S Gratton
email: serge.gratton@enseeiht.fr

N Soualmi
email: nacer.soualmi@cerfacs.fr

L N Vicente

researchers and makes it freely available over the web where possible

Keywords:

Gradient methods Complexity and global rates a b s t r a c t

In some optimization problems found in applications, the derivatives of the objective function can be computed or approximated but at an expensive cost, and it is desirable to know when to use derivative-free methods (such as direct search, for instance) or derivative-based methods (such as gradient or quasi-Newton methods). Derivative-free methods may achieve a steady initial progress for some problems, but after some advance they may also become slower or even stagnate due to the lack of derivatives. It is thus of interest to provide a way to appropriately switch from a derivative-free method to a derivative-based one. In this paper, we develop a family of indicators for such a switch based on the decrease properties of both classes of methods (typically used when deriving worst case complexity bounds).

Introduction and basic concepts

The calculation of functions involved in optimization problems appearing in computational sciences and engineering is frequently based on numerical simulation. The smoothness of the function and the access to whatever form of derivatives vary considerably across applications. While there are problems of totally black-box type where only function values can be computed in a certain (sometimes unknown) feasible region, there are other more structured problems of continuously differentiable type where both function values and gradients can be computed, an example of special interest to us being the acoustic full-waveform inverse problem in Earth imaging [START_REF] Virieux | An overview of full-waveform inversion in exploration Geophysics[END_REF]. In such problems, the calculation of the gradient may come at a cost higher than the one for the function value, and such difference may depend on the dimension of the problem and the accuracy required.

Although Derivative-Free Optimization provides now a theory [START_REF] Conn | Introduction to Derivative-Free Optimization[END_REF] to understand models and families of directions used in the various classes of methods as well as their convergence properties, a question that to our knowledge has never been addressed is when should one switch from a derivative-free method to a derivativebased one, when the gradient can be computed (or possibly approximated by finite differences). Such a general question can be posed in many different ways depending on the methods under consideration, their costs per iteration, the computational budget available, and the final accuracy desired for a solution. Other issues like non-smoothness or local vs global optimization may also play a relevant role when analyzing such an issue.

In this paper, we try to make a first contribution to the topic by considering a continuously differentiable setting where the gradient of the objective function f : R n → R can be computed (or possibly approximated by finite differences), and the minimization of the function is unconstrained. The gradient of f will be considered Lipschitz continuous in R n (or in a level set corresponding to an initial iterate), with constant L ∇f > 0. Our idea to develop an indicator for the switch under consideration will be to form appropriate ratios of lower bounds for the decreases attained in successful iterations (involving the derivative-free method and what would be expected for the derivative-based one). These lower bounds will be the ones used when deriving worst case complexity bounds (WCC) or global rates for such methods, and we will try to take advantage of the mismatches that appear in such bounds with or without using derivatives. Our ultimate goal is to detect a switch when not enough progress is being achieved compared to the one that could be done if derivatives were available, letting the derivative-free method continue otherwise (meaning to do not switch).

To illustrate and test our ideas, we will consider the following simple direct-search method which imposes a sufficient decrease condition based on a quadratic forcing function. We consider an iteration uniquely defined by a poll step which evaluates the objective function using a positive spanning set (PSS), i.e., a set of nonzero vectors that spans R n with non-negative coefficients.

Algorithm 1.1. Direct-search method (polling)

Initialization: Choose a PSS D, an initial point x 0 , and an initial step size α 0 > 0. The constants 0 < β < 1 ≤ γ are specified. Set k = 0. 1. Poll step: Order the set of poll points

P k = {x k + α k d : d ∈ D}.
Start evaluating f at the poll points following the chosen order.

If a poll point

x k + α k d is found such that f (x k + α k d k ) < f (x k ) -α 2 k /2, then set x k+1 = x k + α k d k
and declare the iteration successful. Otherwise, declare the iteration unsuccessful and set

x k+1 = x k .
2. Update iterate and step size: If the iteration was successful, then maintain or increase the step size parameter:

α k+1 ∈ [α k , γ α k ].
Otherwise, decrease the step size parameter α k+1 = βα k . Increment k by one and go to Step 1.

It is well known that such an algorithm is well defined (for functions with Lipschitz continuous gradients) in the sense that a successful iteration is found in a finite number of step-size reductions [START_REF] Kolda | Optimization by direct search: New perspectives on some classical and modern methods[END_REF]. In fact, it is possible to prove [START_REF] Kolda | Optimization by direct search: New perspectives on some classical and modern methods[END_REF] that if the iteration k is unsuccessful, then

∇f (x k ) ≤ cm(D) -1 L ∇f max d∈D d 2 + 1 2 min d∈D d α k , (1)
where cm(D) is the cosine measure of the PSS D, defined as

cm(D) = min 0 =v∈R n max d∈D v ⊤ d v d .
The cosine measure of a PSS is always positive. For instance, the PSS D ⊕ formed by the coordinate vectors and their negatives is such that cm(D ⊕ ) = 1/ √ n, and so is QD ⊕ where Q is an orthogonal matrix. Given an ǫ ∈ (0, 1), it is known that such an algorithm takes at most O(nǫ -2 ) iterations and O(n 2 ǫ -2 ) function evaluations to drive the norm of the gradient below ǫ (see [START_REF] Vicente | Worst case complexity of direct search[END_REF]). 1 The dependence of these WCC bounds on ǫ reduces to ǫ -1 if the function is convex and tolog(ǫ) if the function is strongly convex (see [START_REF] Dodangeh | Worst case complexity of direct search under convexity[END_REF]). These bounds depend quadratically on the Lipschitz constant L ∇f .

For the sake of simplicity, we take the gradient method with backtracking as our derivative-based method.

Algorithm 1.2. Gradient method (backtracking)

Initialization: Choose initial point x 0 . Let c ∈ (0, 1) and b > 0 be specified. Set k = 0. 1. Backtrack: Let α k be the first scalar in b, b/2, b/4, . . . such that

f (x k -α k ∇f (x k )) ≤ f (x k ) -cα k ∇f (x k ) 2 .
(2) It is known that Algorithm 1.2 is well defined in the sense that it is always possible to find α k of the form given in the algorithm such that (2) is satisfied (see, e.g., [START_REF] Nocedal | Numerical Optimization[END_REF]). Moreover, each iteration of Algorithm 1.2 satisfies

Update iterate: Compute

x k+1 = x k -α k ∇f (x k ). Increment k
f (x k ) -f (x k+1 ) ≥ C ∇f (x k ) 2 , (3)
with

C = c max 1 -c L ∇f , b . (4)
1 The notation O(A) will mean a scalar times A, where the scalar does not depend on the iteration counter of the method under analysis (thus depending only on the problem or on algorithmic constants).

It is also well known (see [START_REF] Nesterov | Introductory Lectures on Convex Optimization[END_REF]) that the WCC effort for the gradient method (to reduce the norm of the gradient below ǫ)

is of O(ǫ -2 )
in general, reducing to O(ǫ -1 ) and O(-log(ǫ)) in the convex and strongly convex cases, respectively. Note that these bounds depend linearly on L ∇f . The remaining of this paper is organized in three sections. In Section 2, we will describe our main idea to develop a family of indicators for the switch from derivative-free to derivativebased iterations using direct search and the gradient method as motivation. Two concrete indicators are then proposed in Section 3 and their numerical performance (using Algorithms 1.1 and 1.2) is reported. Finally, in Section 4, we will further discuss the scope of our approach.

Elements for the indicators

At each iteration of a gradient-based method, one typically has

f (x k ) -f (x k+1 ) ≥ G L ∇f ∇f (x k ) 2 , ( 5
)
where G > 0 is a fixed constant independent of f or of the iteration counter. This is the case for the gradient method with line search satisfying both Wolfe conditions (sufficient decrease condition (2) and curvature condition), or for the gradient method with backtracking line search imposing only the sufficient decrease condition (2); (see Algorithm 1.2 and (3)-( 4)).

At each successful iteration of a derivative-free method based on sufficient decrease, one typically has

f (x k ) -f (x k+1 ) ≥ D 1 t 2
k , where D 1 > 0 is a fixed algorithmic parameter independent of f and of the iteration counter. The step-size parameter t k represents the trust-region radius δ k in derivative-free trust-region methods or the step size α k in direct-search methods (see Algorithm 1.1). If ℓ is an iteration where the step-size parameter t ℓ is reduced (for either class of methods), one has

∇f (x ℓ ) ≤ D 2 (n)L ∇f t ℓ , (6)
where D 2 (n) > 0 is a fixed constant independent of f and of the iteration counter (but typically dependent on n); see (1) for the direct-search case and [START_REF] Garmanjani | Trust-region methods without using derivatives: Worst case complexity and the non-smooth case[END_REF] for the trust-region one.

Let k be a given successful iteration and r k the last iteration before k where the step size has been reduced. Let C k be the set of indices corresponding to successful iterations between r k and k where some approximation to the gradient is known.

The decrease produced by a gradient-based method would have been at least

f (r k ) -f (x k+1 ) ≥ j∈C k G L ∇f ∇f (x j ) 2 . (7)
On the other hand, the decrease produced by a derivative-free method would have been at least

f (r k ) -f (x k+1 ) ≥ j∈C k D 1 t 2 j ≥ D 1 |C k | D 2 (n) 2 L 2 ∇f ∇f (x r k ) 2 . ( 8
)
Establishing a ratio between the decreases in [START_REF] Nesterov | Introductory Lectures on Convex Optimization[END_REF] with those in [START_REF] Kolda | Optimization by direct search: New perspectives on some classical and modern methods[END_REF], yields two quantities

D 1 L ∇f j∈C k t 2 j G j∈C k ∇f (x j ) 2 and D 1 |C k | ∇f (x r k ) 2 GD 2 (n) 2 L ∇f j∈C k ∇f (x j ) 2 .
This motivates the introduction of the following two indicators

I 1 k = j∈C k t 2 j j∈C k g j 2 and I 2 k = |C k | g r k 2 j∈C k g j 2 (9)
for the switch from the derivative-free regime to the derivativebased one, where g i represents an approximation for ∇f (x i ). Such approximations can be computed based on previous evaluations of the objective function (or in a last resource using finite differences), as we explain later in the paper. These indicators should be in principle increasing in k. In the I 1 case, this is because the gradient is expected to go to zero faster than the step size (see [START_REF] Griewank | On automatic differentiation[END_REF]). In the I 2 case, this is because the gradient in the derivative case is expected to go to zero faster than in the derivative-free one. A possible switch is therefore when such indicators become larger than a specified positive threshold.

Note that we omitted the constant D 2 (n) 2 in ( 9) but such a constant could have been kept in the indicator I 2 k since it scales with n and thus provides some relevant information. Also, we have omitted the unknown Lipschitz constant, although it should be noticed that this constant appears differently in both indicators.

Two concrete indicators and their numerical performance

A number of issues have to be addressed to make such indicators of practical use.

Two concrete indicators

First, a switch based on the size of an indicator has to be made relatively to its scaling. So, one has to compute an initial indicator of reference I scaling (see Algorithm 3.2) that can be then used as a scaling factor to determine the moment of the switch. Such a switch will occur when I k /I scaling ≥ C threshold , for a certain positive threshold C threshold (see Algorithm 3.1).

Our first concrete indicator is solely based on I 1 k in [START_REF] Nocedal | Numerical Optimization[END_REF]. The rationale here is that I 1 k is derived from a tighter bound in [START_REF] Nesterov | Introductory Lectures on Convex Optimization[END_REF]. The computation of the indicator and of the scaling factor I scaling for this pure I 1 case is given in Algorithms 3.1 and 3.2, respectively. In the definition of I 1 k one takes into account all successful iterations from k until the last unsuccessful one r k . Hence, one has to take provision for the fact that one can make a long series of successful iterations, and that such an event may occur from the initial iteration. Algorithm 3.3 (pure I 1 ) takes these aspects into consideration by using a maximum prespecified number E of successful iterations for backtracking from k.

We assume that we are using Direct Search (Algorithm 1.1) as our derivative-free method (DF-Method) and the Gradient Method (Algorithm 1.2) as our derivative-based one (DB-Method), but the presentation is sufficiently abstract to be independent of the specifics of these methods as long as they conform with the presentation of Section 2. After the algorithm descriptions of the indicators, we discuss the calculation of the gradient approximations mentioned there. Set

r k = index of last previous unsuccess (r k = -∞ if it does not exist). Set C k = {max(r k + 1, k -E), . . . , k}. Set I k = j∈C k t 2 j j∈C k g j 2 . if I k I scaling
E scaling are successful). Set C k = {r k + 1, . . . , E scaling -1}. if C k = ∅ then Set I scaling = j∈C k t 2 j j∈C k g j 2 .
else Run DF-Method until a successful iteration k s is performed.

Set

I scaling = t 2 ks g ks

. end if

In our second concrete indicator (Algorithm 3.3), we use the indicator I 2 k in (9) as much as possible as it relates comparable information with or without using derivatives. However, such an indicator requires explicitly a previous unsuccessful iteration. Also here, one has to take provision for the fact that one can make a long series of successful iterations, and that such an event may occur from the initial iteration. Algorithm 3.3 (hybrid I 1 /I 2 ) takes these aspects into consideration by using a prespecified number E of successful iterations after which we resort to using I 1 k in (9) instead. Set else if the E scaling -th iteration is successful then Set r k as the index of the last unsuccessful iteration.

r k = index of last previous unsuccess (r k = -∞ if it does not exist). if (k > E and r k = -∞) or (k -r k > E) then Set C k = {k -E, . . . , k}. Set I k = j∈C k t 2 j j∈C k g j 2 . else Set C k = {r k + 1, . . . , k}. Set I k = |C k | gr k 2 j∈C k g j 2 .

end if if

I k I scaling ≥ C
Set

I scaling = |C k | gr k 2 j∈C k g j

. else

Run DF-Method until a successful iteration k s is performed.

Set I scaling = g ks -1

2 g ks

. end if end if

These procedures to calculate indicators for the switch to derivative-based optimization may require the use of approximations to the gradient. There are different ways of estimating the gradient when using a derivative-free method. If one applies a trust-region method, one can use the gradients of the models.

When using a direct-search method as we do in our paper, one can compute simplex gradients based on the previous evaluated points. Given a sample set Y = {y 0 , . . . , y p }, a simplex gradient is nothing else than the gradient g of a linear interpolation or regression model m(y; y 0 ) = f (y 0 ) + g ⊤ (yy 0 ) centered at one of the points y 0 ∈ Y (typically y 0 = x k ). Let the interpolating conditions m(y i ; y 0 ) = f (y i ), i = 1, . . . , p be written as L ⊤ g = δf (Y ) where

L = y 1 -y 0 • • • y n -y 0 ⊤ and δf (Y ) =    f (y 1 ) -f (y 0 )
. . .

f (y n ) -f (y 0 )    .
Assume that the columns of L are linearly independent. Then, the simplex gradient is computed as

g = ∇ s f (y 0 ) = L -1 δf (Y ) if p = n (determined case) or as g = ∇ s f (y 0 ) = L - † δf (Y ) if p > n (regression/overdetermined case)
, where L † denotes the left inverse of A. Finally, a practical way to determine a set Y of previously evaluated points in the context of direct search is specified in the next section.

A numerical illustration

We have run Algorithm 1.1 (Direct Search) and Algorithm 1.2 (Gradient Method) on twenty problems from the CUTEr collection [START_REF] Gould | CUTEr, a constrained and unconstrained testing environment, revisited[END_REF] of dimension n = 20. Algorithm 1.1 was applied with β = 1/2, γ = 1, and D = D ⊕ . The poll step was implemented in a cyclic way, where one starts using a direction in D stored one column after the column corresponding to the last direction used in the previous poll step. Algorithm The sample set used to compute simplex gradients was selected as follows. First, we only computed simplex gradients after having at least n + 1 overall evaluations. If there is a need to compute an approximation to a gradient for indicator purposes when there are less than overall n + 1 evaluations, one applies finite differences; such a situation has rarely occurred but was anyhow taken into consideration when counting the number of function evaluations. At each iteration, the overall sample set of evaluated points is ordered by increasing distance to the current iterate. Then, one selects the sample set for the simplex gradient calculation as all the points within a distance of 5α k of the current iterate. If there are less than n points within this distance, we fill the set using the remaining ones until there are at least n. In such a way, we never compute underdetermined simplex gradients. If there are more than n + 1 points in the sample set, we solve the regression system in the least squares sense. In all cases (determined and overdetermined), the linear systems are solved by means of the QR factorization, and a regularization is applied to the diagonal elements of R, replacing them by sign(R ii )10 -8 when |R ii | < 10 -8 .

The results are shown first in Figs. 1 and2 for the hybrid I 1 /I 2 case which performed better than the pure I 1 one. In all cases, the budget of function evaluations for Direct Search was 500, except for problem penalty1 where we went until 3000. The plots of the function values in Fig. 1 are given in a logarithmic scale (log(ff * )

where f * denotes the optimal value). Table 1 contains the relevant data corresponding to Fig. 1.

Although we report numbers of function and gradient evaluations, we remark that the primary goal of our work is not to develop a switched method more efficient or robust than both the derivative-free and the derivative-based underlying ones. Instead we tried to develop a technique that can identify an appropriated moment for the switch with desirable properties such as (i) not to switch when that can be foreseen as worse; (ii) switch when not enough progress is being achieved compared to the one that could be done if derivatives were available.

One can see that a switch occurred in 15 out of the 20 problems. Among the 5 problems where no switch occurred, in four of these (arglina, arwhead, nondia, and tquartic) direct search did better than gradient descent -a manifestation of a feature that is desirable for the indicators in question. The other one (integreq) is somehow the single failure of our approach in the sense that a switch should have occurred.

In an attempt to better balance the relative costs of a gradient calculation, it is reported in Table 2 the effort of the gradient and switched methods in the hybrid I 1 /I 2 case for the problems for which the curve corresponding to switched one passed the one corresponding to gradient one (in the plots of Fig. 1). The effort is measured in two ways: (i) summing the number of function evaluations with the number of gradient evaluations multiplied by a factor of 5 (to match the case where the gradient is computed from automatic differentiation; see [START_REF] Griewank | On automatic differentiation[END_REF]); (ii) summing the number of function evaluations with the number of gradient evaluations multiplied by a factor of n = 20 (for the case where the gradient is approximated by finite differences). In 8 out of these 10 problems, the effort of the switched method was considerably less than that of the gradient method.

In Table 3, we provide more information about the relative effort of these two methods in the hybrid I 1 /I 2 case for the remaining problems, meaning those not listed in Table 2 and for which there was a switch. Problem dqrtic is an example where depending on the way a gradient evaluation is weighted, it can be worth or not worth to use a switched version when compared to a derivativebased one. For the remaining problems (in particular for sparsine and vardim), it seems that the initial derivative-free phase was harmful. We note that numerical convergence would have been obtained for these two problems with a very large budget (see Fig. 4). Similar figures (Figs. 3 and4) and tables (Tables 456) are reported for the pure I 1 case. The major difference compared to the hybrid I 1 /I 2 seems to be a lost of prediction of an appropriate switch. In fact, in the pure I 1 case, there was no switch in two other problems (freuroth and sinquad) where it would be appropriated to do it (see Tables 456).

Final remarks

One way to stop the run of a derivative-free method in the hope to switch to a faster, higher order method would be to look at the behavior in function values. In fact, looking at how the function decreases along the iterations would provide heuristic indicators to such a switch. However, stagnation in function values could occur for a number of reasons not necessarily because of the lack of derivatives. In this paper, we tried to propose a more systematic indicator for this switch grounded on certain theoretical principles related to known decrease properties. Our approach is certainly also of heuristic type as we compare lower bounds on the decreases attained by the derivative-free method to the ones that would be obtained if a derivative-based one would had been used. Note also that the indicators here depend on very few parameters, essentially on C threshold , E, and E threshold . Our experiments have shown us that the results presented here are relatively robust with respect to Fig. 1. Plots of functions values for Direct Search (solid, blue), Gradient Method (dash-dot, red), and Direct Search switched to Gradient Method in the hybrid I 1 /I 2 case (dash, magenta). In the x-axis, we count function evaluations. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Table 1 Detailed results of Fig. 1 for Gradient Method (GM) and the Switched Method (SM) in the hybrid I 1 /I 2 case. Columns 2-3: number of gradient evaluations. Columns 4-5: number of function evaluations. Columns 6-7: final value of f obtained. the changes in these parameters, especially in E and E threshold . The choice of C threshold is understandably more critical and problem dependent but it also gives a user a tool for adjustment. Our approach covers a common situation where the step size decreases in successful iterations in the usage of the derivative-free method, as long as (i) by successful here we mean that the trial point is accepted as the new iterate and (ii) a certain condition is explicitly or implicitly satisfied implying that the gradient at the current point is of the order of the step size. Then, such successful iterations will no longer be considered successful in the language ). The solid (blue) line is plotted for better visualization. The (black) vertical line signals the moment of the switch. Note that in the x-axis we count iterations, not function evaluations, and that all indicator values above 100 were plotted using this value for better visualization. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Table 2 It is reported the effort of the Gradient Method (GM) and the Switched Method (SM) in the hybrid I 1 /I 2 case for the problems for which the curve corresponding to the SM passes the GM one in the plots of Fig. of our proposed approach and will be part of those where the step size is reduced.

#∇f GM #∇f SM #f GM #f SM f GM f SM
In particular, the indicators proposed in this paper for the switch from a derivative-free method to a derivative-based one can also be used if the former is a trust-region method. One would not count in C k iterations where the point and the trust-region radius do not change, like in a model-improvement iteration or in a critical iteration of that type. On the other hand, r k may correspond to an unsuccessful iteration, an acceptable iteration

Table 5

It is reported the effort of the Gradient Method (GM) and the Switched Method (SM) in the pure I 1 case for the problems for which the curve corresponding to the SM passes the GM one in the plots of Fig. (where the ratio between actual and predicted reductions is positive but too small), or a critical iteration of the other type, where in all cases the trust-region radius is reduced (see the details in [START_REF] Garmanjani | Trust-region methods without using derivatives: Worst case complexity and the non-smooth case[END_REF]).

As for the derivative-based method, one can use any gradientbased algorithm, of first or second order type, for which the decrease in functions values can be estimated as a function of the size of the gradient. The derivation of complexity bounds for derivative-based trust-region methods (as done in [START_REF] Gratton | Recursive trust-region methods for multiscale nonlinear optimization[END_REF]) is also based on a condition like [START_REF] Gratton | Recursive trust-region methods for multiscale nonlinear optimization[END_REF].

Table 6

It is given more information about the Gradient Method (GM) and the Switched Method (SM) in the pure I 1 case for the remaining problems (those not listed in Table 5 and for which there was a switch). Columns 2-3: sum of the number of function evaluations with the number of gradient evaluations multiplied by 5. Columns 4-5: sum of the number of function evaluations with the number of gradient evaluations multiplied by n = 20. Columns 6-7: final value of f obtained.

+5

  by one and go to Step 1.

Algorithm 3 . 1 .

 31 Procedure for computing the (pure) indicator I 1 at a successful iteration k ≥ E. It is assumed that the kth iteration of the DF-Method has been performed and was successful. It requires the constants E, I scaling , and C threshold .

≥Algorithm 3 . 2 .

 32 C threshold then Switch to the DB-Method. else Continue the calculation of the indicator in the DF-Method. end if Now, we present a possible way to compute the scaling factor I scaling . Procedure for computing the scaling factor I scaling for the (pure) indicator I 1 . It is assumed that the first E scaling (≤ E) iterations of the DF-Method have been run. Set r k = the last unsuccessful iteration (r k = -1 if all the first

Algorithm 3 . 3 .

 33 Procedure for computing the hybrid indicator I 1 /I 2 at a successful iteration k ≥ E. It is assumed that the kth iteration of the DF-Method has been performed and was successful. It requires the constants E, I scaling , and C threshold .

Algorithm 3 . 4 .

 34 threshold then Switch to the DB-Method. else Continue the calculation of the indicator in the DF-Method. end if Procedure for computing the scaling factor I scaling for the hybrid indicator I 1 /I 2 . It is assumed that the first E scaling (≤ E) iterations of the DF-Method have been run. if the first E scaling iterations are successful then Set C k = {0, . . . , E scaling -1}. Set I scaling =

  1.2 was applied with c = 10 -4 and b = 1. When running Direct Search, we applied Algorithms 3.3 and 3.4 to compute the indicator and trigger the moment of the switch to the Gradient Method, with E = 20 and E scaling = 5. The threshold was set to C threshold = 2. In the hybrid I 1 /I 2 case, we have multiplied I 1 k by 100 which has an effect equivalent to dividing I 2 k by D 2 (n) 2 = 20.

Fig. 2 . 1 k

 21 Fig. 2. Plots of the indicator values I k /I scaling in the hybrid I 1 /I 2 case when they were calculated (points, in green or lighter for I 1 k and in red or darker for I 2 k). The solid (blue) line is plotted for better visualization. The (black) vertical line signals the moment of the switch. Note that in the x-axis we count iterations, not function evaluations, and that all indicator values above 100 were plotted using this value for better visualization. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

1 .

 1 Columns 2-3: sum of the number of function evaluations with the number of gradient evaluations multiplied by 5. Columns 4-5: sum of the number of function evaluations with the number of gradient evaluations multiplied by n = 20. Columns 6-7: value of f at the pass or right after it.

Fig. 3 .

 3 Fig. 3.Plots of functions values for Direct Search (solid, blue), Gradient Method (dash-dot, red), and Direct Search switched to Gradient Method in the pure I 1 case (dash, magenta). In the x-axis we count function evaluations. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 4 .

 4 Fig.4. Plots of the indicator values I k /I scaling in the pure I 1 case when they were calculated (points, in green). The solid (blue) line is plotted for better visualization. The (black) vertical line signals the moment of the switch. Note that in the x-axis we count iterations, not function evaluations, and that all indicator values above 100 were plotted using this value for better visualization. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

3 .

 3 Columns 2-3: sum of the number of function evaluations with the number of gradient evaluations multiplied by 5. Columns 4-5: sum of the number of function evaluations with the number of gradient evaluations multiplied by n = 20. Columns 6-7: value of f at the pass or right after it.

Table 3

 3 It is given more information about the Gradient Method (GM) and the Switched Method (SM) in the hybrid I 1 /I 2 case for the remaining problems (those not listed in Table2and for which there was a switch). Columns 2-3: sum of the number of function evaluations with the number of gradient evaluations multiplied by 5. Columns 4-5: sum of the number of function evaluations with the number of gradient evaluations multiplied by n = 20. Columns 6-7: final value of f

	arglinb	+5 GM 581	+5 SM 301	+20 GM 956	+20 SM 526	f GM 4.634146	f SM 4.634146
	arglinc	447	321	792	471	6.135135	6.135135
	bdqrtic	766	161	1786	311	35.410112	35.409468
	broydn3d	545	904	1070	2419	0.000000	0.000000
	engval1	623	225	1178	420	20.278662	20.278662
	freuroth	844	183	2149	228	2696.789277	2679.938911
	nondquar	804	796	1959	2341	0.009174	0.009108
	sinquad	769	633	1804	1713	0.189190	0.187714
	sparsqur	60	114	165	234	0.000000	0.000000
	powellsg	918	704	2493	2009	0.636923	0.633652
	obtained.
	dqrtic	+5 GM 309	+5 SM 576	+20 GM 1059	+20 SM 861	f GM 0.000000	f SM 0.000000
	penalty1	2421	6408	3861	19188	0.000163	0.000231
	penalty2	677	1088	1367	3293	0.008987	0.009266
	sparsine	759	1032	1734	3027	0.000227	1.100321
	vardim	589	912	934	2457	0.000000	7.714170

Table 4

 4 Detailed results of Fig.3for the Gradient Method (GM) and the Switched Method (SM) in the pure I 1 case. Columns 2-3: number of gradient evaluations. Columns 4-5: number of function evaluations. Columns 6-7: final value of f obtained.

		#∇f GM	#∇f SM	#f GM	#f SM	f GM	f SM
	arglina	14	0	59	81	0.0000	0.0000
	arglinb	25	15	456	226	4.6341	4.6341
	arglinc	23	10	332	271	6.1351	6.1351
	arwhead	14	0	157	39	0.0000	0.0000
	bdqrtic	68	10	426	111	35.4101	35.4091
	broydn3d	35	39	370	461	0.0000	0.0000
	dqrtic	50	19	59	481	0.0000	0.0000
	engval1	37	13	438	160	20.2787	20.2787
	freuroth	87	0	409	500	2696.7893	2706.5165
	integreq	8	0	66	500	0.0000	0.0001
	nondia	29	0	459	37	1.9242	1.0000
	nondquar	77	118	419	382	0.0092	0.0152
	penalty1	96	852	1941	2148	0.0002	0.0002
	penalty2	46	129	447	335	0.0090	0.0088
	sinquad	69	0	424	500	0.1892	0.2890
	sparsine	65	133	434	367	0.0002	1.1003
	sparsqur	7	1	25	81	0.0000	0.0000
	tquartic	7	0	66	83	0.6338	0.6328
	vardim	23	103	474	397	0.0000	7.7142
	powellsg	105	87	393	269	0.6369	0.0581

Acknowledgment

Support for the third author was partially provided by FCT under grants UID/MAT/00324/2013, SFRH/BSAB/114622/2016, and P2020 SAICTPAC/0011/2015.


A Jebnouni

M Chemli

P Lévêque

S Fall

M Majdoub

N Leclerc

Effects of vinylene and azomethine bridges on optical, theoretical electronic structure and electrical properties of new anthracene and carbazole based ߨ ߨ ߨ ߨ-conjugated molecules

Keywords: Anthracene, Carbazole, ߨ-linkers, DFT calculations, Photo-induced electron transfer (PET), OFET

This study aims to synthesize and characterize a new semi-conducting 2,7-distyrylcarbazolebased molecule (M1) and its Schiff-base analog (M2). Those two molecules were synthesized via Wittig-Horner and diamine condensation reactions, respectively. The effects of replacing the vinylene (-CH=CH-) linkage between carbazole donor unit and anthracene acceptor unit in molecule M1 by an azomethine (-CH=N-) bridge in molecule M2 on physicochemical and electrical properties are reported in this work. Thermal characterization shows a semi-crystalline morphology in solid state for the two organic materials and a good thermal stability up to 370°C in the case of M2. The observed non-fluorescence character of the Schiff-base molecule has been related to the photo-induced electron transfer (PET) process from the imine receptor moiety to the anthracene fluorophore unit. The effect of the addition of trifluoroacetic acid (TFA) on optical properties of the Schiff base molecule revealed a significant blue shift of the absorption spectrum and a turn-on of the fluorescence as well as visual changes. Computational analyses by Density Functional Theory (DFT) and time-dependent (TD-DFT) were then used to obtain conformation and the energy distributions of the molecular orbitals of the two compounds. The electrochemical behaviors were investigated by cyclic voltammetry and showed a reduction of the band gap of M2 as compared to M1 du to the electron-withdrawing effect the azomethine (-CH=N-) bond. Finally, the chargecarrier mobility of the two compounds were extracted from single-layer diodes in the space-charge limited current regime (SCLC) and organic field effect transistors (OFETs) demonstrating lower hole mobilities in the case of M2 based thin film devices. The hole effective mobility of M2

Introduction

In the last decades, numerous organic semi-conducting materials have been designed, synthesized and characterized for optoelectronic applications. Those materials are based on the combination of aromatic or quinoïdal building blocks such as benzene, [1] thiophene, [START_REF] Leclerc | Electrical and optical properties of Processable Polythiophene Derivatives: Structure-Property relationships[END_REF] pyrrole, [START_REF] Watanabe | Electronic properties of polypyrrole/n-Si heterojunctions and polypyrrole/metal contacts[END_REF] aniline, [START_REF] D'aprano | Macromolecules Stabilization and characterization of pernigraniline salt: the "acid-doped" form of fully oxidized polyanilines[END_REF][START_REF] Macdiarmid | [END_REF] fluorene, [START_REF] Leclerc | Polyfluorenes: Twenty years of progress[END_REF][START_REF] Yang | Efficient blue light-emitting diodes from a soluble poly (paraphenylene) internal field emission measurement of the energy gap in semi-conducting polymers[END_REF][START_REF] Ranger | New Well-Defined Poly(2,7-fluorene) Derivatives: Photoluminescence and Base Doping[END_REF][START_REF] Neher | Conjugated Liquid-Crystalline Polymers for Bright Blue Emission and Polarized[END_REF] anthracene [START_REF] Park | Synthesis of conjugated polymers containing anthracene moiety and their electro-optical properties[END_REF], benzothiadiazole [START_REF] Bulut | LUMO's modulation by electron withdrawing unit modification in amorphous TAT dumbbell-shaped molecules[END_REF][START_REF] Brenno | 2,1,3-Benzothiadiazole and Derivatives: Synthesis, Properties, Reactions, and Applications in Light Technology of Small Molecules[END_REF], thieno [3,2-b]thiophene [START_REF] Biniek | 3,6dialkylthieno[3,2-b]thiophene moiety as a soluble and electron donating unit preserving the coplanarity of photovoltaic low band-gap copolymers[END_REF] and thieno [3,4-b]thiophene [START_REF] Liang | Highly Efficient Solar Cell Polymers Developed via Fine-Tuning of Structural and Electronic Properties[END_REF] as well as carbazole [START_REF] Grazulevicius | Carbazole-containing polymers: synthesis, properties and applications[END_REF] leading to an infinite variety of organic semi-conducting materials. As a consequence, organic optoelectronic devices have known a fast increase in performances. However, most of the organic conjugated materials suffer from a high degree of complexity that makes them expensive and difficult to elaborate in a large scale.

Consequently, new approaches focusing on simple, cost-effective and easily scalable semiconducting materials have emerged recently [16]. Owing to their well-defined chemical structure, small molecules are ideal candidates in this quest for simplicity and reproducibility [START_REF] Roncali | Molecular Materials for Organic Photovoltaics: Small is Beautiful[END_REF].

In line with this idea, two simple and highly efficient blue emitting molecules (M a [START_REF] Blanchard | Highly efficient deep-blue emitting organic light emitting diode based on the multifunctional fluorescent molecule comprising covalently bonded carbazole and anthracene moieties[END_REF] and M b [START_REF] Zhang | Novel carbazole/anthracene hybrids for efficient blue organic lightemittin) (iodes[END_REF]) have been synthesized recently by attaching anthracene endcaps respectively at the 3,6 and 2,7-positions of a carbazole core (Figure 1).

Anthracene unit has been often used in organic semi-conducting materials as electron donor unit with a high thermal stability and good quantum efficiency in electroluminescence [START_REF] Park | Synthesis of conjugated polymers containing anthracene moiety and their electro-optical pro S.ties[END_REF]. In combination with the carbazole unit, known as an excellent chemical building block for blue light emitting properties and organic solar cells [START_REF] Morin | Blue-Light-Emitting Conjugated Polymers Derived from 2,7-Carbazoles, Jean-François[END_REF][START_REF] Park | Bulk heterojunction solar cells with internal quantum efficiency approaching 100%[END_REF], suitable molecular structures have shown high efficiencies in optoelectronic devices. Despite the expected better physicochemical properties for M b, [23] the optical properties of the two carbazole derivatives are very similar. This is probably due to the direct C-C bond used to connect anthracene residues to the carbazole core with a significant steric hindrance between neighboring conjugated units. Indeed, in addition to the aromatic unit composition, the optoelectronic properties of a specific molecular scaffold are also strongly depending on the nature of the chemical π-linker. The presence of ߨ-bridges between aromatic systems plays an essential role in enhancing thermal, physicochemical, self-assembling and optoelectrical properties of conjugated organic molecules [START_REF] Bulut | Rational Engineering of BODIPY-bridged-Trisindole derivatives for Solar Cell Applications[END_REF]. V. Tamilavan et al. [START_REF] Tamilavan | Park Performances of bis(pyrrolo[3,4-c]pyrrole-1,3-dione)-based wide band gap polymer via the incorporation of an appropriate spacer unit between pyrrolo[3,4-c]pyrrole-1,3-dione units[END_REF] demonstrated that introducing rigid aromatic heterocycle spacers between conjugated units can improve the power conversion efficiencies (PCE) of solar cells. Arthur D. Hendsbee et al [START_REF] Hendsbeea | Unusual loss of electron mobility upon furan for thiophene substitution in a molecular seormickc, or[END_REF] have

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shown that replacing thiophene by furan in a phthalimide-based conjugated molecule leads to a total loss of the electron mobility in organic field effect transistors (OFETs). In the present paper, we further explored the impact of chemical modifications of the π-linkers on the optoelectronic properties of the 2,7-carbazole based derivative discussed above. As new π-linkers, we considered the vinylene and the azomethine functions, as they should decrease the steric hindrance between aromatic units and consequently lead to a conjugation extension. Indeed, the introduction of vinylene bonds along the conjugated backbone, well studied in the literature [START_REF] Boucard | A Soluble Green-Light-Emitting Alternating Copolymer with Acceptor-Substituted Bicarbazyl Units[END_REF][START_REF] Ahn | Highly Photoluminescent and Blue-Green Electroluminescent Polymers: New Silyl-and Alkoxy-Substituted Poly(p-phenylenevinylene) Related Copolymers Containing Carbazole or Fluorene Groups[END_REF][START_REF] Ahn | Highly Photoluminescent and Blue-Green Electroluminescent Polymers: New Silyl-and Alkoxy-Substituted Poly(p-phenylenevinylene) Related Copolymers Containing Carbazole or Fluorene Groups[END_REF], has shown to be an efficient approach to obtain lower steric hindrance, leading to higher backbone planarity and then to improved electrical properties [START_REF] Roncali | Synthetic Principles for Bandgap Control in Linear π-Conjugated Systems[END_REF][START_REF] Song | Highly Efficient Light-Emitting Polymers Composed of Both Hole and Electron Affinity Units in the Conjugated Main Chain[END_REF][START_REF] Hwang | Photoluminescent and Electrochemical Properties of Novel Poly(aryl ether)s with Isolated Hole-Transporting Carbazole and Electron-Transporting 1,3,4-Oxadiazole Fluorophores[END_REF][START_REF] Qing | Modulating carrier transfer abilitylinker effect on thieno[3,4-c]pyrrole-4,6-dione based conjugated polymers[END_REF]. Azomethine-based organic semi-conductors exhibit high thermal and environmental stabilities [START_REF] Kaya | Synthesis and characterization of fluorescent graft fluorene-co-polyphenol derivatives: The effect of substituent on solubility, thermal stability, conductivity, optical and electrochemical properties[END_REF][START_REF] Thomas | Synthesis and Properties of a Soluble Conjugated Poly(azomethine) with High Molecular Weight[END_REF], excellent mechanical properties [START_REF] Kaya | Synthesis and characterization of polyphenols derived from 4fluorobenzaldeyde: The effect of electron-donating group on some physical properties[END_REF] beside their ease of preparation [START_REF] Koole | Charge transport through conjugated azomethine-based single molecules for optoelectronic applications[END_REF] and in some cases, liquid crystalline behavior [START_REF] Marin | Designing thermotropic liquid crystalline polyazomethines based on fluorene and/or oxadiazole chromophores[END_REF]. In addition, azomethine bridges, because of their pH-sensitivity, are widely used in sensors fields such as pH indicators [START_REF] Li | Schiff bases containing triphenylamine and pyrrole units: synthesis and electrochromic, acidochromic properties[END_REF] or fluorescence turn-on recognition of metals [START_REF] Gupta | New "on-off" opticalprobe based on Schiff base responding to Al3+ ions: logic gate application[END_REF].

Thus, in this paper, we report on the synthesis of two new molecules, based on the functionalization with anthracene residues in the 2,7-positions of a carbazole unit via either a Wittig-Horner strategy or a diamine condensation reaction and on their full characterization of optoelectronic properties through among other OFET and diode elaborations.

Results and discussion

Synthesis and structural characterization

As discussed in the introduction, two new small molecules with chemical aromatic units linked either by vinylene or by azomethine ߨ-bridges to give respectively the molecules M1 and M2 have been synthesized. Chemical structures of the studied models are presented in Figure 2.

As shown in Scheme 1 and Scheme 2, the 2,7-bisfonctionnalizedcarbazole derivatives (compound 10 and A4) have been prepared by a multi-step synthetic routes and were used on the synthesis of the two conjugated organic semi-conductors M1 and M2 respectively via Wittig-Horner reaction and condensation between carbonyl compound and primary aromatic amines reaction, respectively (Scheme 2 and Scheme 3). The structures were confirmed with Fourier Transform InfraRed spectroscopy (FT-IR), proton and carbon nuclear magnetic resonance ( 1 H NMR and 13 C NMR) and high resolution mass spectrometry (HRMS).

The N-2-ethylhexyl-2,7-bis(methylphosphonate)carbazole (compound 10, Scheme 1) was carried out according to a reported procedure [START_REF] Drolet | 2,7-Carbazolenevinylene-Based Oligomer Thin-Film Transistors: High Mobility Through Structural Ordering[END_REF] and it was then allowed to react with an excess of under inert atmosphere at 70°C. The target compound (M1) was obtained as a yellow cottony solid by purification of the crude mixture with flash column chromatography (Yield: 30 %).

The FT-IR spectrum of the 2,7-distyrylcarbazole showed the presence of the E vinylic (C=C) absorption peak at 966 cm [START_REF] Wang | A solutionprocessible poly(p-phenylene vinylene) without alkyl substitution: Introducing the cis-vinylene segments in polymer chain for improved solubility, blue emission, and high efficiency[END_REF]. The 1 H NMR revealed one doublet (2H, vinylic HC=CH, J = 16.4 Hz) at 8.21 ppm confirming the (E, E) configuration of the molecule. The HRMS of M1 showed molecular ion peak at m/z=684.357.

The precursor 2,7-dinitrocarbazole (A2) was synthesized according to the procedure outlined in literature [START_REF] Morin | 2,7-Carbazole-Based Conjugated Polymers for Blue, Green, and Red Light Emission[END_REF] and was then alkylated with 1-bromo-2-ethylhexane under basic conditions to give N-2-ethylhexyl-2,7-dinitrocarbazole (A3). Following synthesis route similar to that used in the preparation of N-alkyl-3,6-diaminocarbazole [START_REF] Grigoras | Synthesis and characterization of some carbazole-based imine polymers[END_REF], A3 was then reduced with SnCl 2 in a mixture of ethanol and ethylacetate (1:4) under nitrogen atmosphere at 70°C to give N-(2ethylhexyl)-2,7-diaminocarbazole in a 90% yield. M2 was finally synthesized by condensation reaction between N-(2-ethylhexyl)-2,7-diaminocarbazole (compound A4) and anthraldehyde in anhydrous methanol for 24h under inert atmosphere, the mixture was then cooled down to room temperature and the precipitated orange product was collected by filtration on a Büchner funnel (60% yield).

When comparing the FT-IR spectrum of M2 with that of M1, we note the disappearance of the vinylene band at 966 cm -1 and the appearance of two new bands at 1250 cm -1 and 1590 cm 1. Theoretically calculated vibrational modes are higher than the experimental values. This is at bottom due to the fact that the experimental measurements are conducted in solid state. However, theoretical studies are based on the hypothesis that the compound is in gaseous phase, in such condition, anharmonic effects are neglected. As shown in The synthesized molecules are soluble in common organic solvents such as THF, DMF and halogenated solvents.

Thermal properties

Thermal properties of the synthesized compounds were evaluated using thermogravimetric analysis (TGA) and differential scanning calorimetry (DSC). TGA (Figure 3 (a)) shows that the two molecules exhibited good thermal stability up to 360°C for M1 and 370°C for the imine compound under nitrogen flux, temperatures where 5 % of the models weights were lost. The replacement of vinylene linkage (-CH=CH-) between carbazole and anthracene units by the rigid azomethine (-CH=N-) group in molecule M2 induced a significant increase in the thermal stability of the molecule.

The thermal transitions of the molecules have been studied by DSC and the curves are reported in Figure 3 (b) showing a similar thermal behavior. When the samples of the two materials were heated, an endotherm peak due to melting were observed for each model at 227°C and 182°C respectively for M1 and M2. However, the higher melting enthalpy measured for M2 highlights the higher crystallinity of this imine molecule compared to the more standard vinylic model.

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

UV-vis absorption

Photo-physical properties of M1 and M2 were investigated in chloroform solutions and in thin films by using ultraviolet-visible (UV-vis) and photoluminescence (PL) spectroscopies.

Figure 4 shows UV-Vis absorption spectra of the two materials studied in diluted chloroform solutions and in thin films.

The absorption spectra consists each one of two peaks. The absorption bands appearing in the range of 330-370 nm in chloroform solutions and in the solid films can be attributed to the n-ߨ* transition bands of carbazole. The absorption peaks appearing toward 400 nm are assigned to the ߨ-ߨ* transition of the anthracene segment [START_REF] Sun | An X-shaped solutionprocessible oligomer having an anthracene unit as a core: A new organic light-emitting material with high thermostability and efficiency[END_REF]. Whereas, the n-ߨ* transitions of the azomethine linkage conjugated with aromatic nuclei exhibited characteristic absorption bands in the range of 370 to 455 nm [START_REF] Reddy | Synthesis and characterization of novel polyurethanes based on 4,4'-{1,4-phenylenebis[methylylidenenitrilo]}diphenol[END_REF][START_REF] Anuradha | Synthesis and development of Chitosan anchored copper(II) Schiff base complexes as heterogeneous catalysts for N-arylation of amines[END_REF].

As seen in Figure 4, the synthesized Schiff-base (M2) has a red shifted and a slightly broadened UV-visible absorption spectrum as compared with that of M1. This is due to the azomethine (-CH=N-) bond between carbazole and anthracene in molecule M2 which has an electron-withdrawing character [START_REF] Chang | Comparisons of Optical Properties on Metaand Para-Linkages of Copolymers for Poly(phenylene vinylene) and Poly(phenylene azomethine) with Alkylthio Group at the Side Chain[END_REF]. Its presence contributes to the extension of the conjugation (i.e. increases the electron delocalization) leading to the reduction of the optical gap [START_REF] Segura | The chemistry of electroluminescent organic materials[END_REF]. It's important to notice that both vinylene and azomethine spacers lead to more planar molecules as compared with aryl-aryl cross-coupled molecule . These ߨ-bridges increase conjugation and induce a broadening of the absorption spectrum together with a reduction of the optical band gap from 3.05 eV for M b to 2.69 eV and 2.49 eV for M1 and M2, respectively.

DFT calculations have been performed using The Gaussian 09 program at the B3LYP/6-311 + G* level of theory in vacuum on the final molecules. The calculated frontier molecular orbital's (FMO) and the bandgap energies are reported in Table 2. The optimized geometries for M1 and M2 are depicted in Figure 5, and the frontier energy (HOMOs and LUMOs) distributions are presented in Figure 6. As shown in Figure 5, both molecules adopt (E, E) configurations and a very twisted conformation with high dihedral angles between the planar central-core (carbazole) and the adjacent ring of the anthracene moiety. This dihedral angle is higher in the case of M1 (70°) than that of M2 (60°). The calculated frontier energy distributions of M1 are very extended over the molecular backbone. However, due to the electron effect of the azomethine (-CH=N-) linkage in the molecule M2, the LUMO is much localized on the anthracene moiety and the HOMO is much more extended over the molecular backbone (Figure 6). As it was expected, the introduction of the electron-withdrawing azomethine bridge withdraws electron density from the anthracene moiety.

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This increases significantly the calculated electron affinity (EA) and slightly the ionization potentials (IP) resulting in a reduction of the HOMO-LUMO energy gap (Table 2).

The absorption spectra in thin films are highly broadened and red-shifted as compared with those in diluted solutions (20 nm for M1 and 50 nm for M2), and the onset absorptions of the two materials in thin films are 481nm and 550 nm respectively for M1 and M2. This red-shift was attributed to the ߨ-ߨ stacking, which results from the formation of aggregates in solid state [START_REF] Luponosov | Effects of electron-withdrawing group and electron-donating core combinations on physical properties and photovoltaic performance in D-π-A star-shaped small molecules[END_REF]. We notice that a higher displacement of the absorption onset in thin films was observed in the case of M2 indicating stronger ߨ-ߨ stacking interactions which can be attributed to the more important intermolecular interactions due to the better planarity of the Schiff base molecule as shown by DFT calculations (see below). The absorption data are summarized in Table 2.

Photoluminescence properties

The PL measurements were recorded at room temperature using freshly prepared diluted solutions (2.10 -6 M). Fluorescence spectra are shown in Figure 7. In chloroform solution, M1 PL spectrum shows one broad band in the blue-green region with a maximum at 500 nm which can be Dilute solution

-X- λ λ λ λmax (nm) (a) ε ε ε εmax (10 4 .M -1 .cm -1 ) (b) FWHM (nm) (c) λ λ λ λonset (nm) (d) Eg-opt(eV) (e
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attributed to the anthracene moieties [START_REF] Hriz | Synthesis characterization and optical properties of distyrylanthracene-based polymers[END_REF][START_REF] Pu | Arylamino-9,10-diphenylanthracenes for organic light emitting devices[END_REF]. Unfortunately, no photoluminescence was observed for M2. The chromaticity coordinates of the photoluminescence of M1 in (x = 0.45 and y = 0.46) are presented according to the CIE 1931 color spaces in Figure 7.

The fluorescence quantum yields ݔ݁ɸ were calculated from Eq (1).

ݔ݁ɸ = ɸ݂݁ݎ ref ɳ ɳ Eq (1)
Where "I" is the integral of the corrected emission spectrum, "OD" denotes the optical density at the excitation wavelength and "η" is the refractive index of the medium. Quantum yield of M1 was determined in solution by using cresyl violet as a reference λ exc = 400 nm, ɸ = 0.5 in ethanol and the value was 43% [START_REF] Frath | Vectorization and Intracellular Distribution of a Two-Photon-Absorbing, Near-Infrared-Emitting π-Extended Boranil Dye[END_REF]. As shown previously, the fluorescence emission of the molecule M2 was quenched. This quenching is probably partly due to the isomerization of the C=N double bond in the excited state [START_REF] Wang | Two highly sensitive Schiff-base fluorescent indicators for the detection of Zn2+[END_REF] allowing a highly efficient photo-induced electron-transfer (PET) process [START_REF] Ganguly | Selective fluorescence sensing of Cu(II) and Zn(II) using a simple Schiff base ligand: Naked eye detection and elucidation of photoinduced electron transfer (PET) mechanism[END_REF][START_REF] Maity | A two-pocket Schiff-base molecule as a chemosensor for Al3+ New[END_REF].

In systems allowing the PET process, group containing non-bonding electron pair is usually separated by a short spacer from the fluorophore. The Schiff-base nitrogen atom and the electrondonating groups usually contain a relatively high-energy non-bonding electron pair. In this case, after excitation, an electron of the highest occupied molecular orbital (HOMO) is excited to the lowest unoccupied molecular orbital (LUMO), leaving a hole in the former orbital, that can be filled via a fast intramolecular electron transfer from the non-bonding electron pair of the nitrogen atom to the HOMO of the excited fluorophore. This phenomenon provides a mechanism for non-radiative deactivation of the excited state, resulting in the quenching of the fluorescence signal of the system [START_REF] Culzoni | Rhodamine and BODIPY chemodosimeters and chemosensors for the detection of Hg2+, based on fluorescence enhancement effects Anal[END_REF]. The probable mechanisms of fluorescence emission of M1 and PET process of M2 are shown in Figure 8. This class of fluorophores is commonly used as sensors and act as indicators with changes in its fluorescence intensity upon complexation or dissociation [START_REF] Liu | Coumarin-based fluorescent chemosensor for the selective quantification of Zn2+ and AcO-cOr the selective quantification of Zn2+ and AcZn2+ and Action of Zn2+ and Ac+ and Ac[END_REF]. These sensors can also function as pH indicators [START_REF] Rani | Pyrene-antipyrine based highly selective and sensitive turn-on fluorescent sensor for Th(IV)[END_REF], protonation of the Schiff base annihilates the PET mechanism [START_REF] Ganguly | Selective fluorescence sensing of Cu(II) and Zn(II) using a simple Schiff base ligand: Naked eye detection and elucidation of photoinduced electron transfer (PET) mechanism[END_REF].

Effect of trifluoroacetic acid (TFA) on the optical properties of M2

In order to evaluate the PET mechanism, the optical properties of the M2 molecule have been studied in acidic conditions. The effect of addition of TFA on the absorption spectrum in the diluted chloroform solution of M2 is presented in Figure 9 (Left). An important blue shift of the onset spectrum from 506 nm to 476 nm and a hypsochromic effect of the absorption bond between 357 nm and 530 nm were observed. These shifts can be related to the formation of protonated species. After all, protonating azomethine linkage inhibits the n-ߨ* transitions of the azomethine ). This "Turn-On" system constitutes the basis for a proton (acid) sensing study including additional edges such as selectivity and sensitivity [START_REF] Hwang | A highly selective and sensitive fluorescent turn-on Al3+ chemosensor in aqueous media and living cells: experimental and theoretical studies[END_REF] and the compound is said to be fluorogenic.

Electrochemical Properties

The electrochemical behaviors of N-(2-ethylhexyl)-2,7-bis(vinyl-9-anthracene)carbazole (M1) and its Schiff base analog (M2) were investigated by cyclic voltammetry. The organic films were drop-coated onto a platinum working electrode from chloroform solutions and scanned both at positive and negative potentials at a scan rate of 100 mV s -1 . A Pt wire was used as counter electrode and Ag/Ag+ as reference electrode in a 0.1 mol.L -1 solution of tetrabutylammonium tetrafluoroborate in dichloromethane.

The obtained cyclic voltammograms are shown in Figure 12. The lowest unoccupied molecular orbital (LUMO), the highest occupied molecular orbital (HOMO) and the electrochemical band gap values were calculated from the electrochemical onset potentials and by assuming that the energy level of the ferrocene/ferrocenium is 4.8 eV below the vacuum level as in the literature as follows [START_REF] Bredas | Chain-length dependence of electronic and electrochemical properties of conjugated systems: polyacetylene, polyphenylene, polythiophene, and polypyrrole[END_REF].

E HOMO (IP, ionization potential) = -(V onset-ox -V FOC + 4,8 ) eV Eq (2)
E LUMO (EA, electron affinity) = -(V onset-red -V FOC + 4,8 ) eV Eq (3)

E g-el = (E LUMO -E HOMO ) eV Eq (4)
Where V FOC is the ferrocene half-wave potential (0.405 V), V onset-ox is the model oxidation onset and V onset-red is the model reduction onset, all measured versus Ag/Ag+. The calculated electrochemical values are summarized in Table 4. The oxidation and reduction potentials of M1

and M2 are recorded in the range 0.90 to -2.20 V and 1.10 to -1.90 V, respectively. The obtained results showed that the ionization potential of M2 was slightly increased as compared to that of M1, while, the LUMO level was significantly decreased. As a consequence, the measured electrochemical band-gap is reduced from 2.58 eV for M1 to 2.36 eV in the case of M2, in good agreement with optical data. The M2 electrochemical behavior is related to its Schiff base nature.

Indeed, M2 contains azomethine (-CH=N-) linkage in the structure which is known as an electron-

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withdrawing [START_REF] Tsai | New Thiophene-Linked Conjugated Poly(azomethine)s: Theoretical Electronic Structure, Synthesis, and Properties[END_REF] bond. In fact, the introduction of such electron-withdrawing group in conjugated systems induces an increase in electron affinity and in ionization potential [START_REF] Yang | Efficient blue light-emitting diodes from a soluble poly (paraphenylene) internal field emission measurement of the energy gap in semi-conducting polymers[END_REF][START_REF] Ranger | New Well-Defined Poly(2,7-fluorene) Derivatives: Photoluminescence and Base Doping[END_REF][START_REF] Leclerc | Polyfluorenes: Twenty years of progress[END_REF].

To further understand the electronic structure of the synthesized molecules and to select electrodes materials for the charge injection (metal work function) in organic thin-film transistors [START_REF] Zhong | Effects of bridge units on the properties of indolo[END_REF][START_REF] Fall | Ambipolar charge transport in polymer:fullerene bulk heterojunctions for different polymer side-chains[END_REF][START_REF] Bulut | Thiazole-based scaffolding for high performance solar cells[END_REF] and OLEDs [START_REF] Chemli | Synthesis and characterization of new electroluminescent poly(p-phenylene) derivative[END_REF][START_REF] Cho | Flexible integrated OLED substrates prepared by printing and plating process[END_REF], it is essential to establish the relative positions of the characteristic electronic energy levels of these conjugated materials relative to vacuum and to compare them with those of MEH-PPV, ITO, Ca, Al and Au as presented in the energy diagram in Figure 13.

CV results showed that the two conjugated materials have around -5.10 eV HOMO energy and are p-type conjugated materials [START_REF] Mansour | Synthesis, characterization, optical and electrical properties of bis(phenylvinyl)anthracene-based polymers[END_REF]. and M2 (Table 5). The dynamic resistances (R d ) [START_REF] Zahou | Effect of MWCNT loading on the morphological, optical and electrical properties of Nickel phtalocyanine based composites[END_REF] are extracted and presented also in Table 2.

As shown in the corresponding table, M2 has the highest turn-on voltage and dynamic resistance demonstrating that M2 is more resistant.

The J-V characteristics in a logarithmic scale of the two materials are presented in Figure 14 and were used to determine the conduction mechanisms. The logarithmic plot shows three regions indicating three different regimes for each device:

-Region 1: at low voltage (V < 2.6 V) defining the Ohmic regime where the current density varies linearly with the applied voltage following equation [START_REF] Kao | Electrical Transport in Solids: with particular reference to organic semiconductors[END_REF]:

ܬ ஐ = .ݍ  . ߤ. ܸ/݀ Eq (5)
Where q is the electronic charge, ߤ is the charge mobility,  is the free carrier density and d is the thickness of the film. where the current density is space-charge limited (SCLC regime) and the current density follows [START_REF] Benzarti-Ghédira | Effect of the side-chain size on the optical and electrical properties of confined-PPV derivatives[END_REF]:

Compound E ox (V) E red (V) E HOMO (eV) a E LUMO (eV) b E g-el (eV) c M1 0.
ܬ ௌ = ଽ ଼ . ߝ. ߤ . ܸ ଶ /݀ ଷ Eq (6)
where, Ɛ is the permittivity of the organic material (assumed to 4Ɛ 0 where Ɛ 0 is the permittivity of vacuum). The ߤ eff is the effective carrier mobility and is equal to θߤ, where θ is the fraction of free charge (θ = (/ +  ௧ )) and p and p t are free and trapped charge-carriers densities, respectively).

-Region 3: at higher voltages, a trap filled SCLC regime with J-V m was measured (m = 5 and 6

respectively for M1 and M2). When further increasing the voltage, free carriers that cannot be trapped again are introduced, resulting in a fast increase of the current intensity [START_REF] Jaballah | New bithiophene-containing electroluminescent polymer: Synthesis, characterization, optical and electrical properties[END_REF].

The hole effective mobilities (ߤ eff ) in the M1 and M2 films were calculated respectively as 6.2x10 -5 cm²/V.s and 8.4x10 -6 cm²/V.s (Table 5) by fitting the J-V characteristics to the SCLC model (Eq. ( 6)). The estimated charge-carrier mobilities are comparable to the achieved hole mobility in the MEH-PPV based device (≈ 10 -5 cm²/V.s) [START_REF] Amorim | Determination of carrier mobility in MEH-PPV thin-films by stationary and transient current techniques[END_REF].

Effect of trifluoroacetic acid (TFA) on the electrical properties of [ITO/molecule/Al] device

Solution of compound M2 (2.10 -2 M) was prepared and spin-coated onto ITO/glass substrates. The device was exposed to TFA in vapor for 8 seconds (M2 + TFA) before the deposition of the top Al electrode. The hole effective mobilities (ߤ eff ) in the M2 and protonated M2 in films were calculated using the usual formalism for SCLC devices (Eq. ( 6)) and are summarized in Table 6.

M A N U S C R I P T A C C E P T E D

ACCEPTED MANUSCRIPT Table 6. The hole effective mobilities (ߤ eff ) in the neutral and protonated M2 thin films.

We note that the hole effective mobility in the M2 thin film is substantially increased (almost one order of magnitude higher) after exposure to trifluoricacetic acid vapors for times as short as 8s.

This sensitivity to protic acid makes this azomethine-based molecule a candidate to sensor applications.

Organic Field-Effect Transistors:

The output and transfer plots of the devices exhibited typical p-type OFET characteristics (Figures 16 and17). The hole mobilities (µ) were calculated from the transfer characteristics of the devices in the saturation regime. The OFET characteristics (V T and µ) are summarized in Table 7.

The OFET hole mobilities are in the order of (1.4±0.4) x10 -6 cm²/V.s for M1 and (6.5±0.3)x10 -7 cm²/V.s for M2.

Table 7. Bottom-gate/ Bottom -contact OFET characteristics of M1 and M2 thin films.

Charge mobilities in M1 films derived from OFET or deduced through J-V characteristics in the SCLC regime of single-layer diodes are higher than those of M2 films. Surprisingly, the SCLC charge-carrier mobilities are 10 (M2) to 40 (M1) times higher than the OFET ones while the opposite was anticipated from the usual variation of mobility as a function of charge-carrier density [START_REF] Fall | Tailoring the microstructure and charge transport in conjugated polymers by alkyl side-chain engineering[END_REF]. The observed more than one order of magnitude higher out-of-plane hole mobility than the inplane one suggests a high charge-transport anisotropy for these molecules as sometime observed in dumbbell shaped semi-conducting small molecules [START_REF] Bulut | Thiazole-based scaffolding for high performance solar cells[END_REF].

Conclusion

In

Experimental section

Synthesis and Characterization

UV-visible absorption and emission spectroscopy:

A Schimadzu UV-3600 dual-beam grating spectrophotometer with a 1 cm quartz cell was used on recording Uv-vis absorption spectra in chloroform diluted solutions (5.10 -5 M) and in solid state.

Thin films were drop coated from a dichloromethane (DCM) solution (2.10 -2 M) onto glass substrate and the optical band gaps (E g ) were estimated from the onset absorption spectra using the Tauc relation (αhυ)= A (hυ -E g ) 1/2 , where hυ is the absorbed photon energy and α is the corresponding absorption coefficient [START_REF] Culzoni | Rhodamine and BODIPY chemodosimeters and chemosensors for the detection of Hg2+, based on fluorescence enhancement effects Anal[END_REF]. The E g was obtained from extrapolation to α = 0 of the straight line portion in the (αhυ) 1/2 versus hυ plot. The steady-state fluorescence emission spectra were obtained by using a Horiba S2 Jobin Yvon Fluoromax 4 in chloroform diluted solutions (2.10

General synthetic procedure

Synthesis of N-(2-ethylhexyl)-2,7-bis(methylphosphonate)carbazole

The N-2-ethylhexyl-2,7-bis(methylphosphonate)carbazole (compound 10 Scheme 1) was synthesized according to a synthesis route described in the literature by N. Drolet et al [START_REF] Bulut | LUMO's modulation by electron withdrawing unit modification in amorphous TAT dumbbell-shaped molecules[END_REF].

Synthesis of N-(2-ethylhexyl)-2,7-bis(vinylene-(9-anthracene))carbazole (M1)

A mixture of compound (11) (0.58 g, 1 mmol), 9-anthraldehyde (0.50 g ,2.5 mmol), sodium tert-butoxide (0.38 g, 4 mmol) and dry THF (10 mL) were stirred under nitrogen atmosphere for 24h at room temperature. Then, the solvent was removed under reduced pressure, water was added (10 mL) and the organic layer was extracted with chloroform. After evaporating the solvent in vacuum, the residue was purified by flash column chromatography using a mixture of Ethyl acetate/Hexane (1:9, v/v) as eluent, to give the title product as a cottony yellow solid. M.P. 227°C

(Yield: 30 %).

Synthesis of N-(2-ethylhexyl)-2,7-diaminocarbazole

: M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT
The N-(2-ethylhexyl)-2,7-diaminocarbazole (Scheme 2) were synthesized according to a modified procedure described in the literature by J.F. Morin et al. [START_REF] Liu | Coumarin-based fluorescent chemosensor for the selective quantification of Zn2+ and AcO-cOr the selective quantification of Zn2+ and AcZn2+ and Action of Zn2+ and Ac+ and Ac[END_REF] as follow.

4,4'-dinitro-2-azidobiphenyl (A1): Sodium nitrite (7.98 g, 115.65 mmol) was gradually added to a solution of sulfuric acid (60 mL), 4,4'-dinitro-2-biphenyl (15 g, 57.9 mmol) in a mixture of acetic acid (300 mL) and at 0°C. The solution was stirred for 2h at 5-10°C, 7.50 g of urea followed by 750 mL of ice-water and 7.50 g of activated carbon were added. The solution was then stirred for 20 min and filtrated rapidly through a Buchner funnel into a 2 L flask immersed in ice.

Sodium azide (5.07 g, 117 mmol) in 150 mL of water was added dropwise to the clear yellow filtrate. The resulting mixture was stirred for 1h at 0°C and for 24 h at room temperature. The solution was quenched with 700 mL of saturated aqueous solution of NaHCO 3 and extracted three times with ethyl acetate. The organic layer was dried over sodium sulfate and the solvent was removed under reduced pressure. The crude material was recrystallized in ethanol to provide 12 g of the title product as a yellow solid M.P. 170°C (Yield: 72.5 %).

Synthesis of N-(2-ethylhexyl)-2,7-bis(azomethine-(9-anthracene))carbazole (M2)

In a 50 mL flask, a mixture of anhydrous methanol (15 mL), compound A4 (0.310 g, 1 mmol) and 9-anthraldehyde (0.44 g, 2.2 mmol) were refluxed under nitrogen for 24 h at 60°C. The solution was then cooled down to the room temperature and the orange precipitate was filtered through a Büchner funnel. The crude product was dissolved in chloroform, precipitated two times in anhydrous methanol and dried under vacuum to provide 0.411 g of the desired product as an orange solid product. M.P. 182°C (Yield: 60%).

Fabrication and characterization of the diodes

Solutions of the two small molecules (2. 10 -2 M in chloroform) were spin-casted (2500 rpm) onto indium tin oxide (ITO) glass to obtain a film with a thickness of 60 nm. After annealing at 40°C for 1 hour, a thin aluminum (Al) layer (150 nm) was deposited by thermal evaporation at 3x10 -6 Torr. The current-voltage (I-V) characteristics of the devices were recorded with a Keithley 236 source meter using the ITO as an anode and the Al as a cathode.

Fabrication and characterization of OFETs

The charge transport was extracted from the electrical characteristics of organic field-effect transistors (OFET) configuration (bottom contact, bottom gate configuration).

The OFETs were elaborated on commercially pre-patterned test structures whose drain and source contacts are composed of 10 nm thick Indium Tin Oxide (ITO) bilayers and 30 nm thick gold. A 230 nm thick silicon oxide was used as gate dielectric and highly n-doped (3.10 17 /cm 3 ) silicon crystal as gate electrode. The channel length and width channel were 20 µm and 10 mm, respectively. The test structures were cleaned in swap, acetone and isopropyl alcohol for 15 min in and finally using ultra-violet ozone system respectively. Hexamethyldisilazane (HMDS) was then spin-coated (500 rpm for 5 s and then 4000 rpm for 50 s) and annealed at 130°C for 10 min.

Solutions of the conjugated molecules in anhydrous 1,2-Dichlorobenzene (4 mg/mL) were finally spin-coated (1200 rpm for 60 s and 2000 rpm for 120 s). The substrates were then left under vacuum (<10 -6 mbar) for 14 hours to remove residual solvent traces. The elaboration and characterizations of the OFETs were performed in nitrogen ambient [START_REF] Rani | Pyrene-antipyrine based highly selective and sensitive turn-on fluorescent sensor for Th(IV)[END_REF]. The hole mobilities (µ)

were calculated from the transfer characteristics of the devices in the saturation regime, at V DS = -80

V

  (1.2 equivalent) in a mixture of potassium t-butoxide and tetrahydrofurane (THF)

- 1 attributed

 1 respectively to (H-C=N) in plane bending and (C=N) stretching vibrations of the azomethine bridge. The 1 H NMR spectrum reveals a singlet at 9.88 (2H) ppm corresponding to the azomethine (N=CH) proton. HRMS found for C 52 H 43 N 3 : 686.3558 confirming the proposed structure of M2. Theoretical studies of the vibrational frequencies for the two compounds M1 and M2 were performed by Density Functional Theory (DFT) calculations using B3LYP/6-311+G(d) method. Experimental and theoretical FT-IR spectra of the two molecules are shown in supplementary data (Figure S.16 and Figure S.21). The most important vibrational waves with the proposed assignment are summarized in Table

9 (

 9 M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT linkage conjugated with aromatic nuclei and caused the disappearance of the associated absorption bond. The protonation of the nitrogen atom of the azomethine group prevents also the photoinduced electron-transfer (PET) phenomenon from nitrogen to anthracene fluorophore (Figure 10) inducing a naked eye remarkable color change from colorless to purple (Figure 11 (a)) and a "turn-on" of the fluorescence of the fluorophore with no significant shift of the emission maxima (Figure Right)). Both chloroform diluted solutions of M2 in presence of TFA and M1 have almost the same green emission under UV Lamp with 365 nm wavelength (Figure 11 (b)

1. 7 . Electrical properties 1 . 7 . 1 .

 7171 Charge transport mechanism: To investigate the current-voltage (I-V) characteristics of the two synthesized carbazolebased molecules, two single-layer devices with the [ITO/carbazole-based material/Al] structure were fabricated. The J-V (see Figure S.23) curves measured at room temperature indicate typical diode behavior with relatively low threshold bias voltages of 4.6 V and 4.9 V respectively for M1

Figure 15

 15 Figure 15 exhibits the effect of trifluoroacetic acid (TFA) on current density of the [ITO/M2/Al] single-charge-carrier devices.

  summary, a 2,7-bisfonctionnalized carbazole-based Schiff base (M2) and its distyrylcarbazole derivative (M1) were synthesized and characterized by different techniques. Both molecules have good solubility in common organic solvents. The structures of the compounds were confirmed by FT-IR, NMR, and HRMS. Thermal analysis showed a semi-crystalline behavior for and higher thermal stability in the case of M2. Replacing the vinylene linkage of the molecule M1 by the azomethine bridge which has an electronic withdrawing effect activates conjugation leading to a broader and red-shifted absorption spectrum and to the reduction of the optical gaps. The calculated frontier molecular orbital's (FMO) and the bandgap energies are found to be in good agreement with experimental data. In addition, the introduction of the C=N group induces a significant decrease in the HOMO energy level of the molecule. The fluorescence quantum efficiencies of M1 was 43% however the azomethine bond of M2 turns off (quenches) the photoluminescence by photo-induced electron-transfer (PET) phenomenon. The formation of the protonated species of the Schiff base by addition of TFA has a hypsochromic effect of the absorption spectrum and inhibits the PET process, inducing a turn-on of the fluorescence and a visible color change. Moreover, devices with [ITO/Semi-conductor/Al] configuration have shown relatively low turn-on voltages and three different mechanism conductions for each device. Low charge mobilities were extracted from SCLC regime of singlelayer diodes and from field-effect transistors. An important increase of the hole effective mobility in the M2 thin film (measured by SCLC) was noted after exposure to trifluoricacetic acid opening the field of sensor applications to such azomethine molecule. Future work will focus on improving the electrical properties of the synthesized small molecules and investigating the electroluminescence of M1. Also the commendable sensitivity of molecule M2 towards protic acid indicates the possibility of the development of the technique to further use in sensors research.

  ), toluene, methanol, acetone and tetrahydrofurane (THF) were received from Sigma-Aldrich and were used without further purification and the last five solvents were dried by usual methods and distilled under reduced pressure.The purity and the structure of the synthesized products where confirmed by Fourier Transform InfraRed spectroscopy (FT-IR), Nuclear Magnetic Resonance (NMR) and High Resolution Mass Spectrometry (HRMS). NMR and FT-IR spectra were recorded respectively on a Bruker Avance III 400MHz NMR spectrometer and a Perkin-Elmer Spectrum One FT-IR spectrometer (Thermo Scientific Nicolet IS50ATR) according to the Attenuated Total Reflectance (ATR) technique. HRMS experiments were performed on a Bruker micrOTOF-Q II spectrometer within the service center at the ''Institut de Chimie'' of Strasbourg University.

1 H and 13 C

 13 NMR spectra of all compounds were recorded on Nuclear Magnetic resonance spectrometer Bruker Avance III HD with the magnet Bruker Ascent 400 MHz. TF-IR spectra were recorded on a Perkin-Elmer Spectrum UATR Two FT-IR spectrometer (Thermo Scientific Nicolet IS50ATR) according to the Attenuated Total Reflectance (ATR) technique. HRMS experiments were performed on a Bruker micrOTOF-Q II spectrometer within the service center at the ''Institut de Chimie'' of Strasbourg University. DSC and TGA: DSC measurements were performed using a METTLER Toledo DSC 1 instrument, operated at a scanning rate of 10°C/min. Thermogravimetric analyses (TGA) were conducted using a Hi-Res TGA Q5000 apparatus from TA Instruments under helium atmosphere (flow rate of 25 mL/min). Samples (2-4 mg) were heated from room temperature up to 400°C at a rate of 10°C/min.

- 6 M

 6 ) in order to avoid aggregation and reabsorption effects. Samples were excited with a 450 W Xenon lamp. Cyclic voltammetry: The electrochemical behaviors of the conjugated molecules were investigated by cyclic voltammetry with a BioLogic VSP potentiostat using platinum electrodes. A platinum wire and Ag/Ag+ electrodes were used respectively as counter electrode and reference M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT electrode in a 0.1 mol.L -1 solution of tetrabutylammonium tetrafluoroborate [NBu 4 ][BF 4 ] in dichloromethane. The measurements were performed at room temperature, and the cell was degassed with argon before each scan. Density Functional Theory (DFT) and Time Dependent Density Functional Theory (TD-DFT) : Equilibrium geometry structures and the energy distributions of the molecular orbitals for the two the molecules have been performed in the gas phase using The Gaussian 09 software at the B3LYP/6-311 + G* level of theory in vacuum.

  10.98 (-(CH 2 ) 3 -CH 3 ). FT-IR (cm -1 ): 3080-3030 (w, aromatic and vinylic C-H stretching); 2964-2846 (w, aliphatic C-H stretching); 1456 m, 1620 w (m, aromatic C=C stretching); 1320 (w, C-N stretching); 966 (m, E-HC=CH out-of-plane bending); 730 (s, aromatic C-H out-of-plane bending). w: Week, m: Medium, s: Strong HRMS: Calculated for C50H43N: 683.92 Found: 684.36.

1 .Scheme 2 .

 12 AbbreviationsAl: Aluminum

Figure 1 .

 1 Figure 1. Chemical structures of M1 and M2.

Figure 2 .

 2 Figure 2. Chemical structures of 3,6-di(anthracen-9-yl)-9-ethyl-9H-carbazole (M a ) and 2,7di(anthracen-9-yl)-9-ethyl-9H-carbazole (M b ).

Figure 3 .

 3 Figure 3. ATG (a) and DSC (b) curves of M1 and M2.

Figure 4 .

 4 Figure 4. Normalized UV-Vis absorption spectra of M1 and M2 in chloroform diluted solutions and in thin films.

Figure 5 .

 5 Figure 5. Face view (Left) and side view (Right) of the optimized molecular geometry for the minimal energy conformation for the M1 and M2.

Figure 6 .

 6 Figure 6. HOMOs and LUMOs frontier energy distributions for M1 and M2.

Figure 7 .

 7 Figure 7. (Left) Normalized PL spectra of M1 and M2; in chloroform diluted solutions (2.10 -6 M). (Right) the CIE chromaticity diagram of M1.

Figure 8 .

 8 Figure 8. (a) Proposed fluorescence emission mechanism of M1. (b) The PET mechanism of M2.

Figure 9 .Figure 10 .

 910 Figure 9. Effects of (a) the addition 2 equivalents of acid (TFA) on the absorption profile and (b) of increasing concentration of acid (TFA) on the emission profile of M2 in chloroform solvent. (Sample concentrations are 5x10 -5 M for the absorption and 2x10 -6 for the PL) Figure 10. Switch-on fluorescence mechanism of M2 in presence of TFA.

Figure 11 .

 11 Figure 11. The photograph of (a) M2 alone and in the presence of TFA (10 µM) and (b) the photograph of M2 alone, M2 in the presence of TFA and M1 under UV illumination.

Figure 12 .

 12 Figure 12. Cyclic voltammograms of M1 and M2.

Figure 13 .

 13 Figure 13. Energy diagram of ITO, M1, M2, MEH-PPV, PC 71 BM, Ca, Al and Au.

Figure 14 .

 14 Figure 14. The J-V characteristic for the [ITO/molecule/Al] diode in log-log representation.

Figure 15 .

 15 Figure 15. The J-V characteristics in linear (a) and logarithmic (b) scales of neutral and protonated M2.

Figure 16 .

 16 Figure 16. Typical output curves (a) and transfer plots (b) of the bottom-gate/bottom-contact OFET devices for M1.

Figure 17 .

 17 Figure 17. Typical output curves (a) and transfer plots (b) of the bottom-gate/bottom-contact OFET devices for M2. RUNNING HEAD TITLE: New vinylene and azomethine linked anthracene based ߨ-conjugated molecules.

Table 1 ,

 1 the theoretical calculated wave numbers are in good agreement with the experimental FT-

	IR values.

Table 1 .

 1 Selected experimental and calculated vibrational wavenumbers of M1 and M2 (cm -1 ) and their assignment.

	Compound M1	Compound M2	Assignments
	Founded	Calculated	Founded	Calculated
	3080-3030 w	3236-3174 w-s	3080-3030 w	3267-3174 w-s	ν (C-H) aromatic
		3155 w	-	-	ν (C-H) vinylic
	2964-2846 w	3000-3130 w-m	2960-2846 w	3131-3004 w-s	ν (C-H) aliphatic
	1456 m, 1620 w	1376 m, 1416 w, 1608 w, 1628 w, 1654 w, 1666 w, 1674 w, 1684 w	1453 m, 1620 w	1691-1572 w-s 1496-1456 w-m 1442-1321 w-m	ν (C=C) aromatic
				1691w, 1687 w,
	-	-	1590 w	1680 w, 1649 s, 1642 s, 1630 m,	ν (C=N) symmetric
				1602 m, 1571 w
		1666 w, 1694 w,
	1620 w	1692 w, 1684 w,	-	-	ν(C=C) Vinylic
		1674 w
	1320 w	1408 w, 1403 w, 1389 w, 1376 w	1320 w	1504 m, 1516 m	ν (C-N)
	-	-	1250 w	1315 w	δ (H-C=N)
	966 m	994 w, 1011 w, 1014 w, 1016 w, 1019 w	-	-	γ (E-HC=CH) Vinylene
	730 s	753 s	733 s	752 s, 776 w	γ (C-H) Aromatic
	w: Week, m: Medium, s: Strong. ν :stretching, γ : out of plane bending, δ in plane bending.

Table 2 .

 2 Optical spectroscopic parameters of M1 and M2 in diluted solutions and in thin films.

Table 3 .

 3 HOMO, LUMO and Bandgap energies calculated by DFT.

  )

	M 1 CH	349 ; 405			1.91 ; 3.1	105	460	2.69
	M 2	N	333 (f) ; 348 ; 369 (f) ; 389 (f) ; 421	1.11 ; 1.13 ; 1.26 ; 1.64 ; 1.1	140	500	2.49
						Thin films
			λ λ λ λmax (nm) (a)			λ λ λ λonset (nm)	Egap (eV)
	M 1	CH	344 ; 399		149	481	2.57
	M 2	N	334 ; 445		219	550	2.25
			Compound	E HOMO (eV)	E LUMO (eV)	Bandgap (eV)	μ ሬԦ (debye)
			M1	-4.86		-1.87	2.99	1.80
			M2	-4.95		-2.21	2.75	1.67

FWHM (nm)

(a) Absorption peak maximum, (b) Molar extinction coefficient, (c) Full width at half maximum of the absorption spectrum in solution, (d) Absorption onset, (e) Optical band gap energy, (f) Shoulder

Table 4 .

 4 Electrochemical onset potentials and electronic energy levels of M1 and M2.

a Highest occupied molecular orbital ; b Lowest unoccupied molecular orbital ; c Electrochemical band gap.

Table 5 .

 5 Dynamic

	[ITO/M1/Al]	[ITO/M2/Al]

resistances (Rd), threshold bias voltages (Vs) and the charge carriers values for the [ITO/molecule/Al] diodes.

  (36 mL) were stirred and refluxed for 24 h. The solution is allowed to cool down and then poured into ice. The pH was then made slightly basic (pH 7-8) by addition of saturated aqueous sodium bicarbonate solution before extracting with ethyl acetate. The organic phase was washed several times with brine, treated with charcoal and dried over magnesium sulphate. The

	ACCEPTED MANUSCRIPT
	chromatography using a mixture of ethyl acetate/hexane (1:4, v/v) as eluent to provide 5.74 g of the
	title product as an orange crystalline solid. M.P. 81°C (Yield: 74 %). 1 H NMR (400 MHz, Acetone-
	d 6 , ppm): 8.52 (d, 2H, J = 2.0 Hz, Ar); 8.48 (d, 2H, J = 8.8 Hz, Ar); 8. 14 (dd, 2H, J = 8.8 and 2.0
	Hz, Ar); 4.48 (d, 2H, J = 7.6 Hz, N-CH 2 ); 2.2 (m, 1H, CH-(CH 2 ) 3 ); 1.45 (m, 8H, CH 2 aliphatic);
	0.92 (t, 3H, J = 7.6 Hz, -CH 2 -CH 3 ); 0.83 (t, 3H, J = 7.2 Hz, , -(CH 2 ) 3 -CH 3 ). 13 C NMR (75 MHz,
	Acetone-d 6 , ppm): 148.12 (Ar); 142.74 (Ar); 126.78 (Ar); 123.08 (Ar); 115.52 (Ar); 106.92 (Ar);
	1 H NMR (400 MHz, Acetone-d 6 , ppm): 8.38 (m, 2H); 8.25 (d, 1H, J = 2.4 Hz); 8.17 (dd, 1H, J = 8.4 and 2.4 Hz); 7.90 (m, 2H); 7.80 (d, 1H, J= 8.4 Hz). 13 C NMR (75 MHz, Acetone-d 6 , ppm): 148.70; 147.90; 142.85; 139.35; 136.98; 132.23; 130.83; 123.32; 119.72; 114.47. 2,7-dinitrocarbazole (A2): Compound A1 (7.00 g, 24.5 mmol) was slowly added to 700 48.31 (N-CH 2 ); 40.18 (CH-(CH 2 ) 3 ; 32.59 (CH-CH 2 -CH 2 ); 31.30 (CH 2 -CH 2 -CH 2 ); 24.83 (CH-CH 2 -CH 3 ); 23.58 (CH 2 -CH 2 -CH 3 ); 14.13 (-CH 2 -CH 3 ); 11.12 (-(CH 2 ) 3 -CH 3 ). N-(2-ethylhexyl)-2,7-diaminocarbazole (A4) [44]: In a 250 mL flask, a mixture of compound A3 (1.67 g, 4.53 mmol), tin(II) chloride (SnCl 2 ) (10.22 g, 45.3 mmol), ethanol (27 mL) A N U S C R I P T and ethyl acetate 90 (d, M 2H, J = 7.6 Hz, N-CH 2 ); 3.75 (s, 4H, NH 2 ); 2.0 (m, 1H, CH-(CH 2 ) 3 ); 1.30 (m, 8H, CH 2 aliphatic);
	mL of boiling kerosene previously washed with concentrated sulfuric acid. The solution was stirred and refluxed for 1 h and kept at 0°C for 24 h. The obtained precipitate was filtered through a Buchner funnel. The crude product was washed with petroleum ether and recrystallized then in ethanol to give 5.54 g of the title product. M.P. > 300°C (Yield: 88%). 1 H NMR (400 MHz, Acetone-d 6 , ppm): 11.42 (s, 1H, N-H); 8.57 (s, 2H, Ar); 8.51 (d, 2H, J = 8.4 Hz, Ar); 8.17 (dd, 1H, J = 8.4 and 1.2 Hz, Ar). 13 C NMR (75 MHz, Acetone-d 6 , ppm): 147.23; 141.06; 126.36; 122.11; 114.74; 107.64. N-(2-ethylhexyl)-2,7-dinitrocarbazole (A3): A suspension of sodium hydride (NaH) (60% in oil, 1.60 g, 40 mmol) was gradually added to a solution of compound A2 (5.144 g, 20 mmol) in dimethylformamide (DMF) (60 mL) at 0°C. The mixture was stirred for 1h, 1-Bromo-2-0.90 (m, 6H, J = 7.6 Hz, -CH 3 ). 13 C NMR (75 MHz, CDCl 3 , ppm): 143.85 (Ar); 142.50 (Ar); 119.76 (Ar); 118.72 (Ar); 108.11 (Ar); 94.98 (Ar); 47.39 (N-CH 2 ); 39.08 (CH-(CH 2 ) 3 ; 30.99 (CH-CH 2 -CH 2 ); 28.88 (CH 2 -CH 2 -CH 2 ); 24.43 (CH-CH 2 -CH 3 ); 23.13 (CH 2 -CH 2 -CH 3 ); 14.12 (-CH 2 -A C C E P T E D CH 3 ); 11.00 (-(CH 2 ) 3 -CH 3 ).
	ethylhexane (30 mmol, 5.32 mL) was then added dropwise and the mixture was then allowed to
	react for 24h at room temperature. Ethanol was added to the resulting mixture to quench the
	remaining sodium hydride. The solution was poured into water, extracted four times with
	chloroform, washed several times with water and dried with anhydrous magnesium sulfate
	(Na 2 SO 4 ). After evaporating the solvent in vacuum, the residue was purified by column

evaporation of the solvent leaves 1.26 g of the title product as a pink solid. M.P. 74°C (Yield: 90 %).

1 H NMR (400 MHz, CDCl 3 , ppm): 7.68 (d, 2H, J = 8.0 Hz, Ar); 6.56 (m, 4H, Ar); 3.

  ) 3 ; 30.90 (CH-CH 2 -CH 2 ); 28.65 (CH 2 -CH 2 -CH 2 ); 24.30 (CH-CH 2 -CH 3 ); 23.12 (CH 2 -CH 2 -

	ACCEPTED MANUSCRIPT
	(Ar); 131.43 (Ar); 130.92 (Ar); 130.64 (Ar); 129.10 (Ar); 127.60 (Ar); 127.31 (Ar); 125.46
	(Ar);124.50 (Ar); 121.51 (Ar); 120.81 (Ar); 112.28 (Ar); 102.73 (Ar); 47.84 (N-CH 2 ); 39.24 (CH-
	(CH 2 CH 3 ); 14.04 (-CH 2 -CH 3 ); 10.91 (-(CH 2 ) 3 -CH 3 ). FT-IR (cm -1 ) 3080-3030 (w, aromatic C-H
	stretching), 2960-2846 (w, aliphatic C-H stretching), 1453 m, 1620 w (m, C=C stretching), 1590
	cm -1 (w, C=N symmetric stretching), 1320 (w, C-N stretching), 1250 cm -1 (w, H-C=N in plane
	1 H NMR (400 MHz, CDCl 3 , ppm): 9.88 (s, 2H, -N=CH); 8. 85 (d, 4H, J=8.8 Hz, Ar); 8.60 (s, 2H, Ar); 8.22 (d, 2H, J = 8.0 Hz, Ar); 8.09 (d, 4H, 8.4 Hz, Ar); 7.59 (m, 8H, Ar); 7.53 (s, 2H, Ar); 7.41 (dd, 2H, J= 8.4 and 1.6 Hz, Ar); 4.28 (m, 2H, N-CH 2 ); 2.23 A C C E P T E D (m, 1H, CH-(CH 2 ) M bending), 733 (s, aromatic C-H out-of-plane bending). HRMS: Calculated for C 52 H 43 N 3 : 685.35 A N U S C R I P T Found: 686.3558.

3 ); 1.34 (m, 8H, CH 2 aliphatic); 0.95 (t, 3H, J = 7.6 Hz, -CH 2 -CH 3 ); 0.83 (t, 3H, J = 7.2 Hz-(CH 2 ) 3 -CH 3 ). 13 C NMR (75 MHz, CDCl 3 , ppm): 159.24 (-N=CH); 150.57 (Ar); 142.55

Acknowledgements

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

We are thankful to the Ministry of Higher Education and Scientific Research-Tunisia for its support. We want to express our thanks to Christophe Mélart (ICPEES UNISTRA) for ATG measurements.