A Ture Savadkoohi

C.-H Lamarque
email: lamarque@entpe.fr

C Goossaert
email: celien.goossaert@developpement-durable.gouv.fr

The system is coupled to a nonlinear energy sink. • Fast and slow dynamics are revealed. Nonlinear passive tremor control of human arm

Keywords:

An academic model for passive vibration control of human arm suffering from essential tremors is proposed. The nonlinear uncontrolled system equations are linearised around arbitrary postural positions showing variations of linear natural frequencies as functions of postural positions. The coupled system, i.e. the model of the arm with the attached nonlinear absorber, is treated via a time multiple scale method leading to detection of fast and slow systems dynamics. These dynamics provide design tools for tuning parameters of the nonlinear absorber.

Introduction

The unwanted or large amplitude vibrations of systems seek for appropriate control devices to be able to reduce oscillation levels corresponding to comfort and undamaged states for systems. The control solutions cover large spectrum of categories expanding from active to passive solutions [START_REF] Housner | Structural control: Past, present, and future[END_REF]. The active category of absorbers demand injected (at least initial) energies for their activation while passive systems exploit direct linear or nonlinear interactions between oscillators for reducing energies of main systems. [START_REF] Roberson | Synthesis of a nonlinear dynamic vibration absorber[END_REF] illustrated that via supplementing the linear restoring forcing function of an absorber by a cubic term, the suppression width of the absorber would increase. Since then different types of nonlinear absorbers are developed exploiting different sources of nonlinearities. One of such absorbers is named as nonlinear energy sink (NES) [START_REF] Vakakis | Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems[END_REF] which in its early developments possessed pure cubic nonlinearity, i.e. with no linear part [START_REF] Vakakis | Inducing Passive Nonlinear Energy Sinks in Vibrating Systems[END_REF][START_REF] Vakakis | Energy Pumping in Nonlinear Mechanical Oscillators: Part II-Resonance Capture[END_REF]. The control process by NES is accomplished via nonlinear energy exchanges between oscillators leading to periodic or non periodic regimes (Starosvetsky and Gendelman, 2008b;[START_REF] Savadkoohi | Analysis of the 1:1 resonant energy exchanges between coupled oscillators with rheologies[END_REF]. The NES and its derived technologies are applied to different engineering systems such as acoustics [START_REF] Cochelin | Experimental evidence of energy pumping in acoustics[END_REF], aero-elastic effects on wings [START_REF] Hubbard | Targeted energy transfer between a model flexible wing and nonlinear energy sink[END_REF], chatter instabilities [START_REF] Gourc | Quenching chatter instability in turning process with a vibro-impact nonlinear energy sink[END_REF], buildings [START_REF] Gourdon | Nonlinear energy pumping under transient forcing with strongly nonlinear coupling: Theoretical and experimental results[END_REF][START_REF] Savadkoohi | Targeted energy transfer with parallel nonlinear energy sinks, part ii: theory and experiments[END_REF][START_REF] Wierschem | Experimental testing and numerical simulation of a six-story structure incorporating two-degree-of-freedom nonlinear energy sink[END_REF], cables [START_REF] Weiss | Control of vertical oscillations of a cable by a piecewise linear absorber[END_REF], etc. The idea of the current paper is "passive control" of tremors of human arm due to some disorders such as essential tremors, i.e. action and postural tremor, that is the most frequent movement problems and difficulties which affects daily life patients [START_REF] Espay | Essential pitfalls in "essential" tremor[END_REF]. There are several works and patents which aim at mitigation of the pathological tremors with "active" systems. From them we can name patents of Rosen which all of them incorporate velocity dependent forces in an active manner: for example, inducing linear velocity dependent forces (viscous damping) in two [START_REF] Rosen | Tremor suppressing hand control[END_REF] and four (three translation and one angular movement) degrees-of-freedom (dof) [START_REF] Rosen | Multiple degree of freedom damped hand controls[END_REF]. The device consists of a chamber filled with a viscous fluid and a positioned rotation-sensing actuators for sensing the positions and angles and generating electrical signals accordingly. Later on, a four-dof orthosis was invented which permits the patient to move the midpoint of the forearm in a horizontal plane and to rotate the forearm out of the plane via several rotational joints [START_REF] Rosen | Whole-arm orthosis for steadying limb motion[END_REF]. The orthosis could apply linear (viscous) and square-velocity dependent forces [START_REF] Arnold | Evaluation of a controlled-energy-dissipation orthosis for tremor suppression[END_REF]. [START_REF] Rocon | Design and validation of a rehabilitation robotic exoskeleton for tremor assessment and suppression[END_REF] presented a robotic exoskeleton which exploits two control techniques based on biomechanical loading, namely tremor reduction by impedance control and notch filtering at tremor frequency [START_REF] Rocon | Biomechanical loading as an alternative treatment for tremor: a review of two approaches[END_REF]. [START_REF] Lora-Millán | Mechanical vibration does not systematically reduce the tremor in essential tremor patients[END_REF] studied effects of mechanical vibrations on tremors of the arm and forearm. They used some piezoelectric actuators in order to provide stimuli to the arm and forearm and monitoring tremors by sensors. They spotted that the mechanical vibration sometime could increase amplitudes of tremors. A literature review about different devices for tremor control is reported by [START_REF] Castrillo-Fraile | Tremor control devices for essential tremor: A systematic literature review[END_REF]. The linear passive control of tremors are already studied (see for example [START_REF] Hashemi | Tuned vibration absorber for suppression of rest tremor in parkinson's disease[END_REF]). Our paper exploits the NES technology for passive control of tremors of a simplified model of human arm. The paper is organised as it follows: the simplified nonlinear mathematical model of the locomotion of human arm and detection of its linear natural frequencies are presented in Sect. 2. In Sect. 3 the system is coupled to a NES and its fast and slow dynamics are clarified. Application of the proposed system in passive control of human arm due to essential tremors is provided in Sect. 4. Finally, the paper is concluded in Sect. 5.

The mathematical model of the upper limb and linearisation about a postural potion

The simplified academic model of the locomotion of human arm is illustrated in Fig. 1. In this model some biomechanical effects such as • the elbow locking [START_REF] Jackson | A mathematical model of arm swing during human locomotion[END_REF], i.e. limitation of extension of human arm at elbow joint when arm and forearm are at the same line;

• the elastic activities of muscles;

• three-dimensional movement of the arm, are ignored. Moreover, we neglect the movement of hand via assuming that the forearm and the hand constitute a single piece with the equivalent mass of both pieces which is situated at the centre of their masses. Let us define following parameters and variables:

• and : developed angles in arm and forearm, respectively;

• 1 and 2 : the mass of the arm and the equivalent mass of the forearm and the hand, respectively;

• 1 : length of the arm;

• 2 : distance of the equivalent mass 2 from the elbow joint;

• 1 : distance of the center of the mass of the arm to the glenohumeral joint;

• : the moment of inertia of the upper arm with respect to the glenohumeral joint;

• ′ and ′ : applied torques about the glenohumeral and the elbow joints, respectively.

We use the developed model by [START_REF] Jackson | A mathematical model of arm swing during human locomotion[END_REF] for representing locomotion of human upper limb; it reads:

⎧ ⎪ ⎨ ⎪ ⎩ 11 ̈ + 12 ̈ + 13 -+ = ′ 21 ̈ + 22 ̈ + 23 - + = ′ (1)
Different parameters of Eq. 1 are reported in Appendix A. It should be mentioned that other models exist which take into account more complete factors of the locomotion such as detailed bones and joints behaviours [START_REF] Barhorst | A neuro-muscular elasto-dynamic model of the human arm part 1: Model development[END_REF][START_REF] Moody | A neuro-muscular elasto-dynamic model of the human arm part 2: Musculotendon dynamics and related stress effects[END_REF] or pennate [START_REF] Wojnicz | A two dimensional approach for modelling of pennate muscle behaviour[END_REF] and mono-and bi-articular muscles [START_REF] Zagrodny | Cooperation of mono-and bi-articular muscles: human lower limb[END_REF]. Let us linearise system equations around a provided postural positions represented by 0 and 0 ; i.e.

= 0 + = 0 + (2)
Following system is obtained:

⎧ ⎪ ⎨ ⎪ ⎩ 11 ̈ + 12 ̈ + 11 + 12 + 1 = ′ 21 ̈ + 22 ̈ + 21 + 22 + 2 = ′ (3) θ φ m 1 m 2 l 1 d 1 l 2
Elbow joint

Glenohumeral joint

Figure 1: The simplified academic model of the locomotion of human arm.

All variables of the Eq. 3 are clarified in Appendix B.

The parameters 1 and 2 are fact the the resistive moments in order to maintain the postural position. Therefore, the total excitation due to tremor, ext , is summarised as:

ext = ′ -1 ′ -2 (4)
We assume that ext can be decomposed in the form of Fourier series on the basis of the frequency Ω. To simplify the study, we take the first harmonics of the series and we consider a mono-frequency excitation; we can suppose:

ext = 1 sin(Ω ) 2 sin(Ω ) (5)
Equation 3 can be written as:

⎛ ⎜ ⎜ ⎝ 11 12 21 22 ⎞ ⎟ ⎟ ⎠ ⏟⏞⏞⏞⏞⏞⏞ ⏟⏞⏞⏞⏞⏞⏞ ⏟ ⎛ ⎜ ⎜ ⎝ ̈ ̈ ⎞ ⎟ ⎟ ⎠ + ⎛ ⎜ ⎜ ⎝ 11 12 21 22 ⎞ ⎟ ⎟ ⎠ ⏟⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏟ ⎛ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎠ = 1 sin(Ω ) 2 sin(Ω ) (6)
Variation of natural frequencies of the linear system of Eq. 6 with respect to different postures are illustrated in Fig. 2. The system parameters are reported in Table 1. These figures show that system frequencies are sensitive to initial postural positions. This means that classical linear absorbers, i.e. tuned mass dampers [START_REF] Frahm | Device for damping vibrations of bodies[END_REF], can not be efficient for passive control of tremors due to variations of natural frequencies according to different initial postural positions.

In the next section the two degrees of freedom pendulum will be coupled to a NES and different dynamics of the system will be clarified.

Table 1 System parameters of the arm [START_REF] Drillis | Body segment parameters; a survey of measurement techniques[END_REF] (m.sec -2 )

1 (m) 2 (m) 1 (m) 1 (kg) 2 (kg) (kg.m 2 )
9.81 0.364 0.293 0.155 2.07 1.7 0.076

The mathematical model of the upper limb coupled to a NES

Let us consider a simplified model of the upper limb which is coupled to a NES with the distance of 3 from the elbow joint (see Fig. 3). We assign and as the mass and the generalised displacement of the NES which possesses a general nonlinear restoring forcing function as NES ( ), e.g. NES ( ) =

3 . The damping scenario of the NES is supposed to be linear and its coefficient is represented by . Governing equations of the three-dof systems is written as:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 11 ̈ + 12 ̈ + 13 ̈ + 14 -1 + 1 = ′ 21 ̈ + 22 ̈ + 23 ̈ + 24 -2 + 2 = ′ 31 ̈ + 32 ̈ + 33 ̈ + 34 -3 + 3 + ̇ = 0 (7)
Appendix C describes all variables of Eq. 7. Let us also assume that:

′ = 1 sin(Ω ) ′ = 2 sin(Ω ) (8)
We can perform linearisation of system equations around a postural position described in Eq. 2. Introducing the parameter = 2 and assuming ≪ 1, following system is obtained:

′ ⎛ ⎜ ⎜ ⎝ ̈ ̈ ̈ ⎞ ⎟ ⎟ ⎠ + ′ ⎛ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎠ + ′ + ′ + ′ = ⎛ ⎜ ⎜ ⎝ 1 sin(Ω ) 2 sin(Ω ) 0 ⎞ ⎟ ⎟ ⎠ (9)
with variables which are defined in Appendix D.

Let us suppose that = nint( 2 ( 0 , 0 ) 1 ( 0 , 0 )

), ∈ ℕ * , where nint(… ) stands for the nearest integer function. Moreover, we assume that:

1 = Ω + 1 2 = 1 + ̃ 2 = Ω + 2 (10)
In the modal coordinate, we apply following change of variables:

⎛ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎠ = ⎛ ⎜ ⎜ ⎝ 1 2 3 ⎞ ⎟ ⎟ ⎠ , with = ⎛ ⎜ ⎜ ⎝ 11 12 0 21 22 0 0 0 1 ⎞ ⎟ ⎟ ⎠ (11)
where is a matrix of eigenvectors of ′-1 ′ . To take into account dissipations of the main system, following damping matrix can be introduced to the system 9 as: Then, we set Considering explained change of variables in Eq. 9 following system is obtained:

′ = ⎛ ⎜ ⎜ ⎝ 11 12 0 21 22 0 0 0 0 ⎞ ⎟ ⎟ ⎠ (12)
-1 ′-1 ′ = (13) with = ⎛ ⎜ ⎜ ⎝ 1 0 0 0 2 0 0 0 0 ⎞ ⎟ ⎟ ⎠ (14) θ φ m 1 m 2 l 1 d 1 l 2 Elbow joint Glenohumeral joint l 3 m N u
⎛ ⎜ ⎜ ⎝ ̈ 1 ̈ 2 ̈ 3 ⎞ ⎟ ⎟ ⎠ + ( + Λ) ⎛ ⎜ ⎜ ⎝ 1 2 3 ⎞ ⎟ ⎟ ⎠ + ⎛ ⎜ ⎜ ⎝ ̇ 1 ̇ 2 ̇ 3 ⎞ ⎟ ⎟ ⎠ + ̇ 3 ( + ) + 0 ( 3 )( + ) + ( + ) = (15)
and Λ are 3 × 3 matrices, where , , , , , and are 3 × 1 vectors. All of these, are clarified in Appendix E. In the next section the complexified form of system equations are treated by a time multiple scale method for clarifying its different dynamics.

Complexification of the system and keeping its firs harmonics

Let us introduce complex variables of Manevitch [START_REF] Manevitch | The description of localized normal modes in a chain of nonlinear coupled oscillators using complex variables[END_REF]:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ Ψ 1 exp( Ω ) = ̇ 1 + Ω 1 Ψ 2 exp( Ω ) = ̇ 2 + Ω 2 Ψ 3 exp( Ω ) = ̇ 3 + Ω 3 (16)
We are interested to keep first harmonic of the system (Ture [START_REF] Savadkoohi | Analysis of the 1:1 resonant energy exchanges between coupled oscillators with rheologies[END_REF]. Following equation is obtained:

⎛ ⎜ ⎜ ⎜ ⎝ Ψ1 + ( 1 -1 ) 2 Ψ 1 Ψ2 + ( 2 -2 ) 2 Ψ 2 Ψ3 + ( 1 -1 ) 2 Ψ 3 ⎞ ⎟ ⎟ ⎟ ⎠ + ( + Λ) ⎛ ⎜ ⎜ ⎜ ⎝ -2( 1 -1 ) Ψ 1 -2( 2 -2 ) Ψ 2 -2( 1 -1 ) Ψ 3 ⎞ ⎟ ⎟ ⎟ ⎠ + ⎛ ⎜ ⎜ ⎝ Ψ1 Ψ2 Ψ3 ⎞ ⎟ ⎟ ⎠ + Ψ 3 2 ( + )+( + ) (Ψ 3 , Ψ * 3 ) = ℍ (17)
where the function is defined as [START_REF] Gendelman | Targeted energy transfer in systems with non-polynomial nonlinearity[END_REF]:

(Ψ 3 , Ψ * 3 ) = Ω 2 ∫ 2 Ω 0 0 ( 3 ) exp(-Ω )d = - Ψ 3 2 (|Ψ 3 | 2 ) (18)
where

(|Ψ 3 | 2 )
is a function which depends on the nonlinearity type of the NES. The . * stands for the complex conjugate of the variable. Definition of ℍ is provided in Appendix F.

Fast/slow dynamics of the system

To treat system equations, we use a perturbation technique namely, the multiple scale method [START_REF] Nayfeh | Perturbation Methods[END_REF]). Different scales of the time are defined as 0 = (fast scale), and = , ∈ ℕ + (slow scales). These time scales are coupled to each other via the physical parameter . To clarify different systems dynamics, different orders of the should be considered in Eq. 17.

Fast dynamics: 0 order of system equations

The 0 order of the system 17 leads to:

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ Ψ 1 0 = 0 Ψ 2 0 = 0 Ψ 3 0 + 1 2 Ψ 3 - 2 ( 31 1 Ψ 1 + 32 2 Ψ 2 ) + 0 Ψ 3 2 + = 0 (19)
For a NES with cubic nonlinearity,

(|Ψ 3 | 2 ) = 3 4 3 1 |Ψ 3 | 2 , so = -2 Ψ 3 3 4 3 1 |Ψ 3 | 2 .
Fixed points of the system correspond to system behaviours when 0 → ∞, leading to

Ψ 3 0 → 0 or (Ψ 1 , Ψ * 1 , Ψ 2 , Ψ * 2 , Ψ 3 , Ψ * 3 ) = - 2 1 ( 31 Ψ 1 + 32 Ψ 2 ) + ( 1 2 + 0 2 )Ψ 3 + = 0 (20)
If we write system variables in polar forms as Ψ = exp( ), = 1, 2, 3, then Eq. 20 reads:

2 0 + 2 1 + 2 ( 2 3 ) -2 1 ( 2 3 ) 2 3 = ( 31 1 ) 2 2 1 + ( 32 1 ) 2 2 2 + 2 31 32 2 1 cos( 1 -2 ) 1 2 (21)
The complex function  or the real equation of the system 21 are a geometrical surface corresponding to final destinations of the system which is named as slow invariant manifold (SIM). It is seen that the four-dimensional SIM is a function of amplitudes , = 1, 2, 3 and phase lag between Ψ 1 and Ψ 2 represented by = 1 -2 . Different views of the four-dimensional SIM for some values of the are illustrated in Fig. 4. Some zones of the SIM are unstable.

A stability analysis can be carried out via linear perturbation of variables of the Eq. 20 or 21 for tracing stable and unstable zones of the SIM (see for example Lamarque et al. ( 2017); Hurel et al. ( 2019)). When the system reaches to boundaries of the unstable zones, it bifurcates to its another stable zone. Depending on the characteristics of external forcing terms, the system can be attracted by periodic or non periodic regimes [START_REF] Gendelman | Attractors of harmonically forced linear oscillator with attached nonlinear energy sink i: Description of response regimes[END_REF]Starosvetsky and Gendelman, 2008a). Different possible regimes can be revealed by studying slow dynamics of the system which is explained in the next section.

Slow dynamics: 1 order of system equations

The 1 order of first two equations of the system 17 reads:

⎧ ⎪ ⎨ ⎪ ⎩ Ψ 1 1 - 2 1 (Λ 11 + 2 1 1 )Ψ 1 + Λ 12 Ψ 2 + Λ 13 Ψ 3 + 1 2 Ψ 1 + 1 2 Ψ 3 + 1 = ℍ 1 Ψ 2 1 - 2 2 Λ 21 Ψ 1 + (Λ 22 + 2 2 2 ) Ψ 2 + Λ 23 Ψ 3 + 2 2 Ψ 2 + 2 2 Ψ 3 + 2 = ℍ 2 (22)
Which can be written in a compact form as:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ Ψ 1 1 =  1 (Ψ 1 , Ψ * 1 , Ψ 2 , Ψ * 2 , Ψ 3 , Ψ * 3 ) Ψ 2 1 =  2 (Ψ 1 , Ψ * 1 , Ψ 2 , Ψ * 2 , Ψ 3 , Ψ * 3 ) (23)
The evolution of the SIM (see Eq. 20) at 1 time scale yields to:

⎛ ⎜ ⎜ ⎜ ⎜ ⎝  Ψ 1  Ψ * 1  Ψ 2  Ψ * 2  Ψ 3  Ψ * 3  * Ψ 1  * Ψ * 1  * Ψ 2  * Ψ * 2  * Ψ 3  * Ψ * 3 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ Ψ 1 1 Ψ * 1 1 Ψ 2 1 Ψ * 2 1 Ψ 3 1 Ψ * 3 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ = 0 (24)
which can be reorganised as:

⎛ ⎜ ⎜ ⎜ ⎜ ⎝  Ψ 3  Ψ * 3  * Ψ 3  * Ψ * 3 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟  ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ Ψ 3 1 Ψ * 3 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ = - ⎛ ⎜ ⎜ ⎜ ⎜ ⎝  Ψ 1  Ψ * 1  Ψ 2  Ψ * 2  * Ψ 1  * Ψ * 1  * Ψ 2  * Ψ * 2 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ Ψ 1 1 Ψ * 1 1 Ψ 2 1 Ψ * 2 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⏟⏞⏞ ⏟⏞⏞ ⏟  (25)
Looking at Eq. 23, we can write:

 =  1  * 1  2  * 2 (26)
Equilibrium points of the system (corresponding to periodic regimes of the initial system) are those which satisfy:

⎧ ⎪ ⎨ ⎪ ⎩ = 0  = 0 Det( ) ≠ 0 (27)
while singularities of the system (corresponding to non periodic regimes of the initial system) verify:

⎧ ⎪ ⎨ ⎪ ⎩ = 0  = 0 Det( ) = 0 (28)

A numerical example

Let us consider the parameters of the arm which are presented in Table 1. The system is coupled to a NES which its supplementary parameters are reported in Table 2. All possible equilibrium points of the system for sweeping 1 are presented in Figs. 567. As an example, let us take the case which correspond to 1 = 0.2. Equation 9 is integrated numerically by ode45 function of Matlab. Results for different system amplitudes are depicted in Figs. 8-10, showing that the system is being attracted by equilibrium points which are predicted in Figs. 567. Obtained numerical results are confronted with the three-dimensional view of the SIM in Fig. 11 for the section of = 0. It is seen that (after starting from initial conditions), the system oscillates around the SIM and finally is attracted by the equilibrium point which is already clarified in Figs. 567. Different two-dimensional views of this confrontation are presented in Figs.

12. 2 as a function of the detuning parameter: 1 vs. . Results are obtained by direct numerical integration of Eq. 9 for 1 = 0.2.

Application to passive control of the arm

In order to show the capability of the proposed device in tremor control of the arm, the undamped locomotion equations of human arm, i.e. Eq. 6, are integrated numerically (see Table 1 for parameters) and are compared with those which are equipped with the NES (see Table 3 for parameters). Results are shown in Figs. 13 and14 showing the capability of the NES in controlling the diverging response of the main system. 2 as a function of the detuning parameter: 2 vs. . Results are obtained by direct numerical integration of Eq. 9 for 1 = 0.2. 2 as a function of the detuning parameter: 3 vs. . Results are obtained by direct numerical integration of Eq. 9 for 1 = 0.2.

Conclusions

The mathematical model of the locomotion of human arm which has been developed by [START_REF] Jackson | A mathematical model of arm swing during human locomotion[END_REF] and represented by a simplified two degrees of freedom nonlinear pendulum, is investigated:

• After linearisation of nonlinear system equation about an arbitrary posture, it is spotted that linear natural frequencies vary with the change of the posture. This means that classical linear passive absorbers, which are tuned to a spacial linear frequency, can not be efficient in general for the tremor control.

• The simplified locomotion model of human arm is coupled to a nonlinear energy sink. Detection of fast and slow system dynamics revealed its slow invariant manifold and characteristic points. Then, application of the proposed system in passive control of an academic diverging system is illustrated. It is seen that the proposed absorber is able to control (mathematically) diverging responses of the arm.

Perspective of this work will be a collaboration with the neurological hospital and medicines for having a real data of patients which suffer from essential tremor and to prepare design tools for such data. C. Definition of different variables of Eq. 7

11 = + 2 ( 2 1 + 2 2 + 2 1 2 cos( )) + ( 2 1 + 2 3 + 2 + 2 1 3 cos( ) -2 1 sin( )) 12 = 2 [ 2 2 + 1 2 cos( )] + [ 2 3 + 2 + 1 3 cos( ) -1 sin( )] 13 = [ 1 cos( ) + 3 ] 14 = -2 1 2 (2 ̇ + ̇ ) ̇ sin( ) + [2 ̇ ( ̇ + ̇ ) -1 3 (2 ̇ + ̇ ) ̇ sin( ) -1 ̇ ̇ sin( ) -1 ̇ (2 ̇ + ̇ ) sin( ) -1 (2 ̇ + ̇ ) ̇ cos( )] 21 = 2 [ 2 2 + 1 2 cos( )] + [ 2 3 + 2 + 1 3 cos( ) -1 sin( )] 22 = 2 2 2 + ( 2 3 + 2 ) 23 = 3 24 = -2 ( 1 2 ̇ ̇ sin( )) + [2 ̇ ( ̇ + ̇ ) -1 3 ̇ ̇ sin( ) -1 ̇ ̇ sin( ) -1 ̇ ̇ cos( )] 31 = [ 1 cos( ) + 3 ] 32 = 3 33 = 34 = - 1 ̇ ̇ sin( ) (31)
and 1 = 0 1 = 1 1 sin( ) + 2 ( 1 sin( ) + 2 sin( + )) + ( 1 sin( ) + 3 sin( + ) + cos( + ))

2 = -[ 2 1 2 ( ̇ + ̇ ) ̇ sin( ) + ( 1 3 ( ̇ + ̇ ) ̇ sin( ) + 1 ( ̇ + ̇ ) ̇ cos( ) + 1 ̇ ̇ ( ))] 2 = 2 2 sin( + ) + ( 3 sin( + ) + cos( + )) 3 = [( ̇ + ̇ ) 2 -1 ( ̇ + ̇ ) ̇ sin( )] 3 = sin( + ) + NES ( ) (32

D. Definition of different variables of Eq. 9

Let us define 2 = and 1 2 = .

′ = ⎛ ⎜ ⎜ ⎝ ′ 11 ′ 12 ′ 13 ′ 21 ′ 22 ′ 23 ′ 31 ′ 32 ′ 33 ⎞ ⎟ ⎟ ⎠ (33) ′ 11 = + ( 2 1 + 2 2 + 2 1 2 cos( 0 )) + ( 2 1 + 2 3 + 2 1 3 cos( 0 )) ′ 12 = 2 ( 2 + 1 cos( 0 )) + 3 (cos( 0 ) 1 + 3 ) ′ 13 = (cos( 0 ) 1 + 3 ) ′ 21 = 2 ( 2 + 1 cos( 0 )) + 3 (cos( 0 ) 1 + 3 ) ′ 22 = 2 2 + 2 3 ′ 23 = 3 ′ 31 = (cos( 0 ) 1 + 3 ) ′ 32 = 3 ′ 33 = (34) ′ = ⎛ ⎜ ⎜ ⎝ ′ 11 ′ 12 ′ 13 ′ 21 ′ 22 ′ 23 ′ 31 ′ 32 ′ 33 ⎞ ⎟ ⎟ ⎠ (35) ′ 11 = cos 0 + 0 2 + cos 0 + 0 3 + 1 cos 0 + cos 0 1 + cos 0 1 ′ 12 = cos 0 + 0 2 + 3 ′ 13 = cos 0 + 0 ′ 21 = 0 + 0 2 + 3 ′ 22 = cos 0 + 0 2 + 3 ′ 23 = cos 0 + 0 ′ 31 = cos 0 + 0 ′ 32 = cos 0 + 0 ′ 33 = 0 (36) ′ = ⎛ ⎜ ⎜ ⎝ 0 0 0 ( ) ⎞ ⎟ ⎟ ⎠ (37) with = 2 and 0 ( ) = 3 . ′ = ⎛ ⎜ ⎜ ⎝ 0 0 0 ̇ ⎞ ⎟ ⎟ ⎠ (38) with 0 = 2 . ′ = ⎛ ⎜ ⎜ ⎝ ′ 1 ′ 2 ′ 3 ⎞ ⎟ ⎟ ⎠ (39)
′ 1 = 1 sin( 0 ) + ( 1 sin( 0 ) + 2 sin( 0 + 0 )) + ( 1 sin( 0 ) + 3 sin( 0 + 0 )) ′ 2 = 2 sin( 0 + 0 ) + 3 sin( 0 + 0 ) ′ 3 = sin( 0 + 0 )

(40) 1 = 1 2 2 = 2 2 (41)

E. Definition of different variables of Eq. 15

The non zero arrays of Matrices 3×3 and Λ 3×3 are:

11 = 2 1 22 = 2 2 31 = ( 11 + 21 ) 1 2 + cos( 0 + 0 )-cos( 0 ) cos( 0 ) 1 11 ( 1 + 1 ) ( 2 -3) 2 -(cos( 0 )) 2 1 2 + 1 2 + 32 = ( 12 + 22 ) 1 2 + cos( 0 + 0 )-cos( 0 ) cos( 0 ) 1 12 ( 1 + 1 ) ( 2 -3) 2 -(cos( 0 )) 2 1 2 + 1 2 + (42)
Λ 11 = -(cos( 0 ) 12 1 + 2 ( 12 + 22 )) cos( 0 )( 11 + 21 ) cos( 0 + 0 )-cos( 0 ) 11 1 1 (cos( 0 )) 2 - , , , , , and read:

= ⎛ ⎜ ⎜ ⎜ ⎝ 0 0 0 ⎞ ⎟ ⎟ ⎟ ⎠ (44) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ -(cos( 0 )) 2
12 1 2 2 -1 2 ( 12 + 22 )( 2 -3) cos( 0 )+ 3 12 1 2 + ( 11 22 -12 21 ) 2 2 -(cos( 0 )) 2 1 2 + 1 2 + --(cos( 0 )) 2 11 1 2 2 -1 2 ( 11 + 21 )( 2 -3) cos( 0 )+ 3 11 1 2 + ( 11 22 -12 21 ) 2 2 -(cos( 0 )) 2

1 2 + 1 2 + - 1 2 2 ( 2 -2 3)(cos( 0)) 2 + 3 2 1 2 + 2 2 (cos( 0 )) 2 1 2 -1 2 - ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (45) = ⎛ ⎜ ⎜ ⎜ ⎝ 0 0 1 ⎞ ⎟ ⎟ ⎟ ⎠ (46) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ -(cos( 0 )) 2
12 1 2 2 -1 2 ( 12 + 22 )( 2 -3) cos( 0 )+ 3 12 1 2 + ( 11 22 -12 21 ) 2 2 -(cos( 0 )) 2

1 2 + 1 2 + - -(cos( 0 )) 2
11 1 2 2 -1 2 ( 11 + 21 )( 2 -3) cos( 0 )+ 3 11 1 2 + ( 11 22 -12 21 ) 2 2 -(cos( 0 )) 2

1 2 + 1 2 + - 1 2 2 ( 2 -2 3 )(cos( 0 )) 2 + 3 2 1 2 + 2 2 (cos( 0 )) 2 1 2 -1 2 - ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (47) = ⎛ ⎜ ⎜ ⎝ ′ 1 ′ 2 ′ 3 ⎞ ⎟ ⎟ ⎠ (48) = -1 ′-1 ⎛ ⎜ ⎜ ⎝ 1 sin(Ω ) 2 sin(Ω ) 0 ⎞ ⎟ ⎟ ⎠ (49)
F. Definition of ℍ in Eq. 17

ℍ = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ∕2 -1 2 (( 1 -2 2 ) 12 -2 22 ) cos( 0)-( 12+

Figure 2 :

 2 Figure 2: Variations of system linear frequencies with respect to initial postural positions 0 and 0 . a) 1 b) 2 .

Figure 3 :

 3 Figure 3: The academic model of the locomotion of human arm coupled with a NES with the mass and the generalised displacement as . The NES presents a nonlinear restoring forcing function and a linear damping scenario.

Figure 4 :

 4 Figure 4: Three-dimensional views of the four-dimensional SIM (see Eq. 21) for some fixed phase lag ( = 1 -2 ). System parameters are = 10, 0 = 1, 3 = 0.299, 0 = 0, 0 = 0.

Figure 5 :

 5 Figure 5: Equilibrium points of the system of Table 2 as a function of the detuning parameter: 1 vs. 1

Figure 8 :

 8 Figure 8: Time history of the system of Table2as a function of the detuning parameter: 1 vs. . Results are obtained by direct numerical integration of Eq. 9 for 1 = 0.2.

Figure 9 :

 9 Figure9: Time history of the system of Table2as a function of the detuning parameter: 2 vs. . Results are obtained by direct numerical integration of Eq. 9 for 1 = 0.2.

Figure 10 :

 10 Figure 10: Time history of the system of Table2as a function of the detuning parameter: 3 vs. . Results are obtained by direct numerical integration of Eq. 9 for 1 = 0.2.

Figure 11 :Figure 13 :

 1113 Figure 11: Three-dimensional view of the SIM of the system of Table 2 for = 0. The SIM is accompanied by corresponding results.

Figure 14 :

 14 Figure 14: Time history of the for the system without and with the coupled NES ( 1 = 0).

  Table 2 as a function of the detuning parameter: 1 vs. 1 Equilibrium points of the system of Table 2 as a function of the detuning parameter: 2 vs. 1

	0.9
	0.8
	0.7
	0.6
	0.5
	2
	N
	0.4
	0.3
	0.2
	0.1
	0
	-1	-0.8	-0.6	-0.4	-0.2	0	0.2	0.4	0.6	0.8	1
						1
	Figure 6:

  Equilibrium points of the system of Table2as a function of the detuning parameter: 3 vs. 1

			2.5
			2
			1.5
			3
			N
			1
			0.5
			0
			-1	-0.8	-0.6	-0.4	-0.2	0	0.2	0.4	0.6	0.8	1
								1
	Figure 7: 0	0.2	0.4	0.6	0.8	1	1.2	1.4	1.6	1.8	2
								t(s)
	Table 2
	The coupled system parameters
		0	3	1			2		0		0	1	2
	10 2	0.1	0.299	0.2		0.2		0		0	10 -3	10	10

Table 3

 3 The coupled system parameters

		0	3	1	2	0	0		1	2
	10 2	1	0.299	0	0	0	0	10 -2	0.1	0.1

  ) 12 1 + 2 ( 12 + 22 )) cos( 0 )( 12 + 22 ) cos( 0 + 0 )-cos( 0 ) 12 1 1 (cos( 0 )) 2 -1 1 +

						1 1 +	1
			( 11 22 -12 21 ) 2 -(cos( 0 )) 2	1	2 + 1	2 +	2
	Λ 12 = -	(cos( 0 1 ( 11 22 -12 21 ) 2 -(cos( 0 )) 2 1 2 + 1 2 + 2
	cos( 0 + 0 ) 2 -1 2 -( 11 22 -12 21 ) 2 -(cos( 0 )) 2 Λ 13 = 1 2 ( 12 + 22 ) cos( 0 )+ 12 1 2 + ( 11 22 -12 21 ) 2 2 (cos( 0 )) 2 1 Λ 21 = (cos( 0 ) 11 1 + 2 ( 11 + 21 )) cos( 0 )( 11 + 21 ) cos( 0 + 0 )-cos( 0 ) 11 1 1 (cos( 0 )) 2 -1 1 + 1 2 + 1 2 + 2	1	(43)
	Λ 22 =	(cos( 0 ) 11 1 + 2 ( 11 + 21 )) cos( 0 )( 12 + 22 ) cos( 0 + 0 )-cos( 0 ) 12 1 1 (cos( 0 )) 2 -1 1 + ( 11 22 -12 21 ) 2 -(cos( 0 )) 2 1 2 + 1 2 + 2	1
	Λ 23 = -1 2 ( 11 + 21 ) cos( 0 )+ 11 1 2 + ( 11 22 -12 21 ) 2 2 (cos( 0 )) 2	1	cos( 0 + 0 ) 2 -1 2 -

  22)( 1 -2 ) 2 2 + 2 12 1 2 + ( 11 22 -21 12 ) 2 2 -(cos( 0)) 2 (( 1 -2 2 ) 11 -2 21 ) 2 cos( 0 )-( 11 + 21 )( 1 -2 ) 2 2 + 2 11 1 2 + ( 11 22 -21 12 ) 2 2 -(cos( 0 )) 2

				⎞
	1	2 + 1	2 +	⎟
	-∕2 -1 1	2 + 1	2 +	⎟ ⎟ ⎟	(50)
	0			⎟ ⎠

Acknowledgement

This article is an extended version of a paper originally presented at "Dynamical Systems -Theory and Applications (DSTA2019)". The authors thank The "Ministère de la transition écologique et solidaire" for supporting this research.

Figure 12: Different two-dimensional views of the SIM which is presented in Fig. 11 A. Definition of different variables of Eq. 1

B. Definition of different variables of Eq. 3

2 2 cos( 0 + 0 ) 21 = 2 2 cos( 0 + 0 ) 22 = 2 2 cos( 0 + 0 ) 1 = 1 1 sin( 0 ) + 2 ( 1 sin( 0 ) + 2 sin( 0 + 0 )) 2 = 2 2 sin( 0 + 0 ) (30)


Corentin Pigot
email: corentin.pigot@univ-amu.fr

Guillaume Noirbent

Sebastien Peralta

Sylvain Duval

Thanh-Tuan Bui

Pierre-Henri Aubert

Malek Nechab

Didier Gigmes

Frederic Dumur
email: frederic.dumur@univ-amu.fr

Sébastien Peralta

Frédéric Dumur

An amine-directed synthesis

Keywords:

published or not. The documents may come

New push-pull dyes based on 2-(3-oxo-2,3-dihydro-1Hcyclopenta[b]naphthalen-1-ylidene)malononitrile

Introduction

During the past decades, a great deal of efforts has been devoted to design push-pull chromophores as these structures can find applications in numerous research fields ranging from Non-Linear Applications (NLO) [START_REF] Carlotti | et al Fluorosolvatochromism and hyperpolarizability of one-arm and two-arms nitro-compounds bearing heterocyclic rings[END_REF][START_REF] Cesaretti | Spalletti Photoinduced intramolecular charge transfer and hyperpolarizability coefficient in push-pull pyridinium salts with increasing strength of the acceptor group[END_REF][START_REF] Motiei | Naderali Third-order nonlinear optical properties of organic azo dyes by using strength of nonlinearity parameter and Z-scan technique[END_REF][START_REF] El-Shishtawy | Linear and nonlinear optical properties of a new dimethine cyanine dye derived from phenothiazine[END_REF][START_REF] Chaumel | Kakkar Sol-gel materials for second-order nonlinear optics[END_REF][START_REF] Matsumoto | Nonlinear optical properties of an azo dye attached polymer[END_REF] to Organic Photovoltaics (OPVs) [START_REF] Kim | Kim Synthesis of new conjugated small-molecule-dyes based on 2-(2-methyl-4H-chromen-4-ylidene) malononitrile as the electron-withdrawing group and their application in photovoltaic devices[END_REF][START_REF] Blanchard | Ludwigs Triphenylamine and some of its derivatives as versatile building blocks for organic electronic applications[END_REF][START_REF] Malytskyi | Raimundo Thiophene-based push-pull chromophores for small molecule organic solar cells (SMOSCs)[END_REF][START_REF] Lee | Ho Use of organic materials in dye-sensitized solar cells[END_REF][START_REF] Lee | et al Recent progress in organic sensitizers for dye-sensitized solar cells[END_REF][START_REF] Mishra | Bäuerle Metal-free organic dyes for dye-sensitized solar cells: from structure: property relationships to design rules[END_REF], Organic Fields Effects Transistors (OFETs) [START_REF] Farré | Synthesis and properties of new benzothiadiazole-based push-pull dyes for p-type dye sensitized solar cells[END_REF], Organic Light-Emitting Diodes (OLEDs) [START_REF] Forget | et al Red-emitting fluorescent organic light emitting diodes with low sensitivity to self-quenching[END_REF] and photoinitiators of polymerization [START_REF] Tehfe | et al Push-pull (thio)barbituric acid derivatives in dye photosensitized radical and cationic polymerization reactions under 457/473 nm Laser beams or blue LEDs[END_REF][START_REF] Tehfe | et al New push-pull dyes derived from Michler's ketone for polymerization reactions upon visible lights[END_REF][START_REF] Mokbel | et al Photoinitiating systems of polymerization and in-situ incorporation of metal nanoparticles in polymer matrixes upon visible lights: push-pull malonate and malonitrile based dyes[END_REF][START_REF] Xiao | et al Julolidine or fluorenone based push-pull dyes for polymerization upon soft polychromatic visible light or green light[END_REF][START_REF] Tehfe | Lalevée Blue-to-red light sensitive push-pull structured photoinitiators: indanedione derivatives for radical and cationic photopolymerization reactions[END_REF]. Typically, a push-pull compound is composed of an electron donor connected to an electron acceptor by mean of a conjugated or a non-conjugated spacer but dyes comprising a conjugated spacer are the most popular ones. Among all possible reactions to create a π-conjugated linker, the Knoevenagel reaction consisting in the dehydrative condensation of an aldehyde with an activated methylene group is a reaction of choice for the preparation of push-pull structures and this reaction is most generally carried out under basic conditions. However, when strong electron acceptors are used, their anionic forms are stable under basic conditions, rendering these conditions ineffective to prepare push-pull dyes. In this case, use of acidic conditions are thus required, as exemplified with Knoevenagel reactions carried out in acetic anhydride. When basic conditions are employed, piperidine is without contest one of the most popular bases due to its relatively high pKa (11.1, 25 °C) which is sufficient to deprotonate activated methylene groups, its low cost, its low toxicity and easiness to evaporate [START_REF] Luna | Deprotection reagents in Fmoc solid phase peptide synthesis: moving away from piperidine?[END_REF]. However, to prepare dyes, the selection of the reaction conditions is of crucial importance as the amine can further react with the newly prepared dye, giving rise to the formation of side-products. This drawback is notably observed for all dyes prepared with 2-(3-oxo-2,3-dihydro-1H-inden-1-ylidene) malononitrile as the electron acceptors or its derivatives. A nucleophilic attack of the amine onto the dicyanomethylidene groups of the electron acceptor is occasionally observed. As a result of this, 3-(dialkylamino)-1,2-dihydro-9-oxo-9H-indeno [2,1-c] pyridine-4-carbonitrile derivatives can be obtained (See Fig. 1) [START_REF] Landmesser | A novel route to 1-substituted 3-(dialkylamino)-9-oxo-9H-indeno[2,1-c]-pyridine-4-carbonitriles[END_REF]. Reaction yields ranging from 17 to 51% were obtained upon mixing one equivalent of aldehyde with one equivalent of base and one equivalent of electron acceptor at room temperature for 14 h.

In 2014, a better reaction yield was reported for the synthesis of C5, reaching 96%, by opposing one equivalent of diethylamine to the preformed push-pull dye 2-(2-(furan-2-ylmethylene)-3-oxo-2,3-dihydro-1H-inden-1-ylidene)malononitrile for 30 min at room temperature (see Fig. 2) [START_REF] Helmy | Read de Alaniz Design and synthesis of donor-acceptor Stenhouse adducts: a visible light photoswitch derived from furfural[END_REF]. This unexpected nucleophilic attack on the dicyanomethylidene group was notably observed during the synthesis of a series of photochromes based on Donor-Acceptor Stenhouse Adducts (DASA) [START_REF] Helmy | Read de Alaniz Design and synthesis of donor-acceptor Stenhouse adducts: a visible light photoswitch derived from furfural[END_REF]. More recently, a fluorescent probe for live cell imaging was developed with a triphenylamine using a similar approach and C6 could be obtained in 47% yield after heating for 12 h at 50 °C [START_REF] Gao | et al Photoactivatable aggregation-induced emission probes for lipid droplets-specific live cell imaging[END_REF]. From these different works, it can be concluded that the formation of the push-pull compound arises prior the nucleophilic attack of the amine onto the dicyanomethylidene group and therefore the cyclization reaction.

Also, the following mechanism was proposed to support the formation of 3-(dialkylamino)-1,2-dihydro-9-oxo-9H-indeno [2,1-c]pyridine-4-carbonitrile derivatives (See Fig. 3).

In this work, we report on the synthesis of two new push-pull dyes PP1 and PP2 based on 2-(3-oxo-2,3-dihydro-1H-cyclopenta[b]naphthalen-1-ylidene)malononitrile EA1 as the electron acceptor. To date, EA1 has only been scarcely used to prepare dyes [START_REF] Chang | et al Fine-tuning the side-chains of non-fullerene small molecule acceptors to match with appropriate polymer donors[END_REF][START_REF] Li | Non-fullerene acceptors based on fused-ring oligomers for efficient polymer solar cells via complementary light-absorption[END_REF][START_REF] Feng | An A-D-A type small-molecule electron acceptor with end-extended conjugation for high performance organic solar cells[END_REF] despite its promising electron-withdrawing ability. We also drastically optimized the synthesis of 1H-cyclopenta[b]naphthalene-1,3(2H)-dione EA2 so that this precursor of EA1 could be obtained with reaction yields higher than 90%. To evaluate the benefit of EA1 in the optical properties of PP1 and PP2, their analogues PP3 [START_REF] Singh | Tuning solid-state fluorescence to the near-infrared: a combinatorial approach to discovering molecular nanoprobes for biomedical imaging[END_REF] and PP4 [START_REF] Meshulam | et al 2-D effects in the second-order optical nonlinearity of organic molecules incorporating carbazole[END_REF] using 2-(3-oxo-2,3-dihydro-1H-inden-1-ylidene)malononitrile as the acceptor were prepared and the optical properties compared to those of PP1 and PP2 (see Fig. 4). It has to be noticed that the synthesis of these two dyes (PP3 and PP4) has been previously reported in the literature. Par allel to this, during the course of our investigations, choice of the amine used for the Knoevenagel reaction was determined as being crucial to avoid the nucleophilic attack of this latter on the push-pull structures so that a series of amines were tested as catalysts. The different products obtained during these different reactions and their properties are also reported and discussed.

Results and discussion

1-Synthesis of the different compounds

As mentioned in the introduction section, the first step consisted in preparing EA1 starting from EA2. Compared to previous reports [START_REF] Buckle | Spicer Antiallergic activity of 2-nitroindan-1, 3-diones[END_REF][START_REF] Pigot | Push-pull chromophores based on the naphthalene scaffold: potential candidates for optoelectronic applications[END_REF], the reaction yield to get EA2 was greatly improved, based on several modifications. Notably, a typical procedure to access to EA2 consists in a Claisen reaction of a dialkyl naphthalene-2,3-dicarboxylate with ethyl acetate in the presence of a base (See Fig. 5).

By increasing the quantity of sodium hydride (2.5 eq. instead of 1.45 eq.) for the first step and by elongating the decarboxylation time (1h30 instead of 20 min) for the second step, the reaction yield could be increased up to 91%, higher than the 65% yield reported in the literature. Noticeably, safer conditions were used for the recrystallization step of EA2, in which benzene was replaced by toluene. Finally, the functionalization of EA2 by malononitrile was carried out under standard conditions, using sodium acetate as the base. EA1 could be obtained in 67% yield, as previously described in the literature [START_REF] Feng | An A-D-A type small-molecule electron acceptor with end-extended conjugation for high performance organic solar cells[END_REF]. After the preparation of EA1, the synthesis of the different push-pull dyes described in Fig. 4 were realized, starting with PP1. This synthesis consisted in a Knoevenagel reaction between EA1 and 4-N,N-dimethylaminobenzaldehyde ED1 in the presence of a base, currently piperidine, in ethanol for 30 min.

Unexpectedly, the reaction media turned red and the structure of a new compound, PP6, was confirmed by NMR spectroscopy (See Fig. 6). Nevertheless, this reactivity of secondary amines, like piperidine, has already been highlighted in the literature, in 2008, by Landmesser et al. [START_REF] Landmesser | A novel route to 1-substituted 3-(dialkylamino)-9-oxo-9H-indeno[2,1-c]-pyridine-4-carbonitriles[END_REF]. As no such reaction has been previously described concerning α,β-unsaturated aldehydes, the same procedure was used for PP2, using the same conditions. After 30 min of reaction, if the Nuclear Magnetic Resonance (NMR) analyses confirmed the absence of the targeted chromophore PP2 in the reaction media, a red colour was also observed for this second reaction opposing EA1 to ED2 (See Fig. 6).

Surprisingly, analyses by 1 H NMR spectroscopy in CDCl 3 revealed the electron-donating moiety ED2 not to be connected to EA1. On the opposite and unexpectedly, the covalent linkage of piperidine to EA1 could be easily evidenced by 1 H NMR analyses, with the presence of two sets of signals at 1.85 and 3.86 ppm. Presence of a singlet at 5.87 ppm integrating for one proton could also be detected. Finally, the exact structure of PP5 could be revealed by X-ray diffraction (see Fig. 7).

The single crystals of PP5 suitable for X-ray crystallographic analyses were grown by slow evaporation of CDCl 3 solvent in the NMR tube. The detailed crystallographic data are given in the Electronic Supporting Information.

As evidenced in Fig. 8, the mechanism inducing the formation of PP5 results from the nucleophilic attack of piperidine on the polarized ketone group. After prototropy, a dehydration reaction resulting from the deprotonation of the activated methylene group by piperidine can occur, providing PP5 as a red powder. Interestingly, attempts to prepare PP5 by simply refluxing EA1 in ethanol, in the presence of piperidine failed to produce PP5 and only traces of product were detected by thin layer chromatography (TLC) even after 2 h of reflux (See Fig. 6). Upon addition of ED2, immediately, the reaction colour turned deep red and a complete consumption of EA1 was observed within 20 min. If the exact role of ED2 is not fully understood, ED2 clearly favours the synthesis of PP5. It can be tentatively assigned to a role of base, this electron donor comprising a dimethylaniline moiety probably facilitating the deprotonation occurring in last step.

Considering that PP6 was also obtained as a red solid, that its absorption properties are clearly centered in the visible range with a small Highest Occupied Molecular Orbital (HOMO)-Lowest Unoccupied Molecular Orbital (LUMO) gap [START_REF] Landmesser | A novel route to 1-substituted 3-(dialkylamino)-9-oxo-9H-indeno[2,1-c]-pyridine-4-carbonitriles[END_REF], the design and synthesis of a series of dyes still based on EA1 as the electron acceptor but differing by the electron-donating groups and the amine pendant groups were realized. The first parameter investigated was the influence of the amine on the reaction yields and the optical properties of the resulting products while keeping ED1 as the electron donor. In this aim, five different amines were investigated. Secondly, the influence of the conjugation length between the electron-donating part and the electron-withdrawing group on both the reaction yields and the optical properties was examined. For this purpose, another electron donor, namely ED2 was used (See Fig. 10). A summary of the different product obtained during these syntheses and the corresponding reaction yields are provided in Table 1. Even if no clear conclusion can be made concerning the im

PP10 (5%) + PP10' (25%) PP2 (60%)
pact of the basicity of the amine on this reaction, we clearly demonstrate that higher reaction yields were obtained when cyclic aliphatic amines were used during the synthesis of PP6-PP10 (See Fig. 9). It might be explained by the higher availability of the nitrogen lone pair electrons in this case. In fact, the cycle drastically inhibits the pyramidal inversion of the nitrogen atom.

Concerning the disparity of the reaction yields, two factors might also be considered. Firstly, for morpholine and pyrrolidine, no column chromatography was necessary to get the product in pure form, the dyes directly precipitating by addition of acetone to the reaction media.

In these conditions, the products could be obtained in high yields. As further detailed in the NMR section, a facile protonation of the dihydro-2-azafluorenes in chloroform was clearly evidenced, resulting in the duplication of the NMR signals (see 1 H NMR spectra in SI).

Considering that CDCl 3 is sufficiently acidic to protonate the dyes in solution, it also means that during the purification process, a non-negligible part of the products was lost by adsorption onto the stationary phase (silicagel), chloroform being used as the eluent. Therefore, it may also explain why PP9 or PP10 were isolated in pure form only in low yields. The formation of the aromatized product observed in the case of PP10 and producing PP10' also contributes to reduce the final reaction yield. To confirm the chemical structure of these dihydro-2-azafluorenes, crystals could notably be obtained upon slow evaporation of chloroform. The crystal structure of PP10 in presented in Fig. 9.

Regarding the condensation of ED2 onto EA1 and the subsequent nucleophilic attack of the amine, adducts totally differing in structures from that obtained with ED1 were obtained, evidencing a significant difference of reactivity with the elongated aldehyde ED2. Notably, a spontaneous aromatization resulting in the formation of a pyridine group was notably evidenced with PP11, PP12 and PP13 (See Fig. 10). The crystal structure of PP12 is presented in Fig. 10.

Elongation of the π-conjugation therefore favoured the oxidation (and the aromatization process), providing the pyridine adducts PP11, PP12 and PP13 as the main products of a one-pot synthesis. Formation of such aromatized structures has been previously reported by Landmesser et al. [START_REF] Landmesser | A novel route to 1-substituted 3-(dialkylamino)-9-oxo-9H-indeno[2,1-c]-pyridine-4-carbonitriles[END_REF] While using aldehydes of short conjugation length, by developing a two-step procedure where chemical oxidants (KMnO 4 , chloranil) were used, the access to aromatized structures was possible. Besides, it has to be noticed that Landmesser et al. also detected the coexistence of the aromatized and none-aromatized forms during the course of their investigations, but the aromatized products were only detected as a minor side-product of the reaction. Surprisingly, while using dioctylamine as the amine, the cyclization product was not detected in the reaction media. Instead, PP2 could be isolated in pure form in 60% yield, demonstrating either a steric hindrance of the amine or a better kinetic favouring the Knoevenagel reaction. After investigating the synthesis of the different azafluorenones PP6-PP13, we focused our work on the synthesis of the initially desired push-pull dyes PP1 and PP2 and their analogues PP3 and PP4 (see Fig. 11).

As previously evidenced, it appeared that the main drawback encountered with piperidine was its nucleophilicity. To overcome this drawback, a none-nucleophilic base was selected for the Knoevenagel reactions, namely N,N-diisopropylethylamine (DIPEA). Indeed, the steric hindrance of this tertiary amine prevents its nucleophilic addition onto PP1-PP4. As expected, PP1-PP4 were obtained with reaction yields ranging between 55 and 65% (See Fig. 11). Formation of these products were confirmed by 1 H and 13 C NMR analyses.

Finally, after the successful synthesis of PP1-PP4 by using DIPEA as the base, the possibility to convert these push-pull structures into azafluorenones was examined. As a proof of concept, the conversion of PP1 to PP6 and PP2 to PP12 was examined. In the two cases, slightly higher reaction yields were obtained compared to the direct one-pot synthesis, resulting from the fact that the push-pull dyes PP1 and PP2 are already formed and purified prior to the cyclization reaction occurring in final step. (See Fig. 12).

2-1 H NMR spectroscopy study

As above-mentioned in the text, the different cyclization products proved to be highly sensitive to the protic character of the solvent. This point was notably evidenced by the low isolated reaction yields obtained for the different compounds after chromatographic purification. This also explains that most of the product remaining immobilized on the acidic silica gel during the complex separation of numerous products by column chromatography. Face to this consideration, this point was also explored by NMR spectroscopy. Two different solvents enabling to perfectly solubilize the different compounds (PP6, PP7, PP8, PP9 and PP10) were tested, namely, a protic solvent i.e. chloroform and an aprotic one i.e. DMSO. Fig. 13 displays the NMR data of PP8 in the two selected solvents. If a clear 1 H NMR spectrum corresponding to the targeted molecule was obtained in DMSO-d 6 , the coexistence of two different forms could be found in the 1 H NMR spectrum of PP8 in deuterated chloroform (see Fig. 13). Indeed, the protonated and the none-protonated form could be determined as being in a 31:69 ratio, the none-protonated state constituting the main form in the sample. Examination of the behaviour of these five chromophores in the two solvents revealed PP7, PP8, and PP9 to be the most sensitive compounds of the series, ratios of 21:79 and 28:72 being respectively determined for PP7 and PP9. As specificity for PP7, traces of the protonated form could be even detected in DMSO, evidencing its easy protonation (See 1 H NMR spectra in SI). Conversely, PP6 proved to be insensitive to the acidic character of chloroform, no trace of protonation being detected by NMR. It has to be noticed that all cyclization reactions have been carried out in the presence of an excess of base so that the protonation detected by 1 H NMR can only originate from acidic traces in the deuterated solvents.

3-UV-visible absorption and photoluminescence spectroscopy

All compounds reported in this work are stable under air, even when diluted in solution so that photophysical measurements could be carried out. As evidenced by NMR analyses, if a protonation for some of the dyes in protic solvents could be demonstrated, no photodegradation of the dyes were detected by NMR during the course of our investigations. Especially, no modification of the NMR signals was detected even for NMR tubes exposed to sunlight during several weeks. In this context, the dyes being stable in solution, these latter could be characterized by UV-visible absorption spectroscopy and the different absorption spectra were recorded in three different solvents, namely chloroform, dichloromethane and DMSO. Interestingly, four different but characteristic UV-visible spectra were found for the cyclized compounds, depending of the length of the conjugated spacer between the electron-donating dimethylaminophenyl group and the electron-accepting azafluorenone moiety, but also to the fact that the pyridine group of the azafluorenone moiety is aromatized or not. Thus, the UV/Vis absorption spectra of the dihydro-2-azafluorenones PP6, PP7, PP8, PP9, and PP10 are dominated by an intensive intramolecular charge-transfer (ICT) band extending between 440 and 600 nm irrespective of the solvent (See Fig. 14).

By changing the appended amine, no shift of the ICT band was found for all compounds, evidencing the push-pull effect to be independent of this substituent. The lack of contribution on the push-pull effect were confirmed by theoretical calculations. As illustrated in Fig. 15 and the corresponding Figures in SI, only the nitrogen atom of the pendant amine was determined as contributing to the highest Occupied Molecular Orbital (HOMO) and the Lowest Unoccupied Molecular Orbital (LUMO). No extension over the alkyl chain of the different amines (piperidine, morpholine, dioctylamine, diethylamine or pyrrolidine) was found, justifying the dihydro-2-azafluorenones to be insensitive to this substitution pattern. For all dyes, an absorption maximum peaking at 500 nm was determined for the five chromophores PP6-PP10 and in the three different solvents. A HOMO-LUMO gap of 2.48 eV was calculated for all dyes. Comparison of the experimental absorption spectra with those determined by theoretical calculations revealed the absorption maxima to be red-shifted by about 13 nm relative to that determined theoretically, this latter peaking at 487 nm.

Besides, a good agreement between the experimental and the theoretical values are thus found for the absorption maxima. Apart from the ICT band, three absorption peaks could also be detected between 350 and 430 nm and origins of these transitions were assigned to HOMO-5/-4/-3 → LUMO transitions admixed with HOMO-1 → LUMO+1, HOMO → LUMO+1 transitions (See Table 2).

Finally, the intense high-energy bands in the 300-350 nm region typically correspond to localized aromatic π-π* transitions of the chromophores. Examination of their UV-visible absorption properties in chloroform revealed the absorption spectra of all dyes to be similar to those determined in DMSO or dichloromethane (See Fig. 14), despites the partial protonation of the dihydro-2-azafluorenones in chloroform. Unexpectedly, elongation of the spacer between the electron donor and the electron acceptor in PP11, PP12 and PP13 resulted in a hypsochromic shift of the absorption maxima irrespective of the amine substituents. Indeed, absorption maxima peaking at 480 nm was found for the three dyes, blue-shifted by about 20 nm relative to the previous se ries. Here again, absence of contribution of the amino group to the push-pull effect was evidenced, the three absorption spectra superimposing each other. Conversely and as anticipated, elongation of the spacer resulted in a significant increase of the molar extinction coefficients of the ICT bands for PP11, PP12 and PP13 relative to those of the previous dihydro-2-azafluorenones (PP6-PP10), resulting from an enhancement of the oscillator strength. Here again, comparison of the theoretical and experimental absorption spectra revealed a good agreement between both. Two main transitions were detected in DMSO, in the 350-420 nm and 420-600 nm regions, the second one corresponding to the ICT band of the chromophores. Positions of these transitions are close to those determined theoretically, the simulated spectra displaying two main absorption bands at comparable positions (See Fig. 16).

A difference as small as 14 nm were found between the theoretical (492 nm) and the experimental (478 nm) absorption maxima. Comparison of the theoretical calculations carried out on the dihydro-2-azafluorenones and the 2-azafluorenones revealed the HOMO energy level of dihydro-2-azafluorenones to benefit from both the electron-donating effects of the dimethylaminophenyl group and the amino pendant group. Conversely, as evidenced in Fig. 17, only the dimethylaminophenyl group of 2-azafluorenones contributes as the electron donor, supporting the blue-shift detected for the ICT bands.

Upon aromatization, a clear modification of the absorption spectra could be evidenced between the dihydro-2-azafluorenone PP10 and the 2-azafluorenone PP10′ corresponding to the aromatized version of PP10. As shown in Fig. 18, a blue-shift of the ICT band was observed upon aromatization of the pyridine moiety, the absorption maximum shifting from 500 nm for PP10 to 430 nm for PP10'. These results are consistent with a reduced electronic delocalization upon aromatization, what is notably observed when quinones are converted to hydroquinone derivatives [START_REF] Chen | Swager Synthesis and characterization of fluorescent acenequinones as dyes for guest-host liquid crystal displays[END_REF]. Parallel to this, the two intense high-energy bands in the 300-350 nm region detected for PP10 were not visible anymore in the absorption spectrum of PP10'. While using the same theoretical model, only a slight shift of the ICT band is observed upon aromatization. Notably, absorption maxima peaking at 481 and 448 nm are theoretically determined for PP10 and PP10′ respectively. If a good accordance is found between the experimental and the theoretical absorption maximum for PP10 (496 nm and 482 nm respectively) and PP10' (434 and 447 nm respectively), a complete mismatch between the theoretical and experimental molar extinction coefficients was found. This is attributable to the long alkyl chains that rendered the optimization of structures and the subsequent calculations difficult to be carried out (see Fig. 19).

Finally, the most intriguing absorption spectrum was obtained for PP5, resulting from the nucleophilic addition of piperidine on the electron acceptor EA1. Presence of numerous optical transitions could be found between 300 and 600 nm irrespective of the solvent, making this dye a panchromatic chromophore (see Fig. 18). Based on the theoretical calculations, a few transitions could be assigned. Notably, in the 300-380 nm region, the different transitions can be assigned to an admixture of HOMO-1→LUMO+1 and HOMO-2 → LUMO transitions. Conversely, in the 380-440 nm region, the absorption band is dominated by the HOMO-1→LUMO and HOMO→LUMO+1 transitions. At low energy, a broad peak composed of multiple transitions extending from 450 nm to 600 nm and dominated by the HOMO→LUMO transition was found. An absorption maximum peaking at 506 nm was found for PP5, close to that obtained for the dihydro-2-azafluorenones PP6-PP10 (500 nm).

It must be noticed that the complexity of the transitions detected by absorption spectroscopy can be tentatively assigned to the coexistence of two forms in solution, the neutral form and its corresponding zwitterion, complexifying the absorption spectrum. By examining the UV spectra of PP1-PP4 in chloroform (see Fig. 20), several trends can be As previously reported in the literature, the HOMO of push-pull structure comprises both the electron-donating group and the π-conjugated spacer [START_REF] Dumur | Sécheresse New chelating stilbazonium-like dyes from Michler's ketone[END_REF]. Upon elongation of the spacer, a destabilization of the HOMO level occurs, resulting in a decrease of the HOMO-LUMO gap and a red-shift of the absorption spectrum towards long wavelengths. Theoretical calculations carried out on PP1-PP4 confirmed the order determined in solution (See Fig. 20) and the distribution of the electronic density on the HOMO and LUMO energy levels (See Fig. 21).

Finally, the emission properties of all dyes were examined in dichloromethane as the solvent. Here again, each series displayed a typical fluorescence spectrum, with an emission centered in the visible range (See Fig. 22). Thus, for the dihydro-2-azafluorenones PP6-PP10, an emission peaking between 568 and 570 nm could be determined. The emission peaks for PP11-PP13 were found at 574 nm, close to that obtained for the former series despites the presence of the elongation of the conjugated spacer introduced between the donor and the acceptor. Upon aromatization of dihydro-2-azafluorenones, no significant change was found in the PL spectrum of PP10', displayed an emission maximum superimposing that of the two previous series. Therefore, it can be concluded that the energetic transition involved in these different dyes are similar. Interestingly, PP5 that drastically differs from the others by the absence of the dimethylaminophenyl group could emit at 562 nm.

Stokes shift values of 70 and 95 nm were determined for the dihydro-2-azafluorenones PP6-PP10 and the 2-azafluorenones PP11-PP13 respectively. It can be therefore concluded that a higher electronic redistribution occurs upon excitation of the 2-azafluorenones relative to the dihydro-2-azafluorenones.

4-Cyclic voltammetry

The electrochemical properties of all compounds have been investigated by cyclic voltammetry (CV) in a dilute solution of dichloromethane. All solutions have been deaerated with argon for 10 min prior to CV measurements. Besides this, traces of oxygen could be still detected on certain CV spectra, attributable to a strong interaction between dyes and oxygen. In this context, determination of the reduction potentials for several dyes was affected by the presence of oxygen, underestimating the real energy level of the LUMO orbitals. The redox potentials of the compounds PP1-PP13 are summarized in Table 3 in which the redox potentials are given against the half wave oxidation potential of the ferrocene/ferrocenium cation couple. The values of the HOMO and LUMO energy levels of the different chromophores were also determined using as the standard the value of the oxidation potential of ferrocene (Fc) based on the calculations of Pommerehne et al. (4.8 eV vs. vacuum) [START_REF] Pomrnerehne | Efficient two layer LEDs on a polymer blend basis[END_REF]. The obtained values of E HOMO and E LUMO are summarized in Table 3. Fig. 23 allows a comparison of the redox potentials of PP1-PP4 molecules. Notably, PP1 and PP2 possess the same electron acceptor and the oxidation potentials vary with the contribution of the electron-donating part that includes the electron donor for PP1, but also the conjugated spacer in the case of PP2.

As a result of this, elongation of the conjugated spacer facilitates the oxidation process in PP2 relative to PP1 and a decrease of the oxidation potential of 250 mV was observed (See Fig. 23a). Parallel to this, a modification of the reduction process centered onto EA1 was also detected, a cathodic shift of 150 mV being evidenced for the reduction peak of PP2 relative to PP1. This cathodic shift of the reduction peak upon elongation of the π-conjugated spacer is quite unusual, considering that the same electron acceptor is present in the two molecules. To support this shift, repartition of the respective HOMO and LUMO energy levels of PP1 and PP2 have to be considered. As shown in Fig. 21, the HOMO levels of PP1 and PP2 is mainly centered onto the electron-donating group with a contribution of the spacer. Conversely, the LUMO levels extend over the whole molecule. As a result of this, and due to the contribution of the electron-rich electron donating group to the LUMO level, a reduced electron-withdrawing ability is observed for PP2, shifting the reduction peak at lower potential than that of PP1. While comparing the dyes at identical electron donor, EA1 proved to be a better electron acceptor than 2-(3-oxo-2,3-dihydro-1H-inden-1-ylidene)malononitrile, the irreversible reduction of the electron acceptor in PP1 occurring at less cathodic potentials than for PP3 (-1.27 V vs. -1.35 V). By electrochemistry, an experimental determination of both the HOMO and the LUMO energy levels can be done from the onset of the oxidation and the reduction peaks. An electrochemical bandgap can be thus determined. As shown in Table 3, a good accordance between the theoretical and the optical bandgaps of PP1-PP4 can be found. On the opposite, the electrochemical bandgaps underestimate the values of approximately 0.5 eV for all dyes. A different behaviour was evidenced with PP5 for which an irreversible oxidation and a reversible reduction process could be evidenced (See Fig. 23c). Considering that PP5 is substituted by a piperidine group and that the oxidation process is centered on the nitrogen atom, an oxidation peak is logically detected at a higher potential than that observed for PP1-PP4, the oxidation of aliphatic amines being more difficult than for the aromatic ones.

U N C O R R E C T E D P R O O F

Compared to PP1-PP4, reduction of PP5 is detected at a more cathodic potential, attributable to the lower electron-withdrawing ability resulting from a LUMO energy level extending over the whole molecule and notably the electron-donating piperidine moiety. Fig. 23d presents the voltammograms of PP8 and PP9 which differ only by the nature of the pendant amine. As evidenced by theoretical calculations, only the nitrogen atom of the pendant amines contributes to the HOMO and LUMO energy levels (see Fig. 17). The oxidation processes for PP8 and PP9 are centered on the same fragments of the molecule so that the two compounds exhibit irreversible oxidation processes located at iden Pigot tical potentials. Similarly, the reduction processes are centered on the same part of the molecules so that a reduction at the same potentials is observed.

U N C O R R E C T E D P R O O F C.
As observed for PP8 and PP9, PP12 and PP13 only differ by the nature of the pendant amine. However, contribution of the nitrogen atom is more limited, and restricted to the LUMO energy level (see Fig. 10). The first oxidation process of PP12 and PP13 is centered on the dimethylaminophenyl group of 2-azafluorenone. Since the electron donating moiety is the same for the two molecules, their oxidation potentials are almost similar. The reduction process is located on the 2-azafluorenone part. Considering that only the nitrogen atom of amines contributes to the LUMO level and not the aliphatic part of the amine, PP12 and PP13 also reduce at almost similar reduction potentials (See Fig. 23e). While comparing PP13 and PP10′ that differ by the length of the conjugated spacer and both possesses an aliphatic pendant amine, similar conclusions than that done for PP1/PP2 can be established. Notably, a facilitated oxidation of PP13 due to an elongation of the conjugated spacer is observed. Parallel to this, et due to the delocalization of the LUMO level over the whole molecule, a reduction at more cathodic potentials is observed for PP10' (see Table 3 and Fig. 23f).

A final comparison is possible between PP10 and PP10' (See Fig. 23g). The difference between these two molecules is in the aromatization of PP10′ compared to PP10. This aromatization will have an impact on both the HOMO and LUMO distributions over the two molecules. The aromatization of PP10′ lowers its electron accepting ability compared to PP10. Indeed, as evidenced by theoretical calculations, for PP10′, the LUMO level is located on the 2-azafluorenone part of the molecule with a contribution to the dimethylaminophenyl group due to the coplanarity of the dimethylaminophenyl group with the naphthalene moiety. Conversely, for PP10, contribution of the dimethylaminophenyl group is negligible due to the orthogonality of the dimethylaminophenyl group with the naphthalene moiety (See Figs. 15 and 17). This directly impacts the reduction potential of the molecule, with a reduction occurring at more cathodic potentials for PP10' (-1.81 V vs. -1.56 V for PP10' and PP10 respectively). Similarly, major differences can be found for the oxidation potentials. Indeed, for PP10, HOMO orbital extends over the whole molecule, including the nitrogen amine of the aliphatic amines.

On the opposite, for PP10′, the HOMO is only centered onto the electron-donating group excluding the nitrogen of the pendant amine. As a result of this, oxidation of PP10′ is more difficult than PP10 and an increase of the HOMO-LUMO gap is observed for PP10′ relative to PP10. Consequently, a blue-shift of the ICT band is observed for PP10′ compared to that of PP10 (See Fig. 18). Finally, a comparison between the theoretical and the electrochemical HOMO and LUMO energy levels can be established. As evidenced in Fig. 24, a good agreement can be found between the theoretical and the experimental values of the HOMO energy levels since a difference lower than 0.5 eV can be determined. Conversely, a greater difference can be found for the LUMO energy levels. As already evidenced in Table 3 for the previous series of PP1-PP4, position of the LUMO levels is underestimated by electrochemistry. The same conclusions can thus be established for the present series of dyes PP6-PP13 and PP10'.

Conclusions

To conclude, a new series of dihydro-2-azafluorenones and aromatized 2-azafluorenones have been synthesized with a simple one-pot synthesis. Influence of the different parameters such as the amine or the spacer in the aldehydes have also been investigated. By elongating the length of the spacer between the electron donating aldehyde and the electron acceptor, a spontaneous aromatization occurred and lead to the formation of the 2-azafluorenones. Major difference in the photophysical properties of the different dyes have been evidenced. For instance, an easy protonation has been shown by an NMR study in the case of the dihydro-2-azafluorenones. Finally, the synergy between theoretical calculations, UV-visible absorption and photoluminescence spectroscopy, and cyclic voltammetry have allowed to determine the position of the HOMO and the LUMO energy levels as well as the HOMO-LUMO gap for all these new compounds.

Further investigations concerning these different families of compounds need to be carried out. Notably, azafluorenones could be of interest for biological applications such as lipid droplets s-specific imaging [START_REF] Fam | Collot Recent advances in fluorescent probes for lipid droplets[END_REF]. Similarly, push-pull dyes could be used as photoinitiators of polymerization, active materials for solar cells or NLO applications.

Experimental section

General informations

All reagents and solvents were purchased from Aldrich, Alfa Aesar or TCI Europe and used as received without further purification. Mass spectroscopy was performed by the Spectropole of Aix-Marseille Uni versity. ESI mass spectral analyses were recorded with a 3200 QTRAP (Applied Biosystems SCIEX) mass spectrometer. The HRMS mass spectral analysis was performed with a QStar Elite (Applied Biosystems SCIEX) mass spectrometer. Elemental analyses were recorded with a Thermo Finnigan EA 1112 elemental analysis apparatus driven by the Eager 300 software. 1 H and 13 C NMR spectra were determined at room temperature in 5 mm o.d. tubes on a Bruker Avance 400 spectrometer of the Spectropole: 1 H (400 MHz) and 13 C (100 MHz). The 1 H chemical shifts were referenced to the solvent peak CDCl 3 (7.26 ppm) and the 13 C chemical shifts were referenced to the solvent peak CDCl 3 (77 ppm). 2-(3-Oxo-2,3-dihydro-1H-inden-1-ylidene)malononitrile [START_REF] Cui | An indanone-based alkoxysilane dye with second order nonlinear optical properties[END_REF], 3-(4-(dimethyl-amino)phenyl)acrylaldehyde ED2 [START_REF] Malina | Belyakov Novel green-yellow-orange-red light emitting donor-p-acceptor type dyes based on 1,3-indandione and dimedone moieties[END_REF] were synthesized as previously reported in the literature, without modification and in similar yields. UV-visible absorption spectra were recorded on a Varian Cary 50 Scan UV Visible Spectrophotometer, with concentration of 5 × 10 -3 M, corresponding to diluted solutions. Fluorescence spectra were recorded using a Jasco FP 6200 spectrometer. The electrochemical properties of the investigated compounds were measured in dichloromethane by cyclic voltammetry, scan rate 100 mV/s, with tetrabutylammonium perchlorate (0.1 M) as a supporting electrolyte in a standard one-compartment, three-electrode electrochemical cell under an argon stream using a VSP BioLogic potentiostat. The working, pseudo-reference and counter electrodes were platinum disk (Ø = 1 mm), Ag wire, and Au wire gauze, respectively. Ferrocene was used as an internal standard, and the potentials are referred to the reversible formal potential of this compound. All CV measurements have been acquired upon deaeration of the solutions with argon during 10 min prior to the acquisition of the curves. No measurements have been carried out in glovebox. DFT calculations [START_REF] Kohn | Self-consistent equations including exchange and correlation effects[END_REF] employing the B3LYP functionals [START_REF] Lee | Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density[END_REF][START_REF] Becke | Density-functional thermochemistry. III. The role of exact exchange[END_REF] and 6-311 + G(d,p) basis set were performed with a Gaussian 09 program [START_REF] Frisch | Gaussian 09[END_REF]. The geometry optimizations for all the molecules were carried out without symmetry constraints and were followed by frequency calculations. The spectroscopic properties of the molecules were calculated by mean of a time dependent density functional theory method (TDDFT) [START_REF] Gross | Local density-functional theory of frequency-dependent linear response[END_REF][START_REF] Runge | Gross Density-functional theory for time-dependent systems[END_REF][START_REF] Gross | Density functional theory of many-fermion systems[END_REF][START_REF] Bauernschmitt | Ahlrichs Treatment of electronic excitations within the adiabatic approximation of time dependent density functional theory[END_REF][START_REF] Casida | Salahub Molecular excitation energies to high-lying bound states from time-dependent density functional response theory: characterization and correction of the time dependent local density approximation ionization threshold[END_REF]. Up to 10 excited states were calculated and the theoretical absorption bands were obtained by considering a band half-width at half-maximum of 0.2 eV [START_REF] Feng | An A-D-A type small-molecule electron acceptor with end-extended conjugation for high performance organic solar cells[END_REF]. The assignment of electronic transitions for λ max has been determined with GaussSum 3.0 software [START_REF] Dennington | GAUSSVIEW (version 5)[END_REF][START_REF] O'boyle | Langner cclib: a library for package-independent computational chemistry algorithms[END_REF].

U N C O R R E C T E D P R O O F U N C O R R E C T E D P R O O F C. Pigot

Synthesis of the electron acceptors and push-pull dyes

5.2.1. 2-(3-Oxo-2,3-dihydro-1H-cyclopenta[b]naphthalen-1- ylidene)malononitrile EA1
In a dried two-necked 100 mL flask, 1H-cyclopenta[b]naphthalene-1,3(2H)-dione (1.1 g, 5.6 mmol) and malononitrile (2.2 g, 33.3 mmol) were dissolved in ethanol (25 mL), and then anhydrous sodium acetate (1.84 g) was slowly added while stirring. After stirring for 2 h, the reaction mixture was poured into ice-water, and acidified to pH 1-2 by the addition of concentred hydrochloric acid. The resulting precipitate was collected by filtration and washed with water giving the crude product. It was finally purified with a flash chromatography (eluent: DCM). Yield = 67%. 1

1H-cyclopenta[b]naphthalene-1,3(2H)-dione EA2

Diethyl naphthalene-2,3-dicarboxylate (10 g, 38.5 mmol) was suspended in dry EtOAc (24 mL) and NaH 95% in oil (2.44 g, 96.4 mmol, 2.5 eq) was added. The reaction mixture was refluxed at 105 °C for 5 h. After cooling, the yellow solid was filtered off and thoroughly washed with a mixture of EtOH-Et 2 O 50/50. Treatment of this solid with 200 mL of a 1 M HCl solution under reflux for 1h30 furnished a new solid. After cooling, the solid was filtered off, washed with water and recrystallized in toluene (200 mL) overnight. The product was obtained as a brown solid. Yield = 91%. 1

General procedure for the synthesis of the push-pull dyes PP1-PP4

2-(3-Oxo-2,3-dihydro-1H-cyclopenta[b]naphthalen-1-ylidene)malononitrile EA1 (0.62 g, 2.55 mmol) or 2-(3-oxo-2,3-dihydro-1H-inden-1-ylidene)malononitrile (0.50 g, 2.55 mmol) and the appropriate aldehyde (4-dimethylaminobenzaldehyde (0.38 g) or 3-(4-(dimethylamino)-phenyl)acrylaldehyde (0.45 g), 2.55 mmol, 1. eq.) were dissolved in absolute ethanol (50 mL) and a few drops of piperidine were added. The reaction mixture was refluxed and progress of the reaction was followed by thin layer chromatography (TLC). After cooling, a precipitate formed. It was filtered off, washed several times with ethanol and dried under vacuum.

2-(2-(4-(Dimethylamino)benzylidene)-3-oxo-2,3-dihydro-1Hcyclopenta[b]naphthalen-1-ylidene)malononitrile PP1

Yield = 68%. [START_REF] Meshulam | et al 2-D effects in the second-order optical nonlinearity of organic molecules incorporating carbazole[END_REF] Yield = 45%.

2-(2-(3-(4-(Dimethylamino)phenyl)allylidene)-3-oxo-2,3-dihydro-1H-inden-1-ylidene)malononitrile PP4

2-(3-(Piperidin-1-yl)-1H-cyclopenta[b]naphthalen-1ylidene)malononitrile PP5

4-Dimethylaminobenzaldehyde ED1 (0.77 g, 5.15 mmol) and 2-(3-oxo-2,3-dihydro-1H-cyclopenta[b]naphthalen-1-ylidene)malononitrile EA1 (1.25 g, 5.15 mmol) were suspended in 20 mL absolute ethanol and a few drops of piperidine were added. Immediately, the solution turned deep red. The flask was introduced in an oil bath preheated at 90 °C. After 15 min, the reaction was ended (TLC control). During that time, an extremely insoluble precipitate was formed. After cooling, the precipitate was filtered off, washed several times with ethanol and ether, and dried under vacuum. Yield = 86%; 1 H NMR (400 MHz, CDCl 3 ) δ: 1.85 (brs, 6H), 3.86 (brs, 4H), 5.87 (s, 1H), 7.52-7.57 (m, 2H), 7.77-7.89 (m, 3H), 8.57 (s, 1H); 13 C NMR (100 MHz, CDCl 3 ) δ: 13

General procedure for the synthesis of the push-pull dyes PP6-PP13

The selected aldehyde (1,64 mmol, 1 eq.) and EA1 (1.64 mmol, 1 eq.) were introduced in a preheated solution of ethanol (40 mL). the selected amine (0.5 mL) was added in the former solution while boiling. Monitoring of the reaction was done by TLC and the solution was approximately refluxed for 30 min. Depending of the amine, upon cooling, a precipitate formed in most of the time. This solid was filtered off and discarded. Addition of Et 2 O in the filtrate furnished a precipitate that was filtered off, washed with ether and dried under vacuum. In the case that no precipitation occurred upon addition of ether, the solvent was removed under reduced pressure and the residue was purified by column chromatography (SiO 2 ). [START_REF] Chaumel | Kakkar Sol-gel materials for second-order nonlinear optics[END_REF][START_REF] Matsumoto | Nonlinear optical properties of an azo dye attached polymer[END_REF]indeno [2,1-c] [START_REF] Chaumel | Kakkar Sol-gel materials for second-order nonlinear optics[END_REF][START_REF] Matsumoto | Nonlinear optical properties of an azo dye attached polymer[END_REF]indeno [2,1-c] [START_REF] Chaumel | Kakkar Sol-gel materials for second-order nonlinear optics[END_REF][START_REF] Matsumoto | Nonlinear optical properties of an azo dye attached polymer[END_REF]indeno [2,1-c]

1-(4-(Dimethylamino)phenyl)-11-oxo-3-(piperidin-1-yl)-2,11dihydro-1H-benzo

1-(4-(Dimethylamino)phenyl)-11-oxo-3-(pyrrolidin-1-yl)-2,11dihydro-1H-benzo

3-(Diethylamino)-1-(4-(dimethylamino)phenyl)-11-oxo-2,11dihydro-1H-benzo

1-(4-(Dimethylamino)phenyl)-3-(dioctylamino)-11-oxo-2,11dihydro-1H-benzo[5,6]indeno[2,1-c]pyridine-4-carbonitrile PP10

Yield = 5%. 1 H NMR (400 MHz, acetone-d 6 ) δ: 0.85 (t, 6H, J = 6.7 Hz), 1.26-1.29 (m, 20H), 1.70 (qt, 4H, J = 7.5 Hz), 2.90 (s, 6H), 3.49-3.56 (m, 2H), 3.79-3.86 (m, 2H), 5.51 (d, 1H, J = 4.5 Hz), 6.68 (d, 2H, J = 8.9 Hz), 7.22 (d, 2H, J = 8.9 Hz), 7.50-7.55 (m, 3H), 7.72 (s, 1H), 7.92 (t, 2H, J = 7.1 Hz), 8.29 (s, 1H); 13 C NMR (100 MHz, acetone-d 6 ) δ: 14.4, 23.3, 27.2, 28.9, 29.3, 29.97, 29.99, 32.6, 40.6, 51.8, 53.9, 60.6, 113.3, 118.9, 120.7, 120.9, 127.8, 128.0, 128.6, 130.1, 130.7, 131.3, 134.9, 135.5, 136.2, 137.2, 151.4, 154.6, 160.3, 187.0; HRMS (ESI MS) m/z: theor: 617.4214 found: 617.4206 ([M+H] + detected). [START_REF] Chaumel | Kakkar Sol-gel materials for second-order nonlinear optics[END_REF][START_REF] Matsumoto | Nonlinear optical properties of an azo dye attached polymer[END_REF]indeno [2,1-c]pyridine-4-carbonitrile PP10' Yield = 25%. 1 H NMR (400 MHz, CDCl 3 ) δ: 0.90 (t, 6H, J = 6.5 Hz), 1.31-1.50 (m, 20H), 1.82 (qt, 4H, J = 6.9 Hz), [START_REF] Motiei | Naderali Third-order nonlinear optical properties of organic azo dyes by using strength of nonlinearity parameter and Z-scan technique[END_REF]

1-(4-(Dimethylamino)phenyl)-3-(dioctylamino)-11-oxo-11Hbenzo

(E)-1-(4-(Dimethylamino)styryl)-11-oxo-3-(pyrrolidin-1-yl)-11H

benzo [START_REF] Chaumel | Kakkar Sol-gel materials for second-order nonlinear optics[END_REF][START_REF] Matsumoto | Nonlinear optical properties of an azo dye attached polymer[END_REF]indeno [2,1-c] [START_REF] Chaumel | Kakkar Sol-gel materials for second-order nonlinear optics[END_REF][START_REF] Matsumoto | Nonlinear optical properties of an azo dye attached polymer[END_REF]indeno [2,1-c]

(E)-1-(4-(Dimethylamino)styryl)-3-morpholino-11-oxo-11Hbenzo

(E)-3-(Diethylamino)-1-(4-(dimethylamino)styryl)-11-oxo-11Hbenzo[5,6]indeno[2,1-c]pyridine-4-carbonitrile PP13

Yield = 5%. 1 H NMR (400 MHz, CDCl 3 ) δ: 1.37 (t, 3H, J = 6.9 Hz), 1.39 (t, 3H, J = 6.9 Hz), 2.98 (s, 6H), 3.85 (q, 2H, J = 6.9 Hz), 3.87 (q, 2H, J = 6.9 Hz), 6.65 (d, 2H, J = 8.2 Hz), 7.51-7.56 (m, 4H), 7.88-7.92 (m, 3H), 8.11 (d, 2H, J = 8.2 Hz), 8.78 (s, 1H); 13

Fig. 1 .

 1 Fig. 1. Chemical structures of 3-(dialkylamino)-1,2-dihydro-9-oxo-9H-indeno[2,1-c]pyridine-4-carbonitrile derivatives C1-C4 formed by nucleophilic attack of the amine catalysts.

Fig. 2 .Fig. 3 .

 23 Fig. 2. Chemical structure of C5 prepared from 2-(2-(furan-2-ylmethylene)-3-oxo-2,3-dihydro-1H-inden-1-ylidene)malononitrile and structure of C6.

Fig. 4 .Fig. 5 .Fig. 6 .

 456 Fig. 4. Chemical structures of push-pull dyes PP1 and PP2, their analogues PP3 and PP4 and the electron acceptors EA1 and EA2.

UFig. 7 .

 7 Fig. 7. 1 H NMR spectrum of PP5 done in CDCl 3 (7.26 ppm) (left) and crystal structure of PP5 (right).

Fig. 8 .

 8 Fig. 8. Mechanism inducing the formation of PP5.

UFig. 9 .

 9 Fig. 9. Synthetic route to PP6-PP10 and crystal structure of PP10.

UFig. 10 .

 10 Fig. 10. Synthetic route to PP11-PP13 and crystal structure of PP12.

Fig. 11 .Fig. 12 .

 1112 Fig. 11. Synthetic route to PP1-PP4.

UFig. 13 .

 13 Fig. 13. Top: 1 H NMR spectra of PP8 in DMSO-d 6 . Bottom: 1 H NMR spectra of PP8 in CDCl 3 with the protonated form representing 31% of the sample. (stars indicate the NMR signals corresponding to the product (*) and its protonated form (*).deduced. As described at the beginning of the study, influence of two factors are studied in this paper, namely the impact of the additional phenyl ring on the electron-withdrawing ability of EA1 and the role of the conjugated spacer on the optical properties of PP1-PP4. Concerning the elongation of the aromatic scaffold of EA1, a red-shift absorption by about 30 nm is observed while comparing PP1 and PP3, PP2 and PP4 i.e. at similar electron-donating group and spacer (see Fig.20). It can be thus concluded that a stabilization of the LUMO energy level is observed for EA1 compared to its analogue 2-(3-oxo-2,3-dihydro-1H-inden-1-ylidene) malononitrile. Concerning the extension of the π-conjugation of the spacer in PP2 and PP4, an even bigger red-shift of 80 nm is detected. As previously reported in the literature, the HOMO of push-pull structure comprises both the electron-donating group and the

UFig. 14 .

 14 Fig. 14. UV-visible absorption spectra of dihydro-2-azafluorenones PP6-PP10 in DMSO (a), dichloromethane (b), chloroform (c) and the theoretical UV-visible absorption spectra in dichloromethane (d).

Fig. 15 .

 15 Fig. 15. Optimized geometries and contour plots of the HOMO and LUMO energy levels for the non-aromatized azafluorenones compounds (PP6-PP10).

Fig. 16 .Fig. 17 .

 1617 Fig. 16. Comparison between the theoretical (right) an the experimental (left) UV-visible absorption spectra recorded in DMSO for PP11-PP13.

Fig. 18 .

 18 Fig. 18. UV-visible absorption spectra of PP10 and PP10′ in chloroform (left) and the theoretical ones (right).

Fig. 19 .Fig. 20 .

 1920 Fig. 19. UV-visible absorption spectrum of PP5 in DMSO (red) and chloroform (green). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

Fig. 21 .Fig. 22 .

 2122 Fig. 21. Optimized geometries and contour plots of the HOMO and LUMO energy levels for PP1-PP4.

Fig. 23 .Fig. 24 .

 2324 Fig. 23.

  H NMR (400 MHz, CDCl 3 ) δ: 3.38 (s, 2H), 7.66-7.75 (m, 2H), 8.10-8.13 (m, 2H), 8.52 (s, 2H); 13 C NMR (100 MHz, CDCl 3 ) δ: 46.7, 124.3, 129.7, 130.6, 136.4, 138.2, 197.6; HRMS (ESI MS) m/z: theor: 196.0524 found: 196.0526 ([M] +. detected).

5. 2

 2 .5. 2-(-2-(-3-(4-(Dimethylamino)phenyl)allylidene)-3-oxo-2,3-dihydro-1H-cyclopenta[b]naphthalen-1-ylidene)malononitrile PP2 Yield = 44%. 1 H NMR (400 MHz, CDCl 3 ) δ: 3.15 (s, 6H), 6.72 (d, 2H, J = 9.0 Hz), 7.46 (d, 1H, J = 14.8 Hz), 7.66-7.68 (m, 3H), 17 Cyclic voltammograms of a) PP1 and PP2, b) PP1 and PP3, c) PP5, d) PP8 and PP9, e) PP12 and PP13, f) PP10′ and PP10′, g) PP10′ and PP13 in dichloromethane solutions (10 -3 M) with tetrabutylammonium perchlorate (0.1 M) as the supporting electrolyte. Scan rate: 100 mV/s. U N C O R R E C T E D P R O O F C. Pigot et al. Dyes and Pigments xxx (xxxx) xxx-xxx

  pyridine-4-carbonitrile PP8 Yield = 36%. 1 H NMR (400 MHz, DMSO-d 6 ) δ: 1.85-1.95 (m, 2H), 1.99-2.08 (m, 2H), 2.85 (s, 6H), 3.62-3.70 (m, 2H), 3.71-3.80 (m, 2H), 5.44 (d, 1H, J = 3.6 Hz), 6.68 (d, 2H, J = 8.8 Hz), 7.16 (d, 2H, J = 8.8 Hz), 7.49-7.55 (m, 2H), 7.73 (s, 1H), 7.86 (d, 1H, J = 7.0 Hz), 7.96 (d, 1H, J = 7.0 Hz), 8.12 (d, 1H, J = 4.0 Hz), 8.16 (s, 1H); 13 C NMR (125 MHz, DMSO-d 6 ) δ: 24.9, 49.9, 51.7, 56.0, 58.4, 108.8, 112.4, 119.3, 119.8, 121.3, 126.6, 127.0, 127.7, 129.2, 129.6, 130.9, 133.5, 134.2, 134.4, 135.5, 150.0, 153.4, 155.5, 185.1; HRMS (ESI MS) m/z: theor: 447.2179 found: 447.2179 ([M]+. detected).

  pyridine-4-carbonitrile PP9 Yield = 10%.1 H NMR (400 MHz, CDCl 3 ) δ: 1.19 (t, 6H, J = 6.9 Hz), 2.84 (s, 6H), 3.54-3.69 (m, 4H), 5.[START_REF] Runge | Gross Density-functional theory for time-dependent systems[END_REF] (s, 1H), 6.68 (d, 2H, J = 8.7 Hz), 7.12 (d, 2H, J = 8.6 Hz), 7.49-7.56 (m, 2H), 7.74 (s, 1H), 7.86 (d, 1H, J = 7.4 Hz), 7.96 (d, 1H, J = 7.4 Hz), 8.14 (s, 1H), 8.26 (s, 1H); 13 C NMR (100 MHz, CDCl 3 ) δ: 13.4, 40.0, 44.8, 51.8, 58.4, 112.3, 117.1, 119.4, 120.0, 120.5, 126.6, 127.1, 127.9, 129.1, 129.8, 130.0, 133.5, 134.0, 134.5, 135.4, 149.9, 153.6, 158.0, 185.3; HRMS (ESI MS) m/z: theor: 449.2336 found: 449.2334 ([M+H] + detected).

Table 1

 1 Isolated yields obtained with different amines during the synthesis of azafluorenones using ED1 or ED2 as the donors.

	Amine	ED1	ED2
		PP6 (14%)	PP5 (86%)
		PP7 (79%)	PP11 (6%)
		PP8 (35%)	PP12 (18%)
		PP9 (10%)	PP13 (5%)

Table 2

 2 Summary of simulated absorption characteristics in dilute dichloromethane of a selected set of synthetized compounds. Data were obtained in dichloromethane solution.

	et al.

Table 3

 3 Electrochemical Homo and E Lumo were determined from the formulas: E HOMO (eV) = -4.8 -E Ox(Onset) and E LUMO (eV) = -4.8 -E Red(Onset).

	PP1 PP2 PP3 PP4 PP5 PP6 PP7 PP8 PP9 PP10 PP11 PP12 PP13 PP10′	-1.26 irr -1.15 irr -1.34 irr -1.21 irr -1.48 -1.41 -1.35 -1.84 -1.79 -1.66 -1.74 -1.63 -1.72 -1.81	-1.19 -1.07 -1.27 -1.13 -1.43 -1.15 -1.18 -1.75 -1.70 -1.62 -1.66 -1.56 -1.65 -1.75	0.66 0.42 0.67 0.45 0.83 irr 0.60 irr ; 0.32 irr 0.58 irr ; 0.36 irr 0.63 irr ; 0.32 irr 0.63 irr ; 0.30 irr 0.74 irr ; 0.39 irr 0.64; 0.36 irr 0.65; 0.40 irr 0.63; 0.30 irr 0.57 irr	0.56 0.37 0.59 0.39 0.72 0.23 0.25 0.21 0.21 0.29 0.25 0.30 0.25 0.40	-5.36 -5.17 -5.39 -5.19 -5.52 -5.03 -5.05 -5.01 -5.01 -5.09 -5.05 -5.10 -5.05 -5.20	-3.61 -3.73 -3.53 -3.67 -3.37 -3.65 -3.62 -3.05 -3.10 -3.18 -3.14 -3.24 -3.15 -3.05	1.75 1.44 1.86 1.52 1.82 1.38 1.43 1.96 1.91 1.91 1.91 1.88 1.90 2.15	2.13 1.87 2.22 1.97 2.47 2.50 2.52 3.01 2.52 2.98 2.65 2.56 2.64 2.92	2.14 1.91 2.20 1.97 3.04 3.17 3.18 3.18 3.19 3.19 2.90 2.80 2.87 3.63
	a Onset reduction potential versus ferrocene (E Red(onset) vs. Fc). b Onset oxidation potential versus ferrocene (E Ox(onset) vs. Fc).

characteristics of PP1-PP13 and values of the electrochemical, optical and theoretical bandgaps of PP1-PP13.

Dyes E red (V) E red(onset a (V) E ox (V) E ox(onset) b (V) E HOMO c (eV) E LUMO c (eV) ΔE El (eV) ΔE opt (eV) ΔE th (eV) c E

  H NMR (400 MHz, CDCl 3 ) δ: 3.85 (s, 2H), 7.79 (dd, 2H, J = 6.2 Hz, J = 3.2 Hz), 8.07-8.19 (m, 2H), 8.49 (s, 1H), 9.19 (s, 1H); 13 C NMR (100 MHz, CDCl 3 ) δ: 44.6, 112.3, 112.6, 125.8, 128.1, 130.5, 130.6, 130.7, 130.9, 135.8, 136.3, 136.4, 166.5, 195.3; HRMS (ESI MS) m/z: theor: 244.0637 found: 244.0640 ([M] +. detected).

  1 H NMR (400 MHz, CDCl 3 ) δ: 3.12 (s, 6H), 6.70 (d, 2H, J = 8.9 Hz), 7.38 (d, 1H, J = 14.7 Hz), 7.62 (d, 2H, J = 8.9 Hz), 7.61-7.71 (m, 2H), 7.84-7.87 (m, 1H), 8.50 (d, 1H, J = 11.9 Hz), 8.61-8.68 (m, 2H); 13 C NMR (100 MHz, CDCl 3 ) δ: 40.1, 67.5, 112.1, 115.1, 115.2, 120.2, 122.5, 123.3, 123.9, 125.1, 130.4, 132.5, 133.9, 134.5, 137.3, 139.9, 149.0, 153.3, 156.1, 160.2, 189.7; HRMS (ESI MS) m/z: theor: 352.1444 found: 352.1441 ([M+H] + detected).

  J = 8.5 Hz), 7.40-7.48 (m, 2H), 7.71 (s, 1H), 7.77 (d, 1H, J = 7.6 Hz), 7.82 (d, 1H, J = 7.6 Hz), 8.18 (s, 1H); 13 C NMR (100 MHz, CDCl 3 ): 25.4, 40.2, 40.4, 49.8, 54.2, 80.2, 110.9, 112.4, 120.5, 120.9, 123.5, 124.1, 127.2, 127.9, 128.4, 129.5, 130.1, 131.9, 133.3, 133.7, 134.8, 136.2, 153.6, 157.3, 160.1, 160.2, 187.2; HRMS (ESI MS) m/z: theor: 461.2336 found: 461.2333 ([M+H] + detected).

	2H), 3.62-3.68 (m, 2H), 5.57-5.60 (m, 2H), 6.67 (d, 2H, J = 8.5 Hz), 7.25 (d, 2H, 5.3.2. 1-(4-(Dimethylamino)phenyl)-3-morpholino-11-oxo-2,11-dihydro-1H-benzo[5,6]indeno[2,1-c]pyridine-4-carbonitrile PP7 Yield = 79%. 1 H NMR (400 MHz, CDCl 3 ) δ: 2.92 (s, 6H), 3.47-3.52 (m, 4H), 3.72-3.77 (m, 4H), 5.42 (brs, 1H), 5.63 (d, 1H, J = 3.9 Hz), 6.68 (d, 2H, J = 8.5 Hz), 7.25 (d, 2H, J = 8.5 Hz), 7.44-7.52 (m, 2H), 7.74 (s, 1H), 7.79 (d, 1H, J = 7.8 Hz), 7.84 (d, 1H, J = 7.8 Hz), 8.16 (s, 1H); 1 H NMR (400 MHz, DMSO-d 6 ) δ: 2.85 (s, 6H), 3.44-3.54 (m, 2H), 3.69-3.78 (m, 6H), 5.49 (s, 1H), 6.68 (d, 2H, J = 8.9 Hz), 7.12 (d, 2H, J = 8.9 Hz), 7.50-7.58 (m, 2H), 7.78 (s, 1H), 7.89 (d, 1H, J = 8.6 Hz), 7.98 (d, 1H, J = 8.6 Hz), 8.04 (s, 1H), 8.97 (s, 1H); 13 C NMR (100 MHz, DMSO-d 6 ) δ: 48.0, 49.1, 51.5, 59.3, 65.9, 112.4, 117.5, 119.0, 119.8, 120.2, 127.2, 128.0, 129.0, 129.8, 129.9, 131.9, 133.3, 133.6, 134.4, 135.3, 149.9, 153.3, 160.2, 185.9; HRMS (ESI MS) m/z: theor: 463.2129 found: 463.2126 ([M+H] + detected).
	pyridine-4-carbonitrile PP6 Yield = 14%. 1 H NMR (400 MHz, Acetone-d 6 ) δ: 1.68-1.80 (m, 6H), 2.90 (s, 6H), 3.58-3.87 (m, 4H), 5.59 (d, 2H, J = 4.8 Hz), 6.71 (d, 2H, J = 8.8 Hz), 7.23 (d, 2H, J = 8.8 Hz), 7.48-7.61 (m, 2H), 7.74 (s, 1H), 7.93 (dd, 2H, J = 13.7 Hz, J = , 7.5 Hz), 8.20 (s, 1H); 1 H NMR (400 MHz, CDCl 3 ) δ: 1.72-1.75 (m, 6H), 2.90 (s, 6H), 3.50-3.57 (m,

  .07 (s, 6H), 3.82 (t, 4H, J = 7.7 Hz), 4.26-4.20 (m, 1H), 6.76 (d, 2H, . Pigot et al. Dyes and Pigments xxx (xxxx) xxx-xxx J = 9.0 Hz), 7.53-7.58 (m, 2H), 7.91-7.98 (m, 2H), 8.04 (d, 2H, J = 9.0 Hz), 8.16 (s, 1H), 8.89 (s, 1H); 13 C NMR (100 MHz, CDCl 3 ) δ: 14.1, 22.6, 26.8, 28.6, 29.3, 29.4, 29.7, 31.8, 40.1, 51.2, 110.6, 116.5, 118.7, 123.7, 124.1, 124.2, 128.0, 128.5, 129.4, 130.0, 131.9, 133.6, 133.8, 134.6, 136.0, 152.2, 158.2, 159.6, 160.5, 187.1; HRMS (ESI MS) m/z: theor: 615.8855 found: 615.8853 ([M+H] + detected).

	P R O O F
	U N C O R R E C T E D

C

  pyridine-4-carbonitrile PP11 Yield = 18%.1 H NMR (400 MHz, CDCl 3 ) δ: 2.06-2.10 (m, 4H), 3.05 (s, 6H), 4.04-4.06 (m, 4H), 6.71 (d, 2H, J = 8.8 Hz), 7.56-7.60 (m, 2H), 7.62 (d, 2H, J = 8.8 Hz), 7.94-8.01 (m, 2H), 8.03 (s, 1H), 8.17-8.22 (m, 2H), 8.79 (s, 1H); 13 C NMR (100 MHz, CDCl 3 ) δ: 25.5, 40.2, 49.6, 81.1, 111.9, 116.3, 118.0, 118.8, 123.8, 123.9, 124.4, 128.0, 128.4, 130.0, 130.2, 133.7, 134.1, 134.6, 136.0, 139.8, 151.3, 156.5, 188.5; HRMS (ESI MS) m/z: theor: 471.2179 found: 471.2181 ([M+H] + detected).

  pyridine-4-carbonitrile PP12 Yield = 6%.1 H NMR (400 MHz, CDCl 3 ) δ: 3.05 (s, 6H), 3.93-4.00 (m, 8H), 6.70 (d, 2H, J = 8.6 Hz), 7.56-7.61 (m, 4H), 7.93-7.98 (m, 3H), 8.14 (d, 1H, J = 15.5 Hz), 8.17 (s, 1H), 8.69 (s, 1H); 13 C NMR (100 MHz, CDCl 3 ) δ: 40.2, 48.6, 48.8, 84.0, 110.8, 111.9, 117.5, 117.8, 118.0, 123.9, 124.0, 124.6, 128.3, 128.9, 130.1, 130.2, 130.4, 132.1, 133.5, 133.6, 134.7, 136.0, 140.8, 151.6, 156.1, 188.9; HRMS (ESI MS) m/z: theor: 487.2129 found: 487.2127 ([M+H] + detected).

  C NMR (400 MHz, CDCl 3 ) δ: 13.8, 29.7, 40.2, 45.2, 112.0, 116.6, 118.1, 118.7, 124.0, 124.1, 124.3, 128.1, 128.6, 130.0, 130.1, 130.2, 132.1, 133.0, 133.1, 133.8, 134.1, 134.6, 136.0, 140.0, 151.4, 156.2, 188.2; HRMS (ESI MS) m/z: theor: 473.2336 found: 473.2339 ([M+H] + detected).

Acknowledgements

The authors thank Aix Marseille University and The Centre National de la Recherche (CNRS) for financial supports. The Agence Nationale de la Recherche (ANR agency) is acknowledged for its financial support through the PhD grants of Corentin Pigot (ANR-17-CE08-0010 DUALITY project) and Guillaume Noirbent (ANR-17-CE08-0054 VISICAT project). Said Aboudou is acknowledged as Master student from the Master Nanosciences and Nanotechnologies of Aix Marseille University for its contribution to the photophysical characterizations of the dyes.

Declaration of competing interest

The authors declare no conflict of interest.

Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi. org/10.1016/j.dyepig.2020.108182.