Turing machines rule?

⇒ Community-dependent some personal observations....

• Historians?

• Programmers?

• Engineers?

• CS students in country x?

• Formal methods people?

• Philosophers?

TM -thé model of computing

Theory: Thé ultimate model of computability? [I]t was Turing alone who [...] gave the rst convincing formal denition of a computable function (Soare, 2007) • Appeal to intuition The real question at issue is What are the possible [human (m.i.)] processes which can be carried out in computing a number? (Turing 1936) • Universal Turing Machine concept

• Argument by conuence huge number of formalism turned out to be equivalent Practice: Thé ultimate model of the modern computer?

• Stored program machine concept: ``The universal machine is a storedprogram machine; that is, unlike Babbage's al-purpose machine, the mechanisms used in reading a program are of the same kind as those used in executing them (Gandy 1988) • Interchangeability of hardware and software: [W]e have often simplied the circuit at the expense of the code (Turing 1947) ⇒ Epistemological reasons that cannot be isolated from socio-historical forces TM -the ultimate model of computability? The original...

• Turing's thesis The computable" numbers may be described briey as the real numbers whose expressions as a decimal are calculable by nite means. [...] According to my denition, a number is computable if its decimal can be written down by a [computing] machine.

⇒ No halting problem in classical sense `normal' (Turing)-computation is innite computation

• Basic operations? [...] so elementary that it is not easy to imagine them further divided (Turing 1936) Quintuple notation: q i , S j : S k M q i,S j ⇒ Compound of move and print

• Implicit and explicit conventions/restrictions: Two types of squares (F and A) and two kinds of symbols

The sequence computed (F -squares) cannot be changed The machine will never print a symbol on an empty F -square if the F -square preceding it is also empty One-way innite tape ⇒ spurious Turing convention (Post 1947) TM -the ultimate model of computability?

In the meantime in 1936... Quadruple notation: q i S j Lq ij ; q i S j Rq ij ; q i S j S k q ij ; two-way innite no alternate squares markers are special symbols halts (when a q i S i is reached for which ther is no quadruplet) ⇒ A number of comparisons with [2] will occur to a reader of that note ⇒ An appendix: [I]t seems to the writer that Turing's preoccupation with computable numbers has marred his entire development of the Turing machine. We therefore suggest a redevelopment of the Turing machine based on the formulation given in the body of the present paper Why then are we talking of thé Turing machine and not, for instance, the Post-Turing machine?

TM -the ultimate model of computability?

Why not Post's machine? Epistemological explanations it is often and erroneously written (Kleene [1987b, p. 56], and[1981, p. 61]) that Post's contribution here was essentially the same as Turing's, but in fact it was much less. This derives from the idea of a symbolic logic rather than that of an algorithm, and may be described by saying that each member of the set is at some time generated [Produced, created, in

g k2 P i k 2 . . . g km k P i k m k g k(m k +1)
produce produce produce

g 1 P i 1 g 2 P i 2 . . . g m P i m g (m+1)
abaP 3 b ababaaaaaababab .... and Normal form:

g i P i 1101P i 110111011101000000 produces P i g i P i 001 11011101000000001
Post's thesis I Any nite sequence of symbols that can be generated by some nite process can also be generated by a system in normal form

TM -the ultimate model of computability? Why not Post's machine?

• Appeal to intuition? [for the thesis to obtain its full generality] an analysis should be made of all the possible ways the human mind can

• Technique II (automation): sequencing, loops and discrimination

The advent of large-scale digital computers has added a fth operation of considerable importance, namely discrimination. This, in general terms, is the operation of making a choice of one of several brances of program (or course or procedure), depending on the outcome of a previous calculation. This operation is peculiar to discrete-variable machines since its outcome is not continuous. The purely analogue machine cannot distinguish the larger of two suciently small numbers, or the sign of either. • symbolization of program instructions (used already with Mark 1 and Bell lab machines)

• storage of numbers and programs in an electronic medium corollary of high-speed -does not require logical insight solvable by computing machines, we do not need to consider an innite series of computers of greater and greater complexity but may think only of a single machine. Even more surprising than the theoretical possibility of such a `best possible' computer is the fact that it need not be very complex. The description given by Turing of a universal computer is not unique.

Many computers, some of quite modest complexity, satisfy the requirements for a universal computer. • Generated sets of nite sequences of letters -symbolic reasoning (Post 1921) • Human computation of real numbers (Turing 1936) • Finite combinatory processes that solve problems in a nite number of steps (Post 1936) • ....

• Computation by a machine (cfr Gandy's work)

• Interactive computation

• (Articial) intelligence
⇒ Are these all the same (without opposing CTT)?

= models ⇒ = Uses Some examples from early programming practices A syntactical theory of language:

• Backus-Naur form: It was only in trying to describe ALGOL 58 that I realized that there was trouble about syntax description. It was obvious that

Post's productions were just the thing [m.i.], and I hastily adapted them to that use. (Backus, 1981) <letter string> ::= <letter> | <letter string><letter>

<letter> ::= A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T | U | Y | W | X | Y | Z
• Patterson and Burroughs company: We are interested in your productions and your theory of canconical languages, which appeal to us as possibly the most natural approach to the theory of computabiliy from the standpoint of one interested in syntactical machines.

(letter to Post, Dec. 17, 1952)

• Chomsky and formal language theory: [...] when I came across Post's work (in Davis) it was obvious that this was a good framework for systems of the sub-recursive hierarchy that could be adapted to the study of language, specically context-sensitive and context free grammars (personal communication, 2005) = models ⇒ = Uses Some examples from early programming practices Towards a mathematical science of computation • Enables the reconnecting of dierent pratices and views through history fundamental in computing

• Be realistic about ones own practice: not only driven by pure science but shaped by complex socio-historical process

• One should not get stuck in one (class of ) model(s) it shall and will limit your thinking cfr debates on hypercomputability

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  Each community has their own favourite models and heroes.... ⇒ If we want a more diverse understanding of computing, need to step away from what we are taking for granted... TM -thé model of computing Theory: Thé ultimate model of computability ⇒ Church-Turing thesis, v.1 Any computable function is Turing computable Practice: Thé ultimate model of the modern computer ⇒ Church-Turing thesis, v.2 Anything that can be computed (by a TM) can be computed by a computer = Computability thesis: Equivalence class of formalizations of computability that impose a basic theoretical limit on what can be computed protection against computationalism ⇒ Why the TM and not e.g. λ-calculus? Part I: Why thé model L. De Mol

(

  A) Perform one of the operations (O 1 O 4 ) and go to instruction d j (B) Perform operation O 5 and according as the box the worker is in is marked or unmarked follow direction d j or d j . (C) Stop. TM -the ultimate model of computability? ...and Post's work on Thue's problem (1947) • Use of the Turing machine (cfr Church's review) but in modied form, viz. formulation 1:

  [...] Post did not attempt to prove that his formalism coincided with any other formalism [...] Post gave no hint of a universal Turing machine. Most important, Post gave no analysis, as did Turing, of why the intuitively computable functions are computable in his formal system. (Soare 2009) ⇒ Countered if one knows more about Post's history which was essentially neglected or forgotten... TM -the ultimate model of computability? Why not Post's machine? Post's thesis I • Not published in the early 1920s! • 1920-1921: Post attacks the niteness problem of symbolic logic prove the decidability of formal logic and in particular rst-order logic • Need for the ultimate form of nite symbolic reasoning • Sequence of reductions inspired by PM to normal system via Post production system through generalization and simplication

  set up nite processes to generate sequences.(Post 1965) (∼ Turing's What are the possible processes which can be carried out in computing a number? ) ⇒ How else would Post have ended up with a formalism quasi-identical to Turing's??• Universality? We shall refer to the system K as the complete normal system because, in a way, it contains all normal systems [...] The complete normal system would thus correspond to Turing's universal computing machine. ) (Post 1965) ∼ a uniform process U which, provided with a (putative) specication for a calculable function F and a numerical argument m will calculate the value f m if this is dened(Gandy 1988) cfr also Church's e • Equivalences/conuence? [Post] uses the argument by conuence; a system of one of the sorts considered can be simulated by any of the others + correspondence with Church (July 1936) reduction of λ-denable to canonical forms ⇒ So, again, why Turing machine and not Post machine? TM -the ultimate model of computability? Why not Post's machine? Socio-historical explanations The social context • Post: Jewish immigrant from Poland (cfr anti-semitism and Jewish quota in the US), manic-depressive, handicaped, position at CUNY mostly teaching, ample time for research + issues with getting his work published • Turing: a distinctive family tree of English gentry, far from rich but determinedly upper-middle-class in the peculiar sense of the English class system; permanent position at King's College ⇒ ∼ Ivy league or, say, University of Aachen don't say that social bakcground does not matter! TM -the ultimate model of computability? Why not Post's machine? Socio-historical explanations The 1930s-40s intellectual context • Church's reviews of (Post 1936) and (Turing 1936): [Post] proposes a denition of nite 1-process which is [...] in fact equivalent, to computation by a Turing machine.[...] He does not [...] regard his formulation as certainly to be identied with eectiveness in the ordinary sense, but takes this identication as working hypothesis in need of continual verication.[...] To dene eectiveness as computability by an arbitrary machine [...] would seem to be an adequate representation of the ordinary notion, and if this is done the need for a working hypothesis disappears. ⇔ [Computability by a Turing machine] has the advantage of making the identication with eectiveness in the ordinary (not explicitly dened) sense evident immediately • Recognition of Post's contribution but in a footnoteish manner (cfr e.g. (Kleene 1952) or (Gödel 1964)) • Post's own usage of the (modied) Turing machine in the Thue paper TM -the ultimate model of computability? Why not Post's machine? Socio-historical explanations The 1950s-60s intellectual context • Davis' book (+ lectures): the notion of Turing machine has been made central in the development. // The present formulation [...] follows Post [6]. • The appeal of the universal machine idea as a model for computers "The reason we have chosen this particular model [a multitape Turing machine] ]is that it closely resembles the operation of a present-day computer (Hartmanis and Stearns 1965) ⇒ ... immersion of the Turing machine concept in computer science ⇒ But, clearly, thé Turing machine is also Post's (and probably Kleene's and Church's) ⇒ Not jut a historical correction Turing's notion of computability = Post's solvability ∼ current notion of computability + need to be aware of these socio-historical forces TM -the ultimate model of the modern computer? TM -the ultimate model of the modern computer? Turing, the inventor of the rst computer... • Epistemological claim: The universal Turing machine as thé model of the modern stored-program computer • Historical claim: Turing is the father of the modern computer Several objections ↔ Historical claim: • Stored-program is a vague and ad hoc notion which ignores historical reality (Bromley 1985; Haigh 2014) • Connecting real computers to a model was not obvious immediately; conations between universal machine and stored-program computer (Daylight 2014) ⇒ Here: stored program itself is a corollary of a practice -Theory is not the captain TM -the ultimate model of the modern computer? A dierence in context... • Number-crunching in applied mathematics (eg ring tables; navigation; etc): laborious, lengthy and error-prone processes of computation • Steady process of mechanization and, ultimately, automatic machines (cfr. De Mol and Durand-Richard, forthcoming) • e.g. US military in need of ring tables: Computation of ring tables at BRL and Penn: human computers and two dierential analyzyers Human work too slow, dierential analyzer too inaccurate (other type of slowness): You nally get errors in the nal result of the order of 1 percent or worse. The ballistics work that we were doing required errors ofsomething like 10 times better than that. So what was being done at the time was that things were being run on the analyzer and then they were being smoothed by hand calculations... (Eckert, 1977) ⇒ Pragmatism: Development of techniques and technologies that t the purpose: computations that are (partially) automatic, fast and precise TM -the ultimate model of the modern computer? Introduction of specic techniques • Technique I (precision): digital machines Thus, broadly speaking, the continuous computers are mechanizations of geometry, while the discrete computers are mechanizations of number theory. [...] the physical limitations inherent in measuring processes result in a region of confusion about each number, which means that numbers are never represented unambiguously. [...] the greater the required accuracy, the more advantageous is the discrete or digital calculator. (Stibitz 1945) From this we get...

⇒

  ultimate model of the modern computer? Why the TM and not x? Epistemological reasons II. Reections on programming notations: explicit formulation of interchangeability hardware and software in the context of automatic programming: Simply enunciated [Turing's theorem] [...] states that any computing machine which has the minimum proper number of instructions can simulate any other computing machine, however large the instruction repertoire of the latter [...] All forms of automatic programming are merely embodiments of this rather simple theorem. (Booth 1960) See (Daylight 2015) I and II ⇒ (Epistemic) Realization of the need for a rapprochement between the theory and practice of computing UTM as a tool to make more precise and to explain insights from practice ⇒ (Socio-historical) An academic eld in need of an identity (Tedre 2015;Daylight 2015; Bullynck, Daylight and De Mol 2015) ⇒ Steady conation between what happened after the facts and what actually happened retro-tting the past to current needs (cfr the need for examples and heroes in science) Intermediate conclusion... Yes, the TM is a basic model of computability and computers, BUT: That is as much an eect of a socio-historical and contingent processes, as it is a reection of a signicant epistemological insight Apparent epistemological arguments tend to ignore historical reality with a risk of creating a research bias Historical and social processes (in science) of forgetting, attening and simplifying that contribute to this (cfr Husserl) Not always in function of science (eg political or other agendas) Plenty of other cases in the history of science cfr the case of Gauss ... and a remark about equivalence arguments Central argument for CTT: equivalences between an ever growing number of formal models (cfr Turing 1936; Post 1936; Kleene 1952; Gandi 1988; etc) ⇒ It suces to focus only on the best model ⇒ Yes, but only under a particular notion of equivalence ⇒ Ignores/erases/forgets intensional and fundamental dierences Reconstructions Why should we care (about the dierences)? = models ⇒ = Theses A case from the 1920s and 1930s Post, Church, Curry, Schönnkel etc: dierent problem contextsvery dierent formalisms → Formalists in search of an ultimate form/building blocks of symbolic logic/reasoning (NOT computation) [T]he method of combinatory iteration completely neglects [...] meaning, and considers the entire system purely from the symbolic standpoint (Post 1920) We are led to the idea [...] of attempting to eliminate by suitabe reduction the [...] fundamental notions [...] from those contexts in which we are dealing with completely arbitrary, logically general propositions (Schönnkel 1924) [we] abstract from the meaning of the symbols [which allows to] regard the proving of theorems (of formal logic) as a game played with marks on paper according to a certain arbitrary set of rules. (Church 1932) = models ⇒ = Theses A case from the 1920s and 1930s Partially reect diversity of dierent intuitive notions • Eective calculability (Church 1936)• an abstract (or strictly formalized) theory [...] [that gives] an analysis of those universal principles in accordance with which valid thinking goes on(Curry 1928)

⇒• 11101000000110111011101 =⇒

 11101000000110111011101 Dierent context require dierent models their intensional differences are fundamental from that perspective (see also: Martinit 2020; Daylight forthcoming) ⇒ ... and, possibly, multiplicity of theses instead of just one: SECD machine formalizes our notion of information-processing system = models ⇒ = Capabilities An example from the literature on universality • Small weakly universal Turing machine (Neary and Woods 2009) Universality in Elementary Cellular automata (Cook 2004) ⇒ weak and semi-weak universality allow the blank portion of the Turing machine's tape to have an innitely repeated word to the left, and another to the right. ⇒ Weak and semi-weakness are possible generalizations only for models that rely on an actual innite tape ..... tag systems = models ⇒ = Capabilities Tag systems and weakness Let T P ost be dened by Σ = {0, 1}, v = 3, 1 → 1101, 0 → 00 A 0 = 10111011101000000 ⇒ Primitive assertion 101110111010000001101 1101110100000011011101 = models ⇒ = Capabilities Tag systems and weakness Let T P ost be dened by Σ = {0, 1}, v = 3, 1 → 1101, 0 → 00 A 0 = 10111011101000000 ⇒ Primitive assertion 101110111010000001101 1101110100000011011101 models ⇔ thé model We do not attach any character of uniqueness or absolute truth to any particular system of logic. The entities of formal logic are abstractions, invented because of their use in describing and systematizing facts of experience or observation [...] In building the geometry, the proposed application to physical space serves as a rough guide in determining what properties the abstract entities shall have, but does not assign these properties completely. (Church 1932) Abstraction is always bounded by the reality it abstracts from but never completely captures it ⇒ One abstraction only cannot and should not be allowed to suce = models ⇔ thé model : If models of computation are abstractions from a reality : If we assume that one and only one model suces to capture computation that never assigns the properties of that reality completely : Then we assume that, essentially, reality does not change what was considered computable in the 1930s equates to what is considered computable today : ⇒ CTT in terms of TM establishes a theoretical limit but does not capture what computation means today, nor what can actually be computed. TM -the ultimate model of computability? L. De Mol Discussion • Complexication of our own histories is a good thing, not a bad thing! Shows philosophical and practical richness of the eld.

•

  Church's and Kleene's work on recursive functions, λ-denability and dening eective calculability (Church's thesis)... • ... and Post's thesis II TM -the ultimate model of computability? Finite combinatory processes. Formulation I... • Post's thesis (II) A general problem is solveable in the intuitive sense if, for each instance of the problem, we can set-up a nite-1-process that solves it.(cfr De Mol 2013)

•

  The workspace: Two-way innite sequence of boxes in which the work leading from problem to answer is to be carried out a box is either marked or unmarked (binary alphabet) a worker or problem solver works in one box at a time and is capable of a set of primitive acts :O 1 mark a box; O 2 unmark a box; O 3 move one box to the left; O 4 move one box to the right; O 5 determining whether the box he is in is marked or unmarked

  • A set of directions, which sets up a deterministic process when applied to each speciifc problem, are in one of the following forms:

An epistemological obstacle will encrust any knowledge that is not questioned. (Bachelard 2002(Bachelard /1938) ) Diolch! Thank you!


Violetta Lonati
email: violetta.lonati@unimi.it

Andrej Brodnik
email: andrej.brodnik@upr.si

Tim Bell
email: tim.bell@canterbury.ac.nz

Andrew Paul Csizmadia
email: a.p.csizmadia@newman.ac.uk

Liesbeth De Mol
email: liesbeth.de-mol@univ-lille.fr

Henry Hickman
email: henry.hickman@pg.canterbury.ac.nz

Therese Keane
email: tkeane@swin.edu.au

Claudio Mirolo
email: claudio.mirolo@uniud.it

Mattia Monga
email: mattia.monga@unimi.it

Matti Tedre
email: matti.tedre@uef.fi

Characterizing the Nature of Programs for Educational Purposes

Programming plays a paramount role in many educational policies and initiatives. However, the current focus on coding skills poses a risk of giving pupils an over simplistic and impoverished idea of what programming means and involves. Their experiences would be much more significant if learning were aimed at understanding the richness of the nature of programs.

Programs are strange creatures that escape simple definitions: they are real -they affect our real lives; they are abstract -they process abstract entities; and they are concrete -they take up space in digital devices memory, and can be copied, transferred, corrupted. Thus, understanding the multifaceted nature of programs is crucial knowledge for all citizens of the digital era, and a fundamental * co-leader

BACKGROUND AND RELATED WORK

The focus on programs. Computer programs are part of our daily life, we use them, we provide them with data, they support our decisions, they help us remember, they control machines, etc. Programs are made by people, but in most cases we are not their authors, so we have to decide if we can trust them. Programs enable computers and computer-controlled machines to behave in a large variety of ways. They bring the intrinsic power of computers to life. Programs have a variety of properties that all citizens must be aware of; due to the intangible nature of programs (NoP), most of these properties are unusual and peculiar, but fundamental for understanding the digital world. In other terms, understanding the NoP is a key component of the computing literacy: it is crucial to enable a creative and conscious use of computing devices, and should be one of the main outcomes of computing education -alongside with, e.g., the development of problem solving and computational thinking skills. Moreover it should be part of any effort aimed at bringing digital competences to the general public. An attempt in this direction has been carried out by the WG proponents in the occasion of defining the Programming competence in DigComp 2.2 framework [? ]; the outcome of this work is a preliminary list of knowledge statements and examples about the NoP as reported in [? ].

However, the full understanding of NoP might not be a natural learning outcome of CS activities. For instance, using visual programming environments does not imply that students are able to recognize that the programs they write have the same nature as the "apps" they use on their mobile phones. Similarly, unplugged activities aimed at developing computational thinking skills might be perceived as disconnected from the use of digital devices and programs in everyday life [? ? ]. To overcome these limited perspectives, teachers need to be aware of what NoP is, and use this knowledge to inform their teaching practice.

Programs, programming, and computational thinking. The centrality of programming in CS is reflected in most computing education initiatives1 , which indeed often include some type of programming activity, mainly under the term 'coding'. One can even argue that, for many teachers, CS is just a synonym for coding [? ].

Another fundamental component of computing education revolves around the idea of Computational Thinking (CT) [? ]. Even if there is no its ultimate definition, this idea concerns the ability to address "problems in a way that enables us to use a computer and other tools to help solve them" [? ]. CT encompasses a variety of creative cognitive processes and activities, like modeling reallife situations, representing information in digital form, organizing data, analyzing and generalizing computational solutions, assessing their social impact, etc. In other terms, CT goes far beyond coding and tries to represent and value the greater richness of computing. Since the above mentioned activities play a fundamental role also in the process of designing programs, our WG approach shares a similar scope as CT. A fundamental difference is that CT-based frameworks mostly focus on CT practical and cognitive skills, while here we chose to reflect about the underlying knowledge about the NoP, that CT activities should both promote and stem from.

Historical and philosophical perspective. As one can expect, the NoP has had a relevant role in the broader discussion about the nature of CS. Some very significant contributions on the issue appeared in the '80-90s [? ? ? ], including the triggering question "Is CS a science?". CS and its impact on society has greatly evolved since, see e.g. [? ? ? ] for more recent contributions on the nature of CS, and [? ] for an articulate presentations of a historical perspective. Such a discussion, however, has been conducted by and for CS, philosophy, and history matter experts, and has rarely reached a wider audience. Notable exceptions are [? ? ? ].

Focusing on programs, the ongoing project "PROGRAMme" starts with the premises that the seeminingly simple question "What is a computer program?" has no simple answer today2 . It thus aims at developing "a coherent analysis and pluralistic understanding of 'computer program' and its implications to theory and practice", by taking a historical and philosophical approach. The project plans to consider the various characterizations of programs that derive from different viewpoints and pertain to different historical phases of the development of the discipline (either in academia or industry).

Considering "PROGRAMme" and the mentioned references on the nature of CS, we take a different approach. Instead of analyzing and contrasting the different points of view about the NoP, we want to identify the fundamental tenets that bring together different views. Specifically we target the educational arena with a goal to develop a NoP framework that teachers and educational policy makers can use in framing CS teaching practice and curricula.

The Nature of Programs and the Nature of Science. The expression "Nature of Programs" draws inspiration from "Nature of Science" (NoS), an expression from the '70s that refers to the fundamental characteristics of science knowledge and scientific inquiry, as derived from how it is produced: a necessary knowledge to make informed decisions with respect to the ever-increasing scientificallybased personal and societal issues [? ]. NoS is a significant component of scientific literacy and it is argued that NoS cannot be learned simply by studying science concepts or attending science labs, but it must be addressed explicitly with active reflective practice and discussions among students in their learning contexts [? ]. This also implies that teachers should have a "shared accurate view of NoS" and agree that NoS needs to be taught and assessed explicitly [? ].

CS education would benefit as well from a similar approach, in order to boost its contribution in creating a CS-literate society that is able to make informed decisions on CS-related issues. This working group proposes a first step in this direction, focusing on the Nature of Programs (including the way that they are created and built), due to the role that programs play in CS and all society.

Cf. CS4ALL (https://www.csforall.org/) and Informatics for All Coalition (https:// www.informaticsforall.org).

See the project website https://programme.hypotheses.org


Vincent Savaux
email: vincent.savaux@b-com.com

Flexible Communication System for 6G Based on Orthogonal Chirp Division Multiplexing

Keywords: OCDM, 6G, multi-USER, multi-RAT

This paper deals with the orthogonal chirp division multiplexing (OCDM) modulation, and its capability to generate inherited waveforms, such as the orthogonal frequency division multiplexing (OFDM) or precoded OFDM, covering a large number of use cases. To this end, we first prove that the OCDM signal can be generated by a transmission chain composed of a discrete Fourier transform (DFT) matrix, followed by a multiplication by a chirp, and then an inverse DFT (IDFT). Then, we show that the OCDM transmission chain can be split into 6 sub-blocks from which we can generate numerous inherited waveforms. Thus, the OCDM waveform is an interesting candidate for flexible communication systems such as multiple radio access technology (multi-RAT).

I. INTRODUCTION

The orthogonal chirp division multiplexing (OCDM) waveform [START_REF] Ouyang | Orthogonal chirp division multiplexing[END_REF] is a good alternative to the orthogonal frequency division multiplexing (OFDM), precoded OFDM [START_REF] Sahin | DFT-Spread OFDM with Frequency Domain Reference Symbols[END_REF], or filterbank-based modulations [START_REF] Farhang-Boroujeny | OFDM Versus Filter Bank Multicarrier[END_REF], especially for future generations of wireless communication post 5G or 6G. In fact, the data symbols are carried by chirps in OCDM instead of sinusoids in OFDM, such as illustrated in Fig. 1. This chirp multiplexing allows the receiver to optimally decode the signal in presence of selective channels in both time and frequency domains [START_REF] Bomfin | Low-Complexity Iterative Receiver for Orthogonal Chirp Division Multiplexing[END_REF], whereas the OFDM and precoded OFDM modulations are only sub-optimal against channels that are selective in frequency and time, respectively [START_REF] Bomfin | A Theoretical Framework for Optimizing Multicarrier Systems Under Time and/or Frequency-Selective Channels[END_REF]. This optimal reception is due to the fact that the chirps inherently spread the possible errors induced by the propagation channel over the whole time and frequency grid. This feature is also achieved by the orthogonal time frequency space (OTFS) modulation [START_REF] Hadani | Orthogonal time frequency space modulation[END_REF], but it requires a precoding over a block of OFDM symbols, whereas the timefrequency spreading in OCDM is carried out symbol per symbol, therefore limiting the latency. In addition, the OCDM modulation scheme may be especially relevant for sensing and communication in 6G [START_REF] Wang | Towards Integrated Sensing and Communications for 6G[END_REF], since chirps can be used in radar applications.

The OCDM modulation is based on Fresnel transform instead of Fourier transform in OFDM. It is nevertheless shown in [START_REF] Ouyang | Orthogonal chirp division multiplexing[END_REF] that the OCDM modulation can be processed by means of a usual inverse discrete Fourier Transform (IDFT), preceded and followed by a multiplication by chirps in both frequency and time domains, therefore reducing the complexity of the transceiver compared with Fresnel transform. The authors in [START_REF] Omar | Spectrum design for orthogonal chirp division multiplexing transmissions[END_REF] suggest an alternative modulation process by describing the OCDM as the usual OFDM modulation precoded by a DFT matrix (similar to DFT-s-OFDM) and a multiplication by a chirp. The scheme in [START_REF] Omar | Spectrum design for orthogonal chirp division multiplexing transmissions[END_REF] offers the advantage that the sampling rate at the output of the IDFT can be set as expected whereas the solution in [START_REF] Ouyang | Orthogonal chirp division multiplexing[END_REF] is limited as it generates OCDM signal sampled at Nyquist rate. Furthermore, the authors in [START_REF] Omar | Spectrum design for orthogonal chirp division multiplexing transmissions[END_REF] show that filtered-OCDM can be also designed (similarly to OFDM) in order to reduce the amplitude of the side lobes.

In this paper, on the one hand we mathematically formalize the principle of the OCDM modulation pro-posed in [START_REF] Omar | Spectrum design for orthogonal chirp division multiplexing transmissions[END_REF], whereas the authors in [START_REF] Omar | Spectrum design for orthogonal chirp division multiplexing transmissions[END_REF] introduce their scheme and a posteriori prove that the assumption of orthogonality among chirps holds. On the other hand, we propose a split of the steps of the OCDM modulation into 6 sub-blocks such that enabling or disabling these subblocks leads to a large number of waveforms inherited from OCDM. Among those modulations, we can find the OFDM, the single carrier-frequency division multiplexing (SC-FDM), or the chirp spread spectrum (CSS), which are used in numerous communication technologies and standards. Thus, besides its inherent robustness against time-frequency selective channels [START_REF] Bomfin | Low-Complexity Iterative Receiver for Orthogonal Chirp Division Multiplexing[END_REF] and its capability for sensing and communication applications [START_REF] Wang | Towards Integrated Sensing and Communications for 6G[END_REF], the OCDM modulation scheme enables the design of multi-radio access technology (multi-RAT), which could be implemented in 6G systems while being compatible with other existing technologies.

The remaining of this paper is organized as follows: Section II introduces the OCDM modulation and mathematically formalizes the DFT-based scheme with precoding and multiplication by a chirp. The split of the OCDM modulation into sub-blocks, and how inherited waveforms are obtained from OCDM is described in Section III. Finally, Section IV concludes this paper.

II. OCDM SYSTEM MODEL

We consider the OCDM signal x n sampled at Nyquist rate, where n = 0, 1, .., N -1 and N is the number of subcarriers (in OCDM the subcarriers refers to chirps instead of sinusoids as in OFDM), as the inverse discrete Fresnel transform of the symbols of constellation {C 0 , C 1 , .., C N -1 } [1]:

x n = 1 N N -1 k=0 C k e -j π N (n-k) 2 +j π 4 gk,n , (1)
where we note g k,n = e -j π N (n-k) 2 +j π 4 the chirp carrying the constellation symbol C k . In order to design the modulation scheme based on IDFT such as described in [START_REF] Omar | Spectrum design for orthogonal chirp division multiplexing transmissions[END_REF], it must be noticed that the DFT of the chirp g k,n , that we denote by G k,m , can be straightforwardly expressed by using the quadratic Gauss sum [START_REF] Savaux | Frequency domain preamble-based channel estimation and equalisation in LoRa[END_REF], which leads to:

G k,m = N -1 n=0 g k,n e -2jπ mn N =e j π 4 N -1 n=0 e -j π N (n-k) 2 e -2jπ mn N = √ N e j π N (m 2 -2km) 0 n=0 e jπ(N n 2 -(2m-2k)n) = √ N e j π N (m 2 -2km) , (2)
where m is the index of the frequency samples. Then, by substituting g k,n by the IDFT of G k,m in (1), we obtain:

x n = 1 N N -1 k=0 C k 1 N N -1 m=0 G k,m e 2jπ mn N = √ N N 2 N -1 m=0 N -1 k=0 C k e j π N (m 2 -2km) e 2jπ mn N = √ N N 2 N -1 m=0 e jπ m 2 N N -1 k=0 C k e -2jπ km N e 2jπ mn N , (3)
where we highlight the DFT of the constellation symbols {C 0 , C 1 , .., C N -1 } within bracket, the multiplication by the chirp e jπ m 2 N and the IDFT, such as described in [START_REF] Omar | Spectrum design for orthogonal chirp division multiplexing transmissions[END_REF]. We can generalize (3) to consider a DFT of size K ≤ N corresponding to K symbols of data {C 0 , C1, .., C K-1 } and a IDFT of size N , such that Q = N -K 2 null subcarriers are located at the edge of the band. Then x n is expressed as

x n = √ K KN N -1 m=0 α m e jπ (m-Q) 2 K K-1 k=0 C k e -2jπ k(m-Q) K e 2jπ mn N , (4)
with α m = 0 if the m-th subcarrier is null, and α m = 1 otherwise. The corresponding OCDM transmission chain is illustrated in Fig. 2, where the cyclic prefix has been introduced to avoid intersymbol interference, similarly to OFDM. It is worth emphasized that the oversampling rate at the output of the IDFT is given by the ratio N K . The corresponding OCDM receiver is composed of the inverse sub-blocks to the transmitter, such as illustrated in Fig. 3. It includes the CP removal, the DFT of size N leading to the received samples Y m , the multiplication by the inverse chirp e -jπ m 2 N , and the IDFT of size K. Moreover, the channel estimation and equalization is performed after the DFT of size N . The way the channel is estimated and equalized is out of the scope of this paper, but details can be found in [START_REF] Ouyang | Orthogonal chirp division multiplexing[END_REF], [START_REF] Bomfin | A Theoretical Framework for Optimizing Multicarrier Systems Under Time and/or Frequency-Selective Channels[END_REF], [START_REF] Omar | Spectrum design for orthogonal chirp division multiplexing transmissions[END_REF].

Fig. 4 shows the bit error rate (BER) performance of OCDM compared with OFDM versus the signal to noise power ratio (SNR) in dB. Two channel models are considered: the additive white Gaussian noise (AWGN) channel (a), and Rayleigh channel (b). The latter has been modeled using a uniform power profile, and the maximum delay spread is equal to 1/4 the length of the CP. On the other hand, the OCDM signal is composed of K = N = 256 subcarriers and data symbols are randomly chosen from a 16-QAM constellation. Furthermore, an equalizer based on the minimum mean square error (MMSE) has been used [START_REF] Omar | Spectrum design for orthogonal chirp division multiplexing transmissions[END_REF]. We can observe that the performance of the OCDM and OFDM match in presence of the AWGN channel, but the OCDM modulation scheme outperforms the OFDM over Rayleigh channel for SNR values larger than 20 dB. Such as aforementioned, this result is due to the inherent spreading of the errors induced by multipath channels by the chirps in OCDM, which cannot be achieved with sinusoids in OFDM.

In the following, we suggest a split of the OCDM transmission chain into sub-blocks in order to highlight other modulation schemes by simply enable and disable these sub-blocks. Beyond this suggested split, obtaining numerous inherited modulations from the OCDM waveform makes this modulation a relevant solution for flexible multi-technology systems, besides its inherent robustness against time and frequency selective channels.

III. GENERATING OTHER WAVEFORMS FROM OCDM

Such as illustrated in Fig. 2, we suggest a split of the OCDM transmission chain into 6 sub-blocks noted 'a', 'b', .., 'f'. After a brief description of each of the blocks, we list (non exhaustively) the different modulation schemes that can be obtained by enabling and disabling these sub-blocks, covering a large number of communications technologies, therefore proving the capability of the OCDM waveform to design flexible communication systems such as multi-RAT transceivers.

The aforementioned sub-blocks are summarized as follows:

• a : the binary to symbol conversion (or mapping b2c). • b : the mapping of these symbols to subcarriers.

• c : the DFT of size K and the parallel to series conversion. • d : the multiplication by the chirp e jπ m 2 N . • e : the IDFT of size N .

• f : the CP addition. From these 6 sub-blocks, we can define a large number of inherited modulation schemes. Among others, we describe the three waveforms that are usually used in numerous communication standards and technologies, such as: 1) a + b + c + e + f: the OFDM precoded with a DFT matrix, such as the SC-FDM or the DFTs-ODFM. They are used in particular in uplink in LTE, LTE-M, NB-IoT [START_REF]3GPP TS 36.211[END_REF], or in 5Gnr [START_REF]3GPP TS 38.211[END_REF]. In that case, the sub-block 'a' corresponds to a conversion of binary packets to symbols of constellation of types X-PSK or M-QAM. Such as indicated in introduction, it is possible to design an optimal receiver for the precoded OFDM, but only for frequency selective channel [START_REF] Bomfin | A Theoretical Framework for Optimizing Multicarrier Systems Under Time and/or Frequency-Selective Channels[END_REF]. 2) a + b + e + f: the OFDM, used in plenty of cellular technologies (LTE, 5Gnr, NB-IoT, etc.), in WiFi, as well as in digital video and audio broadcasting (DVB, DAB, DRM). Once again, 'a' corresponds to a binary conversion of binary packets to symbols of constellation of types X-PSK or M-QAM. Moreover, such as indicated in introduction, a suboptimal received only is achievable in OFDM for frequency selective channels [START_REF] Bomfin | A Theoretical Framework for Optimizing Multicarrier Systems Under Time and/or Frequency-Selective Channels[END_REF]. 3) a + b + c + d + e: the CSS waveform, such as used in LoRa [START_REF]Lorawan TM specification[END_REF]. In that case, the sub-block 'a' is a conversion of binary packets to an integer symbol corresponding to the index of the unique non-null subcarrier in sub-block 'b'. It is worth mentioning that we here omit the CP addition to match the usual LoRa technology, but it is possible to consider a CSS modulation with CP addition. Moreover, note that the scheme 'a + b + c + d + e' enables the modulation of chirps symbols at any desired sampling frequency, which could be used in sensing and communication applications for 6G. These three examples cover a large number of applications from low data rate solutions (NB-IoT, LoRa) up to (ultra-)high data rate systems (4 & 5G). Furthermore, it can be noticed that the OCDM transmission chain is generic enough to cover use cases dedicated to specific technologies as well. For instance, we can highlight the chain 'b+e+f' where 'b' is a pseudo-random choice of the subcarrier index. In that case, 'b+e+f' corresponds to the transmission of sinusoids with pseudo-random frequency hopping, used to generate NB-IoT preambles during the random access procedure. Otherwise, the chain 'd+e+f' with an adaptation of step 'd' (m 2 is substituted by u.m(m+1) where u is an integer number) leads to a Zadoff-Chu (ZC) sequence, which is used for synchronization signals and pilots in 4&5G, NB-IoT, and LTE-M. The authors of [START_REF] Basar | A new technique for OFDM: OFDM-index modulation[END_REF] suggested the OFDM waveform with index modulation (OFDM-IM), where the binary data stream is converted in complex symbols from a constellation, but also defines the active subcarriers. The OFDM-IM can then be designed using the chain 'a + b + e + f' where the sub-blocks 'a' and 'b' are adapted to the index modulation. Recently in [START_REF] Bemani | AFDM: A Full Diversity Next Generation Waveform for High Mobility Communications[END_REF], a new waveform generalizing the OCDM has been proposed, where the chirp e jπ m 2 N in the sub-block 'd' can be substituted by e jπf (m) with f (m) a generic quadratic function of m. This modulation is called affine frequency division multiplexing (AFDM). Thus, we have shown that the OCDM modulation, through its general transmission chain, enables the generation of numerous inherited waveforms, and can actually used to design multi-RAT systems.

IV. CONCLUSION

In this paper, we have presented the OCDM system model and mathematically formalized the signal to highlight the transmission chain based on a precoding composed of a DFT matrix and the multiplication by a chirp, followed by an IDFT. Then, we have split the OCDM transmission chain into 6 sub-blocks from which we can generate numerous inherited waveforms, by enabling or disabling these sub-blocks, such as the OFDM, precoded OFDM, or CSS. It results that the OCDM modulation, besides being a good candidate for 6G communications, enables the design of flexible multi-RAT transceivers compatible with numerous existing standards and technologies. The future works will consist in dealing with the pending technical challenge related to the OCDM.

Fig. 1 .

 1 Fig. 1. Variation in frequency (Y axis) of the data symbols subcarriers in function of the time (X axis) in OFDM (sinusoids) and OCDM (chirps).

Fig. 2 .^Fig. 3 .

 23 Fig. 2. OCDM transmission chain.

Fig. 4 .

 4 Fig. 4. BER performance of OCDM and OFDM versus SNR (dB) for AWGN channel (a) and Rayleigh channel (b).


Alberto Adan-Mas

Jacob Olchowka

Lydie Bourgeois

Philippe Legros

Patrizia Paradiso

Fátima Montemor

Liliane Guerlou-Demourgues

Delamination of Nickel-Cobalt Oxyhydroxides for Electrochemical Energy Storage Applications

Keywords:

Nickel-cobalt oxyhydroxide has been delaminated by tetrabutylammonium (TBA + ) intercalation in aqueous media. The electrochemical performance of the different materials obtained during delamination has been evaluated, with special emphasis on the effect of the intercalated species and their influence on the behavior of the delaminated material, toward electrochemical energy storage applications. Delamination in TBAOH, reported for the first time for nickel-cobalt oxyhydroxide in aqueous media, is an excellent route to increase the number of electrochemically active sites and to enhance the capacity of nickel-cobalt oxyhydroxide. Results show an increase in capacity from 112 mA•h•g -1 in 1 M LiOH at 1 A•g -1 for the nondelaminated precursor to 165 mA•h•g -1 after exfoliation and restacking. Thus, carefully designed exfoliation is a promising route to enhance the performance of electrode materials for electrochemical energy storage.

Introduction

The electrochemical performance of energy storage devices depends on the selection of electrode materials, which are usually optimized to obtain maximum capacity/capacitance, specific power, or cycling stability, depending upon application. Nonetheless, active surface area is always a critical parameter. Related to charge storage, chemical and morphological stability, electrode-electrolyte interaction, and electron and ion transport; a maximized surface area is of extreme importance in the development of novel materials, especially for high-power applications. For that reason, nanostructured materials, in particular two-dimensional layered materials, have become a field of great interest. [START_REF] Pomerantseva | Energy Storage: The Future Enabled by Nanomaterials[END_REF] Layered double hydroxides (LDHs), also known as anionic clays, have been pinpointed as promising pseudocapacitive materials because of their high redox activity, low cost, and environmental friendliness. LDHs are two-dimensional ionic lamellar materials consisting of positively charged brucite-like layers with an interlayer region containing either solvent molecules or anionic species. [START_REF] Wang | Recent Advances in the Synthesis and Application of Layered Double Hydroxide (LDH) Nanosheets[END_REF] However, layered double hydroxide materials present a major limitation: reduced accessibility to the inner surface of the host layers by the electrolyte, which leads to reduced active surface area for energy storage applications. For that reason, many efforts have been made to delaminate such materials. In addition, if the material is successfully delaminated, the positively charged layers can work as building blocks to produce functional materials by integrating them with negatively charged species. Consequently, many attempts have been done to exfoliate different types of LDHs. [START_REF] Wang | Recent Advances in the Synthesis and Application of Layered Double Hydroxide (LDH) Nanosheets[END_REF] The most common procedure consists of the intercalation of molecules/ions in the interlamellar space and the subsequent delamination in an appropriate solvent. Many solvents have been used, such as butanol, [START_REF] Adachi-Pagano | Delamination of Layered Double Hydroxides by Use of Surfactants[END_REF] acrylates, [START_REF] O'leary | Delamination of Layered Double Hydroxides in Polar Monomers : New LDH-Acrylate Nanocomposites †[END_REF] toluene, [START_REF] Jobbágy | Delamination and Restacking of Hybrid Layered Double Hydroxides Assessed by in Situ XRD[END_REF] or formamide, [START_REF] Hibino | New Approach to the Delamination of Layered Double Hydroxides[END_REF][START_REF] Ma | Synthesis and Exfoliation of Co2+-Fe3+ Layered Double Hydroxides: An Innovative Topochemical Approach[END_REF] among others. However, from an environmental and practical perspective, it is of great interest to achieve delamination in water. The first reported aqueous-based delamination was performed by Hibino and Kobayashi [START_REF] Hibino | Delamination of Layered Double Hydroxides in Water[END_REF] for Mg-Al-lactate. The method presents three main advantages compared to delamination in organic solvents, namely, the exclusive use of water, the higher concentration achieved in the delaminated colloidal suspension, and, finally, the fact that the synthesis process does not require heat. Henceforward, other materials have been delaminated in a similar fashion, such as Zn-Al-lactate. (9) Nickel (hydr)oxide and cobalt (hydr)oxide exfoliations are of special interest for electrochemical energy storage applications. [START_REF] Ida | Synthesis of Hexagonal Nickel Hydroxide Nanosheets by Exfoliation of Layered Nickel Hydroxide Intercalated with Dodecyl Sulfate Ions[END_REF][START_REF] Vadivel | Regulating the Cationic Rearrangement of Ni-Rich Layered Oxide Cathode for High-Performance Li-Ion Batteries[END_REF][START_REF] Chen | Design and Preparation of NiCoMn Ternary Layered Double Hydroxides with a Hollow Dodecahedral Structure for High-Performance Asymmetric Supercapacitors[END_REF] On the one hand, nickel-based materials, predominantly NiO and Ni(OH)2, exhibit a well-known electrochemical activity due to faradaic processes but their high resistivity limits their performances. [START_REF] Wang | A Review of Electrode Materials for Electrochemical Supercapacitors[END_REF] Cobalt-based materials (Co3O4 and Co oxyhydroxides) are also promising candidates due to their excellent reversible redox reactions that translate into good cycling stability [START_REF] Wang | A Review of Electrode Materials for Electrochemical Supercapacitors[END_REF][START_REF] Olchowka | Ionic Liquids to Monitor the Nano-Structuration and the Surface Functionalization of Material Electrodes: A Proof of Concept Applied to Cobalt Oxyhydroxide[END_REF] as well as their high conductivity, especially if they contain mixed-valence Co 3+ /Co 4+ . [START_REF] Invernizzi | Controlled Nanostructuration of Cobalt Oxyhydroxide Electrode Material for Hybrid Supercapacitors[END_REF] However, they present either low-capacity values or short potential ranges and, for that reason, its combination with nickel (hydr)oxides is of great interest. Consequently, nickel-cobalt oxyhydroxides were considered for delamination and electrochemical energy storage electrode applications in this work. Nickel hydroxide, cobalt hydroxide, and nickel-cobalt hydroxides were successfully delaminated in water by the intercalation of lactates by Lang et al., [START_REF] Schneiderová | Nickel Hydroxide Ultrathin Nanosheets as Building Blocks for Electrochemically Active Layers[END_REF][START_REF] Schneiderová | Electrochemical Performance of Cobalt Hydroxide Nanosheets Formed by the Delamination of Layered Cobalt Hydroxide in Water[END_REF][START_REF] Schneiderová | Nickel-Cobalt Hydroxide Nanosheets : Synthesis, Morphology and Electrochemical Properties[END_REF] and their electrochemical performance was studied by means of cyclic voltammetry. Another known route is the exfoliation of oxide and oxyhydroxide materials by means of cation intercalation. [START_REF] Liu | Swelling and Delamination Behaviors of Birnessite-Type Manganese Oxide by Intercalation of Tetraalkylammonium Ions[END_REF] A common feature of layered oxides is their cation-exchange property, which enables the modification of the interslab space, ultimately leading to exfoliation. As Sasaki and Ma pointed out, [START_REF] Ma | Nanosheets of Oxides and Hydroxides: Ultimate 2D Charge-Bearing Functional Crystallites[END_REF] there is a well-established route for the exfoliation of layered metal oxides, which consists of first a solid-state calcination synthesis of precursor materials with posterior acidification to produce their hydrated protonic form. Then, these can be modified with positively charged bulky organic ions (TMA + , TBA + , etc.) that increase the interlayer distance, decrease the electrostatic interaction between the layers, and enable their delamination. For instance, Tang et al. recently used this approach to exfoliate birnessite manganese dioxide and cobalt oxyhydroxide in order to perform a controlled restacking to obtain homogeneous Mn-Co composites with enhanced electrochemical performance. [START_REF] Tang | Prediction of Isoelectric Point of Manganese and Cobalt Lamellar Oxides : Application to Controlled Synthesis of Mixed Oxides[END_REF][START_REF] Tang | Synergy of Mn and Co in Slab-Based Nanocomposites for Hybrid Supercapacitors: Impact of Restacking Process on Electrochemical Properties[END_REF] Several materials have been exfoliated in that manner, such as MnO2, [START_REF] Liu | Swelling and Delamination Behaviors of Birnessite-Type Manganese Oxide by Intercalation of Tetraalkylammonium Ions[END_REF] TaO3 [START_REF] Fukuda | Colloidal Unilamellar Layers of Tantalum Oxide with Open Channels[END_REF] or HCoO2. [START_REF] Kim | Soft-Chemical Exfoliation Route to Layered Cobalt Oxide Monolayers and Its Application for Film Deposition and Nanoparticle Synthesis[END_REF] However, to the best of the author's knowledge, neither delamination of nickel nor mixed nickel-cobalt oxyhydroxide has been proposed for electrochemical energy storage applications. The different materials obtained during the delamination process of nickel-cobalt oxyhydroxide materials in water, by means of TBA + intercalation, are characterized and studied here as potential electrode materials for charge storage in alkaline media. Finally, the influence of different cations in the electrolyte, namely, K + and Li + , is considered during the charge-discharge of the different active materials and its role discussed. The process used for the delamination of nickel-cobalt oxyhydroxide nanoslabs in tetrabutylammonium hydroxide is exemplified in Figure 1 and detailed hereinafter. It is worth noting that, in all experimental steps, the metal (oxyhydr)oxide layer remains unchanged and is characterized by MO6 edge sharing octahedral units. During the delamination, M-O distances could slightly be modified depending on the M n+ oxidation state, but changes mainly occur in the interlayer spacing, where ion substitutions occur.

Experimental Section

Synthesis of α-Nickel-Cobalt Hydroxide

Typically, 5 g of nickel nitrate hexahydrate and 10 g of cobalt nitrate hexahydrate were dissolved in 100 mL of distilled water. 40 mL of 2M NaOH containing 1.5 mL of H2O2 was quickly poured in the solution, forming a green-brown precipitate of α-Ni-Co(OH)2•(CO3 2-, 2NO3 -)y. This solution was aged for 24 h. It was then centrifuged at 4000 rpm for 5 min and washed with H2O and ethanol 6 times until neutral pH to remove impurities. Approximately 7 g of the α phase, named as NCOH {nickel cobalt hydroxide}, was recovered after the powder was dried at 40 °C for 24 h.

Oxidation Reaction to γ-Nickel-Cobalt Oxyhydroxide 2.6 g of NCOH was suspended in a mixture of 20 mL of 10M NaClO (Prolabo) with 80 mL of 5M KOH and kept under stirring for 24 h. The resulting suspension was centrifuged at 4000 rpm for 5 min. Once the powder was recovered, it was washed 5 times with distilled water and several times with ethanol until neutral pH. 2.2 g of the resulting γ-phase was recovered and named as NCOx-HNK {as in nickel-cobalt oxide intercalated with protons (H), sodium (Na), and potassium (K)}.

Protonation of γ-Nickel-Cobalt Oxyhydroxide

1.8 g of NCOx-HNK was suspended in 150 mL of 10 mM HNO3 and stirred overnight. The resulting product is a protonated version of the γ phase in which the sodium and potassium ions intercalated in the interslab space of material are exchanged by protons. After washing with distilled water and ethanol several times until neutral pH is reached, 1.39 g of product was recovered and designated as NCOx-H. It is worth mentioning that this step is critical since oxyhydroxides easily dissolve in acid. Higher concentrations and other acids, such as HCl, were also tested, resulting in the dissolution of the product.

Delamination and Restacking of γ-Nickel-Cobalt Oxyhydroxide Nanosheets in TBAOH

To maintain a 1:1 H + :TBA + ratio, 0.5 g of NCOx-H was suspended in a mixture of H2O:40% TBAOH (1.5M solution in water, obtained from Acros) containing 2.5 mL of TBAOH and 257.5 mL of H2O. The solution was maintained under stirring for 90 days at room temperature. The resulting black suspension was then centrifuged at 5000 rpm for 10 min to remove impurities. Then, the supernatant solution was heated at 50 °C for 10 days to remove the solvent and induce restacking of the suspended nanoparticles. The final product (0.1 g) is designated as NCOx-TBA.

Physicochemical Characterization

Powder X-ray diffraction patterns were obtained using a PANalytical X'Pert Pro powder diffractometer in the Brag-Brentano geometry using a monochromator with Kα1 line (λ = 1.5406 Å) at room temperature, with 2θ range from 8 to 80° with a 0.0170° measuring step. The material was previously ground to fine powder and measured on a steel powder specimen holder. Scanning electron microscopy direct imaging was performed at the Plateform Aquitaine de Caractérisation des Matériaux (PLACAMAT) by means of a SEM-FEG HR (JEOL 6700F microscope). Transmission electron microscopy imaging was performed with a TEM JEOL JEM 1400 Plus microscope, while high-resolution imaging was obtained by means of a TEM-FEG HR (JEOL 2200FS microscope) working at 200 kV coupled to an EDX JEOL Si-Li detector with an image resolution of 0.19 nm and an EDS point resolution of 1.5 nm. The images for non-exfoliated samples were obtained by suspending a portion of the powder in ethanol and dropping it on top of a carbon-coated copper grid. In the case of exfoliated samples, an aliquot was taken of the exfoliated suspension and the carbon-coated grid was simply immersed in the solution.

Fourier transform infrared spectrometry was performed with Thermo Electron Corp. Nicolet 5700 equipment in specular mode for wave numbers from 400 to 4000 cm -1 . The powders were previously mixed with KBr at a 1:100 ratio and pressed into disks.

Raman measurements were performed by means of a Yvon Horiba LabRAM HR-800 microspectrometer with a 514.5 nm excitation wavelength (Ar + laser) and a power adjusted to ca. 50 μW to avoid any degradation of the sample or a LabRAM HR Evolution Raman spectrometer (HORIBA, Jobin Yvon, Edison, NJ, USA) with an operating laser at 532 nm.

Nitrogen adsorption/desorption isotherms were recorded using an ASAP 2020 (Micromeritics) at 377 K. Prior to the measurements, degasification at 80 °C for 300 min and at 110 °C for 200 min under high vacuum was performed. The Brunauer-Emmett-Teller specific surface area was computed using adsorption data in the relative pressure (P/Po) range of 0.18-0.30. The pore size distribution was derived from the amounts adsorbed in the relative pressure range from 0.01 to 0.99 by means of the adsorption branch using the Barrett-Joyner-Halenda (BJH) method.

Elemental analysis (carbon, hydrogen, nitrogen, and sulfur content) was determined by combustion at 920 °C in a tin foil in excess of oxygen by means of a Thermo Flash EA 111E Series apparatus. Inductively coupled plasma-optical emission spectroscopy (ICP-OES) was used to obtain the concentrations of cobalt, nickel, sodium, and potassium. Results were obtained by dissolving 10-20 mg of the powder into HCl, diluted to a concentration in the 1-200 mg•L -1 range and analyzed by means of a Varian 720ES ICP-OES spectrometer.

The mean oxidation state of metal ions has been determined by iodometric titration. First, approximately 15 mg of the (hydr)oxide material was dissolved in 5 mL of H2O, 10 mL of KI at 10 g/L, and, finally, 5 mL of HCl (37%). KI is added to reduce all of the transition metal ions that have an oxidation state superior to 2+, leading to the oxidation of I -into I2.

Then, the solution is heated for total dissolution and I2 is titrated with 0.1M Na2S2O3. The mean oxidation state (x) is calculated using eq 1:

where Veq is the volume of sodium thiosulfate added at the equivalence. This procedure is repeated at least three times. Atomic force microscopy (AFM) was used to measure the thickness of suspended delaminated nanoparticles. The aqueous suspension of tetrabutylammonium-intercalated nickel-cobalt oxyhydroxide was centrifuged at 14600 rpm for 10 min. The resulting supernatant solution was isolated and diluted to double their volume. A 1 mL aliquot of the diluted solution was drop-cast on top of a mica surface and dried at 40 °C for 7 days. A Nanosurf AFM in tapping mode with a silicon pointprobe-plus Nanosensor cantilever (PPP-NCLAuD-10, gold coated with a resonance frequency of 315 kHz) was used to obtain the delaminated sample topography. The average nanoparticle thickness was determined from at least 4 regions in 2-4 μm 2 AFM images.

Electrochemical Characterization

The electrode material consisted of 80% active material, 15% active carbon, and 5% of PTFE (Sigma-Aldrich) as binder.

The mass of the components was determined by means of a microbalance Sartorius MC5-0CE with an accuracy of 0.01 mg. Once the electrode material was formed, a round area of 1 cm 2 was pressed on a stainless-steel grid by applying 7 tons by means of a hydraulic cylinder press. The mass of active material varied between 1.5 mg and 9.5 mg. All electrochemical measurements were performed with a three-electrode electrochemical cell by means of an Interface IFC1000-07087 potentiostat (Gamry Instruments). Platinum foil of 2.5 cm × 2.5 cm was used as counter electrode, while a SCE electrode was used as reference. 1M LiOH and 1M KOH were used as electrolytes. Cyclic voltammetry was carried out at different scan rates (1-20 mV•s -1 ) in a potential window ranging from -0.5 V to 0.65 V (vs SCE) and an initial activation of 5-10 cycles. Galvanostatic charge-discharge was also carried out at different current densities (1-10 A•g -1 ) in a similar potential range. Lifetime cycling tests were performed during 5000 charge-discharge cycles at 10 A•g -1 .

Results and Discussion

Physicochemical Characterization

The procedure used in the delamination of nickel-cobalt oxide nanosheets is exemplified in Figure 1. Essentially, α-Ni-Co(OH)2•(CO3 2-,2NO3 -)x•(H2O)y {NCOH} is synthesized by coprecipitation. This is followed by oxidation in NaClO/KOH into γ-(Ni-Co)O2•(H + ,Na + ,K + )x•(H2O)y {NCOx-HNK} and protonation into γ-(Ni-Co)O2•(H + )x•(H2O)y {H-NCOx-H}. Then, the powder is suspended in a TBA-OH solution, where it gets exfoliated in small nanoslabs of γ-(Ni-Co)O2 that are suspended in solution. By means of solvent evaporation the material becomes restacked {NCOx-TBA}.

X-ray Diffraction

XRD results are presented in Figure 2. In accordance to literature, the patterns were indexed with a hexagonal cell. [START_REF] Faure | Characterization of a Turbostratic α-Nickel Hydroxide Quantitatively Obtained from an NiSO4 Solution[END_REF] Supporting Information Figure S1 presents X-ray diffraction patterns of nickel hydroxide and cobalt oxyhydroxide as reference. The diffraction peak at the lowest angle, indexed as (001) reflection, has its inter-reticular distance that corresponds to the interslab distance, i.e., the distance between two adjacent metal planes, while the (110) reflection corresponds to half of the metal-metal distance within the slab [START_REF] Faure | Characterization of a Turbostratic α-Nickel Hydroxide Quantitatively Obtained from an NiSO4 Solution[END_REF] and is therefore strongly influenced by the average metal oxidation state. In the pristine material, carbonate and nitrate anions are intercalated to maintain electroneutrality due to the excess charge introduced by Co(III), resulting in an interslab distance close to 7.93 Å, as reported by Delmas et al. [START_REF] Faure | Preparation and Characterization of Cobalt-Substituted α-Nickel Hydroxide Stable in KOH Medium Part II. α-Hydroxide with a Turbostratic Structure[END_REF] The metal-metal slab distance (ahex parameter value) obtained for the α phase is 3.04 Å, in accordance with literature [START_REF] Faure | Characterization of a Turbostratic α-Nickel Hydroxide Quantitatively Obtained from an NiSO4 Solution[END_REF] and with the value expected for Ni 2+ ions coexisting with a certain number of Co 3+ ions. After oxidation in NaClO/KOH (Figure 2b-d), the (110) reflection is shifted toward a higher angle and the inter-reticular distance obtained is close to 1.40 Å, which is consistent with the oxidation of nickel and cobalt ions to an average oxidation state of 3.2+, as confirmed by iodometric titration, presented in Table S1. The resulting material (Figure 2b) γ-(Ni-Co)O2•(H + ,Na + ,K + )x(H2O)y, exhibits the (001) diffraction peak at 6.92 Å. This interslab distance decreases when compared to the pristine material, due to the modification of the interslab composition resulting from the decarbonization, which provokes the intercalation of alkali cations present in the medium (Na + ,K + ) to maintain electroneutrality. The material is then acidified (Figure 2c) to substitute the alkali ions in the interslab space for protons, since H + ions have better affinity to TBA-OH ions and exfoliation can only be achieved with this intermediate step. [START_REF] Gao | Preparation of Nanometer-Sized Manganese Oxides by Intercalation of Organic Ammonium Ions in Synthetic Birnessite OL-1[END_REF] The substitution of alkali cations in the interslab space to the smaller protons shortens the interslab distance ((001) reflection) of NCOx-H to 6.53 Å. During the subsequent steps, TBA + intercalation and exfoliation, the (110) line is almost not shifted, which shows that the metal oxidation state reached is maintained. Finally, once the material is exfoliated and restacked by TBA + intercalation and solvent evaporation (Figure 2d), the material becomes restacked with an interslab distance of 6.51 Å, indicating that the nature of the material is preserved during the exfoliation process.

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In summary, nickel-cobalt oxyhydroxide slabs (with oxidation state of 3.2+, as evidenced by iodometric titration) are separated by intercalated protons, potassium and sodium ions, and water molecules. During the acid treatment, sodium and potassium ions are substituted by protons. Considering their smaller size, the insterslab distance slightly decreases to 6.53 Å. Then, the material is introduced in a solution of TBA + , for protons to be substituted by TBA + , whose bigger molecular size increases the interslab distance to reach delamination. Considering that the delaminated material is in suspension, the material was precipitated by solvent evaporation. During solvent evaporation, the slabs restacked, preserving their nature and resulting in a material similar to the predelaminated precursor that has protons intercalated (widely available in the aqueous solvent), thus yielding a similar, yet slightly lower, interslab distance as the protonated material (6.51 Å). This process is depicted in Figure 1.

Elemental Analysis

The results of elemental chemical analysis, displayed in Table S2, indicate at first that cobalt and nickel are present in a 0.65/0.35 atomic ratio, which is maintained during the entire process, exemplifying that the reactions are topotactic and preserve slab integrity. Once the pristine hydroxide material (NCOH) is oxidized to oxyhydroxide (NCOx-HNK), significant amounts of potassium and sodium are present within the material, as expected from literature. [START_REF] Faure | Preparation and Characterization of Cobalt-Substituted α-Nickel Hydroxide Stable in KOH Medium Part II. α-Hydroxide with a Turbostratic Structure[END_REF] During protonation (NCOx-H), there is an overall decrease on the amount of intercalated cations. Nonetheless, their substitution for proton ions is not complete. Finally, after exfoliation and restacking in TBAOH, the carbon signal dramatically augments and nitrogen is still present, while the relative weight percentages of nickel and cobalt diminish, showing that TBA + ions are still within the material. Additional purification steps are therefore required to successfully remove TBA + since rinsing with ethanol, acetone, and water cannot completely remove the organic material.

Scanning and Transmission Electron Microscopy

The evolution of the material's morphology throughout the synthesis procedure is studied by means of SEM and TEM. The size of aggregates is reduced with each step of the synthesis. Initially, NCOH consists of big aggregates of densely packed particles (>10 μm) (Figure 3a). After oxidation, NCOx-NKH presents a very similar morphology (Figure 3b). However, the domain size of the particles is slightly reduced. Acidification further decreases the size of the aggregates to particles of 5 μm or smaller size, with a very heterogeneous distribution (Figure 3c). After TBAOH exfoliation and restacking by evaporation (Figure 3d), the particle size distribution becomes more homogeneous and particles decrease their domain size to approximately 1 μm diameter, although bigger particles can still be found. Thus, porosity and surface area seem to increase at every step of the exfoliation procedure. TEM results, shown in Figure 4, provide additional morphological information. NCOx-HKN consists of big aggregates (>200 nm) that present randomly oriented nanostructured domains. Thus, nanoparticles of approximately 1-10 nm aggregate in random directions, resulting in the formation of secondary microparticles that should correspond to crystallites, considering the broad diffraction peaks observed in Figure 2. A similar structure can be observed once the material is acidified (Figure 4b). However, both the aggregates (>500 nm) and the coherent domains within the particles are bigger in this case, ranging from 15 to 25 nm size. Once exfoliated by TBA + intercalation, dispersed particles of 10-20 nm diameter size are obtained (Figure 4c). Given the thin nature of these nanoslabs, it is challenging to obtain TEM images with good resolution. However, it seems that the nanoslabs also consist of smaller coherent domains of 1-20 nm, maintaining the nature of the bulk material. It is worth mentioning that, in addition to these smaller sized particles, bigger aggregates can also be found, which shall be separated by centrifugation at higher speeds (>15000 rpm). Finally, once the material is restacked, a granulated morphology similar to the starting material is obtained (Figure 4d). Images of these materials at different magnifications can be found in the Supporting Information (Figure S2).

Fourier Transform Infrared and Raman Spectroscopy

When FTIR results are considered, NCOH (Figure 5a) is characterized by a large broad signal at 3446 cm -1 and a large band and 1646 cm -1 (stretching and bending vibrations of water molecules), a strong narrow peak at 1376 cm -1 assigned to vibrations of monodentate carbonate and nitrate anions and, finally, a broad band centered at 673 cm -1 and a narrow peak at 522 cm -1 that corresponds to stretching vibrations of metal-oxygen bonds (M-O). When the material is oxidized (Figure 5b), the signals associated with vibration modes of water become mitigated, while the peaks associated with carbonate anions disappear. On the other hand, a double peak at 570 and 632 cm -1 appears that corresponds to M-OH and M-O stretching vibrations. When the sample is protonated (Figure 5c), this low-wavenumber peak becomes intensified, while signals associated with water molecules become weaker. Finally, after exfoliation and restacking (Figure 5d), FTIR spectrum presents many additional peaks that are associated with TBA + molecules since ethanol and acetone rinsing is not enough to eliminate them. There are two main relevant features: (i) the signal for the bands associated with water molecules is increased (more water molecules intercalated in the interslab space of the restacked material are in accordance with the increased interslab distance observed by XRD, although they could be brought by the residual TBA + );

(ii) M-O and M-OH vibrations, visible at 500-700 cm -1 , are maintained and become more significant than in the α phase, in which they are barely visible. [START_REF] Xu | Interconversion of Brucite-like and Hydrotalcite-like Phases in Cobalt Hydroxide Compounds[END_REF][START_REF] Hu | Synthesis of α-Cobalt Hydroxides with Different Intercalated Anions and Effects of Intercalated Anions[END_REF][START_REF] Yuan | The Rapid Exfoliation and Subsequent Restacking of Layered Titanates Driven by an Acid-Base Reaction[END_REF][START_REF] Ishijima | Two Exfoliation Approaches for Organic Layered Compounds: Hydrophilic and Hydrophobic Polydiacetylene Nanosheets[END_REF][START_REF] Faure | Preparation and Characterization of Cobalt-Substituted α-Nickel Hydroxides Stable in KOH Medium Part I. Α′-Hydroxide with an Ordered Packing[END_REF] Details on the 400-1000 cm -1 ranges can be found in Figure S3. Raman spectra, shown in Figure 6, provide additional evidence on the phase transformation processes during exfoliation. NCOH presents two main bands, at 464 and 528 cm -1 , assigned to M-OH and M-O vibration modes. When the material is oxidized, these two bands are present at approximately 451 and 527 cm -1 with different relative intensities, and a shoulder is observed around 580 cm -1 . This additional signal can also be detected in NCOx-HNK and may be associated with structural defects. Finally, a broad band from 400 to 700 cm -1 with 3-4 peaks is observed when the sample is exfoliated and restacked, corresponding to Co-O and Ni-O vibrations which might be influenced by the presence of TBA + . However, this specific effect requires further study, which is out of the scope of the present work. [START_REF] Zhong | Co3O4/Ni(OH)2 Composite Mesoporous Nanosheet Networks as a Promising Electrode for Supercapacitor Applications[END_REF][START_REF] Diaz-Morales | The Importance of Nickel Oxyhydroxide Deprotonation on Its Activity towards Electrochemical Water Oxidation[END_REF]

Nitrogen Adsorption Isotherms

The specific surface areas of NCOx-H and NCOx-TBA were analyzed by BET. Results, presented in Figures S4 andS5, show that both materials display a type IIB isotherm, characteristic of nonporous or macroporous materials, with a narrow hysteresis cycle, which indicates the presence of some mesopores. NCOx-H presents a specific surface area of 190 m 2 •g -1 with a relevant number of mesopores, while NCOx-TBA displays a specific surface area of only 94 m 2 •g -1 . Results suggest that TBA + molecules intercalated in the interlayer space of the final material, as clearly observed by FTIR, reduce the N2 adsorption surface area, which may result in a lower accessible active surface area and worsened electrode-electrolyte interaction in their evaluation as energy storage materials. In any case, a certain effect in the electrochemical behavior is expected from BET results.

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Atomic Force Microscopy AFM results for delaminated TBA + -intercalated nickel-cobalt oxide in distilled water, diluted and drop-cast on top of mica, are presented in Figure 7. Results display a homogeneous dispersion of nanoparticles with different diameters, ranging from 30 to 175 nm. On the other hand, smaller particles present a thickness around 1.5-3 nm, while the biggest ones reach values up to 85.6 nm. Solvent molecules, aggregation during sample preparation for AFM measurements and trapped TBA + may increase the expected thickness of the nanolayers. Nonetheless, results present a great number of nanoparticles with a thickness of ∼3 nm, associated with 2-3 monolayer stacks of nickel-cobalt oxyhydroxide, when compared to the theoretical value for a nickel-cobalt oxyhydroxide monolayer (0.52 nm) with adsorbed TBA + molecules, 0.84 nm in size, on its surface, as depicted in Figure 7. [START_REF] Omomo | Redoxable Nanosheet Crystallites of MnO2 Derived via Delamination of a Layered Manganese Oxide[END_REF][START_REF] Oliva | Review of the Structure and the Electrochemistry of Nickel Hydroxides and Oxy-Hydroxides[END_REF][START_REF] Tang | Exfoliation and Restacking of Manganese and Cobalt Based Lamellar Oxides for Supercapacitor Electrodes[END_REF]

Electrochemical Characterization

This section aims at comparing the electrochemical behavior of nickel-cobalt oxyhydroxide during its delamination process to determine the effect of the intercalated species on their performances.

Study in 1M KOH Electrolyte

Cyclic voltammetry curves at 5 mV•s -1 are shown in Figure 8a in the potential range from -0.3 to 0.55 V (vs SCE). For the three materials, one anodic peak and its cathodic counterpart are observed, centered at approximately 0.3 V and -0.05 V respectively, as expected for mixed nickel-cobalt hydroxides being reversibly oxidized to oxyhydroxides. [START_REF] Delmas | The Effect of Cobalt on the Chemical and Electrochemical Behaviour of the Nickel Hydroxide Electrode[END_REF][START_REF] Wang | In Situ Anchoring Massive Isolated Pt Atoms at Cationic Vacancies of α-NixFe1-x(OH)2 to Regulate the Electronic Structure for Overall Water Splitting[END_REF] Galvanostatic discharge curves, Figure 8b, are characteristic of "battery" materials with faradaic processes. The best capacity results are obtained for the material after exfoliation-restacking (118.7 mA•h•g -1 at 1 A•g -1 ) against 106.1 mA•h•g - 1 and 76.8 mA•h•g -1 respectively for NCOx-HNK and protonated NCOx-H oxyhydroxides before exfoliation. It is worth mentioning that all of the materials exhibit high Coulombic efficiency, with 96%, 94%, and 98% for NCOx-HNK, NCOx-H, and NCOx-TBA, respectively, revealing their high energy efficiency. These results are likely caused by a better surface accessibility and electrode-electrolyte interaction by the delaminated material. The cycling behavior of the materials is presented in Figure 8d, with a detailed view of the first 200 cycles presented in Supporting Information Figure S6. The capacity for NCOx-HNK oxyhydroxide is reduced during the first 20 cycles and maintained henceforth. On the other hand, during the first 100 cycles, NCOx-TBA shows a strong increase in capacity. This evolution may be explained by the fact that, initially, the interference of TBA + impedes an optimal electrode-electrolyte interaction. After activation, TBA + is removed from the surface of the active material and there is an increase of the number of electrochemically active sites. However, defining the exact mechanism that induces an activation and increase in capacity for the delaminated material requires further investigation. Moreover, once activated, the capacity decay of this material is similar to the performance displayed by the parent oxyhydroxide materials. It is also worth considering the possible influence of TBA + ions to nickel-cobalt nanoslabs affecting the degradation of the material. After 5000 cycles, the capacity retention of the different materials is 62%, 74%, and 113% for NCOx-HNK, NCOx-H, and NCOx-TBA, respectively. Finally, when higher current densities are applied (Figure 8c), the best rate capability is displayed by the exfoliated NCOx-TBA that retains up to a 57% of capacity, while a less promising performance is displayed for NCOx-HNK and NCOx-H with only 43% and 14% capacity retention, respectively, at the same applied current density. These results are likely caused by a better surface accessibility and electrode-electrolyte interaction by the delaminated material. Results concerning galvanostatic discharges at different applied current densities and cyclic voltammetries at different scan rates are displayed in detail in Figure S7.

Study in 1M LiOH Electrolyte

The electrochemical performance of the materials shown in Figure 9 has also been evaluated in 1M LiOH. The performances of NCOx-HNK and NCOx-H are slightly influenced by the change of electrolyte, with similar cycling performance and capacity values. Values of 112.6 and 91.0 mA•h•g -1 are obtained at 1 A•g -1 for NCOx-HNK and NCOx-H, respectively, with a capacity retention of 63% and 67% after 5000 cycles in 1M LiOH. The main difference resides on the capacity retention obtained at higher applied current densities as displayed in Figure 9c. There is an evident enhancement in performance for the restacked TBA + -intercalated nickel-cobalt oxyhydroxide in LiOH compared to KOH, resulting in capacity values of 165.1 mA•h•g -1 at 1 A•g -1 and 112.5 mA•h•g -1 at 10 A•g -1 . Moreover, the preactivation step required for NCOx-TBA in KOH reduces in LiOH at the first 10 cycles, as observed in Figure S8. A capacity retention of 60% for NCOx-TBA is obtained after 5000 cycles at 10 A•g -1 . The Coulombic efficiency is similar to the results obtained in KOH electrolyte, although overall higher current efficiencies for the oxidized materials are obtained, with 97%, 98%, and 98% for NCOx-HNK, NCOx-H, and NCOx-TBA, respectively. In this case, the capacity of the exfoliated and restacked material is nearly 1.5 times the initial capacity displayed by the non-exfoliated oxyhydroxide at 1 A•g -1 . Details on the performance of the different materials at different scan rates and applied current densities are provided in Figure S9.

On the one hand, whether in KOH or in LiOH, the significant increase in capacity between the exfoliated material and the parent oxyhydroxide may be explained by the increased electrode-electrolyte interaction and ease for electrolyte penetration within the material once it is exfoliated and restacked. Although BET indicates a more reduced surface area for NCOx-TBA, this may not correspond to the real electrochemically active surface of the material. TBA + initially blocks the access to the electrolyte and reduces the surface area. However, when immersed in electrolyte and after a certain activation process, TBA + is removed from the surface and a more intimate contact between active material and electrolyte can be achieved, increasing the number of active sites. Moreover, it is observed by TEM and SEM an apparent decrease on the microcrystalline structure (Figures 3 and4). Possibly, when the material is exfoliated and restacked, the interaction between the nanoslabs is reduced, promoting electrolyte penetration and consequently an enhanced capacity response. Therefore, two factors can be considered as main contributors for the capacity increase of the material: reduction of the microcrystalline domain and weakened interslab interaction that promotes electrolyte penetration. Moreover, the use of LiOH, as compared to KOH, may result in better faradaic charge-transfer phenomenon and better alkali ion penetration. As reported in literature, [START_REF] Delmas | M Nickel Hydroxide and Derived Phases Obtained by Chimie Douce from NaNiO2[END_REF] LiOH is likely to inhibit oxygen evolution and therefore to favor material oxidation. This is evident when cycling stability is evaluated in LiOH, since the material displays a more stable electrochemical response, with higher capacity that only requires 10 cycles to activate, in contrast to the activation observed when KOH is used.

In conclusion, the exfoliation of nickel-cobalt oxyhydroxide by means of TBA + intercalation proves that delamination is a successful route for the enhancement of the electrochemical properties of the material and that TBA + intercalation does not hamper the electrochemical response of the active material. Despite the preactivation step, especially required in 1M KOH, an optimized performance is observed in which an intimate active material-electrolyte contact is achieved. Thus, exfoliation routes of layered materials to obtain few-layer nanostructures have a positive impact on the electrochemical properties of the electrode material. A comparison with similar Ni-Co composite electrode materials from literature also validates the efficiency of the exfoliation-restacking approach, leading to materials that possess among the highest specific capacities (see Table 1).

In principle, obtaining few-layer nanosheets of a lamellar material by delamination has two expected effects in the electrochemical performance. On the one hand, it serves as a precursor route to create composite materials that interact at an atomic level, combining the benefits of different materials. On the other hand, when a delaminated material is restacked, as in this case, there is an enhanced electrode-electrolyte interaction that leads to a better electrochemical performance. The results here [START_REF] Adán-Más | Advanced metal graphene composite electrodes for a new generation of electrochemical energy storage devices[END_REF] presented for the first time exemplify the potential application of delamination of layered materials for the synthesis of Ni-Co-Mn composites as an example.

Conclusions

Delamination of nickel-cobalt oxyhydroxide by intercalation of tetrabutylammonium in its interslab space has been proved as a successful route to enhance the electrochemical performance of nickel-cobalt oxyhydroxide by increasing the electrochemical active surface area and electrode-electrolyte interaction after an electrochemical activation process to remove residual TBA + cations. Results show a rise in capacity from 112 mA•h•g -1 in 1M LiOH at 1 A•g -1 for the nondelaminated precursor to 165 mA•h•g -1 after exfoliation and restacking, phenomena associated with better electrodeelectrolyte interaction. Moreover, the influence of the electrolyte cation has been studied, resulting in a shorter activation period for lithium than for potassium hydroxide, which has been associated with better alkali ion penetration, as a consequence of lower ion hydration sphere. Overall, results are greatly dependent on the intercalated ions in the interslab, which is directly correlated to the electrochemical performance of the material.

Figure 1 .

 1 Figure 1. Schematic of the exfoliation process for γ-NixCo1-xO2 nanoslabs that includes (a) α-nickel-cobalt hydroxide, (b) γ-nickel-cobalt oxyhydroxide, (c) protonated γ-nickel-cobalt oxyhydroxide, (d) delaminated γ-nickel-cobalt oxyhydroxide, and (e) restacked γ-nickel-cobalt oxyhydroxide.

Figure 2 .

 2 Figure 2. X-ray diffraction patterns for (a) NCOH used as reference, (b) NCOx-HNK, (c) NCOx-H, and (d) restacked NCOx-TBA. The numbers correspond to the inter-reticular distances calculated on the basis of indexation with a hexagonal cell.

Figure 3 .

 3 Figure 3. Scanning electron microscopy images of (a) NCOH, (b) NCOx-HNK, (c) NCOx-H, and (d) restacked NCOx-TBA after delamination in TBAOH and restacking by solvent evaporation.

Figure 4 .

 4 Figure 4. Transmission electron microscopy images of (a) NCOx-HNK and (b) NCOx-H, (c) exfoliated oxide slabs in TBAOH (NCOx-TBA), and (d) restacked NCOx-TBA.

Figure 5 .

 5 Figure 5. FTIR results for (a) NCOH, (b) NCOx-HNK, (c) NCOx-H, and (d) restacked NCOx-TBA.

Figure 6 .

 6 Figure 6. Raman results for (a) NCOH, (b) NCOx-HNK, (c) NCOx-H, and (d) restacked NCOx-TBA.

Figure 7 .

 7 Figure 7. Atomic force microscopy topography (a) 2D image, (b) 3D image, and (c) height profiles of delaminated tetrabutylammonium-intercalated nickel-cobalt oxyhydroxide obtained in tapping-mode. Each dotted line corresponds to a height profile with the same numeration. (d) Schematic representation of theoretical values for a nickel-cobalt oxyhydroxide bilayer.

Figure 8 .

 8 Figure 8. Comparison of the electrochemical response by (a) cyclic voltammetry, (b) galvanostatic discharge, (c) capacity retention at different applied current densities, and (d) capacity retention evaluated by galvanostatic continuous charge-discharge cycling during 5000 cycles at 10 A•g -1 for the different composites obtained during delamination of nickel-cobalt oxyhydroxide by TBA + intercalation in 1 M KOH.

Figure 9 .

 9 Figure 9. Comparison of the electrochemical response by (a) cyclic voltammetry, (b) galvanostatic discharge, (c) capacity retention at different applied current densities, and (d) capacity retention evaluated by galvanostatic continuous charge-discharge cycling during 5000 cycles at 10 A•g -1 for the different composites obtained during delamination of nickel-cobalt oxyhydroxide by TBA + intercalation in 1M LiOH.

Exfoliation of Nickel-Cobalt Oxyhydroxide

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Acknowledgments

This work is dedicated to the memory of Patrizia Paradiso. This work has been performed in the scope of the International Doctoral Program in Functional Materials (IDS-FunMat, Erasmus Mundus) and funded by the European Union. We express our gratitude to the Fundação para a Ciência e Tecnologia (FCT) for the funding under Contracts M-ERA.NET/0004/2014, PTDC/QUI-ELT/28299/2017, and UID/QUI/00100/2019. We also acknowledge Laura Ilharco for her assistance in BET measurements and interpretation, Auguste Fernandes in his aid for FTIR spectra acquisition, Kush Kumar Upadhyay for electrochemical discussion, Laetitia Etienne for CHNS and ICP measurements, Eric Lebraud for performing the XRD experiments, Catherine Denage, Sabine Goma, and Philippe Dagault for their general assistance, and Marion Gayot and Sonia Buffière for assisting with TEM analysis at PLACAMAT (Plateforme Aquitaine de Caractérisation des Matériaux UMS 3626 CNRS-Université de Bordeaux, Pessac, France).

Supporting Information

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsaem.2c01905. XRD diffraction patterns, iodometric titrations, chemical analyses, TEM images, FTIR results, BET isotherms, 200 initial cycles continuous charge-discharge comparison, cyclic voltammetries, and galvanostatic charge-discharge curves (PDF) -Notes : The authors declare no competing financial interest.


Gaëlle Saade
email: gaelle_saade@outlook.com

Eva Bogaerts

Sophie Chiavassa

Guillaume Blain

Grégory Delpon

Manon Evin

Youssef Ghannam

Ferid Haddad

Karin Haustermans

Charbel Koumeir

Ellina Macaeva

Lydia Maigne

Quentin Mouchard

Noël Servagent

Edmond Sterpin

Stéphane Supiot

Vincent Potiron

Reduction of radiotoxicity by ultra-high dose rate protontherapy

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

1-Introduction

Electron beams are widely used for Ultra-High Dose Rate Radiotherapy (UHDR-RT) studies.

The novelty of UHDR-RT using protons still requires a lot of knowledge of the parameters involved in its response.

Protons : -Spread a little along the trajectory.

-Have a precise targeting.

2-Materials and Methods

3-Results

4-Conclusion Zebrafish embryos

No experimental authorization is required This model allows to study rapidly several parameters allowing a better understanding of proton UHDR-RT.

Effect of proton UHDR-RT on the morphology of zebrafish embryos vs conv RT :

No difference of the curvature

What are the requirements that allow obtaining a protective effect using proton UHDR-RT ?

Effect of the mean dose-rate on the protection of zebrafish embryos :

Significant protection at 7700 Gy/s.