Melissa Yeung

David Cohen-Steiner

Mathieu Desbrun

Material Coherence from Trajectories via Burau Eigenanalysis of Braids

In this paper, we provide a numerical tool to study material coherence from a set of 2D Lagrangian trajectories sampling a dynamical system, i.e., from the motion of passive tracers. We show that eigenvectors of the Burau representation of a topological braid derived from the trajectories have levelsets corresponding to components of the Nielsen-Thurston decomposition of the dynamical system. One can thus detect and identify clusters of space-time trajectories corresponding to coherent regions of the dynamical system by solving an eigenvalue problem. Unlike previous methods, the scalable computational complexity of our braid-based approach allows the analysis of large amounts of trajectories.

Studying two-dimensional flows and their induced transport and mixing properties is key to geophysical studies of atmospheric and oceanic processes. However, one often has only sparse tracer trajectories (e.g., positions of buoys in time) to infer the overall flow geometry. Fortunately, topological methods based on the theory of braid groups have recently been proposed to extract structures from such a sparse set of trajectories by measuring their entanglement. This braid viewpoint offers sound foundations for the definition of coherent structures. Yet, there has been only limited efforts in developing practical tools that can leverage topological properties for the efficient analysis of flow structures: handling a larger number of trajectories remains computationally challenging. We contribute a new and simple computational tool to extract Lagrangian structures from sparse trajectories by noting that the eigenstructure of the Burau matrix representation of a braid of particle trajectories can be used to reveal coherent regions of the flows. Detection of clusters of spacetime trajectories corresponding to coherent regions of the dynamical system can thus be achieved by solving a simple eigenvalue problem. This paper establishes the theoretical foundations behind this braid eigenanalysis approach, along with numerical validations on various flows.

I. Introduction

Studying two-dimensional flows and their induced transport is key to geophysical studies of atmospheric and oceanic processes. In particular, identifying regions of similar dynamical fate within experimental or numerical temporal data often leads to a better understanding of the overall flow geometry, from which material transport and mixing properties can be accurately quantified. Such an analysis can also be crucial in short-term forecasting of pattern evolution in complex 2D dynamical systems. While classic dynamical systems theory has established solid foundations (such as fixed points, orbits, and stable manifolds) to study structures in steady and timeperiodic fluid flows, these tools become inadequate to capture the complexity of aperiodic large-scale flows: tracers in even a very simple Eulerian velocity field can undergo very different and intricate trajectories-a phenomenon often referred to as Lagrangian chaos, to distinguish it from its Eulerian counterpart (turbulence) [START_REF] Amon | Lagrangian chaos, Eulerian chaos, and mixing enhancement in converging-diverging channel flows[END_REF] . Consequently, numerous definitions of the notion of coherent regions in flows have been formulated to help in this endeavor, from which algorithms can be developed to automatically generate a simplified skeleton of the overall dynamics of the system from the input data.

A. Deformation-based coherent structures.

A large majority of approaches to material coherence discovery involve quantifying how the velocity field induces, in time, a deformation of the original space (see 2 and 3 for two recent reviews). Instantaneous Eulerian diagnostics tools [START_REF] Weiss | The dynamics of enstrophy transfer in two-dimensional hydrodynamics[END_REF][START_REF] Yang | Chaotic transport and mixing by ocean gyre circulation[END_REF] were first proposed to find coherent features in instantaneous velocity fields based on the eigenvalues of their gradient tensors. Later, the notion of Lagrangian coherent structures (a term originally coined by Haller and Yuan 6 ) was introduced to describe material curves that delineate regions with qualitatively different tracer dynamics. These boundaries between coherent regions are, to a certain extent, finite-time analogs of invariant manifolds and can be derived from the finite-time Lyapunov exponent field of the flow [START_REF] Haller | Lagrangian coherent structures from approximate velocity data[END_REF][START_REF] Shadden | Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional[END_REF] . Ensued a series of computational approaches with various theoretical guarantees and varying computational efficacy, each offering a segmentation of the flow into regions of coherent material transport [START_REF] Voth | Experimental measurements of stretching fields in fluid mixing[END_REF][START_REF] Mathur | Uncovering the Lagrangian skeleton of turbulence[END_REF][START_REF] Kasten | Lagrangian feature extraction of the cylinder wake[END_REF][START_REF] Lekien | The computation of finite-time Lyapunov exponents on unstructured meshes and for non-euclidean manifolds[END_REF] . More recently, these separatrices were reformulated through local geometric properties of the flow map and its Cauchy-Green strain tensor field as material curves with locally maximal repelling, attracting, or shearing impact on neighboring fluid elements [START_REF] Haller | A variational theory of hyperbolic Lagrangian coherent structures[END_REF][START_REF] Farazmand | Computing Lagrangian coherent structures from their variational theory[END_REF] . More global geometric definitions, such as least stretching material lines or stationary curves for the tangential stretching, were also shown to reveal further structures that include weakly but coherently stretching material vortices and Lagrangian jet cores [START_REF] Haller | Geodesic theory of transport barriers in two-dimensional flows[END_REF] . Finally, authors have also classified directly coherent regions (as opposed to the separators between regions) via ergodicity-based [START_REF] Budišić | Geometry of the ergodic quotient reveals coherent structures in flows[END_REF] , observer-based [START_REF] Mezić | Analysis of fluid flows via spectral properties of the Koopman operator[END_REF] , and probabilistic 18 approaches.

B. Topology-based coherent structures.

While there is no universally agreed upon definition for coherent structures, most of the definitions and computational methods mentioned above assume a fine knowledge of the Eulerian velocity field in space and time to deduce a good approximation of the flow map. However, oceanic flows are often known only as a set of sparse particle trajectories in time (typically, buoys). Such a sparse sampling of the dynamical system does not lend itself well to a geometric analysis of transport, rendering previous approaches unable to provide qualitative insights of the underlying flow. Fortunately, topological methods have recently been proposed to extract structures from a sparse set of trajectories by measuring their entanglement [START_REF] Thiffeault | Braids of entangled particle trajectories[END_REF] . Topology and group-theoretic tools have been commonplace in the study of fluid dynamics since the early work of Leray 20 and Arnold [START_REF] Arnold | Sur la géométrie differérentielle des groupes de lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits[END_REF] ; in the case of particle trajectories, the theory of braid groups, a classical area of topology, was shown to be relevant to the analysis of mixing in two-dimensional flows [START_REF] Thiffeault | Measuring topological chaos[END_REF] -not only for periodic systems, but for aperiodic, finite-time systems as well. A set of trajectories (whether these are trajectories of particles in the flow or trajectories of stirring rods that induce a motion in the surrounding fluid) has a topological structure that can be represented using Artin's braid group [START_REF] Artin | Theory of braids[END_REF] . The resulting braid viewpoint offers a precise and versatile tool to distinguish different regimes of dynamics, to provide a lower bound on topological entropy, and in the case of positive topological entropy, to reveal the presence of topological chaos. Allshouse and Thiffeault [START_REF] Allshouse | Detecting coherent structures using braids[END_REF] introduced such a topological method for detecting material coherence from a small set of particle trajectories. By examining the growth of loops that surround sets of particles trajectories, they identify coherent structures as loops that have negligible growth. Coherent regions are then defined as containing particles that possibly mix with other particles within the region itself but do not mix with particles outside the region; the set of trajectories arising from the particles within a coherent region forms a coherent bundle.

C. Computational challenges.

Even if the use of braid groups offers sound foundations for the definition of coherent structures, there has been only limited efforts in developing practical computational tools for the efficient analysis of flow structures. A significant exception is the work of Thiffeault and Budišić [START_REF] Thiffeault | BRAIDLAB: a software package for braids and loops[END_REF][START_REF] Budišić | Finite-time braiding exponents[END_REF] which allows for the construction of an algebraic braid from trajectories by tracking the crossing of particles in space-time with respect to the projection onto an arbitrary line. Resulting braids can then be analyzed, allowing one to estimate the topological entropy of the flow, as well as compute the growth of loops under the action of the braid. However, finding coherent sets of trajectories that travel together but do not entangle other trajectories require an exhaustive search through loops enclosing pairs of particles to find the least growing ones. Dramatic improvement in efficiency was offered in [START_REF] Allshouse | Detecting coherent structures using braids[END_REF] by limiting the exploration to "pair-loops" of nearby punctures, turning an exponential search into a quadratic one without significant artifacts-although it may end up merging multiple coherent regions into one single larger region. Yet, handling a larger number of trajectories remains computationally challenging. Allshouse and Thiffeault [START_REF] Allshouse | Detecting coherent structures using braids[END_REF] examine up to 20 trajectories, while the more recent work of Francois et al. [START_REF] Francois | Braid entropy of two-dimensional turbulence[END_REF] examine up to 100 trajectories.

D. Contributions.

In this paper, we present a scalable and efficient approach for detecting coherent sets from sparse tracer trajectories of a two-dimensional dynamical system. We adopt the trajectories-as-a-braid viewpoint pioneered by Thiffeault et al., to which we contribute a new and simple computational tool to extract Lagrangian structures by noting that the eigenstructure of the Burau matrix representation of a braid of particle trajectories can be used to reveal coherent regions of the flows. 28 More precisely, we show that the Burau matrix of a reducible braid possesses eigenvectors that are almost piecewise constant over coherent sets. Thus, we deduce that one can detect and identify clusters of space-time trajectories corresponding to coherent regions of the dynamical system by solving an eigenvalue problem. This paper both establishes the theoretical foundations behind this braid eigenanalysis approach and presents numerical validations on different flows. In particular, we apply our approach to study Aref's blinking vortex flow [START_REF] Aref | Stirring by chaotic advection[END_REF] and a modified Duffing oscillator [START_REF] Allshouse | Detecting coherent structures using braids[END_REF] . In the analysis of Aref's blinking vortex flow, our method distinguishes chaotic regions from KAM surfaces, over a range of flow strengths. In the analysis of the modified Duffing oscillator, we also detect two additional limit cycles in addition to detecting the two dominant regions of mixing detected by Allshouse and Thiffeault [START_REF] Allshouse | Detecting coherent structures using braids[END_REF] due to our ability to handle a significantly larger amount of trajectories. We also discuss the computational complexity of our approach and contrast it with existing approaches which seek individual slow-growing loops to identify coherent regions.

II. Braids and Dynamics

Throughout this paper, we assume that a flow on a twodimensional domain D ⊂ R 2 is known via a set of n disjoint particle trajectories {x i (t)} i=1..n , where each trajectory x i : [0, 1] → D records the path of the ith particle in time. Each of the trajectories can be viewed as a strand in space-time. Since these strands can be arbitrarily intertwined, the set of all trajectories resembles a "physical" space-time braid of entangled strands (see Fig. 1(a)). A topological analysis of this braid-like structure provides, remarkably, important dynamical information of the sampled flow.

A. Braid terminology

We now define a few notions related to braids that will be relevant in our exposition for characterizing flows.

Geometric and pure braid.

From a mathematical standpoint, we say that the collection {x i (t)} i=1..n of strands forms a geometric braid when the set of final positions x i (1) are given by a permutation of the initial positions x i (0). If this permutation is the identity, the braid is said to be a pure braid.

Topological braid.

One can further discard geometric information (e.g., distances between particles) to form a more concise, purely topological description of a geometric braid: a topological braid is an equivalence class of geometric braids, where two geometric braids are considered equivalent when one can be deformed continuously into the other without having strands crossing in space-time. The product of two topological braids on n strands can then be formed by concatenation, defining a group B n called the braid group.

B. Encoding trajectories as a topological braid

A topological description of a set of particle trajectories can be determined by projecting the strands to a fixed spacetime plane R×[0, 1] and only retaining information about how strands pass over one another [START_REF] Thiffeault | BRAIDLAB: a software package for braids and loops[END_REF] . More specifically, we select an arbitrary projection line, and enumerate the strands i = 1, • • • , n according to their ordering along the projection line (Fig. 1(b)). As particles move in time, and their strands cross one another, the projection of the strands onto the projection line will change, and the enumeration is updated accordingly. For 1 ≤ i < n, we let σ i denote the braid consisting of a single crossing given by passing the ith strand behind the (i + 1)st strand (Fig. 2(a)); one can thus think of the braid σ i as a simple clockwise half-twist interchanging strands i and i + 1. Conversely, σ -1

1 denotes the single crossing given by passing the ith strand in front of the (i + 1)st strand. Then the sequence of crossings in time can be recorded by a concatenation (product) of σ i . (Following the standard practice in braid literature, we adopt the convention of composing our braids from left to right.) This sequence specifies the topological braid σ corresponding to the intertwined input trajectories. This composition of braids σ i is a convenient and wellstudied description of a topological braid, namely, the σ i are the generators used in Artin's presentation [START_REF] Artin | Theory of braids[END_REF] of the braid group B n . An additional observation we will use is that if a set of particle trajectories forms a geometric braid, then a change in the projection line only changes the resulting topological braid by conjugation [START_REF] Thiffeault | Braids of entangled particle trajectories[END_REF][START_REF] Thiffeault | BRAIDLAB: a software package for braids and loops[END_REF][START_REF] Boyland | Topological methods in surface dynamics[END_REF] .

C. Braid dynamics

We now point out two important relationships between topological braids and the dynamical system they encode.

Action on closed curves.

For each braid on n strands, we can associate an isotopy class of homeomorphisms on an n-punctured disk that fix the boundary pointwise [START_REF] Farb | A Primer on Mapping Class Groups[END_REF] , where each generator σ i is associated to a homeomorphism from D n to D n that has support on a twice-punctured disk and that is described on this support by a half-twist. Simple closed curves in D n encompass-ing two or more punctures may thus be subject to stretching and folding under the associated homeomorphism. Consequently, the braid group induces a (right) action on the set of isotopy classes of simple closed curves. This action can be exploited to study the flow: for instance, the growth rate of a loop around punctures under the action of a braid provides a lower bound on the topological entropy of the surrounding flow [START_REF] Thiffeault | Braids of entangled particle trajectories[END_REF] .

Reducible braids.

Nielsen-Thurston theory [START_REF] Farb | A Primer on Mapping Class Groups[END_REF] distinguishes a special class of braids, called reducible braids, which are particularly relevant to the study of dynamical systems. A reducible braid α on n strands preserves a family C of disjoint simple closed curves, each enclosing more than one but fewer than n punctures. The family C is referred to as a reduction system for α.

In general, the reduction system of a reducible braid may be very complex geometrically, but every reducible braid is conjugate to a braid whose reduction system consists of a family of round curves (geometric ellipses) [START_REF] Bernardete | A combinatorial approach to reducibility of mapping classes[END_REF][START_REF] Bernardete | Braids and the nielsen-thurston classification[END_REF][START_REF] Birman | Braids: A survey[END_REF] , which we call a round reduction system. A reducible braid with a round reduction system can be described by a collection of tubes (i.e., tubular braids) and strands: each tube is traced out by the path of a simple closed curve in C , and each tube may enclose other tubes or strands-see Fig. 2 for an illustration. We call particle trajectories enclosed by a tube a coherent bundle, and we call their initial positions (at time t = 0) a coherent region (or coherent set), following Thiffeault's nomenclature [START_REF] Thiffeault | Braids of entangled particle trajectories[END_REF] .

x i x i+1

D. Burau representation

A convenient tool for studying a braid, called the Burau representation [START_REF] Burau | Über zopfgruppen und gleichsinnig verdrillte verkettungen[END_REF] , is given by associating to each Artin generator

σ i the n x n block matrix ∀1 ≤ i ≤ n, σ i -→     I i-1 0 0 0 0 1 -s s 0 0 1 0 0 0 0 0 I n-i-1     , (1)
parameterized by a (real or complex) variable s, where I k denotes the k x k identity matrix. For conciseness, we will sometimes express the Burau representation of σ i as

B[σ i ] (s) := I i-1 ⊕ 1 -s s 1 0 ⊕ I n-i-1
where we express the block structure of the matrix representation as a direct sum ⊕. A concatenation of generators corresponds to a sequence of multiplications of the aforementioned block matrices, resulting in an n x n Burau matrix whose elements are polynomials in s with coefficients in Z. This Burau representation is thus a homomorphism from the braid group B n to the general linear group of degree n. While the Burau representation is known to be not faithful for n≥5 [START_REF] Long | The Burau representation is not faithful for n ≥ 6[END_REF][START_REF] Bigelow | The Burau representation is not faithful for n = 5[END_REF] (with a rather large kernel for n ≥ 6 [START_REF] Church | Infinite generation of the kernels of the Magnus and Burau representations[END_REF] ), this matrix representation does nonetheless allow us to examine the dynamics of a motion of particles through a study of its eigenvectors.

The reduced Burau representation is a variant obtained by taking the quotient of the unreduced representation by the invariant subspace of the matrices B[σ i ](s) generated by the constant vector (1, . . . , 1).

III. Coherence from Burau eigenanalysis

We now state and prove the main theoretical result of this paper, which will be at the core of our computational approach to finding coherent structures from a set of trajectories.

Main Result. Let β ∈ B n be a reducible pure braid on n strands with reduction system C β consisting of k simple closed curves. For each ε > 0, there exists δ > 0 such that for s ∈ C with modulus 1 and argument at most δ , the reduced Burau matrix B[β ](s) has k eigenvectors whose span makes an angle at most ε with the space of piecewise constant vectors on (D n , C β ) modulo constant vectors.

In the remainder of this section, we prove this result in three main stages.

A. Decomposing a reducible braid

We begin with the claim that a reducible braid α ∈ B n that preserves a family C of round curves can be written as a product of tubular braids with trivial interior braiding, followed by a product of interior braids, with trivial braiding between tubular braids.

η 1 η 2 η 2 η 1
We follow the notation of Band and Boyland [START_REF] Band | The Burau estimate for the entropy of a braid[END_REF] and refer to the braid that moves the group of η 1 consecutive strands starting at strand i behind the group of η 2 consecutive strands starting at strand i + η 1 (see inset) as:

σ i,η 1 ,η 2 = (σ i+η 1 -1 • • • σ i+η 1 +η 2 -2 ) (σ i+η 1 -2 • • • σ i+η 1 +η 2 -3 ) • • • (σ i • • • σ i+η 2 -1 ) . (2)
Lemma 1. Let α ∈ B n be a reducible braid that preserves a finite non-empty family C of non-nested round curves. For each puncture p r of D n not enclosed by any curve of C , extend C by adding a round curve C r enclosing the single puncture p r . Let k be the number of curves in the extended family C .

Then there exists a finite sequence of tuples (i( ), η 1 ( ), η 2 ( )) of non-negative integers and corresponding exponents ε ∈ {-1, +1} for the generators of the tubular braids, as well as a finite sequence of (interior) braids β j , 1 ≤ j ≤ k, each supported on a punctured disk with boundary given by a reduction curve C j ∈ C , such that the braid α can be expressed as:

α = L ∏ =1 σ ε i( ),η 1 ( ),η 2 ( ) • k ∏ j=1 β j .
Proof. Let E j denote the punctured disk enclosed by the curve C j ∈ C . Let

D k = D n \ k j=1 E j
denote the n-punctured disk with k disks (each containing punctures) removed. Then the braid σ induces an automorphism σ on D k . The induced automorphism σ describes the tubular braid structure of σ with respect to C .

If we collapse each hole of D k to a puncture, we can consider D k as a k-punctured disk. Therefore, σ can be given by a sequence of Artin generators for the braid group B k :

σ = ∏ σ ε a ,
where 1 ≤ a < k and ε = ±1 for all . Viewed from above, each generator σ a corresponds to a clockwise half-twist interchanging the holes obtained by removing E a and E a +1 . Thus, each σ ε a specifies a tuple (ε , i( ), η 1 ( ), η 2 ( )), given by the direction of the half-twist, the minimum index of the points contained in E a , the number of punctures in E a , and the number of punctures in E a +1 , respectively. Defining the braids

β j = 1 ∏ =L σ -ε i( ),η 1 ( ),η 2 ( ) • σ | E j
completes the decomposition.

B. Piecewise-constant vectors

Next, we show that the Burau representation B[α](s) of a reducible braid α with a round reduction system maps a piecewise constant vector v that respects the structure of α to a piecewise-constant vector. More precisely, we show that if v is constant on the interior of every curve of a round reduction system C , then the image B[α](s) v is a piecewise constant vector, which is constant on the interior of every curve of α(C ) = C . Definition 2 (piecewise constant vectors). Let C be a collection of non-nested pairwise-disjoint simple closed curves in the n-punctured disk D n . We say that a vector

v = (v 1 , • • • , v n ) is piecewise constant on (D n , C ) if v r = v r
whenever the punctures p r and p r are enclosed by the same C j r ∈ C .

The tubular braid σ i,η 1 ,η 2 described in Eq. ( 2) has an obvious round reduction system C . The following observation is the basis of our method:

Lemma 3. If v is a vector that is piecewise constant on (D n , C ), the image B[σ i,η 1 ,η 2 ](s)v is also piecewise constant on (D n , σ i,η 1 ,η 2 (C )) for all s.
Proof. Because the braid σ i,η 1 ,η 2 has a grid structure with all its crossing of the same sign, the product of the Burau matrices of the involved generators is easily computed and yields that σ i,η 1 ,η 2 has Burau matrix

I i-1 ⊕    1 -s s -s 2 • • • s η 2 -1 -s η 2 s η 2 • • • 0 . . . . . . . . . . . . 1 -s s -s 2 • • • s η 2 -1 -s η 2 0 • • • s η 2 I n 2 0 n 2 ,n 1    ⊕ I n-i-η 1 -η 2 +1 ,
From the Burau matrix structure, it is clear that piecewiseconstant vectors are sent to piecewise-constant vectors.

We can easily generalize Lemma 3 and show that the image of a piecewise constant vector under any reducible braid that preserves a finite non-empty family of round curves is also piecewise constant: Lemma 4 (invariance of piecewise-constant vectors). Let α be a reducible braid with a reduction system C consisting of k round curves. Let v be a vector in C n that is piecewise constant on (D n , C ). Then the image B(α)(s)v is a vector that is piecewise constant on (D n , C ) = (D n , α(C )) for all s.

Proof. Write σ in the form

σ = m ∏ =1 σ ε i( ),η 1 ( ),η 2 ( ) • k ∏ j=1 β j ,
where (i( ), η 1 ( ), η 2 ( )), ε , and β j are given in the proof of Lemma 1. Since Burau matrices fix constant vectors, then for each 1 ≤ j ≤ k we have that B[β j ](s) v = v for all vectors v that are constant on the punctured disk E j enclosed by the curve C j . Furthermore, by Lemma 1, the Burau matrices of the braids σ ε i( ),η 1 ( ),η 2 ( ) also send piecewise constant vectors to piecewise constant vectors. The claim follows.

In the case of pure braids, both spaces of piecewise constant vectors appearing in the above lemma coincide. We can thus expect the existence of piecewise constant eigenvectors. To do so, we use the fact [START_REF] Squier | The Burau representation is unitary[END_REF] that reduced Burau matrices preserves a specific sesquilinear form, which is known to be positive definite when the Burau parameter s has unit modulus and is close enough to 1 (more precisely, when the argument is less than 1/n). The same is true for the reduced Burau matrix restricted to the space of piecewise constant vectors modulo constant, which is invariant by the above lemma. Hence this restriction is conjugate to a unitary matrix, implying that it is always diagonalizable. We thus deduce: Proposition 5. Let α be a reducible pure braid with round reduction system C . If s has modulus 1 and argument less than 1/n, the space of piecewise constant vectors on (D n , C ) modulo constant vectors has a basis consisting of eigenvectors of the reduced Burau matrix B[α](s).

C. Putting it all together

Now, recall that every reducible braid β ∈ B n is conjugate to a reducible braid α that preserves a family of round curves, and we can write β = γαγ -1 , for some γ ∈ B n . Their corresponding reduced Burau matrices can thus be written as

B[β ](s) = B[γ](s) B[α](s) B[γ -1 ](s).
When s = 1, B[γ](s) is a permutation matrix. Since the Burau representation B[γ](s) is continuous in s, then when s is close to 1, the Burau matrix B[γ](s ≈ 1) is close to a permutation matrix. Thus, if B[α](s ≈ 1) has an eigenvector that is piecewise constant on its components, then B[β ](s ≈ 1) has an eigenvector that is almost piecewise constant on its components, yielding in the main result. More formally:

Proof of the Main Result. Let β be a reducible pure braid on n strands with reduction system C β . There exists a braid α conjugate to β in B n , with β = γαγ -1 for some γ ∈ B n , such that α has a round reduction system C = γ -1 (C β ), whose curves are given by γ -1 (C j ), for C j ∈ C β 41 . We consider γ as an automorphism of the punctured disk as necessary.

Let their corresponding reduced Burau matrices be denoted

A(s) = B[α](s) B(s) = B[β ](s) C(s) = B[γ](s), Then B(s) = C(s)A(s)C(s) -1 .
By Proposition 5, since α is a reducible pure braid with a round reduction system C , there exists δ A > 0 such that for all s with modulus 1 and satisfying |s -1| < δ A , the reduced Burau matrix A(s) has a set of eigenvectors forming a basis of the space E of piecewise-constant vectors on

(D n , C ). If v(s) is such an eigenvector, C(s)v(s) is an eigenvector of B(s) since B(s)C(s)v(s) = C(s)A(s)C(s) -1 C(s)v(s) = C(s)A(s)v(s) ∝ C(s)v(s)
Also because C( 1) is induced by the permutation matrix given by the action of γ on the punctures, C(1)E is the space of piecewise constant vectors on (D n , γ(C )) = (D n , C β ). Now since C(s) is continuous in s at s = 1, there exists δ C > 0 such that for all s satisfying |s -1| < δ C , we have that

C(s)C(1) -1 -I < ε. (3) Since C(s)E = C(s)C(1) -1 C(1)E,
the claim follows.

IV. Numerical implementation

Based on the previously discussed eigen-properties of the Burau representation of a pure reducible braid, we now describe an algorithmic approach to detecting coherent regions from a set of trajectories.

A. From trajectories to braid

Given a 2D domain D and a set of n particle trajectories x j (t k ) in D for 1 ≤ j ≤ n and for time step indices 1 ≤ k < l we first convert the input into a topological braid by artificially closing each trajectories so that a pure braid can be constructed as explained in Sec. II.

Closure of trajectories.

So far, we have discussed the braid-theoretic interpretation of a set of disjoint particle trajectories that form a pure braid, whose properties reflect the underlying dynamics of a flow or a mechanical system. In a pure braid, the set of initial positions is the same as the set of final positions. However, in most physical systems, particle trajectories are not periodic, and the set of final positions are not typically the same as the set of initial positions. Consequently, the resulting algebraic braids formed by changing the projection line in Sec. II will not necessarily be conjugate [START_REF] Thiffeault | Braids of entangled particle trajectories[END_REF][START_REF] Boyland | Topological methods in surface dynamics[END_REF] . In order to fix this issue, we form a pure braid by appending the set of initial positions to the end of the set of particle trajectories; i.e., for each trajectory

(x i (t 0 ) , • • • x i (t l-1
)), we append the initial position to form the trajectory

(x i (t 0 ) , • • • x i (t l-1 ) , x i (t l ) ≡ x i (t 0 )) .
Note that this artificial closure may change the reducibility of the braid; but we will show in Sec. V that this theoretical shortcoming does not affect our ability to detect coherent sets in practice: when trajectories are sufficiently long, this small modification of the trajectories does not alter the overall dynamics significantly. Trajectory closures do, however, guarantee that the modified trajectories result in a pure braid, by construction.

Braid construction.

We use BRAIDLAB [START_REF] Thiffeault | BRAIDLAB: a software package for braids and loops[END_REF] to compute the algebraic braid {σ i } i=1..n corresponding to the motion of the particles, with respect to some arbitrary projection line. We note that if our initial positions happen to be on a regular grid, one should perturb the initial positions and/or choose a projection line that is not aligned with the grid lines: this will help resolve some coincident particles on the projection line, allowing the topological braid to be computed. The resulting pure braid is then encoded as a sequence σ of length L of Artin generators σ i corresponding to the L crossings of the piecewise-linear trajectories with respect to the projection line.

B. From braid to Burau matrix

The sequence σ of Artin generators specifies a Burau matrix corresponding to the braid of particle trajectories: for each generator σ i of the topological braid, its corresponding Burau matrix B[σ i ](s) is given by Eq. ( 1), and the product ∏ B[σ a ](s) of these matrices in the order of the sequence σ is the Burau matrix B of the braid.

Choice of Burau parameter.

Based on our results from Sec. III, we must pick a value s such that |s| = 1 and s ≈ 1 to construct the Burau matrix for the flow. We should also choose s not to be too close to 1 so that B[σ ](s) is not too numerically close to the identity matrix B[σ ](1). Besides the guarantee of diagonalizability afforded by Proposition 5, one of the advantages of picking a value s with a magnitude of one is that it prevents the rapid growth of the Burau matrix coefficients, which could grow exponentially over long durations otherwise. In practice, choices of s with |s -1| between 10 -4 and 10 -12 worked consistently well. In all our examples, we used s = exp(10 -8 i).

C. Finding coherent regions

We then proceed to an eigendecomposition of the Burau matrix B. While the guarantees of Sec. III were proved for reduced Burau matrices, we found that using unreduced Burau matrices worked as well in practice. It is in fact not difficult to see that the theoretical guarantees can be extended to unreduced Burau matrices except in certain cases that are unlikely to happen; we omit further discussions on this point here, and focus instead of numerical details.

Eigensolver.

There are a number of numerical libraries (MATLAB, LA-PACK, etc) that can be used to compute eigenvectors of a dense non-symmetric matrix like B. In particular, one can exploit parallelism by finding on separate cores the eigenvectors corresponding to different search intervals of eigenvalues [START_REF] Gates | Accelerating computation of eigenvectors in the dense nonsymmetric eigenvalue problem[END_REF][START_REF] Kestyn | FEAST eigensolver for non-Hermitian problems[END_REF] . These numerical methods thus allow us to handle braids with tens of thousands of strands on a personal computer. Note that one should not expect each relevant eigenvector to correspond to each coherent region: instead, a relevant eigenvector will typically be nearly piecewise constant over multiple coherent regions, see Figures 5 and7 for examples.

Visualization.

We visualize eigenvectors of the Burau matrix as follows. We first compute the Voronoi diagram [START_REF] Boissonnat | Algorithmic geometry[END_REF] of the initial positions {x i (0)} 1≤i≤n , which tessellates the domain D into disjoint convex cells {V i } 1≤i≤n . For each eigenvector, we then simply color each cell based V j on the associated entry of the eigenvector of matrix B. Note that when the initial positions are on a regular grid as we will use in Sec. V, the resulting visualization forms a pixelation of the domain (e.g., see Fig. 5).

Further processing.

We note that further processing can be done to help a user with finding material coherence. First, many of the eigenvectors found with our methods are quite easy to discard automatically through image processing; see Fig. 3 for examples of eigenvectors that do not depict meaningful material coherence. Second, we can also extract a series of potential coherent regions as follows: since each value of a relevant eigenvector v corresponds to a value at a particle at the first timestep t 0 , we can compute the levelsets L θ = {i | |v iθ | < ε} consisting of all particles whose corresponding ith value in v is approximately θ . These coherent sets can then be observed as they evolve in time. Finally, boundaries of a coherent sets correspond to the material loops studied by Allshouse and Thiffeault [START_REF] Allshouse | Detecting coherent structures using braids[END_REF] . We can thus consider the isotopy class of loop(s) that separate L θ from the rest of the domain: applying the method described by Allshouse and Thiffeault [START_REF] Allshouse | Detecting coherent structures using braids[END_REF] , we can track the growth of the material loop via Dynnikov coordinates and verify that the boundaries of the levelsets we found are indeed slow-growing to confirm coherence. Note that in Section V B, we will examine a modified Duffing oscillator and compare the coherent sets found in our method (Figure 7) with the coherent sets found by Allshouse and Thiffeault (Figure 8).

D. Discussion

Since our numerical probing of coherent sets only involves an eigenvalue problem, our exploration of material coherence from trajectories scales very well. In particular, we can handle braids arising from thousands or tens of thousands of particles on low-end personal computers, far beyond what was reported in Allshouse and Thiffeault [START_REF] Allshouse | Detecting coherent structures using braids[END_REF] and Francois et al. [START_REF] Francois | Braid entropy of two-dimensional turbulence[END_REF] .

Our approach is not without limitations. In particular, no strong guarantee is offered that this eigenanalysis will uncover all relevant coherent sets: our use of trajectory closures and the unfaithfulness of the Burau representation are two of the theoretical hurdles that may prevent a proper detection. Our approach should thus be used as a detection tool to narrow in on potential coherent sets. We demonstrate next that numerical experiments behave remarkably well on practical flows.

V. Experiments on flows

In the following section, we demonstrate the relevance of our contribution on non-trivial flows. We focus on two different continuous dynamical systems: the blinking vortex flow, and the (modified) Duffing oscillator. Starting from a discretization of the domain D (a square in all our examples) through particles forming a regular grid, each particle is advected in time by the flow, generating an associated trajectory in space-time. The set of trajectories is then processed as described in Sec. IV, and compared to what is known in the literature.

A. Blinking vortex flow

The blinking vortex flow was introduced by Aref as an idealization of stirring [START_REF] Artin | Theory of braids[END_REF] . The flow is given by a pair of vortices separated by a finite distance, blinking on and off periodically in an alternating fashion in an incompressible, inviscid fluid.

We consider a modified version of this flow in an unbounded domain (modeled on the complex plane). The velocity field due to a single point vortex located at x = a on the x-axis is given by ṙ

= 0 θ = Γ 2πr ,
where Γ is the strength of the vortex, and r = (xa) 2 + y 2 is the distance to the center of the vortex. The mapping, in dimensionless form [START_REF] Ottino | The kinematics of mixing: stretching, chaos, and transport[END_REF] , induced by identical two vortices at ξ i = ±a, each acting for time T , is given by the twist map

x y → ξ i + (x -ξ i ) cos ∆θ -y sin ∆θ (x -ξ i ) sin ∆θ + y cos ∆θ ,
where ∆θ = µ r 2 , with µ = ΓT 2πa 2 , and r = (xξ i ) 2 + y 2 . The parameter µ is the flow strength, and its value controls the behavior of the system. We make distances dimensionless with respect to a and place the vortices at ξ i = ±1. When both vortices act simultaneously (T = 0, µ = 0), the system is integrable. We perturb the system by increasing µ from zero and check the Poincaré sections t = kT , k ∈ Z. Chaotic regions appear for all µ > 0 [START_REF] Doherty | Chaos in deterministic systems: strange attractors, turbulence, and applications in chemical engineering[END_REF] . For small values of µ, small chaotic regions exist near the elliptic and hyperbolic points. As µ increases, the size of the chaotic regions grow, destroying confining KAM surfaces as the chaotic regions merge. See a few Poincaré sections for various values of µ in Fig. 4. Using our numerical approach, we give eigenvectors for the blinking vortex flow with various values of µ in Fig. 5 for a set of 900 trajectories of particles, which were initially placed on a regular grid covering the domain. Coherent regions appear, indeed, as isolevels of the eigenvectors as expected.

B. Modified Duffing oscillator

In this example, we study a modified Duffing oscillator, given by ẋ = y + α cos (ωt)

ẏ = x 1 -x 2 -dy + γ cos (ωt) (4)
with α = 0.1, γ = 0.14, d = 0.08, ω = 1. This compressible system is also studied by Allshouse and Thiffeault [START_REF] Allshouse | Detecting coherent structures using braids[END_REF] with the same parameters as an example a system with two primary Trajectories of the modified Duffing oscillator (4) belong to one of three types, each illustrated in Figure 6(right) with the corresponding colors. The yellow region corresponds to the yellow limit cycle and its basin of attraction. The green regions (eventually) attract to one of the two green limit cycles. The blue region attracts to neither the yellow limit cycle nor green limit cycles during the time period studied.

Using the computational method described in this paper, we obtain the Burau eigenvectors for the modified Duffing oscillator depicted in Figure 7. In addition to the two initial conditions found by Allshouse and Thiffeault [START_REF] Allshouse | Detecting coherent structures using braids[END_REF] , illustrated in Figure 8, the two additional limit cycles (green in Figure 6(left)) are also present.

We note that since the modified Duffing oscillator is a com-pressible system, particle positions can in fact coincide. In order to form a well-defined braid, we select a sufficiently sparse sampling of the domain and a time window so that the dynamics are reasonably well-captured yet no two particles coincide at any time. Allshouse and Thiffeault [START_REF] Allshouse | Detecting coherent structures using braids[END_REF] argue that the ability of the braid theoretic approach to detect coherent sets even in an incompressible flow is a testament to the wide applicability of the method.

VI. Conclusions

Our approach based on topological braids is especially advantageous when the input data are sparse, since it does not require nearby trajectories or derivatives of the velocity field. However, the braid approach is not without limitations. Accuracy is limited by the length of trajectory histories, and braid methods are only adapted to time intervals where particles never intersect at the observation scale. Fortunately, many geophysical flows, when observed at sufficiently large scale, are nearly incompressible, rendering this last limitation only mildly restrictive.

Faithfulness.

In this work, we have chosen to use the Burau representation for the analysis of spatially sparse particle trajectories. The Burau representation is not faithful but confers computational advantages (n vs. n 2 ) in both space and time requirements compared to the faithful Lawrence-Krammer

FIG. 1 .

 1 FIG. 1. From trajectories to topological braid. (a) A set of spacetime trajectories forms a physical braid as the trajectories entangle in time; (b) by tracking the order of the particles along a projection line as time goes by, one can construct a topological braid encoding the trajectories.

FIG. 2 .

 2 FIG. 2. Braids and dynamics. (a) The topological braid of a set of trajectories can be encoded through a composition of Artin generators σ i (where trajectory x i goes behind trajectory x i+1 as depicted) and their inverses; (b) a schematic example of 7 particles evolving in time (starting at the top) where the resulting braid is reducible as there are three tubular braids, formed by pairs or a triplet of particles.

FIG. 3 .

 3 FIG. 3. Irrelevant eigenvectors. Eigenvectors of Aref's blinking vortex flow for µ = 50. Most of the eigenvectors of a Burau matrix are nearly constant (left), or only show faint parts of coherent structures (middle/right); here, eigenvector 12, 426, and 574. Compare these results with Fig. 5(bottom right) showing an eigenvector that clearly reveals dynamical structures.

5 FIG. 4 .

 54 FIG. 4. Poincaré sections for Aref's blinking vortex flow. From left to right, top to bottom: µ = 1, 5, 10, 20, 30, 35, 40, 50.

5 FIG. 5 .

 55 FIG. 5. Burau Eigenvectors for Aref's blinking vortex flow. From left to right, and top to bottom: µ = 1, 5, 10, 20, 35, 50. Nine hundred particles were used, started on a 30 × 30 regular grid.

3 FIG. 6 .

 36 FIG. 6. Modified Duffing oscillator. Left: a three-coloring of initial particle positions by their position at time t = 250; Right: Phase portrait for trajectories belonging to each of the three regions.

3 FIG. 7 .FIG. 8 .

 378 FIG. 7. Two Burau eigenvectors for modified Duffing oscillator. Two of the eigenvectors of the Burau matrix computed from trajectories of the modified Duffing oscillator (1225 particles); the three main dynamical regions are present as expected.

Acknowledgments

MD gratefully acknowledges the Inria International Chair program during which he started this work, and all the members of the TITANE team for their support.

representation [START_REF] Bigelow | The Lawrence-Krammer representation[END_REF] . If a greater level of topological and dynamical detail is preferred, one may wish to consider the Lawrence-Krammer representation instead.

Parallelism.

The algorithm that we describe in this chapter lends itself naturally to parallelism. We do not discuss the details here, but we note that the matrix chain multiplication required by our algorithm is amenable to optimization. In particular, in addition to the usual considerations of the matrix chain multiplication problem [START_REF] Hu | Computation of matrix chain products. part I[END_REF][START_REF] Hu | Computation of matrix chain products. part II[END_REF] , we remark that by taking the sequence of braid generators into account, then depending on the flow, we can potentially partition the braid β = ∏ L σ b into subsequences of neighboring generators (e.g., each subsequence consists only of generators σ i-j , σ i-j+1 , • • • , σ i+ j , for small j). This effectively partitions the sequence of matrix multiplications into subsequences that each consist of (mostly) sparse matrix multiplications, thereby reducing the computational requirements of our analysis. Future work may wish to formalize these and other computational considerations: such a computational optimization may be crucial, in particular if a (n 2 × n 2 ) Lawrence-Krammer representation is used.


Dr Rita Meunier-Prest
email: rita.meunier-prest@u-bourgogne.fr

Dr Guillaume Gruntz

Dr Frédéric Castet

Dr Yohann Nicolas
email: yohann.nicolas@enscbp.fr

Amélie Wannebroucq

Marcel Bouvet

Thierry Toupance

Pr M Bouvet

Pr Thierry Toupance

Electrochemical and Spectroelectrochemical Behavior of a Tetracyanotriphenodioxazine in Solution and Thin-Films

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Introduction

Recent research efforts in the field of electron-poor -conjugated compounds mainly concerns n-type organic semiconductors (OSC) which are combined with p-type materials in heterojunctions. Whereas the latter afforded high performances in many different electronic devices, n-type OSCs still remain rare for numerous applications. Thus, most of studies on organic photovoltaic devices bring into play fullerene derivatives as n-type active OSCs. On the other hand, the development of new n-type OSCs is driven by the possibility in organic electronics to build complementary circuits, with the ability to switch organic fieldeffect transistors (OFET) with both positive and negative gate voltages [1] , as it has been already achieved with inorganic materials in complementary metal-oxide-semiconductors (CMOS).

To design an efficient n-type OSC for these devices, the stabilization of its anionic form is required. Indeed the reduced form of the material, which bears the charge carrier, should not react chemically with its environment or with itself. A solution consists in shifting the reduction potential towards less negative values to obtain less reactive anions. Thus, the introduction of electron-withdrawing substituents on aromatic cores leads to more stable charge carriers by lowering the energy of the delocalized Lower Unoccupied Molecular Orbital (LUMO) and increasing the reduction potential. For example, fluoro or cyano substituents introduced on a phthalocyanine ring [2] or on oligothiophenes [3] stabilize the reduced states as well as carbonyl groups in perylene. [4] Another way to transform a p-type material into a n-type analogue is to introduce nitrogen instead of carbon atoms directly inside the aromatic core. Thus, whereas pentacene exhibits a p-type behavior, tetraazapentacenes that contain four nitrogen yield efficient n-type OSCs. [5] Similarly, triphenodioxazines (TPDO) containing nitrogen and oxygen atoms and endowed with cyano groups afforded stable n-type OFETs. [6] Additionally, the quinonoid structure at the center of the pentacycle in TPDOs is responsible for enhancing the stability of their anionic form. [7] Recently, we extended their potentialities of n-type OSCs to another organic electronic device -the conductometric chemical sensors. [8] Thus, we reported a new class of n-type sublayer in Molecular Semiconductor -Doped Insulator heterojunctions (MSDI) [8] made of a tetracyano triphenodioxazine bearing two triisopropylsilylethynyl moieties (TiPS-TPDO-tetraCN), combined with a highly conductive molecular material, the lutetium bisphthalocyanine, LuPc2. In the course of this investigation, an unexpected behavior was highlighted. When the devices were prepared by vacuum evaporation of TiPS-TPDO-tetraCN, a high and competitive MSDI chemical sensor was obtained in contrast to devices processed by spin-coating. We demonstrated that it was due to the formation of the diprotonated molecule during the vacuum evaporation process. However, in the triphenodioxazine series, only few examples of diprotonated structures were reported [9] and no data about their redox properties. Taking into account the quinonediimine structure of TPDOs, it is worth reminding the electrochemical behavior of parabenzoquinones to understand this of TPDOs. Due to their important role in biological processes, [10] quinones have been widely studied. It is well-known that benzoquinones (BQ) can be reduced into dihydroquinones (H2Q), but the involved mechanisms and thermodynamics depend on the quinones environment. Thus, a molecule adsorbed on a working electrode exhibits electrochemical processes different from these of molecules in solution, due to intermolecular interactions and solvation energies. When the quinones are incorporated into a self-assembled monolayer, the global two electronstwo protons electrochemical reduction, in aqueous media, can be described on the basis of a nine-member square scheme. [11] However, the reaction proceeds through proton-coupled electron transfer in aqueous media, [12] but not in unbuffered media. [13] In the solidstate, in addition to the reduced and oxidized forms, another structure, called quinhydrone, can be stabilized, which combines both redox species. Quinhydrones are well-known charge transfer complex between the electron-acceptor (BQ) and the electrondonor (H2Q)  systems. They are stabilized by  interactions inside the donor-acceptor stacks while infinite molecular chains formed through hydrogen bonds between neighboring columns. [14] To get a deeper insight in the chemical structure and the formation mechanism of the diprotonated TiPS-TPDO derivative during the vacuum evaporation of TiPS-TPDO-tetraCN and in the redox properties of thin films used in MSDI devices, we hereafter report on its electrochemical reduction behavior both in solution and solid-state. In particular, a spectroelectrochemical study was carried out in acetonitrile solutions, with and without water, followed by simulations of the voltammograms, to characterize the different reduced species, to study the reversibility of the electrochemical reaction and to better understand the reaction mechanism. The key point was to identify if the diprotonated species was also formed in solution. Finally, to highlight the role of protons, electrochemical responses of films prepared by vacuum evaporation of TiPS-TPDO-tetraCN onto ITO substrates were recorded in aqueous medium.

Results and Discussion

TIPS-TPDO-tetraCN in solution

Electrochemical behavior of TiPS-TPDO-tetraCN in CH2Cl2. The cyclic voltammogram of TiPS-TPDO-tetraCN recorded in dichloromethane (CH2Cl2) (Fig. 1) presented two quasi-reversible reduction steps at -185 mV and -618 mV / SCE with a difference between anodic and cathodic peak potentials larger (Ep = 100 mV) than what it was theoretically expected (58 mV), due to the resistivity of the medium. For both systems, the ratio of the anodic to cathodic peak currents was equal to one and the voltammogram recorded on ultramicroelectrode, under stationary conditions, showed two waves of equal intensity each one corresponding to the transfer of one electron. It matches the following EE mechanism leading first, to the radical anion and then to the dianion (𝑬 𝟐 𝟎 < 𝑬 𝟏 𝟎 ).

(1)

Electrochemical behavior of TiPS-TPDO-tetraCN in MeCN.

In acetonitrile, the electrochemical behavior was slightly different. Both reduction steps were still observed. The systems were fully reversible (Ep = 58 mV) with a little offset more positive potentials (E 0 1 = -112 mV and E 0 2 = -487 mV) compared to those obtained in CH2Cl2. The difference between both redox systems, (E 0 1 -E 0 2), was smaller (375 mV) than that observed in CH2Cl2 (433 mV). This difference depends on many factors such as delocalized or localized charge in the ions, ion pairing with the supporting electrolyte, structure of the molecule (linear, presence of unsaturated bonds or aromaticity) [15] and also the nature of the solvent. [16] In particular, it has been shown that this difference (E 0 1 -E 0 2) increases with decreasing the solvent polarity, [16] as we have observed when switching solvent from acetonitrile to dichloromethane. In the present case, peak currents varied linearly with v 1/2 indicating that the reaction is diffusion controlled. Another difference appeared on the second reduction system where the cathodic peak (Epc2) was smaller than the first one and was preceded by a small shoulder (Epcw). The explanation for this shoulder was partly provided by the same study in presence of water (Fig. 2). When increasing amounts of water were added, evolution of the second system was observed: the anodic peak (Epa2) decreased while the cathodic one was shifted towards positive values up to the potential corresponding to the shoulder, at -450 mV (Epcw), which grew with the increasing concentration of water. The figure 2B shows the evolution of the cathodic peak potentials as a function of the logarithm of the water concentration. As it was evidenced in the reduction mechanism of quinones, this potential displacement can be assigned to fast hydrogen-bonding equilibria [17] inducing stabilization of the dianion by water molecules. Finally, a last difference has been observed. While both redox systems had the same intensity in CH2Cl2, the second system was smaller than the first one in MeCN. Consequently, a more complex mechanism should occur that consumes part of the radical anion and provides the initial product back. To investigate more deeply this mechanism, a spectroelectrochemical study was conducted.

Spectroelectrochemistry of TiPS-TPDO-tetraCN in MeCN.

The UV-visible spectrum of TiPS-TPDO-tetraCN in acetonitrile (Fig. 3 before electrolysis) was first recorded and exhibited two maxima at 554 and 522 nm and a shoulder at 484 nm (Table 1). The absorption spectrum simulated using time-dependent density functional theory (TD-DFT) at the M06-2X/6-311+G (d) level (see the computational section for details) is provided in supporting information (Fig. S1), as well as the shape of the frontier molecular orbitals and the electron density difference between the ground state and the lowest-energy excited state (Fig. S2). As shown in the computed spectrum, the first absorption band, which is attributed to a HOMO-to-LUMO electronic excitation, is red-shifted by 44 nm (0.16 eV) compared to experiments.  (M -1 cm -1 ) 3300 6200 6600

Then, an electrolysis of TiPS-TPDO-tetraCN (10 -3 M) was performed to determine the number of electrons involved in the global reduction process. A potential of -0.6 V was imposed, i.e. after the second reduction step (Fig. 2). After consumption of 2 electrons per molecule of TiPS-TPDO-tetraCN, the current decreased to zero and the electrolysis was stopped (96 mC) as expected from earlier studies. This confirms the involvement of 2 electrons in the global reduction step.

To get an insight on the nature of the reduced species and on the reversibility of the electrochemical reaction, we then carried out two spectroelectrochemical experiments on the solution of neutral TIPS-TPDO-TetraCN. In the first one, we imposed a potential of -0.2 V corresponding to the first reduction step. After consumption of 48 mC corresponding to an exchange of one electron per molecule, i.e. half of the amount of electricity consumed at -0.6 V, the experiment was stopped. It took 1800s.

The progress of the electrolysis was followed by UV-visible spectroscopy (one spectrum every 6 s, i.e. 300 spectra). Fig. 3A presents the evolution of the spectra. The absorption bands of TiPS-TPDO-tetraCN at 484, 522 and 554 nm decreased while new absorption bands at 628 and 680 nm appeared. It is important to note that after consumption of one electron per molecule, half of the initial amount of TiPS-TPDO-tetraCN remained present in the cell. This topic will be addressed in details in the paragraph "Discussion on the reduction mechanism in acetonitrile".

In the second spectroelectrochemical experiment, we imposed a potential of -0.6 V corresponding to the second reduction step.

The experiment was stopped when the current tended to zero, after 3600 s and consumption of 96 mC. The consumption of two electrons per molecule of TiPS-TPDO-tetraCN led to complete disappearance of the absorption bands of the initial product (Fig 3B). At the same time, three absorption bands at 574, 628 and 680 nm (Table 2) grew rapidly. Moreover, the spectra showed an isosbestic point at 586 nm, which reveals the existence of only two chromophores in the solution. The reaction is perfectly reversible since the spectroelectrochemical reoxidation of this solution at + 0.1 V gave back to the absorbance spectrum of TiPS-TPDO-tetraCN (Fig. 3C-D) with the same intensity as that of the initial product. At a first glance, we assigned the new absorption bands emerging at large wavelengths to the (TiPS-TPDO-tetraCN) 2-dianion, which is the expected reduced species in organic solvent. However, this assumption was contradicted by TDDFT calculations, which evidenced that the 2e-electrolysis of TiPS-TPDO-tetraCN should give rise to blue shift of 23 nm (0.08 eV) of the absorption maximum (Fig. S1), instead of the red shift observed experimentally. It must be also pointed out that, on the time scale of the spectroelectrochemical experiment, the formed dianionic species remains perfectly stable.

In light of these results, the spectrum obtained after one electron reduction per molecule at -0.2 V corresponds to the superimposition of the spectrum of the initial compound, TiPS-TPDO-tetraCN and of a 2e -reduction product, different from (TiPS-TPDO-tetraCN) 2-. To rationalize this finding, the existence of a chemical reaction involving the radical anion (TiPS-TPDO-tetraCN) •-and leading to TiPS-TPDO-tetraCN and a dianionic compound should be considered. Since 𝐸 2 0 < 𝐸 1 0 (eq.1) and owing to the computational results, a disproportionation (DISP) mechanism can be ruled out. [18] To check the involvement of the radical anion in the formation of TiPS-TPDO-tetraCN and dianion, a spectroelectrochemical experiment at -0.2 V in the presence of (2,2,6,6-Tetramethylpiperidin-1-yl)oxyl (TEMPO)a radical scavenger [19] -, was performed. First, the absorption spectrum and the cyclic voltammogram of TEMPO were recorded to ensure that its responses were out of the range studied in our experimental conditions. Actually, the UV-visible spectrum presented one band at 254 nm and a weak band at 640 nm and the voltammogram showed only a redox system at 0.6 V / SCE. Fig. 4 depicts the evolution of the absorption spectra during the spectroelectrochemistry in the presence of an excess of TEMPO (CTEMPO = 5 x CTiPS-TPDO-tetraCN). In this region, the absorption of TEMPO was negligible compared to this of TPDO derivatives. A decrease of the absorption bands corresponding to TiPS-TPDO-tetraCN was observed as above. But, contrary to the experiment in the absence of TEMPO, there was almost no emergence of the bands at 628 and 680 nm. This confirms the key role of the radical anion (TiPS-TPDO-tetraCN) .-in the chemical process. The very small band at 680 nm is probably due to a competition between this chemical reaction and the trapping of the radical anion by TEMPO.

Spectroelectrochemistry of TiPS-TPDO-tetraCN in presence of water. The spectroelectrochemical experiment of TiPS-TPDO-tetraCN was repeated in acetonitrile containing 2% H2O v/v i.e. 𝑐 𝐻 2 𝑂 = 1 𝑀 as previously done for cyclic voltammetry. This corresponds to the addition of 100 µl of water in the experiment of Fig. 2. The evolution of the absorption spectra was similar to that observed in absence of water. The only difference is the position of the shoulder at 574 nm that is slightly shifted to 564 nm in presence of water. It should be noted that the UV-visible spectrum at the end of the electrolysis remained the same as without water and still was different from that of the diprotonated compound (TiPS-TPDO-tetraCN)H2. [8][9] It corresponded only to that of the dianionic compound (Fig. 3 and Table 2). Kinetics of formation and disappearance of the different products were measured according to the potential imposed during the electrolysis. Therefore, the absorbance at 522 nm and 680 nm was monitored to follow the concentration of TiPS-TPDO-tetraCN and this of the dianionic product, respectively (Fig. 5). The band at 522 nm was preferred to that at 554 nm to avoid the absorption spectra's overlap of TiPS-TPDO-tetraCN and the new dianionic species. During the electrolysis at -0.6 V, absorbance at 522 nm (Fig. 5A) decreased until a value of 0.1, which corresponds to the foot of the band at 564 nm due to the formation of the dianion. Reoxidation of the solution led back to the initial concentration in TiPS-TPDO-tetraCN (Absorbance of 0.44). At the same time, absorbance at 680 nm grew rapidly and disappeared during reoxidation (Fig. 5B). Both kinetic curves of absorbance of the band at 522 nm during electrolysis at + 0.1 V (Fig. 5A) and that at 680 nm during electrolysis at -0.6 V (Fig. 5B) exhibited the same shape, which shows the perfect reversibility of the reaction. However, when a potential of -0.2 V was imposed, the absorbance at 680 nm remained very low and did not increase contrary to that was obtained in the absence of water (Fig. 3). Reoxidation of the solution gave back only partially the initial product.

Comparison of kinetics at -0.6 V in absence and in presence of water was performed and the results are shown in Fig. 6. The small differences in concentration of TiPS-TPDO-tetraCN across the different experiments carried out have induced small differences in absorbance. In order to correct these differences, we divided all the absorbances of a same experiment by that of the band at 522 nm in the initial spectrum (A522). Fig. 6 shows the presence of water did not affect kinetics of formation of the dianion (band at 680 nm) while kinetics of disappearance of TiPS-TPDO-tetraCN was much faster in presence of water.

Discussion on the reduction mechanism in acetonitrile. The above results revealed the main features on the reduction mechanism of TiPS-TPDO-tetraCN in acetonitrile: it is reduced in two successive one-electron steps (Eq. 1 with E 0 1 > E 0 2), the second system being smaller than the first one, as shown by cyclic voltammetry. A 1 e -per molecule electrolysis, at -0.2 V (first reduction step (𝐸 1 0 )), has led to half of the initial concentration in TiPS-TPDO-tetraCN and dianionic compound as shown by spectroelectrochemical studies. The same experiment performed in presence of a radical scavenger, no longer leads to TiPS-TPDO-tetraCN and the dianionic product. The existence of a chemical reaction involving the radical anion (TiPS-TPDO-tetraCN) .-and leading to TiPS-TPDO-tetraCN and an unknown dianionic compound must be considered. It cannot be a disproportionation mechanism (DISP) corresponding to the homogeneous electron transfer reaction that, from B, gives back to A and C because 𝑬 𝟐 𝟎 < 𝑬 𝟏 𝟎 ). [18] Similarly, dimerization is not foreseeable because it is a 0.5 e -process. Therefore, a more complex process that consumes part of the radical anion and gives the initial product back must be envisaged. Let us consider a mechanism (bottom rectangle in scheme 2) where the radical anion (TiPS-TPDO-tetraCN) .-(B in scheme 2) and the dianion TiPS-TPDO-tetraCN 2-(C in scheme 2) are both involved in first order reactions such as rearrangement, ring opening, ring reorganization or conformation changes.

For example, planar and rigid -conjugated systems with a pentacene like frame where two aromatic rings containing nitrogen atoms and having vicinal disubstitution of aryl/aryl groups undergo a cleavage of the N/aryl bond that leads to non-planar molecules. [20] A second example is that of quinacridone analogues, which exhibit conformation dynamic changes triggered electrochemically. [21] Structural changes accompanying electron transfer have already been reported to render disproportionation (DISP) to be less unfavorable. [22] They could occur as a chemical reaction preceding or following the electron transfer. Due to the delocalization of negative charges bore by the nitrogen atoms till the cyano moieties in ortho and para positions, the reduction of TIPS-TPDO-tetraCN (A in scheme 3) leads to chemical structures negatively charged on these atoms, as exemplified by the 2e -reduced TIPS-TPDO-tetzraCN 2-(C in scheme 3). Let us consider chemical structures compatible with such a DISP mechanism. At first, it is worth mentioning that diarylamino-1,4-benzoquinone can be chemically oxidized into dioxazine derivatives (Scheme 4). [START_REF] Hunger | Industrial Dyes[END_REF] Thus, such a quinone structure could be assumed as the result of the reduction of TiPS-TPDO-tetraCN (A). However, it requires breaking simultaneously two chemical bonds, which is generally not kinetically favored. But, starting from the doubly reduced TiPS-TPDO-tetraCN 2-(C), it is possible to write a mesomeric form that involves a 1,2benzoquinonelike structure (E in scheme 3), which necessitates to break only one chemical bond. In a similar way, such orthoquinone structure (D) can be obtained by rearrangement, from the 1e -reduced (TiPS-TPDO-tetraCN) .-. Its 1e -reduction (E 0 3) leads to the 1,2-benzoquinonelike dianion (E in scheme 3). The structures (D) and (E) are proposed here as examples of restructured species to illustrate the phenomena observed in the spectroelectrochemical experiment.

Scheme 3. Various possible intermediates involved in the DISP mechanism.

For sake of clarity, the TIPS groups have been omitted and the non bonding electron pairs are only shown on the atoms involved in the process. [START_REF] Hunger | Industrial Dyes[END_REF] Scheme 4. Chemical oxidation of diarylamino-1,4-benzoquinone into dioxazine derivatives. [START_REF] Hunger | Industrial Dyes[END_REF] The existence of this dynamic changes triggered electrochemically could induce the regeneration of the initial product, A, by two new DISP pathways (Bottom rectangle in scheme 2). Both are homogeneous electron transfer reactions of B and D to form respectively A and C (reaction (1)) or A and E (reaction ( 2)). After solving the system equations, the disproportionation constant of reaction (1) remains too small to bring back to TiPS-TPDO-tetraCN in such a large proportion as in the spectroelectrochemical experiments. Another possibility consists in reaction ( 2) of scheme 2 involving the radical anions, B and D that disproportionate into the initial product, A, and the compound E, obtained after modification of the dianion. It occurs only if E 0 3 is higher than E 0 1 and E 0 2. As indicated in Scheme 2, disproportionation of B and D leads to the regeneration of A and to the 2e -reduced form E. This balance equation occurs with no modification of the number of bonds, while rupture/creation of bonds takes place during the rearrangement of B in D (scheme 3).

We decided to simulate this mechanism to check if it could explain the smaller intensity of the second system compared to that of the first one in the cyclic voltammetry study. At the same time, influence of addition of water on the evolution of the second system has also to be taken into account in the global mechanism (Top rectangle in scheme 2). Indeed, the anodic peak decreased while the cathodic one was shifted towards positive values up to the potential corresponding to the shoulder, at -450 mV. Such a behavior was already reported when quinones are reduced in the presence of water: Quan et al [17e] emphasized the role of hydrogen bonding between water molecules and the dianion that was stated to exist as a mixture of various proton states due to its acid-base properties. F and G correspond to complexes of radical anions, B, and dianions, C, with water molecules. The radical anion complex is more easily reduced than the non-complexed species and its potential E 0 4 is related to E 0 2, K4 and K5. When a relatively large concentration of water (cH2O > 0.8 M i.e. when the amount of water is higher than 80 µL in Fig. 2) is introduced in the cell, partial destruction of TiPS-TPDO-tetraCN was observed.

Reactions 𝐴 ↔ 𝐼 and 𝐺 ↔ 𝐻 were introduced to reflect this phenomenon. It should be noted that neither the DISP mechanism (red bottom rectangle in scheme 52) nor the mechanism involving reactions with water (top blue rectangle in scheme 25) can alone reflect the experimental results. The global mechanism in scheme 2 need to be considered to fit properly the experimental voltammograms. In particular, if the DISP mechanism is omitted, the anodic peaks decreased too much when the concentration in water was increased and, on the other hand, if the mechanism caused by water is missed, the potential shift obtained when increasing water concentration cannot be observed. The description of the approach used during the digital simulations was presented in the supporting information.

The comparison between experimental and simulated curves is presented in Fig. 7 for two different concentrations in water (1 M and 2 M). All the parameters are the same in both simulations except the water concentration. When possible, the values entered in the simulations, such as E 0 1 and E 0 2, are derived from experimental data. Other constants were adjusted by fitting related parameters, for example, E 0 4 was adjusted by fitting the equilibrium constants K4 and K5.

The good agreement between experimental and simulated curves in the presence of H2O strongly supports the formation of complexes with water in anion stabilization and of the involvement of structural changes accompanying electron transfer that make disproportionation (DISP) to be more favorable. M (open circles) curves were calculated using scheme 2 with E 0 1 = -0.113 V; ks1 = 0.04 cm s -1 ; E 0 2 = -0.49 V; ks2 = 0.01 cm s -1 ; E 0 3 = -0.105 V; ks3 = 0.02 cm s -1 ; ks4 = 0.02 cm s -1 ; K1 = 0.3 ; kf1 = 15 s -1 ; kf2 = 1 s -1 ; K3 = 0.5 M; kf3 = 0.06 M -1 s -1 ; K4 = 600 M; kf4 = 3 M -1 s -1 ; K5 = 1500 M ; kf5 = 2 M -1 s -1 ; K6 = 100; kf6 = 2560 s -1 ; kfDISP = 10 M -1 s -1 . In all electrochemical reactions, the transfer coefficient,  was taken as 0.5, the diffusion coefficient was set at 10 -5 cm 2 s -1 for all species and a resistance of 1 k and a capacitance of 6 µF were introduced.

Films of vacuum evaporated TiPS-TPDO-tetraCN

We previously demonstrated [8] that films obtained by vacuum evaporated TiPS-TPDO-tetraCN lead to the formation of the diprotonated molecule. Fig. 8 depicts the absorption spectrum of a film of vacuum evaporated TiPS-TPDO-tetraCN deposited on an ITO electrode. Compared to that of TiPS-TPDO-tetraCN in solution (Fig. 3 and Table 1) or to that of a film deposed by spin coating, [8] additional bands appeared at 460, 585 and 638 nm which have been attributed to the diprotonated compound (TiPS-TPDO-tetraCN)H2. [8] The reduced form is certainly stabilized in the solid-state via - interactions. Since the absorption bands of TiPS-TPDO-tetraCN were remained visible on the evaporated films, it can be suggested that TiPS-TPDO-tetraCN and (TiPS-TPDO-tetraCN)H2 coexist in the solid-state. The charge transfer interactions could stabilized it as it was observed in the case of quinhydrone charge transfer complexes obtained from an equimolar mixture of quinone and dihydroquinone. [START_REF] Bouvet | [END_REF] When the film was dissolved into CH2Cl2 (Fig. 8), the absorption spectrum evolved to lead to those of TiPS-TPDO-tetraCN (compare Fig. 3 and Fig. 8). In details, the bands attributed to (TiPS-TPDO-tetraCN)H2 disappeared attesting the destruction of the diprotonated molecule.

The electrochemical behavior of the film was studied in aqueous solution according to pH. Degradation of the film occurred rapidly, therefore, a thin layer of nafion was dropped on the film. However, in that case, the anodic peak potentials were shifted towards more positive values compared to that obtained on the electrode without nafion, probably due to the low rate of transport of the counterions, whereas the cathodic ones remained unchanged. Therefore, only the cathodic peak potentials were reported according to pH (Fig. 9). From a mechanistic point of view, the reduction of TiPS-TPDO-tetraCN in aqueous medium involves an exchange of two electrons and one or two protons, according to pH. The mechanism can be described using a nine-member square scheme (Scheme 5). [11,25] If the protonations are fast (at equilibrium), which is the case in buffered medium, the square scheme is kinetically equivalent to two successive reactions. [11, 25c] Scheme 5. The reaction scheme. In order to lighten the chemical formulae, TIPS groups have been omitted. The reductions are written horizontally from the left to the right and the protonations downwards. Indexing of potentials are different from those used by Laviron, [11, 25c] to be coherent with schemes 1 and 2.

The slopes of the graph Epc = f(pH) (Fig. 9) define the number of protons involved during the reduction. For pH lower than 4.5, TiPS-TPDO-tetraCN was reduced in two successive one electron steps. The slopes of the straight lines were respectively -55 and -59 mV / pH unit, which indicates the uptake of one proton per electron. The first redox system has led to the protonated radical, which was then reduced in another (1e -, 1H + ) process yielding the fully protonated compound, (TiPS-TPDO-tetraCN)H2. Above pH 4.5, the two stages overlapped and the process was equivalent to a simple 2e -reaction with a slope equal to -34 mV / pH unit. It corresponds to a (2e -, 1H + ) process leading to the monoprotonated anion, (TiPS-TPDO-tetraCN)H -. The changes in slopes at pH 4.5 correspond to pKa5 (Scheme 5). The evolution of the reduction of as a function of pH provides another proof of TIPS-TPDO-TetraCNH2 formation. Finally, the value of pKa5 is close to the pKa one of aromatic amines, which justifies the chemical structure drawn in (Scheme 5).

Conclusions

The electrochemical reduction of TiPS-TPDO-tetraCN was studied both in solution and solid-state after vacuum evaporation on ITO electrodes. In CH2Cl2 solution, it occurs in two successive 1e -reduction steps. But, in MeCN, the process is more complex: the intensity of the second system is lower than the first one and meanwhile a shoulder appears before the second cathodic peak. It has been attributed to the formation of a complex with water that facilitates the reduction of the radical anion. On the other hand, the spectroelectrochemical study demonstrated that the first 1e - reduction leads to an equimolar mixture of the initial product, TiPS-TPDO-tetraCN, and a dianionic species (formed during the second reduction step). Firstly, it has been proven this result cannot stem from a DISP mechanism implicating only TPDO-type species (E 0 1 > E 0 2). Instead, a DISP reaction involving the radical anion and a counterpart molecule derived from structural changes to give back TiPS-TPDO-tetraCN and a restructured dianion has been successfully envisaged. Indeed, voltammograms which were simulated based on these mechanisms involving structural changes of the radical anion, a new DISP reaction and complexation with water, properly fit with the experimental results, even with different water concentrations. Secondly, the study of TIPS-TPDO-teraCN thin films obtained by vacuum evaporation provides a proof that they are composed of diprotonated molecules. Indeed, the potential variations according to the pH showing that reduction occurs when pH < 4.5 in two successive (1e -, 1H + ) steps and above pH 4.5 in a single (2e -, 1H + ) stage were additional evidences of hydrogen-nitrogen bonds in evaporated TIPS-TPDOtetraCN materials. Moreover, this study has also made possible the experimental pKa determination of the diprotonated compound (TiPS-TPDO-tetraCN)H2. Finally, we demonstrated that (TiPS-TPDO-tetraCN)H2 is unstable and decomposes in solution while its straightforward formation in solid-state yields a stable material.  and charge transfer interactions contribute to this stabilization as it was shown for the quinhydrone charge transfer complexes. Consequently, this study suggests that the active thin film coated in MSDI devices consists in a complex between diprotonated and unprotonated TIPS-TPDO-tetraCN. It brings a crucial element to understand the electronic properties of TPDO -based devices.

Experimental Section

Chemicals. All chemicals were used as received. Activated alumina, 2,2,6,6-tetramethyl-1-piperidinyloxy free radical (TEMPO) and tetrabutylammonium hexafluorophosphate (Bu4NPF6) were purchased from Sigma-Aldrich. Acetonitrile (MeCN, RPE) and CH2Cl2 (technical grade) were purchased from Carlo Erba. They were percolated on activated alumina. Britton-Robinson buffers (BR) [26] were prepared with acetic acid 99.7% (0.04 M), boric acid 99.99% (0.04 M) and phosphoric acid 99.999% (0.04 M) from Aldrich and with NaCl (0.2 M) from Carlo Erba as supporting electrolyte. The pH was adjusted to the desired value by addition of NaOH 1 M and distilled water. TiPStetracyanotriphenodioxazine (TiPS-TPDO-tetraCN) was synthesised according to a previously reported procedure. [6a] Thin films of TiPS-TPDO-tetraCN (50 nm) were prepared by sublimation under secondary vacuum (ca. 10 -6 mbar) in a UNIVEX 250 thermal evaporator (Oerlikon, Germany), at a rate of ca. 0.3 Å.s -1 , by heating in a temperature range of 320-350 °C, on Indium Tin Oxide (ITO) electrodes. For electrochemical experiments, a thin layer of nafion was deposed on the film and dried at 80 °C during one hour.

Electrochemistry. All electrochemical experiments were performed with a PGSTAT302 N (Metrohm) potentiostat connected to a PC and the data collected were analyzed using the Nova® 1.11 Software. Cyclic voltammetry (CV) was carried out by means of a three-electrode configuration consisting of the Pt disk (1.6 mm in diameter, Bioanalytical Systems) or the Indium Tin Oxide (ITO) modified electrode as working electrode, a platinum wire as counter-electrode and an SCE as reference. In organic solutions, the reference electrode was separated from the solution by a glass chamber with a porous vycor tip filled up with a saturated solution of Bu4NPF6 in the organic solvent (MeCN or CH2Cl2). Potentials were reported versus the saturated calomel electrode, SCE.

The working Pt electrode was soaked for 10 min in KOH (2 M), polished with 0.1 µm alumina, etched for 10 min in concentrated sulfuric acid (2 M) and sonicated 10 min in water, and then in absolute ethyl alcohol.

The solutions were deoxygenated for 10 min with argon, and a positive overpressure of argon was maintained above the electrolyte during the entire measurement performed at room temperature.

Digital simulations of cyclic voltammograms were simulated with DigiElch 8.

Spectroelectrochemistry.

The in situ spectroelectrochemical experiments were performed with a BioLogic cell having an optical path of 1 mm. The working electrode was a Pt grid, the reference electrode an SCE and the counter electrode a Pt wire. The spectroelectrochemical experiments were carried out with a PGSTAT302 N (Metrohm) potentiostat. The spectra were recorded every 6 seconds by a UV-visible diode array spectrometer (KINSPEC II/MMS-16 VIS, BIOLOGIC ) running in the kinetic mode with external trigger. The 150 W Xe lamp and the spectrometer were connected to the cell by optical fibers.

Computational chemistry. Molecular structures were optimized using density functional theory (DFT) at the B3LYP/6-31+G (d,p) level. Each structure was characterized as a minimum of the potential energy surface based on its real harmonic vibrational frequencies. Vertical excitation energies and excited state properties were determined using timedependent (TD) DFT with the M06-2X exchange-correlation functional (XCF) and the 6-311+G (d) basis set. This XCF was shown to provide good estimates of the transition energies of charge transfer states of medium and large-size organic molecules. [27] Solvent effects (acetonitrile) were included in all geometry optimizations and computations of optical properties by using the polarizable continuum model in its integral equation formalism (IEF-PCM). [28]

Scheme 1 .

 1 Scheme 1. Schematic view of the TIPS-tetracyano-triphenodioxazine (TiPS-TPDO-tetraCN)

Figure 1 .

 1 Figure 1. Voltammograms of TiPS-TPDO-tetraCN 1.9x10 -3 M in CH2Cl2 + Bu4NPF6 0.1 M: A, on a Pt disk electrode (diameter 1.6 mm) at a scan rate of 100 mV s -1 and B, on an ultramicroelectrode of 50 µm at a scan rate of 40 mV s -1 .

Figure 2 .

 2 Figure 2. A, evolution of the voltammograms of TiPS-TPDO-tetraCN 2.10 -4 M in MeCN + Bu4NPF6 0.1 M, on a Pt disk electrode (diameter 1.6 mm) at a scan rate of 100 mV s -1 , when increasing amounts of water are introduced into the cell (5 ml of solution) and B, cathodic peak potentials of the second reduction step as a function of the logarithm of the concentration of water added to the solution.

Figure 3 .

 3 Figure 3. Spectroelectrochemical experiments performed on 10 -3 M TiPS-TPDO-tetraCN in MeCN + Bu4NPF6 0.1 M at (A) -0.2 V during 1800 s, (B) -0.6. V during 3600 s, (C) after reoxidation at + 0.1 V and (D) comparison of the last spectrum of each electrolysis with the initial spectrum.

Figure 4 .

 4 Figure 4. Spectroelectrochemical experiments performed on 10 -3 M TiPS-. TPDO-tetraCN in MeCN + Bu4NPF6 0.1 M in presence of 5x10 -3 M TEMPO.

Figure 5 .

 5 Figure 5. Evolution of the absorbance of bands at: A. 522 nm and B. 680 nm, during the different spectroelectrochemical experiments in the presence of 2% H2O (1 M). The imposed electrolysis potentials are indicated on the graph.

Figure 6 .

 6 Figure 6. Comparison of the kinetics of appearance of the dianion (band at 680 nm) and disappearance of TiPS-TPDO-tetraCN (band at 522 nm) in absence and in presence of water during electrolysis at -0.6 V

Figure 7 .

 7 Figure7. Cyclic voltammograms of TiPS-TPDO-tetraCN 2.10 -4 M in MeCN + Bu4NPF6 0.1 M, on a Pt disk electrode (diameter 1.6 mm) at a scan rate of 100 mV s -1 , when water was introduced at a concentration of 1 M (dashed curve) and 2 M (solid line). The corresponding simulated curves of the 1 M ( open squares) and 2 M (open circles) curves were calculated using scheme 2 with E 0 1 = -0.113 V; ks1 = 0.04 cm s -1 ; E 0 2 = -0.49 V; ks2 = 0.01 cm s -1 ; E 0 3 = -0.105 V; ks3 = 0.02 cm s -1 ; ks4 = 0.02 cm s -1 ; K1 = 0.3 ; kf1 = 15 s -1 ; kf2 = 1 s -1 ; K3 = 0.5 M; kf3 = 0.06 M -1 s -1 ; K4 = 600 M; kf4 = 3 M -1 s -1 ; K5 = 1500 M ; kf5 = 2 M -1 s -1 ; K6 = 100; kf6 = 2560 s -1 ; kfDISP = 10 M -1 s -1 . In all electrochemical reactions, the transfer coefficient,  was taken as 0.5, the diffusion coefficient was set at 10 -5 cm 2 s -1 for all species and a resistance of 1 k and a capacitance of 6 µF were introduced.

Figure 8 .

 8 Figure 8. Absorption spectra of a film of vacuum evaporated TiPS-TPDO-tetraCN (50 nm) deposed on an ITO electrode dipped in a Britton-Robinson buffer (blue solid line) and of the same film immediately after its dissolution in CH2Cl2 (black dashed line) and after 3 days (red dotted line).

Figure 9 .

 9 Figure 9. Variation of the cathodic peak potentials of an ITO electrode modified by a film of vacuum evaporated TiPS-TPDO-tetraCN according to pH. The electrode was dipped in a Britton Robinson buffer.

Table 1 .

 1 Absorption bands of TiPS-TPDO-tetraCN in acetonitrile

	max (nm)	484 (sh.)	522	554

Table 2 .

 2 Absorption bands of the dianionic species obtained after 2e - electrolysis in acetonitrile

	max (nm)	574	628	680
	 (M -1 cm -1 )	2000	4000	9000

Scheme 2. The DISP mechanism. Bottom (circled in red): Disp mechanism; top (circled in blue): mechanism involving water molecules.

Acknowledgements

The authors acknowledge the Agence Nationale de la Recherche for funding through the projects CAP-BTX ANR-BLAN-2010-917-02, FMOCSOLE ANR-BLAN-2010-938-01 and OUTSMART ANR-2015-CE39-0004-03 and the MENESR for a PhD grant (A. W.). Financial support from the European Union (FEDER) and the Conseil Régional de Bourgogne through the FABER and the PARI SMT 08 and CDEA programs is gratefully acknowledged. We would like to thank the European Union for funding through the COST action TD1105 EuNetAir. The University of Bordeaux (G.G. PhD support), the CNRS and the institute of polytechnique of Bordeaux are acknowledged for financial support.

Keywords: TPDO • quinone diimine • square scheme • proton coupled electron transfer • disproportionation