArSe) 2 (10mol%) NFSI, Base, 100°C rac-Allenylamines R 1 Se H H R 2 N(SO 2 Ph) 2 LG Ar -BH -ArSe-LG Active catalyst B R 2 R 1 H H . R 1 H N(SO 2 Ph) 2 R 2 (Ar*

Asymmetric allenylations using vinylic Se(IV) intermediates

Stable & Isolable chiral vinylselenoxides

1 [a] 1 Benzene 0.1 M / 20% 51% 2 CHCl3 0.1 M / 34% 47% 3 CHCl3 0.01 M / 68% 21% 4 [b] CHCl3 0.01 M 1 equiv. 98% - d 5 [c] CHCl3 0.01 M 1 equiv. - d - d
[a] Isolated yield, [b] o-dianisyldiselenide was recovered in 65% yield, [c] reaction run at room temperature over 7 days, [d] not observed.

Synthesis of chiral diselenides reagents

With both the amidoselenylation and allenylation reactions optimized, we then focused on the development of an asymmetric version of this transformation. Inspired by the impressive applicability of iodoarenes bearing lactacte(s) or lactamide(s) motif(s) 20 in the field of hypervalent iodine chemistry, we decided to adapt this strategy to diselenide chemistry. Four diselenides bearing ortho-lactate-or lactamidebased functional groups were then synthesized in 5 to 7 steps with overall yields ranging from 49% to 71% (Scheme 4). This was notably done by the initial selenylation of in situ generated organomagnesium intermediates bearing mono-o-or bis-o-MOM-protected phenols with PMBSeCN, yielding 8a and 8b in 78 and 98%, respectively. These arylselenides intermediates were then MOM-deprotected before the installation of the lactacte(s) motif(s) was done thanks to a (bis)Mitsunobu reaction providing the chiral arylselenides 10a or 10b in 88% and 90% yield. Once isolated, these intermediates were finally PMBdeprotected using PIFA, and furnished the chiral C1-or C2symmetrical diselenides 11a and 11b. Interestingly, the PMBprotected intermediates 10a and 10b also served as points of synthesis divergence as they could be saponified, activated by isobutylchloroformate, and amidified with 2,4,6-trimethylaniline to yield the corresponding mono-or bis-lactamic arylselenides 12a and 12b with 78 and 72%. As for their ester counterparts, these arylselenides bearing lactamides motifs were finally PMBdeprotected under oxidative conditions to yield the desired mono-or bis-lactamide aryldiselenides 13a and 13b with yields above 90%.

Scheme 4. Synthesis of chiral diselenides bearing lactates or lactamides motifs. a) from 6a : MOMCl (1.5 equiv.), DIPEA (2 equiv.), CH2Cl2, 0°C, 2h; b) from 6b : MOMCl (2.5 equiv.), DIPEA (4 equiv.), CH2Cl2, 0°C, 2h; c) from 7a : iPrMgCl (1.1 equiv.), Et2O, 0°C, 1h, followed by PMBSeCN (1.2 equiv.), 0°C, 1h; d) from 7b : nBuLi (2.2 equiv.), THF, -40°C to 0°C, 2h, followed by PMBSeCN (1.2 equiv.), r.t., 2h; e) from 8a : i-Et3SiH (3 equiv.), TFA (20 equiv.), CH2Cl2, r.t., 3h, ii-9 (2 equiv.), DIAD (2 equiv.), Ph3P (2 equiv.), THF, 0°C to r.t., 12h; f) from 8b : i-Et3SiH (6 equiv.), TFA (40 equiv.), CH2Cl2, r.t., 3h, ii-9 (4 equiv.), DIAD (4 equiv.), Ph3P (4 equiv.), THF, 0°C to r.t., 12h; g) from 10a : i-NaOH (2N), THF/MeOH, r.t., 2h, ii-N-methylmorpholine (4 equiv.), isobutylchloroformate (2 equiv.), CH2Cl2, 1h, followed by 2,4,6-trimthylaniline (3 equiv.), r.t., 12h; h) from 10b : i-NaOH (2N), THF/MeOH, r.t., 2h, ii-Nmethylmorpholine (8 equiv.), isobutylchloroformate (4 equiv.), CH2Cl2, 1h, followed by 2,4,6-trimthylaniline (6 equiv.), r.t., 12h; i) PIFA (0.5 equiv.), CH2Cl2, r.t., 1h.

Asymmetric allenylations of alkynes

The novel chiral diselenides 11a, 11b, 13a, 13b were then engaged in the aminoselenoxidation of 4-octyne using our optimized conditions (Table 2). The corresponding vinylselenoxides were then formed in yields ranging from 80 to 89% with low to moderate diastereoselectivities. In the case of the vinylselenoxides possessing lactates substituents, both 14 and 15 were isolated as mixtures (Table 2, Entries 1 &2). On the contrary, the vinylselenoxides bearing lactamides chiral appendages (Table 2, Entries 3 & 4) were perfectly separable on simple normal-phase silica gel chromatography, yielding each (R se )-and (S se )-diastereomers of 16 and 17 with the exact same yields of 41%, thus indicating no diastereoselectivity control of the selenoxidations step in these cases. A remarkable observation was then made on the isolated vinylselenoxides 17: the perfect enantiomeric stability 17 of the selenium absolute configuration. Indeed, no epimerization was noticed after their isolation, and so even after prolonged storage (up to 12 months) of the powderous isolated compounds. Next, in view of determining the precise absolute configuration at the selenoxide stereocenter, several attempts were undertaken to try to obtain exploitable single crystals from each respective pure for 11a : 56% (5 steps) 11b : 71% (5 steps) 13a : 49% (7 steps) 13b : 55% (7 steps)

N(SO 2 Ph) 2 Se O O . H N(SO 2 Ph) 2 N(SO 2 Ph) 2 Se O + rac-4 3 1 Solvent, DABCO 95°C, 2h -ArSeOH 5 R 1 Se MOMO R 1 R 2 MOMO R 1
(90%) (72%) (91%) (84%)
selenoxydes. Unfortunately, none of these efforts led to suitable X-ray structures, and only electronic circular dichroism experiments allowed a formal attribution of the absolute configurations beard by the selenium atom on each isolated diastereoisomer of 17 (See SI for more information). [a] Isolated yield, [b] Separation of both diastereisomers was performed on normal phase silica gel chromatography.

With this configurationally stable selenoxides in hands, their allenylations by a β-syn-elimination were investigated (Scheme 3). [START_REF]When heated in our best conditions, the mixture of monolactates 14 yelds 71% of the allene 18 with 12% e.e[END_REF] With this higher level of stereocontrol attained using the bislactamides-based vinylselenoxides, we were then eager to decipher the precise mechanism explaining such enantioselective β-syn-elimination step, and we were confident that this important question could be addressed by modeling the reaction mechanism using Density Functional Theory (DFT).

As expected, such calculations showed a 5-center concerted βelimination-type mechanism leading to the formation of the allenylamides 4 through well-organized Transition States (TSssealed by H-bonding type interaction with the selenoxide motif), along with the extrusion of the corresponding arylselenenic acids (See Figure 1). Moreover, depending on the chirality displayed by the selenium atom and the substitutents (i.e. mono-or bislactamides), TSs with significantly different energies were obtained when considering the β-elimination step, depending on the abstracted proton (Ha or Hb). As can be seen on Figure 1 for the vinylsenoxide (R Se ,R,R)-17, these energy differences are in fact explained by the positioning of the propyl chain, which can fit into a less sterically demanding pocket for TS-A (favored) than for TS-B [START_REF]DFT calculations starting from (R Se ,R)-16 or (S Se ,R)-16[END_REF] (disfavored due to steric clashes with one of the lactamide unit). Thereby, from (R Se ,R,R)-17, a 4,29 kcal/mol difference was calculated for the deprotonation of Hb vs. Ha, which implies that this (R Se ,R,R)-17 leads majoritarily to the allenylamide (R)-4, as experimentally observed. Similarly, a significant difference (1,68 kcal/mol) was obtained when starting from (S Se ,R,R)-17, whose selenium-β-elimination strongly favors the allenylamide (S)-4 (not shown on Scheme 4, for more information see SI). Moreover, if these DFT results strongly support the major influence of the chirality at the Se(IV) atom on the H-abstraction, a slight positive match effect of the (R)lactamides motifs was thus also determined towards the formation of the (R)-allene 4 [and a miss-match for the (S)-allene 4]. As for 4-octyne, the diastereoselectivity of the amidoselenoxidation was low to moderate on the alkynes tested. Of note, in the case of substrates 18d and 18k, only one diastereomer could be observed. Once isolated, each diastereomerically pure vinylselenoxides was then heated in CHCl3, in the presence of DABCO at 95°C for 2h. Complete conversions were observed in all cases, delivering allenes 20a-k in yields ranging from 51 to 99%. [START_REF]VCD studies performed on allenes (-)-20a and (-)-20i have show that their absolute configurations were (R)-configuration, which is in perfect accordance with (-)-(R)-4. Thereby, by extension, every laevorotary allenes (-)-20 along this sequence were assigned to be (R), and every dextrorotary[END_REF] The level of enantiocontrol was found to be excellent, especially on the substrates bearing ortho-substituted aromatic moities, such as 18a (88% & 84% e.e.), 18b (84% & 82% e.e.), 18c (96% & 92% e.e.), 18e (99% & 82% e.e.), 18i (92% & 82% e.e.), and 18k (82% e.e.). Intriguingly, the result on the p-methoxynapthyl derivative 18k was very different to the one gathered on the o-methoxynaphthyl substate 18j (4% & 8% e.e). This kind of motif present on 18j thus appears to correspond to a limitation of the method, probably due to too hindered o-and o'-positions. A second limitation identified in the course of this scope was the detrimental effect of the meta-functionalization of the aromatic moiety. This could be clearly seen on alkyne 18d, as the corresponding allene 20d was isolated in 98% yield, and with only 47% e.e. Finally, a last indication can finally be taken out of this substrate scope: a slight match effect towards the (R)allenes when the lactamide bears a (R)-configuration. This confirms the tendency identified by DFT calculations when starting from (R Se ,R,R)-17.

To conclude, this work describes a method to generate enantioenriched amidoallenes from simple alkynes using novel chiral diselenide reagents under oxidative conditions. The success of our approach is based on the precise design of the chalcogenic reagent, and more specifically on the installation of chiral secondary amide substituents enabling the formation of configurationally-stable chiral selenoxide intermediates. To the best of our knowledge, such H-bonding-based stabilization of a selenoxide chirality is unprecedented, and thus opens new opportunities for future applications in asymmetric synthesis. Once isolated, these enantiopure selenoxides (which could be stored several months after isolation) could be engaged in thermal β-syn-elimination reactions leading to the corresponding allenes in good yields and with high enantiomeric excesses. Importantly, the diselenide reagents could be partly recovered at the end of each reaction, which minimizes the atomic cost of the method. Efforts to extend the use of these selenium-based reagents towards a catalytic asymmetric version 13 of this reaction are currently underway in our laboratory, and will be reported in due course.

alkyne

Ar-Yield [b] 19 Yield [b] 19' (-)-(R)-20 (+)-(S)-20 18a 40% 45% 79% yield [b] 88% e.e. [c] 93% yield [b] 84% e.e. [c] 18b 45% 28% 70% yield [b] 84% e.e. [c] 64% yield [b] 82% e.e. [c] 18c 43% 26% 69% yield

[a] 96% e.e. [c] 95% yield [b] 92% e.e.

[c] 18d

Not observed 17% -98% yield [b] 42% e.e. [c] 18e 30% 33% 82% yield

[a] 99% e.e. [c] 74% yield [b] 82% e.e. [c] 18f 36% 29% 99% yield [b] 82% e.e. [c] 57% yield [a] 82% e.e. [c] 18g 42% 41% 61% yield [b] 86% e.e. [c] 63% yield

[a] 82% e.e. [c] 18h 46% 38% 67% yield [b] 72% e.e. [c] 69% yield [a] 36% e.e. [c] 18i 44% 44% 84% yield

[a] 92% e.e. [c] 83% yield [b] 82% e.e. 4% e.e. [c] 51% yield [b] 8% e.e. [c] 18k

Not observed 35% -66% yield [b] 82% e.e. [c] [a] CHCl3, DABCO (1 equiv.), 95°C (sealed tube), [b] Isolated yield, [c] Determined by chiral HPLC.

Scheme 3 .

 3 Scheme 3. Facile amidoselenoxydation of 4-octyne.

Uemura

  Scheme 3. Stereocontrolled β-syn-elimination towards enantioenriched allenes.

Figure 1 .

 1 Figure 1. DFT-calculated TS structures associated to the abstraction of either Ha or Hb starting from selenoxide (R Se ,R,R)-17.

Table 1 .

 1 Optimization of the allenylative vinylselenoxide-β-syn-elimination conditions.

	Entry	Solvent	Concentration	DABCO	Yield 4	[a]

Table 2 .

 2 Chiral selenoxides syntheses using diselenides 11a,b and 13a,b.

	Entry	R1	R2	Yield	[a,b]
	1		H	80% (14a and 14b, 42/58 ratio, unseparable)
	2			89% (15a and 15b, 31/69 ratio, unseparable)
	3		H	16a (41%)+ 16b (41%)

4

17a (41%) + 17b (41%)

Table 3 .

 3 Scope of substrates using chiral diselenide 13b.

Acknowledgements

Financial support from the Agence Nationale de la Recherche (ANR-19-CE07-0008-01 -HYPE4ALL), the Ministère de la Recherche, and the CNRS, including doctoral & post-doctoral research assistantships for R.S. and R.N., is gratefully acknowledged. This work has benefited from the analytical facilities and expertise of the CESAMO platform at the University of Bordeaux. The Centre de Calcul Intensif d'Aix-Marseille is also acknowledged for granting access to its high performance computing resources.

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