Isabela D Rodrigues
email: idrodrigues@usp.br

Americo Cunha
email: americo.cunha@uerj.br

André Teófilo

André T Beck
email: atbeck@sc.usp.br

Optimization of a 2D reinforced concrete frame considering a seismic load via cross-entropy method

Keywords: Risk optimization, reinforced concrete frame, seismic actions

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

Optimization of a 2D reinforced concrete frame considering a seismic load via cross-entropy method [START_REF] Abnt | NBR 15421: Projeto de estruturas resistentes a sismos -Procedimentos[END_REF]

Introduction

Earthquakes represent a concern for several countries, since they have the potential to cause a great number of casualties and damages in structures. Brazil is a mid-plate country located in the South America tectonic plate, which is considered a stable region when compared to places near the boundaries of tectonic plates. For the sake of comparison, a seismic event of magnitude 5 occurs in Brazil once in five years on average, while in the Andean region an earthquake of this magnitude happens on average twice a week [START_REF] Assumpc ¸ão | Terremotos no brasil: preparando-se para eventos raros[END_REF]. Even though Brazil is located inside a tectonic plate, it presents a considerable history of small to moderate earthquakes, also including two events with moment magnitude (M) higher than 6. Studies also report damages occurred in the João Camara earthquake, in Rio Grande do Norte state and Itacarambi earthquake, in the state of Minas Gerais [START_REF] Takeya | The 1986-1988 intraplate earthquake sequence near joão câmara, northeast brazil-evolution of seismicity[END_REF], [START_REF] Chimpliganond | The intracratonic caraíbas-itacarambi earthquake of december 09, 2007 (4.9 mb), minas gerais state, brazil[END_REF]. Such damages are explained in places with small to moderate hazards with the definition of risk, which considers hazard, exposure, vulnerability, and consequences. One should note, therefore, that a low hazard does not imply low seismic risk in a region [START_REF] Reiter | Earthquake hazard analysis: issues and insights[END_REF], especially if buildings are not properly designed to withstand seismic loads.

In order to evaluate the risk of building damage and collapse, the concept of Performance-based earthquake engineering (PBEE) has been developed over the years [START_REF] Haselton | Calibration of model to simulate response of reinforced concrete beam-columns to collapse[END_REF]. According to Krawinkler [START_REF] Krawinkler | Challenges and progress in performance-based earthquake engineering[END_REF], it corresponds to the design, evaluation and construction of structures whose performance under extreme loads responds to the needs and objectives of owners-users and society. As a preliminary step, design professionals, owners and other stakeholders identify the desired building performance, and, as the design decisions are made, it is necessary to evaluate if the final building can achieve the indicated performance.

Improving the structural behavior can also be done with optimization techniques. In this respect, the optimization process under uncertainties has many advantages over deterministic ones, since deterministic optimization considers uncertainties in an indirect way. When the cost of failure is incorporated in the objective function, the optimization problem becomes a Risk Optimization, or also Life Cycle Cost Optimization [START_REF] Beck | A comparison of deterministic, reliability-based and risk-based structural optimization under uncertainty[END_REF]. So, this paper aims to apply the cross-entropy optimization technique to minimize the cost of a 2D reinforced concrete frame subjected to a seismic action.

Optimization problem and Methodology

Problem definition

The objective of this work is to find a set of parameters for column and beam cross section that minimizes the total expected repair costs and the probability of collapse of a 2D reinforced concrete frame subjected to El Centro earthquake. The design random variables for the problem are column cross section width and height (columns are considered to be squared) (c h ) and beam height (b h ).

Determine: d* = {c ⇤ h , b ⇤ h }; That minimizes: f (d) = h c c 2 ⇥ c 2 h + b h ⇥ b w + P m,f ail k=1 c f,k ⇥ p f,k i Subjected to: 0.19  c h  0.40; 0.30  b h  0.50 (in meters) With: M = {C h , B h }; C h ⇠ T N (c h , 0.04, 4.27, 0.97) ; B h ⇠ T N (b h , 0.04, 4.53, 0.47) . (1)
Where: b w corresponds to the beam width; c c represents the initial cost of construction; c f corresponds to the expected cost associated with the failure mode considered; m, f ail corresponds to the modes of failure considered herein by the Damage Limit States. It is important to make clear that c h and b h are independent parameters. The initial cost of construction will be considered herein as the Brazilian "Basic Unity Construction Cost" (CUB) for the state of Sao Paulo, converted to dollar. For the month of June, the cost is estimated as R$1894.49/m 2 , and the dollar exchange rate is R$5.34, which means that the CUB value in dollar is $354.77/m 2 .

A representation of the 2D frame studied herein is presented in Figure 1a, with units in meters. Figure 1b shows the time-series for El Centro earthquake. The evaluation of the dynamic behavior of the 2D frame was done using OpenSees software [START_REF] Mckenna | Nonlinear finite-element analysis software architecture using object composition[END_REF], and details of the Finite Element Model used is presented in section 2.4. The design of the 2D frame considers the two columns containing 6 longitudinal reinforcement bars with diameter of 12.5 mm and a transverse reinforcement composed of one stirrup with two shear legs of 6.3 mm with 15 cm spacing. Beams have a fixed width (b w ) of 19 cm, and contain 2 longitudinal reinforcement bars with 12.5 mm diameter for tensile strength, and 2 longitudinal bars with 20 mm diameter for compression. The transverse reinforcement if done considers one stirrup with two shear legs of 5 mm diameter with 12 cm spacing. The compressive strength of concrete is 23 MPa, steel used is CA-50 and the concrete cover is 2.50 cm. Expected gravity loads are applied in the structure as an uniformly distributed load on the beam and are used to define seismic masses on the model. These loads include 1.05 times the dead load and 0.5 kN/m 2 for live load, which consists of 25% of the 2.0 kN/m 2 maximum live load for residential buildings established in ABNT NBR 6120 [START_REF] Abnt | NBR 6120: Cargas para o cálculo de estruturas de edificac ¸ões[END_REF]. P-Delta effects are considered in the columns.

In this paper, collapse is defined with the consideration of structural capacity (C) presented in Wen et al. [START_REF] Wen | Vulnerability function framework for consequence-based engineering[END_REF], corresponding to the maximum response a structure can withstand without reaching a limit state. This work considers the qualitative and quantitative definitions of the Damage Limit States proposed in Hazus manual [START_REF]Earthquake Loss Estimation Methodology: Advanced Engineering Building Module (AEBM), Technical and User's Manual[END_REF] for the building type reinforced concrete moment resisting frames (C1), being: Slight (SSD), Moderate (MSD), Extensive (ESD) and Complete (CSD) Structural Damage. Quantitative values for each Limit State considering a low-rise structure (1 to 3 pavements) and a pre-code classification, which means that structures do not take seismic actions in the design phase, are adopted considering values of median interstory drift (which means interstory displacement divided by story height) capacity (Sc), represented in Table 1.

All Damage Limit States are evaluated in this work, with an associated probability of failure for each one of them and an associated cost of failure. The reference of the cost is taken from Del Vecchio et al. [START_REF] Del | Repair costs of reinforced concrete building components: from actual data analysis to calibrated consequence functions[END_REF], that evaluated the actual repair cost of a database of 120 RC residential buildings damaged by the 2009 earthquake in L'Aquila, Italy. The costs were normalized and them calculated in dollars, making it possible to use the values for different regions from the study. The Damage States definition used by the authors are associated with the definitions from Hazus manual [START_REF]Earthquake Loss Estimation Methodology: Advanced Engineering Building Module (AEBM), Technical and User's Manual[END_REF], and the associated costs of failure (c f ) are summarized in Table 1 In each dynamic evaluation of the building, the algorithm check if any of the Limit States is reached, and save the information on the Index Function. Latter, using RWAS algorithm (described in section 2.3), the probability of failure (p f ) for each LS is calculated and, multiplied by the cost of failure, are summed in the objective function. An overview of the proposed methodology is presented in Figure 2, with all the implementation done using Matlab software [17]. All the steps of the methodology are described in details on the following sections.

Cross-entropy optimization method

This paper uses the cross-entropy method to find the optimum parameters of the 2D frame considering seismic actions. The full presentation and discussion of the method is done by Cunha [START_REF] Cunha | Enhancing the performance of a bistable energy harvesting device via the cross-entropy method[END_REF]. The main idea of the method used herein is to transform the non-convex optimization problem into an equivalent problem to estimate a rare event, which can be efficiently treated as a Monte Carlo like algorithm. It is only necessary that the problem has a single solution. The feasible region is sampled considering a chosen probability distribution, and mean and standard deviation of the samples are used to update the optimum point estimation.

The two steps of the process are defined as sampling and learning. In the sampling step, the feasible region is sampled considering the chosen probability distribution (truncated Gaussian in this case), and the objective function is evaluated in each one of the samples. Next, on the learning step, a subset of these samples, named as elite sample set, is defined considering the samples that produces the highest values for the objective function.

After that, the distribution is updated using statistics from this elite sample set, modifying the given distribution in a way to make it as close as possible to a Dirac delta centered on the global optimum. The distribution mean value gives an approximation to the global optimum, and its update is done to move the center of the distribution toward the optimization problem optimum, while decreasing the standard deviation and "shrinking" the distribution around its central value [START_REF] Cunha | Enhancing the performance of a bistable energy harvesting device via the cross-entropy method[END_REF]. Based on the steps described, the algorithm for the computational implementation of the method, based on Cunha [START_REF] Cunha | Enhancing the performance of a bistable energy harvesting device via the cross-entropy method[END_REF], is described below:

Step 1: Define the number of samples N s and the number of elite samples N e , where N e < N s ; define the convergence tolerance tol, the maximum of iteration levels t max , a family of probability distributions (this work uses Gaussian distribution, and the equations presented are valid for this distribution) and an initial vector of the parameters of the model;

Step 2: Generate the N s independent and identically distributed samples to be evaluated;

Step 3: Evaluate the objective function in all N s samples, sort the results and define the elite samples N e with the points that better performed;

Step 4: Update the estimators of the mean value (µ t ) and standard deviation ( t ) with aid of the elite samples set, as shown in Equation 2.

µ t = ↵µ t + (1 ↵) µ t 1 t = t t + (1 t ) t 1 (2)
Where µ t and t are the estimators with the aid of the elite sample on the actual iteration, ↵, and t are smooth parameters, and t is given by t

= ⇣ 1 1 q ⌘ q .
The parameters are such that 0 < ↵  1, 0.8   0.99 and 5  q  10.

Step 5: Repeat steps 2 to 4 until the stop criterion is met. Herein, max ( ) < tol.

Improved weighted average simulation (RWAS)

The improved weighted average simulation is a technique developed by Okasha [START_REF] Okasha | An improved weighted average simulation approach for solving reliability-based analysis and design optimization problems[END_REF] to solve structural reliability problems, based on the weighted average simulation method (WASM) proposed by Rashki et al. [START_REF] Rashki | A new efficient simulation method to approximate the probability of failure and most probable point[END_REF]. Its main goal is to determine the probability of failure by generating uniformly distributed samples and applying the probability density value as the weight index at each sample. After that, the probability of failure is computed by dividing the sum of the weight indices of all samples [START_REF] Rashki | A new efficient simulation method to approximate the probability of failure and most probable point[END_REF] [START_REF] Okasha | An improved weighted average simulation approach for solving reliability-based analysis and design optimization problems[END_REF]. The modification proposed allows the evaluation of the probability of failure with a small number of performance functions evaluation, since the p f converges faster to the final result with only a fraction of the generated samples [START_REF] Okasha | An improved weighted average simulation approach for solving reliability-based analysis and design optimization problems[END_REF].

At the beginning of the method, it is necessary the evaluation of the Index function I(i) for all the samples generated using uniform distribution (N us ). This step is necessary to distinguish if the samples are located in the failed region (g i < 0) or in the safe region, (g i 0), based on the definition of failure of the problem. The Index function is represented in Equation 3. Since the problem evaluated herein consists on design values considered also as random values, the evaluation of the Index function can be performed only once, since a change in the mean value consists in only evaluate again the weight index.

I(i) = ( 1, if g i < 0 0, if g i 0 (3)
The next step is to evaluate the PDF value (f X (x n )) of each random variable, based on the mean value for each variable considered in the step of the cross-entropy optimization method. With the PDF value, the weight indices of the samples are calculated as The modifications proposed in RWAS consist on sorting the generated samples in a descending order according to the values of their weight indices, with the id number of each sample assigned to the rank r i according to the place of this sample in the weight index sorting process [START_REF] Okasha | An improved weighted average simulation approach for solving reliability-based analysis and design optimization problems[END_REF]. The probability of failure is than calculated incrementally in the order of the ranks, considering Equation 4.

W (n) = f X (x n ) ⇥ f Y (y n ) ⇥ f Z (z n ),
P k f = P k i=1 I (r i ) • W (r i ) P k i=1 W (i) (4)
The incremental process can be terminated once a convergence criterion is reached. To evaluate this convergence criterion, it is necessary to evaluate the probability of failure already accumulated at increment k, as defined in Equation 5, which represents the upper limit value for the contributions of the remaining samples at increment k, assuming all of them are located in the failed region [START_REF] Okasha | An improved weighted average simulation approach for solving reliability-based analysis and design optimization problems[END_REF].

R k f = P N j=k+1 W (r j ) P N i=1 W (i) (5)
By doing this, the result is the highest probability of failure that can be predicted at increment k, even with the assumption that the rest of the samples all fail. The probability of failure cannot exceed the upper limit value of p f , given by Equation 6 [START_REF] Okasha | An improved weighted average simulation approach for solving reliability-based analysis and design optimization problems[END_REF].

P k f U = P k i=1 I (r i ) • W (r i ) + P N j=k+1 W (r j ) P N i=1 W (i) (6)
The convergence criteria can be considered as the step where the difference between the upper limit value of p f and the value of the accumulated probability of failure of the kth increment is smaller than a specified tolerance (T OL), which means

⇣ P k f ⌘ U P k f < T OL.

Lumped Plasticity Model

The Finite Element Model used herein consists on a lumped plasticity model developed in OpenSees software [START_REF] Mckenna | Nonlinear finite-element analysis software architecture using object composition[END_REF] and its own library to create beams and columns elements. Figure 3a represents the model in a 2D frame. The model is used to simulate the nonlinear hysteretic response of reinforced concrete (RC) beams or columns under large deformation and is also developed to enable simulation of the nonlinear dynamic response of RC frame buildings under earthquake ground motions.

To properly model the inelastic behavior of beams and columns elements, a nonlinear spring model developed by Ibarra et al. [START_REF] Ibarra | Hysteretic models that incorporate strength and stiffness deterioration[END_REF] is used. The material, named uniaxialMaterial IMKPeakOriented on OpenSees library, is applied to a zero-length element represented by the springs on Figure 3a. Joints are represented by an elastic element with the length of the joint and infinity stiffness. The rest of the element is modeled also with elasticBeamColumn element with its area and Young's Modulus of resistance of the material. To account for the degradation of strength and stiffness associated with large deformations, suitable geometric transformations, and a leaning (P-) column are used in the analysis. The effects of foundation flexibility have not been considered at this part of the model development.

The nonlinear spring model consists of a monotonic backbone curve and hysteretic degradation rules to capture post-peak in-cycle softening which are associated with concrete crushing and reinforcing bar buckling at large cyclic deformations [START_REF] Haselton | Calibration of model to simulate response of reinforced concrete beam-columns to collapse[END_REF]. Figure 3b represents the monotonic curve by an idealized trilinear end moment (M) versus chord rotation (✓) response of an equivalent cantilever column. The curve is defined considering five parameters: yield moment capacity M y ; initial elastic secant stiffness to yield point K e ; maximum moment capacity M c ; plastic chord rotation from yield to cap point ✓ cap,pl ; post-capping plastic rotation capacity ✓ pc . The flexural yield strength M y generally is computed using strain compatibility approach. It is assumed that sections remain plane and uses an equivalent rectangular compressive stress distribution under ultimate loads with a concrete crushing strain of 0.003 [START_REF] Haselton | Calibration of model to simulate response of reinforced concrete beam-columns to collapse[END_REF]. The eqautions for the model parameters can be found on [START_REF] Haselton | Calibration of model to simulate response of reinforced concrete beam-columns to collapse[END_REF], [START_REF] Haselton | Beam-column element model calibrated for predicting flexural response leading to global collapse of RC frame buildings[END_REF] and [START_REF] Rodrigues | Avaliac ¸ão da vulnerabilidade sísmica de edificac ¸ões regulares em concreto armado no brasil através da elaborac ¸ão de curvas de fragilidade[END_REF].

Results and Discussion

The solution of the optimization problem is presented. The domain is randomly sampled considering N s = 50 points, considering a truncated Gaussian distribution. The elite samples are selected considering N e = round (N s /10), the maximum number of iteration is t max = 150, and the convergence criteria is set as tol = 5⇥10 4 . The smoothing parameters are ↵ = 0, 7, = 0.8 and q = 5. The number of uniformly distributed samples for the RWAS method is N us = 5000 samples.

Figure 4 shows the domain sampling at different iterations of the algorithm, illustrating the CE method. The evolution of the algorithm is also represented in Table 2, where each line displays the iteration, the total cost obtained by the objective function (f (d)), mean and standard deviation of c h and b h . In this simulation, the optimum value obtained is $128.99 with the optimum dimensions of c h = 35.49 cm and b h = 40.32 cm. To check the accuracy of the results, the process is repeated 5 times, all with the same algorithm set up, to check the values of the objective function and dimensions of the frame. The summary of the process is shown in Table 3. One should notice that the results presented in Figure 4 and Table 2 correspond to simulation number 5. The results obtained in the 5 simulations are very close to each other, with the minimum cost of, approximately, $129.00. For simulation 5, the cost of construction corresponds to $116.14 (90.04% of total cost) and the cost of failure is $12.86 (9.96% of total cost). One can also notice that an increase on the mean value of the column cross section makes the beam height decrease to the minimum value, as observed in simulation 03. The performance of the algorithm can be evaluated by the time necessary to complete all 5 simulations, which are 17.84 hours in a Desktop Intel Core i5-10400 2.90 GHz, with RAM 16GB. The step that consumes most of the time on the simulations is the dynamic evaluation of the frame on OpenSees to calculate the Index Function for the RWAS, which also indicates the need to have a Finite Element Model that captures precisely the response of the building with a good processing time, justifying the choice for the Lumped Plasticity Model.

Conclusions

The results of this paper show the accuracy of the cross-entropy optimization method, associated with the improved weighted average simulation (RWAS) to calculate the probability of failure of a 2D reinforced concrete frame subjected to a seismic load. All 5 simulations performed converged to the same minimum of the objective function, making the algorithm suitable to find the optimum dimensions of column and beam cross section, considering a Risk Optimization process. This is an important step for the development of a optimization process of reinforced concrete structures with seismic actions, considering the PBEE.

CILAMCE-2022

Proceedings of the XLIII Ibero-Latin-American Congress on Computational Methods in Engineering, ABMEC Foz do Iguac ¸u, Brazil, November 21-25, 2022

Figure 1 .

 1 Figure 1. Representation of the 2D RC frame studied herein and the time series of El Centro earthquake

Figure 2 .

 2 Figure 2. Flowchart of the methodology

  with X, Y and Z being the random variables considered as statistically independent.CILAMCE-2022Proceedings of the XLIII Ibero-Latin-American Congress on Computational Methods in Engineering, ABMEC Foz do Iguac ¸u, Brazil, November 21-25, 2022

Figure 3 .

 3 Figure 3. Representation of the Finite Element Model and Idealized trilinear end moment versus chord rotation

Table 1 .

 1 . Damage Limit States considered, capacity (S C ) and cost of failure (c f ) associated (in dollars) Damage Limit State S c (%) c f ($)

	Slight Damage	0.40	66.56
	Moderate Damage	0.64	89.45
	Extensive Damage	1.60	108.91
	Complete Damage	4.00	140.34

Table 2 .

 2 Evolution of the CE algorithm t f(d) ($) µ c h (m) µ b h (m)

	c h (m)	b h (m)

Table 3 .

 3 Results of all the simulations performed

	Simulation f (d) ($) µ c h (m) µ b h (m) Number of iterations
	01	128.91	0.3556	0.3866	20
	02	128.36	0.3525	0.4060	20
	03	128.19	0.3640	0.3002	10
	04	129.41	0.3558	0.3950	14
	05	128.99	0.3549	0.4032	16

Acknowledgements. This study was financed by the Sao Paulo Research Foundation (FAPESP) -Finance Codes 2020/14072-7 and 2019/13080-9; Brazilian Higher Education Council (CAPES); Brazilian National Council for Research (CNPq)-Finance Code 309107/2020-2; Fundac ¸ão Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ) -Finance Codes 210.167/2019, 211.037/2019 and 201.294/2021. The opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors only and do not necessarily reflect the views of the sponsors or affiliates Authorship statement. The authors hereby confirm that they are the sole liable persons responsible for the authorship of this work, and that all material that has been herein included as part of the present paper is either the property (and authorship) of the authors, or has the permission of the owners to be included here.


Cyril Poriel

Joelle Rault-Berthelot

Dihydroindenofluorene Positional Isomers

1

Dihydroindenofluorene positional isomers

Cyril Poriel* and Joëlle Rault-Berthelot* Univ Rennes, CNRS, ISCR-6226, F-35000 Rennes, France Email: Cyril.poriel@univ-rennes1.fr, joelle.rault-berthelot@univ-rennes1.fr

CONSPECTUS.

Bridged oligophenylenes are very important organic semiconductors (OSC) in Organic Electronics (OE). The fluorene unit, which is a bridged biphenyl, is the spearhead of this class of materials and has led in the last twenty years to fantastic breakthroughs in Organic Light-Emitting Diodes (OLEDs). The dihydroindenofluorenes belong to the family of bridged terphenyls and can be viewed as the fusion of a fluorene unit with an indene fragment. The dihydroindenofluorenes have also appeared as very promising building blocks for OE applications. In the dihydroindenofluorene family, there are five positional isomers, with three different phenyl linkages (para/meta/ortho) and two different ring bridges arrangements (anti/syn). We focused our attention on the concept of positional isomerism. Indeed, the structural differences of the dihydroindenofluorenyl cores lead to unusual electronic properties, which have been described by our group since 2006 thanks to the five DiSpiroFluorene-IndenoFluorene positional isomers (dihydroindenofluorenes substituted on the bridges by fluorenyl units, Figure 1). 6,12-dihydroindeno [1,2-b]fluorene (the para anti isomer) is constructed on a para terphenyl core and possesses an anti geometry. If this isomer has been widely investigated over the last twenty years, studies of the four others remain very scarce. Thus, 11,12-dihydroindeno[2,1a]fluorene (the para syn isomer) is also built on a bridged para terphenyl but possesses a syn geometry. This particular geometry has been advantageously used by our group to drastically tune the electronic properties and this isomer has emerged as a promising scaffold to obtain a stable blue emission arising from conformationally-controlable intramolecular excimers. These preliminary studies have shown the crucial influence of the geometry on the electronic properties of the dihydroindenofluorenes.

The modification of the phenyl linkages from a para linkage to a meta linkage provides the 'meta isomers', namely 7,12-dihydroindeno[1,2-a]fluorene (the meta anti isomer) and 5,7dihydroindeno[2,1-b]fluorene (the meta syn isomer). With these two regioisomers, the strong impact of both the linkage and the geometry on the electronic properties were particularly highlighted over the years. The last positional isomer of the family is 5,8-dihydroindeno[2,1c]fluorene, which possesses a central ortho terphenyl backbone and a syn geometry. This isomer offers a unique example due its ortho linkage which induces a particular helicoidal turn of the dihydroindenofluorenyl core.

Thanks to a structure-properties relationship approach, we describe in the present account the molecular diversity of the five DiSpiroFluorene-IndenoFluorene positional isomers, and its consequences both in terms of organic synthesis and of electronic properties. This account shows how positional isomerism can be a powerful tool to tune the electronic properties of OSCs.

PROJECT ORIGINS

Regioisomerism, also called positional isomerism, is a central concept in organic chemistry which can have remarkable consequences on the properties of molecules. A simple structural modification can drastically influence the electronic and physical properties of an OSC and hence the performance of the corresponding electronic device. With that in mind, our group has started in 2007 to investigate the impact of regioisomerism on the electronic properties of dihydroindenofluorene derivatives. This account focuses on this feature with the study of the five DiSpiroFluorene-IndenoFluorene regioisomers (DSF-IF, Figure 1).

Figure 1. DiSpiroFluorene-IndenoFluorene and TerPhenyl regioisomers

The dihydroindenofluorene family contains five regioisomers (Figure 1), with different phenyl linkages (para/meta/ortho) and different ring bridges arrangements (anti/syn). Although 6,12-dihydroindeno [1,2-b]fluorene (para anti isomer) has been described for the last twenty years, studies on the four other isomers remain very scarce. Thus, 11,12dihydroindeno[2,1-a]fluorene is also constructed on a bridged para terphenyl but possesses a syn geometry. The modification of the phenyl linkages from a para to a meta linkage provides 7,12-dihydroindeno[1,2-a]fluorene and 5,7-dihydroindeno[2,1-b]fluorene with an anti and a syn geometry respectively. The last isomer is 5,8-dihydroindeno[2,1-c]fluorene, which possesses a central ortho terphenyl and a syn geometry. In the dihydroindenofluorene nomenclature, the letter is assigned to the edge of the indene/fluorene fusion. It should be mentioned that the antiaromatic counterparts, indenofluorenes, have also attracted remarkable attention notably from the Haley group. [START_REF] Frederickson | Explorations of the Indenofluorenes and Expanded Quinoidal Analogues[END_REF][START_REF] Fix | Indenofluorenes and Derivatives:Syntheses and Emerging Materials Applications[END_REF] The present story begins in 2005 when our group started a new research field linked to materials for OE. Inspired by the work of Müllen [START_REF] Setayesh | Bridging the Gap between Polyfluorene and Ladder-Polyp-Phenylene:Synthesis and Characterization of Poly-2,8-Indenofluorene[END_REF] and Scherf [START_REF] Reisch | Poly(Indenofluorene) (PIF), a Novel Low Band Gap Polyhydrocarbon[END_REF] on dihydroindeno [1,2-b]fluorene and the work of Salbeck on spirobifluorene, [START_REF] Saragi | Spiro Compounds for Organic Optoelectronics[END_REF] the first example of a DiSpiroFluorene-IndenoFluorene, 1a, constructed with the 6,12dihydroindeno [ 9,9'-spirobifluorene (SBF) is the association of two fluorenes through a shared spiro carbon and is obtained from the coupling of 2-iodobiphenyl 6 with fluorenone 7 followed by an intramolecular cyclisation of resulting fluorenol 8 (Scheme 1, Top). [START_REF] Saragi | Spiro Compounds for Organic Optoelectronics[END_REF] As 1a can be considered as the fusion of two SBF, a similar strategy was imagined either from indeno[1,2-b]fluorene-6,12-dione 9 (Route A, Scheme 1, Bottom-Right) or from diiodoterphenyl 11 (Route B, Scheme 1, Bottom-Left). These two routes are drastically different in terms of regioselectivity. If route A is regioselective, exclusively leading to 1a, route B is not and has led, in 2008, to the first example of a [2,1-a] regioisomer (2b, Figure 2). [START_REF] Poriel | New Dispiro Compounds:Synthesis and Properties[END_REF] This notion of regioselectivity vs non-regioselectivity is an important concept in this account. Indeed, a regioselective approach appears in general more secure as it leads to a single regioisomer.

Although more risky, a non-regioselective approach may lead to different regioisomers, which can be an appealing feature if their separation is possible.

Scheme 1. Retrosynthetic analysis towards 9,9'-spirobifluorene (Top) and 1a (Bottom)

Route A (Scheme 2) is based on the coupling reaction between 9 8-10 and 6. This reaction provides diol 10 further involved in an intramolecular electrophilic cyclisation leading to 1a. [START_REF] Cocherel | New 3π-2Spiro Ladder-Type Phenylene Materials:Synthesis, Physicochemical Properties and Applications in OLEDs[END_REF] This route is efficient and regioselective, exclusively providing 1a. It should be noted that such types of intramolecular cyclisations are efficient and versatile tools to generate spiro carbons.

Scheme 2. Synthesis of 1a-Route A At first sight, the second strategy (Route B, Scheme 3) 6,10 looked as straightforward as the first (Route A, Scheme 2). However, this route involves in its last step the intramolecular bicyclisation of the diol 12 providing as expected 1a but also a second regioisomer 2a.

Unfortunately, we failed to detect in 2006 [START_REF] Horhant | Dispirofluoreneindenofluorene (DSFIF):Synthesis, Electrochemical, and Optical Properties of a Promising New Family of Luminescent Materials[END_REF] the presence of 2a, which was only revealed two years later. [START_REF] Poriel | New Dispiro Compounds:Synthesis and Properties[END_REF] We did not expect, at that early stage, both the solvent and the temperature of the final cyclisation to be key parameters influencing the ratio of isomers formed.

Scheme 3. Synthesis of 1a and 2a-Route B From a mechanistic point of view, the bicyclisation of 12 (Figure 2, Left) unfolds sequentially. The first cyclisation provides the 2-substituted spirobifluorene. The second cyclisation can occur either on the opposite side leading to the anti isomer 1a or on the same side of the first cyclisation, leading to the syn isomer 2a. In 2009, we observed that the ratio of the anti/syn isomers could be tuned as a function of three parameters: solvent, temperature and substituents borne by the fluorenes (Figure 2). [START_REF] Poriel | Encumbered DiSpiro[Fluorene-IndenoFluorene]:Mechanistic Insights[END_REF] For example, the 1a/2a ratio can be modified from 99/1 in CH 2 Cl 2 at room temperature to 91/9 in CH 3 CN at reflux. The effect on the 1b/2b ratio is more pronounced shifting from 74/26 in CH 2 Cl 2 at room temperature to 34/66 in CH 3 CN at reflux. These effects are consistent with the stabilisation of the carbocation and to a less hampered interconversion between conformers. [START_REF] Poriel | Encumbered DiSpiro[Fluorene-IndenoFluorene]:Mechanistic Insights[END_REF] In addition to these solvent and temperature effects on the selectivity, another important element that affects the isomers distribution is the nature of the R groups borne by the fluorenes. In fact, as R becomes more encumbering (from R=H in 1a/2a to R=3,5-di-tert-butylphenyl in 1e/2e), the distribution gradually shifts towards the formation of [2,1-a] isomers at the expense of [1,2-b] isomers. The reaction becomes even regioselective with R=3,5-di-tert-butylphenyl exclusively providing 2e. This observation was at first confusing as 2a-2e were less energetically stable compared to 1a-1e. [START_REF] Poriel | Encumbered DiSpiro[Fluorene-IndenoFluorene]:Mechanistic Insights[END_REF] The mechanistic investigations have shown that the increased selectivity towards [2,1-a] isomers as a function of the temperature, solvent and substituent, involved the relative locking of the encumbered rotamers in the more stable pro-2 conformation due to a high energy barrier for the interconversion to the pro-1 conformation. Thus, the synthesis can be directed towards the formation of [1,2-b] isomers with: (i) low polarity solvents, (ii) low temperature and (iii) small R groups. The opposite leads to the [2,1a] isomers. As above mentioned, when R=H, the maximum yield obtained for 2a was low (9%) rendering very difficult its separation by column chromatography from its isomer 1a. We thus started in 2010 to investigate a regioselective route towards 2a involving the dione 14 (obtained from the intramolecular cyclisation of 13, [START_REF] Thirion | 2,1-a)-Indenofluorene Derivatives:Syntheses, X-ray Structures, Optical and Electrochemical Properties[END_REF] Scheme 4). The key feature of this approach is to build the dihydroindeno[2,1-a]fluorenyl core prior to the final cyclisation step of 15 in order to avoid the formation of the other isomer. In 2013, the meta isomers 3a (syn isomer) and 4a (anti isomer) represented the second generation of DSF-IFs (Figure 1). [START_REF] Romain | Dependence of the Properties of Dihydroindenofluorene Derivatives on Positional Isomerism: Influence of the Ring Bridging[END_REF] Their respective dihydroindenofluorenyl cores were nevertheless already described in 1955. [START_REF] Chardonnens | Synthèses dans la Série des Indéno-Fluorènes IV. Cis-Fluorènacène (Indéno-(2′,1′:2,3)-Fluorène) et Trans-Fluorènaphène (Indéno-(1′,2′: 1,2)-Fluorène)[END_REF] Theoretically, these meta isomers 3a/4a can be synthesized following similar approaches than those exposed above for para isomers, following either a regioselective route A (Scheme 5) or a non-regioselective route B (Scheme 6). Thus, 3a was regioselectively synthesized from a short and efficient route involving the synthesis of dione 17 (obtained from meta-substituted terphenyl 16), followed by the intramolecular cyclisation of indenofluorenol 18 (Scheme 5). [START_REF] Romain | The Structure-Property Relationship Study of Electron-Deficient Dihydroindeno[2,1b]fluorene Derivatives for n-Type Organic Field-Effect Transistors[END_REF][START_REF] Romain | Thioxanthene and Dioxothioxanthene Dihydroindeno[2,1-b]Fluorenes: Synthesis, Properties and Applications in Green and Sky-Blue Phosphorescent OLEDs[END_REF] Scheme 5. Synthesis of 3a-Route A

The same regioselective approach was unfortunately not successful for the synthesis of 4a, which was instead synthesized from the non-regioselective Route B (Schemes 6 and 7). Two different synthetic strategies towards 3a/4a were developed in this context. The first (Scheme 6) was directly adapted from that of 1a/2a and involved diodoterphenyl 19 as precursor of 3a/4a (ratio: 65/35). [START_REF] Romain | ortho-, meta-, and para-Dihydroindenofluorene Derivatives as Host Materials for Phosphorescent OLEDs[END_REF] This route is versatile as many DSF-IF isomers substituted on the fluorenes can be synthesised by simply changing the fluorenone during the coupling with 19 (coupling with fluorenone 7 provides diol 20 and coupling with fluorenone 21 provides diol 22). In 2011, we used this route to synthesise 3b/4b substituted on the fluorenes with tertbutyl groups. [START_REF] Poriel | A Robust Pure Hydrocarbon Derivative Based on the (2,1-b)-Indenofluorenyl Core with High Triplet Energy Level[END_REF] Scheme 6. Synthesis of 3a/4a and 3b/4b-Route B-Strategy 1

In all the non-regioselective synthetic routes based on an intramolecular bicyclisation, the diols were key compounds. Another route towards the methoxy protected diol 27 was therefore developed in 2013 (Scheme 7). [START_REF] Romain | Dependence of the Properties of Dihydroindenofluorene Derivatives on Positional Isomerism: Influence of the Ring Bridging[END_REF] The first step involved the synthesis of 9-(2bromophenyl)-9H-fluoren-9-ol 24 from a Grignard reaction between 23 and 7. Compound 24 was then protected with a methyl group (25) in order to avoid the formation of the palladacycle 26 during the cross-coupling which provided 27. The intramolecular bicyclisation of 27 finally lead to the formation of the two isomers 3a/4a (50/50). The last isomer 5a derived from the ortho-terphenyl core may also theoretically be obtained through Route A (from 28/29) or Route B (from 30/31), Scheme 8. However, none of these two routes were used to prepare 5a notably because both 28 and 30 appeared to be difficult to synthesize. Another synthetic approach towards 5a (Scheme 8, Route C) involving an orthosubstituted spirobifluorene (32) [START_REF] Poriel | Structure-Property Relationship of 4-Substituted-Spirobifluorenes as Hosts for Phosphorescent Organic Light-Emitting Diodes:An Overview[END_REF] was therefore developed in 2015. This route is regioselective (Scheme 9). [START_REF] Romain | ortho-, meta-, and para-Dihydroindenofluorene Derivatives as Host Materials for Phosphorescent OLEDs[END_REF] A phenyl carboxylate unit was first attached to C4 of the SBF backbone (33) before regioselectively constructing the dihydroindeno[2,1-c]fluorenyl core by an intramolecular cyclisation leading to the ketone 34. Then, the anchoring of a biphenyl unit on 34 provided indenofluorenol 35 further cyclised to provide 5a.

Scheme 9. Synthesis of 5a-Route C

We have shown in these studies that a non-regioselective synthesis can be a powerful tool to obtain two regioisomers in a single step. Although all these molecules were initially designed for electronics, the synthetic approach itself became, from an organic chemistry point of view, highly appealing to explore. Due to the different linkages and bridges, the dihydroindenofluorenyl and fluorenyl fragments possess different structural characteristics (Figures 3 and4) which have important consequences on the electronic properties described below. For example, the meta linkages in 3a/4a [START_REF] Romain | Dependence of the Properties of Dihydroindenofluorene Derivatives on Positional Isomerism: Influence of the Ring Bridging[END_REF] and the ortho linkage in 5a [START_REF] Romain | ortho-, meta-, and para-Dihydroindenofluorene Derivatives as Host Materials for Phosphorescent OLEDs[END_REF] lead to a contraction of the dihydroindenofluorenyl core compared to 1a' 11 /2a [START_REF] Thirion | 2,1-a)-Indenofluorene Derivatives:Syntheses, X-ray Structures, Optical and Electrochemical Properties[END_REF] (see dihydroindenofluorene length in Figure 3). It should be noted that the literature only reports the X-Ray structure of the diisopropyl analogue 1a' [START_REF] Cocherel | New 3π-2Spiro Ladder-Type Phenylene Materials:Synthesis, Physicochemical Properties and Applications in OLEDs[END_REF] and not 1a itself. The dihydroindenofluorene contraction is particularly visible in 3a with its crescent moon shape and in 5a with its particular helicoidal turn. However, the meta linkages of the dihydroindeno[1,2-a]/[2,1-b]-fluorenyl cores are only partially responsible for this contraction. In the case of 4a, the contraction is accentuated by the anti geometry. A similar contraction effect caused by the ring bridging is also found for para isomers, the contraction being this time accentuated by the syn geometry (dihydroindenofluorenyl core of 2a is shorter than that of 1a').

Physico-Chemical Properties: Influence of the linkages and of the ring bridges arrangements

The ring bridging also has important consequences on the flatness and the deformations of the dihydroindenofluorenyl core. For example, in the meta isomers series the dihydroindeno The spiro-connected fluorenes in the syn isomers 2a, 3a and 5a are also strongly impacted by positional isomerism (Figure 4). As the distance between the spiro carbons is significantly shorter for 2a (3.44 Å) compared to that of 3a (5.27 Å), due to the different linkages (Figure 3), a different arrangement of the cofacial fluorenes is detected. The fluorene are staggered in 2a and eclipsed in 3a (Figure 4). Thus, in 2a, there is an interaction between the two cofacial fluorenes (which will be used to tune the optical properties of aryl substituted [2,1-a] isomers, see below) and not in 3a. The ortho linkage of the last syn isomer in the series, 5a, leads to an even stronger elongation of the Cspiro-Cspiro distance (5.76 Å) and to a different arrangement of the fluorenes.

To conclude, the bridging and the linkages have strong repercussions on the structural properties of both dihydroindenofluorenes and fluorenes, which have themselves important consequences on the electronic characteristics described below. Table 1 provides a summary of the electronic properties of 1a-5a. The five isomers display different electrochemical behaviour (Figure 5), which highlights the impact of isomerism on the electron transfers. 1a presents two oxidation waves with maxima at 1.47 V and 1.87 V/SCE whereas 2a possesses three waves at 1.36, 1.69 and 1.99 V (Figure 5, Left). [START_REF] Thirion | 2,1-a)-Indenofluorene Derivatives:Syntheses, X-ray Structures, Optical and Electrochemical Properties[END_REF] The first monoelectronic oxidation wave has been assigned for 1a/2a to the oxidation of the dihydroindenofluorenyl core. Compared to 1a, two differences are pointed out for 2a: (i) the first oxidation at 1.36 V is shifted towards less anodic potentials and (ii) an additional oxidation process is surprisingly observed at 1.69 V. The shift observed between the first oxidation potentials of 1a and 2a cannot be ascribed to the different geometry of the two dihydroindenofluorenyl cores but to the specific arrangement of the two fluorenyl units. [START_REF] Thirion | 2,1-a)-Indenofluorene Derivatives:Syntheses, X-ray Structures, Optical and Electrochemical Properties[END_REF] In fact, in 2a, the dihydroindenofluorenyl core is subjected to a less important withdrawing effect of the cofacial fluorene dimer compared to that of the two non-interacting fluorenyl units in 1a. The additional oxidation observed at 1.69 V for 2a was also at first sight surprising and ascribed to the oxidation of this cofacial fluorene dimer. Thus, the different ring bridges arrangement of 1a and 2a and the resulting different geometry induce different electrochemical properties. As both molecules are constructed on an identical para-terphenyl core, this demonstrates that the linkage is not the only factor involved in the electronic properties.

Electrochemical Properties

Electrochemistry of the meta isomers 3a and 4a provides a different example (Figure 5, Middle). Both compounds possess two successive oxidation waves (3a: 1.53/1.94 V, 4a: 1.57/1.88 V). The first monoelectronic oxidation has been assigned to the oxidation of the dihydroindenofluorenyl core and the second multielectronic wave to the concomitant oxidation of fluorenyl units and dihydroindenofluorenyl radical cation. Again, the effects of the phenyl linkages and of the ring bridging on the electrochemical properties need to be pointed out. The shift of the first oxidation potential to more positive values from 1a/2a [START_REF] Thirion | 2,1-a)-Indenofluorene Derivatives:Syntheses, X-ray Structures, Optical and Electrochemical Properties[END_REF] to 3a/4a [START_REF] Romain | Dependence of the Properties of Dihydroindenofluorene Derivatives on Positional Isomerism: Influence of the Ring Bridging[END_REF] has been assigned to a linkage effect (para vs meta linkages) and translates a shorter π-conjugation pathway in meta isomers compared to para ones.

There is also a slight shift in the first oxidation potential between 3a and 4a despite their identical meta linkages. This small difference was assigned to the contraction and the deformation of the dihydroindeno[1,2-a]fluorenyl core of 4a (vs the dihydroindeno[2,1b]fluorenyl core of 3a), which induce a poorer delocalisation of π-electrons and hence a more anodic oxidation potential. Isomer 5a displays again a unique behaviour (Figure 5, Right). The first oxidation is only detected as a shoulder at 1.6 V, providing a HOMO level lying at -5.87 eV, almost identical to that of 4a (-5.86 eV) but deeper than those of 1a-3a (Table 1). However, the HOMO of 5a (mainly located on the fluorenes) is noticeably different than those of all the other isomers (located on the dihydroindenofluorene), Figure 6. The ortho terphenyl linkage of 5a causes the deformation of the dihydro[2,1-c]indenofluorenyl core leading to an inversion of the HOMO and HOMO-1 compared to the other isomers. [START_REF] Romain | ortho-, meta-, and para-Dihydroindenofluorene Derivatives as Host Materials for Phosphorescent OLEDs[END_REF] These electrochemical investigations clearly show the diversity of these regioisomers. 13,14,18,21- 23 The linkage has important consequences on the π-electrons delocalization but is far from being the only parameter driving the electronic properties. The bridging is also important and has a double role: (i) it modifies the geometry (syn vs anti) and can force the two spiro-linked fluorenes to interact (such as in 2a) and/or (ii) can increase the deformation of the dihydroindenofluorenyl core.

Optical Properties

The positional isomerism of the dihydroindenofluorenyl core also has strong repercussions on the optical properties. In THF, 1a-5a present almost identical low-energy absorption maxima (Table 1) with nevertheless an impressive difference in the intensity of this band (Figure 7, Left). 1a and 2a possess similar and well resolved absorption spectra, which is a characteristic of highly rigid structures. With respect to 1a, the UV-vis spectrum of 2a exhibits a main absorption band slightly hypsochromically shifted. This blue shift has been assigned to a better delocalisation of π-electrons in 1a compared to 2a. Oppositely to the electrochemistry, 1a and 2a display a similar behaviour in absorption spectroscopy and the cofacial fluorenes of 2a only have a limited influence on the electronic transitions. The spectrum of 3a is also similar in shape but is red shifted compared to the para syn homologue 2a. Therefore, para substituted derivatives do not always possess longer π-conjugated pathways compared to meta substituted derivatives and this feature shows that the linkages are not the only parameters involved in the optical properties and that the ring bridging is also of chief importance. [START_REF] Sicard | Spirobifluorene Regioisomerism:A Structure-Property Relationship Study[END_REF] This is one of the most important findings we learned from these isomers. The different spectrum of 4a displaying very weak bands at 333 and 342 nm confirms the key role played by the ring bridging on the transitions. Indeed, isomers 1a-3a all possess intense low-energy absorption bands. Compared to 1a-3a isomers, 4a has no symmetry axis/plane, and its constrained structure leads to a deformation of the dihydroindeno[1,2-a]fluorenyl core, which has a strong influence on the partially allowed/forbidden character of its first -* transition. Isomer 5a, which also possesses a highly distorted dihydroindenofluorenyl backbone displays an identical characteristic with a weak band at 338 nm. [START_REF] Romain | ortho-, meta-, and para-Dihydroindenofluorene Derivatives as Host Materials for Phosphorescent OLEDs[END_REF] Studying the corresponding non-bridged terphenyls (para-Terphenyl pTP, meta-Terphenyl mTP and ortho-Terphenyl oTP, Figure 1) has allowed a better understanding of the impact of the bridges on the electronic properties of dihydroindenofluorenes. There is indeed an impressive red shift between the absorption maxima of terphenyls (Figure 8) and those of their corresponding dihydroindenofluorene analogues. This feature has been attributed to the presence of the bridges in the latters. In fact, the bridges avoid any torsion between the constituted phenyl units in the dihydroindenofluorenes, feature responsible for the disruption of the -conjugation in the terphenyls. In addition, the lowest energy band of each terphenyl corresponds to a HOMO/LUMO transition, which is not the case for 4a and 5a. Therefore, the presence of the bridges has changed the nature of the main transition in 4a and 5a compared to their terphenyl analogues mTP and oTP but not for the three other isomers 1a-3a. This is an unexpected behaviour induced by the bridge effect in dihydroindenofluorenes. Regarding the fluorescence properties, 1a-3a possess well resolved spectra (Figure 7, Right) with a very small Stokes shift, both features being characteristic of highly rigid structures. 2a possesses the largest Stokes shift which has been linked to structural changes by theoretical calculations. [START_REF] Thirion | 2,1-a)-Indenofluorene Derivatives:Syntheses, X-ray Structures, Optical and Electrochemical Properties[END_REF] Indeed, at the ground state, 2a shows a staggered conformation of the fluorenes whereas in the first singlet excited state a cofacial eclipsed arrangement is detected (Figure 9). [START_REF] Thirion | 2,1-a)-Indenofluorene Derivatives:Syntheses, X-ray Structures, Optical and Electrochemical Properties[END_REF] Although this geometry change has no significant effect on the fluorescence spectrum of 2a, it is at the origin of the spectacular excimer emission which will be discussed below for 2c-e. Compared to all the other isomers, the quantum yield of 4a was found to be significantly lower and its fluorescence decay more complex (two lifetimes vs one lifetime), Table 1. Such unusual emission properties are the signature of unconventional processes, such as the presence of several emissive states or partially decoupled excited states. These studies have revealed the unique photophysical properties of the dihydroindeno[1,2-a]fluorenyl backbone compared to the other isomers and the remarkable impact of the regioisomerism on the photophysics. Again, the fluorescence spectra of terphenyls have been precious tools in the understanding of the emission properties. There is indeed a strong similitude between the emission spectra of 1a/2a and 3a/4a and those of their corresponding terphenyls pTP and mTP whereas their absorptions were significantly different (Figures 7 and8). Despite being built on a different linkage, these terphenyl isomers surprisingly display identical emission spectra. There is therefore a breakdown of the mirror image symmetry in their absorption/emission spectra assigned to the planarization of the terphenyl core in the excited state. [START_REF] Heimel | Breakdown of the Mirror Image Symmetry in the Optical Absorption/Emission Spectra of Oligo(para-Phenylene)s[END_REF] This planarization of pTP and mTP in the excited state leads to a geometry analogous to that of 1a/2a and 3a/4a, which are already planarized at the ground state. This important structural feature explains why the fluorescence spectra of dihydroindenofluorenes at room temperature remain roughly similar and almost independent of the linkages. Emission spectra at low temperature (77 K) have led to different conclusions. Emission spectra at 77 K (Figure 10) allow to determine the triplet state energy (E T ), which is a key property for host phosphors in phosphorescent OLEDs (PhOLEDs). [START_REF] Romain | Thioxanthene and Dioxothioxanthene Dihydroindeno[2,1-b]Fluorenes: Synthesis, Properties and Applications in Green and Sky-Blue Phosphorescent OLEDs[END_REF][START_REF] Romain | ortho-, meta-, and para-Dihydroindenofluorene Derivatives as Host Materials for Phosphorescent OLEDs[END_REF] As the two para isomers 1a/2a and the two meta isomers 3a/4a present almost identical spectra, the phosphorescent contributions are fully driven by the linkages with a very weak influence of the bridging. The E T is therefore identical for 1a/2a (2.52 eV) and for 3a/4a (2.76 eV). The E T of the ortho isomer 5a is reported at 2.63 eV. Interestingly, the corresponding terphenyls pTP, mTP and oTP (E T =2.55, 2.82 and 2.67 eV respectively) possess almost identical values than their corresponding dihydroindenofluorenes. Oppositely to the conclusions drawn above, this study of the terphenyls confirms that the nature of the linkage fully drives the E T . This is an important feature that enables the tuning of the triplet energy without strongly modifying the singlet energy. This can be advantageously used for example in the field of Thermally Activated Delayed Fluorescence materials. [START_REF] Yang | Recent Advances in Organic Thermally Activated Delayed Fluorescence Materials[END_REF]

Properties modulation

The positional isomers offer many possibilities in terms of molecular design but only a few have been investigated yet. Basically, in such structures, one can modify (i) the bridges and/or (ii) the substituents attached to the dihydroindenofluorenyl core (Scheme 10).

Scheme 10. Examples of substituted dihydroindenofluorenes

In 2010, we reported the incorporation of xanthenyl units on the bridges of a dihydroindeno [1,2-b]fluorenyl core (36, Scheme 10-Left). This efficient emitter (Φ=0.63) has been incorporated in non-doped blue-emitting OLEDs displaying high performance. [START_REF] Poriel | Incorporation of Spiroxanthene Units in Blue-Emitting Oligophenylene Frameworks:A New Molecular Design for OLED Applications[END_REF] Following a similar design strategy, the bridges of the dihydroindeno[2,1-b]fluorenyl core were substituted by thioxanthene and dioxothioxanthene (37-38). The incorporation of the sulphur atom allows tuning the HOMO/LUMO levels, retaining nevertheless the high E T of the dihydroindeno[2,1-b]fluorenyl core (2.76 eV), key feature for an application as host in blue PhOLEDs. [START_REF] Romain | Thioxanthene and Dioxothioxanthene Dihydroindeno[2,1-b]Fluorenes: Synthesis, Properties and Applications in Green and Sky-Blue Phosphorescent OLEDs[END_REF] Another appealing modification was presented by Marks et al, [START_REF] Usta | Synthesis, and Characterization of Ladder-Type Molecules and Polymers. Air-Stable, Solution-Processable n-Channel and Ambipolar Semiconductors for Thin-Film Transistors via Experiment and Theory[END_REF] who have incorporated dicyanovinylene units on the bridges of a dihydroindeno[1,2-b]fluorene (39). As dicyanovinylene units are strong electron-withdrawing groups, the electronic properties of the dihydroindeno [1,2-b]fluorene was significantly modified. 39 is weakly fluorescent and displays a low LUMO level whereas its analogue 1a is an efficient fluorophore with a high LUMO level (Table 1). In 2015, we reported a similar design with the incorporation of dicyanovinylene units on the bridges of a dihydroindeno[2,1-b]fluorene (40). [START_REF] Romain | The Structure-Property Relationship Study of Electron-Deficient Dihydroindeno[2,1b]fluorene Derivatives for n-Type Organic Field-Effect Transistors[END_REF] This molecule possesses a low LUMO level (-3.81 eV vs -1.75 eV for 3a) and was successfully used in nchannel OFETs with excellent electrical stability. This stability has allowed to incorporate these OFETs in "pseudo-complementary metal oxide semiconductor" inverters. During this work, compound 41 possessing differently substituted bridges was also reported.

Interestingly, 41 presents the electronic characteristics of the different functional groups borne by the bridges, ie dicyanovinylene and ketone. Such dihydroindenofluorenes possessing different bridge substitutions are almost absent from literature and could constitute an appealing future class of functional materials.

The second design strategy is related to the substitution of the dihydroindenofluorenyl core (Scheme 10, middle). Camerel et al reported the attachment of protomesogenic fragments on the dihydroindeno[1,2-b]fluorenyl core (42) to generate liquid crystalline phases. [START_REF] Thiery | Luminescence Modulation in Liquid Crystalline Phases Containing a Dispiro[fluorene-9,11′-Indeno[1,2-b]fluorene-12′,9′′-fluorene[END_REF] Temperature-dependent solid-state luminescence measurements have notably revealed that this compound displays interesting thermochromic luminescence properties. Another symmetric dye (43) incorporating electron-donating trimethoxyphenyl units on the dihydroindeno[1,2-b]fluorenyl core was also reported. [START_REF] Thirion | Intramolecular Excimer Emission as a Blue Light Source in Fluorescent Organic Light-Emitting Diodes:A Promising Molecular Design[END_REF] Due to the -conjugation extension, 43 is an efficient blue emitter (391/412 nm) displaying a very high quantum yield (0.9). These examples show how the electronic properties of the dihydroindenofluorenyl backbone can be easily tuned by molecular design to reach functional materials.

The first example of a non-symmetric DSF-IF (44) for OLEDs was reported in 2011. [START_REF] Thirion | Synthesis and Properties of a Blue Bipolar Indenofluorene Emitter Based on a D-π-A Design[END_REF] This molecule incorporates diarylamino (hole-transporting) and phenylbenzimidazolyl (electrontransporting) groups at each extremity of a dihydroindeno [1,2-b]fluorenyl core. This substitution gives to 44 very different properties than those of its unsubstituted analogue 2b. For example, 44 is a blue emitter ( max =435 nm) whereas its constituting building block 2b is a near-UV/violet emitter ( max =348 nm, Figure 11). As 44 possesses a high quantum yield (0.85) and bipolar properties, it has been successfully incorporated in single-layer OLEDs with performance impressively improved compared to 2b. [START_REF] Thirion | Synthesis and Properties of a Blue Bipolar Indenofluorene Emitter Based on a D-π-A Design[END_REF] Finally, both the bridges and the dihydroindenofluorenyl core have been modified, although this strategy is difficult from a synthetic point of view. The first bipolar molecules based on dihydroindeno[2,1-b]/[1,2-a]fluorenes 45/46 (Scheme 10, Right) appeared in 2015 displaying very different electronic properties compared to those of 3a/4a. [START_REF] Romain | Donor/Acceptor Dihydroindeno[1,2-a]fluorene and Dihydroindeno[2,1-b]fluorene:Towards New Families of Organic Semiconductors[END_REF] When incorporated as host in blue PhOLEDs, 46 displays higher performance than 45 (External Quantum Efficiency of 7.9% and 2.6% respectively) which is attributed to a significant charge-carrier mobility difference (10 -4 cm 2 /V.s and 10 -6 cm 2 /V.s respectively). This highlights the crucial role played by the geometry profiles in the design of host materials for PhOLEDs.

We cannot report herein all the modifications made to the dihydroindenofluorenes. An exhaustive review is currently in preparation. If all the previous design strategies are rather classic, the last with which we would like to conclude is much more original (Figure 11) and draws benefit from the particular syn geometry of 2a. [START_REF] Thirion | Violet-to-Blue Tunable Emission of Aryl-Substituted Dispirofluorene-Indenofluorene Isomers by Conformationally-Controllable Intramolecular Excimer Formation[END_REF][START_REF] Thirion | Intramolecular Excimer Emission as a Blue Light Source in Fluorescent Organic Light-Emitting Diodes:A Promising Molecular Design[END_REF][START_REF] Poriel | Blue Emitting 3 π-2 Spiro Terfluorene-Indenofluorene Isomers:A Structure-Properties Relationship Study[END_REF] Thanks to this geometry, the dihydroindeno[2,1-a]fluorenyl core can hold two fluorescent units in a cofacial arrangement leading to an emission arising from intramolecular excimers. As above exposed, the couples 1a-2a/1b-2b display very similar optical properties as the main emission is induced by the corresponding dihydroindenofluorenyl core (Figure 11). However, when the fluorenes are substituted by aryl units (1c-d/2c-d, see structures in Figure 2), drastically different emission properties are obtained. Indeed, if the aryl-substituted 1c-d are violet emitters with classic emission properties, the 2c-d isomers are blue emitters with emission arising from the interactions of the cofacial "aryl-fluorene-aryl" arms (Figure 11, Bottom-Left). [START_REF] Thirion | Violet-to-Blue Tunable Emission of Aryl-Substituted Dispirofluorene-Indenofluorene Isomers by Conformationally-Controllable Intramolecular Excimer Formation[END_REF] Finally, 2c-d also possess remarkable tuneable optical properties since the emission wavelength can be modulated through the steric hindrance between the substituted phenyl rings leading to conformationally-controlable intramolecular excimer formation (Figure 11, Bottom-Right). When the substituent borne by the phenyl unit is bulky such as a tert-butyl group in 2c, the two "aryl-fluorene-aryl" arms move away from each other reducing their mutual interactions, resulting in a blue shift of the emission compared to 2d. [START_REF] Thirion | Violet-to-Blue Tunable Emission of Aryl-Substituted Dispirofluorene-Indenofluorene Isomers by Conformationally-Controllable Intramolecular Excimer Formation[END_REF] These families of dyes can therefore cover a large wavelengths range from near-UV to sky-blue.

Conclusion

The concept of positional isomerism in the design of OSCs is more and more used and dihydroindenofluorene isomers have undergone a significant development in the last ten years. We show in this account how positional isomerism can be an efficient tool to tune the electronic properties of dihydroindenofluorenes. As the two main molecular parameters involved, ie bridges and linkages, can be combined in many different ways, they offer powerful design possibilities. However, except the dihydro [1,2-b]indenofluorene, the design of the other isomers for OE is poorly developed and provides an interesting future direction. Another appealing design strategy concerns the modification of the π-systems as recently exemplified with dihydroindacenodithiophene regioisomers. [START_REF] Peltier | Modulating the Physical and Electronic Properties over Positional Isomerism:The Dispirofluorene-Dihydroindacenodithiophene (DSF-IDT) Family[END_REF] Finally, if the use of dihydroindenofluorenes was mainly directed to electronics, it is clear that other applications will be developed in the future. The recent work on predicted ordered materials based on the dihydro[1,2-b]indenofluorene is an example of a new direction for these compounds. 35 34 It most likely won't be the only one.

  1,2-b]fluorenyl backbone was reported in 2006. 6 Synthetic approaches a. Para isomers: dispiro[fluorene-9,6'-indeno[1,2-b]fluorene-12',9''-fluorene] 1a and dispiro[fluorene-9,11'-indeno[2,1-a]fluorene-12',9''-fluorene] 2a

Figure 2 .

 2 Figure 2. Proposed mechanism for the bicyclisation of difluorenols 12a-e (Left). Molecules 1a-e/2a-e (Top-Right). Ratio of regioisomers formed (Bottom-Right)1 2

Scheme 4 .

 4 Scheme 4. Synthesis of 2a-Route A

Scheme 7 .

 7 Scheme 7. Synthesis of 3a and 4a-Route B-Strategy 2

Figure 3 .

 3 Figure 3. ORTEP drawing of 1a' and of 2a-5a

  [2,1- b]fluorenyl core of 3a is almost flat whereas the dihydroindeno[1,2-a]fluorenyl core of 4a presents more pronounced deformations due to the strong contraction imposed by the bridging.[START_REF] Romain | Dependence of the Properties of Dihydroindenofluorene Derivatives on Positional Isomerism: Influence of the Ring Bridging[END_REF]

Figure 4 .

 4 Figure 4. ORTEP drawing of the syn isomers 2a, 3a and 5a

Figure 5 .

 5 Figure 5. Cyclic voltammetry of 1a-5a (CH 2 Cl 2 -Bu 4 NPF 6 , 0.2 M)

Figure 6 .

 6 Figure 6. Calculated LUMO (Top) and HOMO (Bottom) of 1a-5a

Figure 7 .Figure 8 .

 78 Figure 7. UV/Vis absorption (Left) and emission spectra (Right) of 1a-5a (THF)

Figure 9 .

 9 Figure 9. X-Ray structure (Left), optimised geometry in the ground state (Middle) and in the first singlet excited state (Right) of 2a1 3

Figure 10 .

 10 Figure 10. Emission spectra at 77K of 1a-5a (methylcyclohexane/2-methylpentane)

Figure 11 .

 11 Figure 11. Top: Emission spectra in THF of 1a-d (Left) and 2a-d (Middle), X-Ray structure of 2d (Right). Bottom: Photograph of the light emitted by 1d/2d, exc =365 nm (Left) and schematic representation of the phenyl substitution in [2,1-a] isomers (Right).

Table 1 .

 1 Selected electronic properties of 1a-5a a. in THF, b. quinine sulphate as reference, c. in methylcyclohexane/2-methylpentane, d. in CH 2 Cl 2 -Bu 4 NPF 6 , e. in DMF-Bu 4 NPF 6 , f. in THF-Bu 4 NPF 6 .

Acknowledgments

The authors thank all the collaborators in the group involved in the DSF-IF project. L.J. Sicard is especially acknowledged for many careful readings of the manuscript. We also thank the CINES (2018-A0040805032-GENCI-Montpellier), the ANR (11-BS07-020-01/14-CE05-0024), the Région Bretagne and the ADEME.

BIOGRAPHICAL INFORMATION

Cyril Poriel received his PhD in 2003 from the University of Rennes 1. After a postdoctoral stay at the University of Exeter (UK), he joined the CNRS (Rennes) in 2005, where he is currently senior researcher. His main research interest deals with the design of π-conjugated architectures for Organic Electronics: blue fluorophores for OLEDs, high-triplet hosts for PhOLEDs and electron deficient semiconductors for n-type OFETs. He is author/co-author of more than 80 publications, reviews and book chapters.

Joëlle Rault-Berthelot received her PhD in 1986 from the University of Rennes 1, where she is currently CNRS research director. She has been working for 35 years in electrochemistry. Since 2005, she is involved in the design of π-conjugated systems for Organic Electronics and is author/co-author of more than 130 publications, reviews and book chapters.