Myriam Orquéra
email: orquera@univ-tln.fr

Sébastien Campocasso

Dominique Millet

Some principles to optimize an additively manufactured multi-component product Journal of Engineering Design

Keywords: Product design, Topological optimization, Additive manufacturing

Depending on the requirements, the design of a product may result in very different solutions. Topological optimization is a mathematical tool that can be used to obtain lighter parts without decreasing their stiffness. As the optimized part shapes are often too complex to be manufacturable, additive manufacturing is generally used. Usually, the topological optimization is performed on a single part. However, for a product, the decreasing of mass and inertia in each individual part has an impact on the loads applied onto all the parts of the system. Moreover, different paths can be used to optimize a product, e.g. optimizing all the parts simultaneously or part after part. Therefore, the aim of this paper is to propose, based on a case study, some optimization principles to find the best optimization path to design an additively manufactured product. In order to do this, the concept of topological optimization loops is first proposed. Then, several optimization paths are compared. In comparison with a usual single topological optimization, the proposed method leads to an additional gain in mass of up to 35% for the case study. Finally, some optimization principles are suggested to choose the most adapted path according to the designer objectives.

Introduction and motivations

A mechanical system design must fulfill the functional needs which are commonly, for example: the product must be light; the product must withstand mechanical stresses ... In order to meet these expectations, topological optimization (TO) can be used [START_REF] Bendsøe | Optimization of Structural Topology, Shape, and Material[END_REF], but generally leads to an unmanufacturable design concept using conventional processes. Thus, as the shapes may be very complex [START_REF] Barbieri | A design strategy based on topology optimization techniques for an additive manufactured high performance engine piston[END_REF], the choice of additive manufacturing (AM) to manufacture topological optimized parts is often wise [START_REF] Liu | Current and future trends in topology optimization for additive manufacturing[END_REF]. In recent years, many additive manufacturing processes have been developed. Layer by layer, shape by shape or line by line, these fairly recent processes allow the material to be deposited where it is needed [START_REF] Brackett | Topology optimization for additive manufacturing[END_REF][START_REF] Thompson | Design for additive manufacturing: trends, opportunities, considerations, and constraints[END_REF]. Multiple studies have shown the gains of the TO on the re-design of a part by comparing it with that obtained for conventional processes. Thus [START_REF] Hällgren | Re)Design for Additive Manufacturing[END_REF][START_REF] Ren | Topology optimisation for steel structural design with additive manufacturing[END_REF] proposed the re-design of an existing conventional manufactured part for AM. The results show a very significant gain on the final part mass as well as on the buy-to-fly ratio, while maintaining stable and efficient mechanical behaviour. Global design methods for AM using TO have been developed such as those proposed by [START_REF] Ponche | A novel methodology of design for Additive Manufacturing applied to Additive Laser Manufacturing process[END_REF][START_REF] Vayre | Designing for additive manufacturing[END_REF], but published design methods often remain applied to a single mechanical part, with few articles dealing with the design optimization of a product. However, when topological optimization is used for mechanical assembly, the boundary conditions are modified by the decrease in the mass and inertia of the optimized parts, thus opening a way to perform an additional optimization. The purpose of this article is to address to the following four issues:

(1) How to take into account the new boundary conditions during the topological optimization of a multicomponent-product? (2) By which part(s) should the optimization start?

(3) Does this choice have an influence on the result? (4) And if so, would it be possible to establish optimization principles?

To answer these questions, this article first introduces an organization chart presenting a method of loop optimization in order to take into account new boundary conditions. Then, different paths for managing the impact of inertia are shown. The impact of the choice of design space at each step of this iterative scheme is also discussed. Finally, a path ranking taking into account the objectives of the designer is proposed and some optimization principles are suggested.

Previous methods of design for additive manufacturing (DfAM) using topology optimization

Generalities on topology optimization

According to [START_REF] Guo | Recent development in structural design and optimization[END_REF] topology optimization "aims to find the optimal way of material distribution in the structure". A simple and generic formulation of an optimization problem can be written as follows [START_REF] Zhou | An integrated approach to topology, sizing, and shape optimization[END_REF] :

𝑚𝑖𝑛 Γ∈Ω 𝑓(Γ) Subject to 𝑐(Γ) ≥ 0 Equation (1)
With:

 Γ variable  Ω predefined design domain  𝑓(Γ) objective function  𝑐(Γ) constraint function
Many TO algorithms exist like ground structures or genetic algorithms, each of them having different advantages and drawbacks [START_REF] Tang | A survey of the design methods for additive manufacturing to improve functional performance[END_REF]. Thanks to its simplicity and nice results, the SIMP model (Simple Isotropic Material with Penalization) is implemented in numerous commercial finite element codes for topology design [START_REF] Eschenauer | Topology optimization of continuum structures: A review[END_REF][START_REF] Kim | On the development of structural optimization and its relevance in engineering design[END_REF][START_REF] Saadlaoui | Topology optimization and additive manufacturing: Comparison of conception methods using industrial codes[END_REF] and adapted for additive manufactured part design [START_REF] Aremu | Practical Aspects of Structural Optimization -A study guide[END_REF][START_REF] Krishna | Topology optimization using solid isotropic material with penalization technique for additive manufacturing[END_REF].

The SIMP material model covers the complete range of density values from 0 to 1 [START_REF] Bendsøe | Optimization of Structural Topology, Shape, and Material[END_REF][START_REF] Bendsøe | Material interpolation schemes in topology optimization[END_REF][START_REF] Rozvany | Generalized shape optimization without homogenization[END_REF][START_REF] Zhou | The COC algorithm, Part II: Topological, geometrical and generalized shape optimization[END_REF], i.e. the TO result is a density distribution. Thus, according to the formulation of minimum compliance, the density  is introduced as a continuous variable (such as 0≤≤1), like a porous material with microscale voids. To achieve a 0/1 density distribution, the method assumes that the stiffness of the i-th element is determined by:

𝐸 𝑖 (Γ) = 𝜌 𝑖 𝑝 (Γ). 𝐸 * Equation (2)
Where 𝐸 * is the full stiffness of the isotropic material, 𝜌 𝑖 (Γ) the density of the i-th element and 𝑝 the penalty factor (1 ≤ 𝑝 ≤ 4). Its role is to penalize intermediate density (𝑝 ≥ 3 is usually required).

In order to get closer to the real structural behaviour, several researchers have recently proposed improvements in the SIMP model [START_REF] Chu | Topology optimization of multi-material structures with graded interfaces[END_REF][START_REF] Micheletti | Topology optimization driven by anisotropic mesh adaptation: Towards a free-form design[END_REF][START_REF] Zhang | Multiscale concurrent topology optimization for cellular structures with multiple microstructures based on ordered SIMP interpolation[END_REF].

Design for AM using TO

According to [START_REF] Pradel | A framework for mapping design for additive manufacturing knowledge for industrial and product design[END_REF], during the design process for additive manufacturing, one step must be the structure optimization. Applying the SIMP model -used throughout this articlewith a commercial software (Optistruct from Altair) implies following the workflow detailed in Figure 1.

For [START_REF] Rodrigue | An assembly-level design for additive manufacturing methodology[END_REF], the design space (DS) is the volume in which the material can be assigned during the optimization. [START_REF] Tang | Design method for lattice-skin structure fabricated by additive manufacturing[END_REF] defined the design space as the volumes which are used to bind the functional shapes and assist them in fulfilling their functional roles. As advised by (Goelke, 2016), the DS should be the largest one, allowing "to explore leading into the most design improvement opportunities". Functional shapes (FS) are defined as surfaces which can fulfill some functional requests [START_REF] Tang | Design method for lattice-skin structure fabricated by additive manufacturing[END_REF]. They can be surfaces needed for connecting joints, or where loads are applied.

A non-design space (NDS), even though part of the optimization problem, will not be modified during the optimization (Altair University, 2015). The use of NDS is optional when performing the optimization. NDS is:

 used to give volume to functional shapes,  used to represent standard parts like bearings or the minimum thickness of fluid channels.

The DS and optional NDS are integrated into the optimization software. Boundary conditions, load steps (defined for each stage of the part or of the product life), the material, optimization objective and constraints are defined to obtain the material density distribution. The TO result, which is a density distribution, has to be interpreted. Thus, some parameters have to be chosen by the designers (M.-H. [START_REF] Hsu | Interpreting three-dimensional structural topology optimization results[END_REF][START_REF] Hsu | Interpreting results from topology optimization using density contours[END_REF] leading to a particular result [START_REF] Morretton | Impacts of the settings in a design for additive manufacturing process based on topological optimization[END_REF]. The parameters are, for example, the design space definition, the density threshold, or the penalty factor. Numerous articles or industrial applications illustrate design for AM by exploiting TO. For example [START_REF] Tomlin | Topology optimization of an Additive Layer Manufactured (ALM) aerospace part[END_REF] demonstrated that different DS and NDS lead to different design and mechanical behaviour of a bracket. [START_REF] Walton | Design and development of an additive manufactured component by topology optimisation[END_REF]) realized an optimised rear upright manufactured by electron beam melting. More recently [START_REF] Munk | On the benefits of applying topology optimization to structural design of aircraft components[END_REF] used topology optimization to find the minimum weight structures for a land gear.

Topological optimization of a multi-component product

As previously said, most of the AM articles have been written about the design of a single part [START_REF] Barbieri | A design strategy based on topology optimization techniques for an additive manufactured high performance engine piston[END_REF][START_REF] Han | Additive design and manufacturing of jet engine parts[END_REF][START_REF] Hodonou | Material-design-process selection methodology for aircraft structural components: application to additive vs subtractive manufacturing processes[END_REF][START_REF] Mancuso | Topological optimization of a structural naval component manufactured in FDM[END_REF][START_REF] Munk | On the benefits of applying topology optimization to structural design of aircraft components[END_REF][START_REF] Obaton | Fabrication additive : état de l ' art et besoins métrologiques engendrés[END_REF][START_REF] Ren | Topology optimisation for steel structural design with additive manufacturing[END_REF][START_REF] Tomlin | Topology optimization of an Additive Layer Manufactured (ALM) aerospace part[END_REF][START_REF] Walton | Design and development of an additive manufactured component by topology optimisation[END_REF], whereas the design optimization of additively manufactured multi-component products has rarely been studied. [START_REF] Rodrigue | An assembly-level design for additive manufacturing methodology[END_REF]) designed a planetary gear system. After a step of parts consolidation, a topological optimization was carried out on each rigid body. The purpose of the study conducted by [START_REF] Sossou | An additive manufacturing oriented design approach to mechanical assemblies[END_REF] was to manufacture additively a nonassembly topologically optimized vice. The necessary clearance and the most adapted orientation were chosen before applying a TO on each part. [START_REF] Orquéra | Design for additive manufacturing method for a mechanical system downsizing[END_REF] studied a multicomponent system (an oscillating cylinder engine). The goal of this study was to functionally improve the engine by using AM capabilities. To achieve this, TO was applied on each component together with functional improvements of the joints. Generally, mechanical systems are products whose parts have relative motions. Some parts and connecting joints undergo inertia loads. Topological optimization of moving parts allows to lightweight them and thus to reduce the loads due to inertia. For example, in the Figure 2, the part 1' which is the topologically optimized part 1, gives new boundary conditions (BC) on the part 2'. So, part 2' may be lighter with a new TO. The so optimized part denoted 2'' has an impact on the BC of the optimized part 3' which can be optimized again, and so on.

In light of this clearly observable research gap, the purpose of this article is to consider mass and inertia decrease in the designing of an optimized multi-component product.

Method

Optimization paths

The first step of the method is to determine the first part or set of parts that has to be optimized.

As this optimization sequence planning may influence the design, different optimization paths have to be identified and compared according to certain criteria.

The criteria would correspond to data relative to the mechanical behaviour (denoted M b ) of the system. It can be, for example, the mass of the system. For a pump or a propulsion system, the input or output torque may be the selected data. Following this general idea, different optimization paths can be identified, directly linked to the chosen criteria or not, as for instance:

Path 1-The part whose inertia (or mass) has the greatest impact on the mechanical behaviour is optimized first, Path 2-Beginning the optimization with the part that has a parameter involved in the highest number of equations related to the mechanical behaviour, Path 3-All parts are optimized simultaneously, Path 4-Realizing derivate paths, for example mixing Path 1 with Path 2,

Path 5-The part that has the most connecting joints to the other parts of the system is optimized first.

To determine which part has the greatest impact on the feature M b , the reduced sensibility coefficient S* [START_REF] Petit | Techniques inverses et estimation de paramètres. Partie 1[END_REF] is used as shown in the Equation (3).

S * (𝑀 𝑏 /𝐶 𝑖 ) = 𝐶 𝑖 .

𝜕𝑀 𝑏 ∂ 𝐶 𝑖

Equation (3)

M b represents the mechanical feature and C i the characteristic of the part i. The part whose characteristic leads to maximize S * (𝑀 𝑏 /𝐶 𝑖 ) is optimised first for the Path 1.

For the Path 2, the static or dynamic mechanical equations have to be written in order to determine which part has a parameter involved in the highest number of equations. For the Path 5, connecting joints have to be counted. In a next section three paths and two derivate one are applied to a case study in order to estimate the influence of the path choice.

Optimization impact estimation

The optimization impact estimation consists of estimating the need for a new topological optimization. For instance, in the Figure 2, the product has an output torque T 0 . After the first TO the new torque is T 1 . If both torques are identical or almost identical, further optimization is not useful. On the contrary if they are not identical, a new TO is necessary. An important issue is to decide from which T 1 value should a new TO be realized. For this purpose, the mechanical behaviour ratio (denoted R Mb ) is proposed and calculated according to Equation (4).

𝑅 𝑀𝑏 = 𝑀𝑏 𝐿-1 𝑀𝑏 𝐿 Equation (4)
𝑀𝑏 𝐿-1 and 𝑀𝑏 𝐿 are the respective values of the mechanical behaviour criteria, previously chosen by the designer, before and after the optimization. L (for loop) corresponds to the number of the considered TO loops. A parameter ε called convergence coefficient is also established. If the result of the R Mb ratio is lower than 1-ε or higher than 1+ε, the impact is estimated as significant.

So, new iterations have to be done while the mechanical ratio does not belong to [1-ε, 1+ε]. The lower ε is, the more the impact of TO on the mechanical behaviour is taken into account. But this will influence the number of loops and therefore the calculation time. In this article, the impact of the ε value is discussed based on a case study.

The topological optimization loop

Based on the previous explanations, a organization chart showing the concept of Topological Optimisation Loop (TOL) for a multi-component is proposed in Figure 3. Depending on the path choice, the TOL is applied first on the part which should be first optimised (for paths 1, 2 and 5), or on all the parts simultaneously for the path 3.

As explained in the topology optimization workflow (Figure 1), a design space and an optional non-design space are CAD designed for each "p" part and integrated into the optimization software. They are denoted by (𝐷𝑆 + 𝑁𝐷𝑆) 𝐿=1,𝑝𝜖[1,𝑛] for the first loop. L and n are respectively the loop and the part number. Boundary conditions, load steps, material, optimization objective and constraints are also initially defined.

The first loop is a classic application of the topological optimization (using the TO workflow shown in Figure 1). The result is a material density distribution. After interpretation, a new CAD-design of the part is used as the design space (with the same non-design space as before, as explained in a next section) for the next loop. Denoted (𝐷𝑆 + 𝑁𝐷𝑆) 𝐿=2,𝑝∈ [1,𝑛] , it now has a new mass and inertia which impact the other parts.

As explained previously, if the result of the R Mb ratio is lower than 1-ε or higher than 1+ε, the impact is estimated as significant. So, new iterations have to be done. Although this new organization chart is helpful to optimize a multi-component product, it should be noted that it has to be applied part by part.

Discussion about the design space

Generally the design space is chosen as the least restrictive to allow the software to find the optimum solution. So, normally the DS volume has to be the largest one. At the end of the first loop of the TO, as explained before, the mass and inertia are decreased so the loads intensities are smaller than those of the first TO loop. A second TO has to be done in which two DS may be used:

 the result of the first loop,  the initial design space.

In order to allow more possibilities for the software, it seems wise to impose the new boundary conditions on the initial design space. But, because the loads are lower, using the initial design space leads to a mass higher than using the DS re-designed from the previous TO result, as shown in Figure 4. The algorithm decreases the mass and verifies if the constraints are violated or not, until they are validated (see Figure 4). Moreover, if the first optimization loop gives a density distribution result under consequent loads, this same result will necessarily withstand the new lower loads. So using the DS of the previous loop has a great probability of converging with a lower mass and a shorter calculation time.

In conclusion, for the first TO the design space should be the least restrictive as possible. Then, for the next loops, using the previous DS with lower loads achieves better results.

Optimization path principle for the designers

Each path will lead to a specific result with a CPU time, a total mass, an estimation of the mechanical feature, etc. The main purpose of this article is, based on the paths results, to suggest optimization path principles in order to help designers to choose the best path considering their design objective. Hence, in the next section different paths, using the TOL, are applied to a case study in order to make some comparisons, to assess the improvement of the system (thanks to the presented method) and to conclude on the optimization path principle.

Case study

System presentation

The subject of this case study is a robot arm tool holder of a sharpening machine proposed by [START_REF] Bône | Mécanique générale: cours et applications[END_REF]. As shown in Figure 5, it is composed of a rotating support 1 on which is mounted a pivoting arm 2. The tool holder 3 has a cylindrical joint along z 2 with the arm 2. Design spaces and non-design spaces were CAD designed for each part, as shown in Figure 6. The total mass of the non-optimized robot is equal to 16.2 kg and needs a torque to turn around z 0 equal to 634 N.mm.

Paths identification

During its work, the robotic structure is permanently subjected to several types of loads like the grinding forces, or the inertial forces of moving masses [START_REF] Mejri | Dynamic characterization of machining robot and stability analysis[END_REF]. The deflections of the structure under these loads are a source of defects for the grinding work that should be minimized. That is why the objective of this study is to use the TOL in order to reduce inertia. Three optimization paths are first proposed:

(1) The part whose inertia (or mass) has the greatest impact on the mechanical behaviour is optimized first. (2) Beginning the TOL with the part that has a parameter involved in the highest number of equations.

(3) All parts are optimized simultaneously.

The input torque T 01 (see Equation ( 8) in the appendix) is chosen to be the mechanical behaviour data M b used for the comparison of the several optimization paths. The mathematical model used is proposed by [START_REF] Bône | Mécanique générale: cours et applications[END_REF]. T ij and F ij represent respectively the torque and the force between part i and part j. I p(L,m) depicts the inertia of part p at point L around the axle m. All the equations are located in the appendix. As shown in Table 1, the component 3 impacts the highest number of equations. The input torque (see Equation (8) in the appendix) has been evaluated for initial characteristics. To determine which part has the highest impact, the reduced sensibility coefficient S* of the Equation ( 3) is determined for each characteristic as shown in Table 2. The mass of arm 2 has the most important influence on the torque T 01 . To summarize, the first Path begins with part 2, and the second Path 2 with part 3. Figure 8 graphically summarizes the three paths.

For this study, the convergence coefficient ε is imposed at 0.1. The results are compared in terms of robot total mass, input torque T 01 , and time used by the software to perform all the topological optimization iterations.

Other paths can be derived from these first three, like the two other possibilities shown in Figure 9. Path 4 is inspired from both Path 1 and Path 2, and consists in applying the TOL on parts 2 and 3 simultaneously and then on part 1. As the mass of part 3 can be neglected, it has been chosen to not optimize this part in Path 5.

Details of the topological optimization paths

The topological optimization and CAD software used are respectively Optistruct and Inspire from Altair. For each component, the TO objective is to minimize the mass with stress and displacement constraints. A static displacement constraint has been defined at points A, B and C for each part (the points are visible in Figure 7). This total displacement must not exceed 0.3 mm. For parts 1 and 2, (x 1 , z o ) is used as a symmetric plane (the axle is visible in the Figure 5). The tool holder 3 turns around the axle z 2 . As the loads follow the same direction, a circular repetition around the axle (7 repetitions) has been imposed. It was determined to set the optimization parameters to 3 for the penalty factor and 6 for the related sensitivity filter called "mindim". The mesh size is equal to 2 mm (except for part 3 due to its smaller size, where the mesh size is equal to 1 mm). Finally, all the topological optimizations were performed on the same cluster. It is composed of two processors Xeon E5-2690/v2, each using ten cores of 25 Mo L3 with a frequency equal to 3 GHz. The RAM memory is equal to 256 Go.

Path 1: beginning with the most influential part (part 2)

The results are summarized in Table 3.  TOL on part 2

The first loop has converged and all constraints have been satisfied after 44 iterations. With the initial design space, the input torque T 01 is equal to 634 N.mm. The first optimization loop decreases this value to 299 N.mm. The resulting ratio equal to 2.12 has exceeded the limits specified in the previous section. So, a new loop is performed.

As explained previously, the design space used for the second loop is the result of the first loop.

The second loop has converged and all constraints have been satisfied after 39 iterations. The ratio between the previous and new mechanical behaviour, equal to 1.05, is within the limits specified in the previous section. The optimization loops for part 2 have thus been finished. It can be noted that the re-design mass value is 12% less than the re-design mass value result with the initial design space, confirming the conclusions of the previous section. Then, new loads induced by mass and inertia of part 2 at the end of loop 2 are mandatory for the TOL of part 3.

 TOL on part 3

The first loop generates a ratio R Mb equal to 2.43, which is beyond the limits. The ratio of the second loop remains within the limits.

 TOL on part 1

The ratio R Mb required two loops to be within the limits.

Path 2: beginning with the component which impacts the highest quantity of equations (part 3)

All optimization loops have converged and all constraints have been satisfied. Taking into account the mechanical behaviour ratio, each part has required two loops, as shown in Table 4.

For each part, the design space obtained after the first loop is used for the second loop.

Path 3: all parts simultaneously optimized

The results of Path 3 are summarized in Table 5. To obtain a value of R Mb within the boundaries, three loops are necessary.

Results analysis

Comparison of the results

As presented before, the results are different for each path. Figure 10 shows the re-designed products for each path and the results are summarized and compared in Table 6. Without the TOL proposed previously, only one TO loop would have been performed. To verify the gains made by performing TOL, Table 6 shows the differences between a single topological optimization and the TOL results. Results prove that TOL produces significant gains in mass compared with one TO loop, and this, within the same amount of time in both instances. It can be noted that the complex shapes of the robot arm so obtained cannot be manufactured by conventional processes. Using the Analytic Hierarchy Process (AHP) method proposed by [START_REF] Mançanares | Additive manufacturing process selection based on parts' selection criteria[END_REF], a powder bed fusion (PBF) process should be used as the most adapted AM process. The total time is lower for Path 3 compared with Paths 1 and 2. Indeed, as the parts have been simultaneously optimized, the total CPU time is not the sum of the software time for each part optimization, but the sum of the maximum CPU time for each loop, as explained in Figure 11.

About the convergence coefficient ε

The convergence coefficient used for this case study is equal to 0.1. This means that the TOL scheme is ended if the mechanical behaviour of the new loop exceeds plus or minus 10% of the previous one. The results of the different paths show that this value is wise and gives good results with few iterations.

Choosing a convergence coefficient close to zero will drastically increase the quantity of loops without effective results. Moreover, the low decrease in loads during the next iterations will reduce the potential gain at each new iteration. On the contrary, choosing a high coefficient of convergence will reduce the quantity of loops, but the results may not be optimum. For example, using ε equal to 0.3 for the third path leads to only two loops (Table 7) but with a total robot mass increased by 11%. Likewise, the input torque has increased by 7%. In return, there is only a 3% savings in time. Figure 12 shows changes in the total mass and the R Mb ratio as a function of the loop number for the path 3. The lower ε is, the more the R Mb and the total mass tend towards an asymptotic (R Mb =1 in orange on the graph in Figure 12). However, there is a significant gain when the slope is high. In conclusion, it is necessary to choose a reasonable value of ε to obtain a significant gain without loss of time. Thus it seems that the convergence coefficient should be included in the range [0.1; 0.2].

Principles to select a path for an additively manufactured multi-component product optimization

Each path has led to different results. Considering the designer objectives, the choice of the path can be different. A classification of the paths is suggested in Table 8. First, it can be noticed that Path 1 has nearly no advantage. On the contrary, Path 3 seems to have the best numbers, especially to lead to the lowest input torque. Path 4 has one tenth of a gram more than Path 2; this is low enough to equalize them. Both enable the greatest decrease in mass. Lastly, if saving time is the primary objective, Path 5 has to be used. Thanks to these results, the most effective path can be chosen to achieve the designer objectives. For this reason, three optimization principles can be concluded (for mechanisms without a closed-loop kinematic chain and to minimize the mass with stress as objective and displacement as constraints):

(1) To optimize a product with a minimum use of time:

 The part(s) with the least influence on the mechanical behaviour must not be optimized.  The most influential part has to be optimized first.

(2) To optimize a product in order to obtain the best mechanical behaviour:

 All parts which impact the mechanical behaviour have to be optimized simultaneously.

(3) To optimize a product in order to obtain the lowest mass:  The part whose mass is involved in the highest number of equations has to be optimized first.  To reduce the length of time, the heaviest parts can be optimized simultaneously.

Conclusion

The case study presented in this article has resulted in the design of an optimized robot arm. More generally, the topological optimization loop (TOL) concept proposed in this article takes into account the decrease in mass and inertia during the topological optimization of a mechanical product. Even if the topology of the results may differ a little, the presented method is not software dependent. Indeed, the TO loop can be applied in the same way and the conclusions will be identical. To perform an optimization on a multi-component product, the topological optimizations may be applied on the parts in different orders. Several optimization paths have been tested, leading to different results. Finally, for the case study examined in this report, using the TOL framework together with the best optimization path leads to an additional 35% gain in mass in comparison with a single topological optimization. Integrating the TOL framework in a Design for Additive Manufacturing (DfAM) method, will provide a rational and helpful tool for obtaining a mechanical system in which:

 material savings is achieved,  the mass is lower,  the dynamic behaviour is better,  the joints between rigid bodies withstand lower loads,  the input power can be decreased…

The choice of the design space for the successive optimization loops has also been studied. Using the density distribution result of the previous loop as the new design space seems to improve the results. Finally, as the choice of the path must be made to achieve the designer objectives, three optimization principles have been stated, depending on the objective: minimizing the calculation time, obtaining the best mechanical behaviour or obtaining the lowest mass. Thus, following these recommendations will help designers to optimize a multi-component product. The presented method is a convenient tool to design static or dynamic systems. However, it must be used for multi-component products in which mass and inertia have an impact on the mechanical behaviour. To complete this research, future work should focus on studying complex systems with closed-loop kinematic chains. Another perspective could be to conduct a similar study for other TO objectives and constraints.

Appendix

In this section the equations used for the case study are detailed. As explained previously, the mathematical model used is proposed by [START_REF] Bône | Mécanique générale: cours et applications[END_REF]. T ij and F ij represent respectively the torque and the force applied between part i and part j. I p(L,m) depicts inertia of part p at point L around axis m (Figure 5).

(𝑥 ⃗ 0 , 𝑥 ⃗ 1 ) = (𝑦 ⃗ 0 , 𝑦 ⃗ 1 ) = 𝛼 Equation ( 5)

(𝑧 ⃗ 0 , 𝑧 ⃗ 2 ) = (𝑥 ⃗ 1 , 𝑥 ⃗ 2 ) = 𝛽 Equation ( 6) (𝑥 ⃗ 2 , 𝑥 ⃗ 3 ) = (𝑦 ⃗ 1 , 𝑦 ⃗ 3 ) = 𝛾 Equation ( 7)

𝑇

Figure 1 .

 1 Figure 1. Traditional topology optimization workflow

Figure 2 .

 2 Figure 2. Comparison between one usual TO and multiple TO considering new BC on a multi-component design

Figure 3 .

 3 Figure 3. Topological optimization loop (TOL) organization chart considering inertia and mass decrease

  DS: Design space NDS: Non-design pace L: Loop number p: Part marker n: Total number of optimized parts CAD of the design and non-design space of the part p First quantification of the mechanical feature previously chosen by the designer Estimation of the optimisation impact If the impact is important, a TO is done on the part p TO results are CAD designed and become the new design and non-design space of the part p Loop indexation If the impact is not significant the TOL is finished Mechanical behaviour : MbL (𝐷𝑆 + 𝑁𝐷𝑆) 𝐿,𝑝∈[1,𝑛] Initial design space and non-design space : (𝐷𝑆 + 𝑁𝐷𝑆) 𝐿=1,𝑝∈[1,𝑛] ; Mechanical behaviour : MbL=1 Quantification of the new mechanical feature A TO is done on the part p

Figure 4 .

 4 Figure 4. Impact of the design space choice for the second loop

Figure 5 .

 5 Figure 5. Case study: robot arm[START_REF] Bône | Mécanique générale: cours et applications[END_REF]

Figure 6 .

 6 Figure 6. (a) Robot arm CAD designed; (b) Support; (c), Pivoting arm; (d) Tool. Design spaces are in green, and non-design spaces in orange.

Figure 7 .

 7 Figure 7. Boundary conditions applied on (a) Part 1, (b) Part 2, (c) Part 3

Figure 8 .

 8 Figure 8. The three paths studied to optimize the product

Figure 9 .

 9 Figure 9. Derived paths to optimize the product

Figure 11 .

 11 Figure 11. Comparison of the CPU times for the different paths

Figure 12 .

 12 Figure 12. Mass and RMb evolution as a function of the number of loops for the Path 3

Design and non- design space definition Material features definition Parts meshing Boundary conditions and load steps Variables, constraints, objective and parameters Result analysis and density threshold choice STL or STP export and re- design FE analysis Optimization

	Design space
	Non-design space

Table 1 .

 1 Feature which impacts the highest number of mechanical equations

	Feature	Number of equations using the feature		Concerned equations
	I1(O,zo) (kg.m²)		1				T01
	I2(H,x2) (kg.m²)		2				T12;T01
	I2(H,z2) (kg.m²)		2				T12;T01
	m2 (kg)		2				F12;T01
	m3 (kg)		4			F23;F12;T12;T01
	I3(G,x3) (kg.m²)		2				T12;T01
	I3(G,z2) (kg.m²)		3				T23;T12;T01
	Table 2. Inertia and mass sensibility evaluation
	Characteristic Ci	m2	m3	I1(O,zo)	I2(H,x2)	I2(H,z2)
	S * (𝑇 01 /𝐶 𝑖 ) = 𝐶 𝑖 .	𝜕T 01 ∂ 𝐶 𝑖	0.34 0.21 0.03 0.05 0.03

Table 3 .

 3 Optimization results for Path 1

		TOL on	TOL on	TOL on
		part 2	part 3	part 1
	Initial mass (g)	8000	1264	6970
	Loop number	loop1 loop2 loop1 loop2 loop1 loop2
	R Mb	2.12	1.05	2.43	1.04	1.23	1.02
	Re-design mass value (g) 1055	768	274	246	1466	1130
	Torque T01 (N.mm)	299	285	117	112	91	89
	Loop time (h:min)	8:37	0:26	19:28 0:45	8:58	1:02
	Total time (h:min)			39:19

Table 4

 4

		. Optimization results for path 2
		TOL on	TOL on	TOL on
		part 3	part 2	part 1
	Initial mass (g)	1264	8000	6970
	Loop number	loop1 loop2 loop1 loop2 loop1 loop2
	RMb	1.36	1.01	3.84	1.08	1.23	1.04
	Re-design mass value (g)	274	246	926	749	1434	912
	Torque T01 (N.mm)	466	461	120	112	91	88
	Loop time (h:min)	19:28 0:45	10:20 0:32	7:48	1:09
	Total time (h:min)			40:04

Table 5 .

 5 Optimization results for path 3

		Initial Loop 1 Loop 2 Loop 3
	RMb		5.74	1.20	1.07
	TOL on part 1. Mass (g) 6970	1616	1287	1133
	TOL on part 2. Mass (g) 8000	1055	801	735
	TOL on part 3. Mass (g) 1264	274	246	234
	Torque T01 (N.mm)	634	110	92	86
	Total time (h:min)		20:54

Table 6 .

 6 Results of each path and comparison between using a single topological optimization and using the TOL

		Optimization without TOL	Path 1 Path 2 Path 3 Path 4 Path 5	Gain between Path 3 and without TOL
	Torque T01 (N.mm)	110	89	88	86	89	91	22%
	Total mass (g)	2945	2143	1906	2102	1925	2668	29%
	Total time (h:min)	19:28	39:19	40:04	20:54	28:37	18:52	-7%
	Number of loops	1	2	2	3	2	2

Table 7 .

 7 Impact of the convergence coefficient for Path 3

	Path 3	ε=0.3	ε=0.1	ε=0.05
	Torque T01 (N.mm)	92	86	85
	Total mass (g)	2334	2102	2051
	Total time (h:min)	20:15		21:05
	Number of loops	2	3	4

Table 8 .

 8 Ranking of the paths considering the objectives of the designer

			The objective is to reduce the
		mass	torque	time
	Path 1	3	3	4
	Path 2	1	2	5
	Path 3	2	1	2
	Path 4	1	3	3
	Path 5	4	4	1
	1 represents the best path, 5 represents the worst.

  𝑂,𝑧 ⃗ 0 ) + (𝐼 2(𝐻,𝑥 ⃗ 2 ) + 𝐼 3(𝐺,𝑥 ⃗ 3 ) ). 𝑠𝑖𝑛²𝛽 + (𝐼 2(𝐻,𝑧 ⃗ 2 ) + 𝐼 3(𝐺,𝑧 ⃗ 2 ) ). 𝑐𝑜𝑠²𝛽 + 𝑚 2 . ℎ² + 𝑚 3 . (ℎ + 𝑟. 𝑠𝑖𝑛𝛽)²}. 𝛼̇+ 𝐼 3(𝐺,𝑧 ⃗ 2 ) . 𝛾. 𝑐𝑜𝑠𝛽] Equation (8)𝑇12 = (𝐼 2(𝐻,𝑦 ⃗⃗ 2 ) + 𝐼 3(𝐺,𝑥 ⃗ 3 ) + 𝑚 3 . 𝑟 2 ). 𝛽 ̈+ {(𝐼 2(𝐻,𝑧 ⃗ 2 ) + 𝐼 3(𝐺,𝑧 ⃗ 2 ) -𝐼 2(𝐻,𝑥 ⃗ 2 ) -𝐼 3(𝐺,𝑥 ⃗ 3 ) ). 𝛼̇. 𝑐𝑜𝑠𝛽 + 𝐼 3(𝐺,𝑧 ⃗ 2 ) . 𝛾̇}. 𝛼̇. 𝑠𝑖𝑛𝛽 -𝑚 3 . 𝑟. [(ℎ + 𝑟. 𝑠𝑖𝑛𝛽). 𝛼̇2. 𝑐𝑜𝑠𝛽 + 𝑔. 𝑠𝑖𝑛𝛽] 𝐹 23 = 𝑚 3 . [𝑔. 𝑐𝑜𝑠𝛽 -𝑟. 𝛽 2 -(ℎ + 𝑟. 𝑠𝑖𝑛𝛽). 𝛼̇2. 𝑠𝑖𝑛𝛽]

	01 =	𝑑 𝑑𝑡	[{𝐼 1(Equation (9)
	𝑇 23 =	𝑑 𝑑𝑡	[𝐼 3(𝐺,𝑧 ⃗ 2 ) . (𝛾̇+ 𝛼̇. 𝑐𝑜𝑠𝛽)]	Equation (10)
					Equation (11)
	𝐹 21 = (𝛼̈. 𝑡) 2 . ℎ. ( 3.𝑚 2 2	+ 2. 𝑚 3 )	Equation (12)
	𝑂𝐻 ⃗⃗⃗⃗⃗⃗⃗ = ℎ. 𝑥 ⃗ 1		Equation (13)
	𝐻𝐺 ⃗⃗⃗⃗⃗⃗⃗ = 𝑟. 𝑧 ⃗ 2		Equation (14)

h and r are constant.x


Dr Myriam Roy

Dr Veronika Berezhnaia

Dr Marco Villa

Dr Nicolas Vanthuyne

Dr Michel Giorgi

Jean-Valère Naubron

Dr Salomé Poyer

Dr Valérie Monnier

Prof. Dr Laurence Charles

Dr Yannick Carissan

Jean- Valère Naubron

Dr Denis Hagebaum-Reignier

Prof. Dr Jean Rodriguez

Prof. Dr Marc Gingras
email: marc.gingras@univ-amu.fr

Dr Yoann Coquerel
email: yoann.coquerel@univ-amu.fr

Dr J.-V Naubron

Stereoselective Syntheses, Structures and Properties of Extremely Distorted Chiral Nanographenes Embedding Hextuple Helicenes

come   L'archive ouverte pluridisciplinaire

Introduction

Two-dimensional polycyclic aromatic hydrocarbons (PAHs) are diverse in sizes and shapes, starting from small naphthalene or triphenylene units to graphene sheets and/or flakes having both dimensions larger than 100 nm. Large PAHs can exhibit exceptional electronic and physical properties governed by their size and the nature of their periphery. Thus, considerable efforts are currently underway to synthesize well-defined large 2D PAHs such as graphene nanoribbons and nanographenes by organic synthesis (bottom-up approach) in solution or on a surface and to investigate their properties and applications. [1] Because PAHs are somewhat flexible and stretchable molecules, it allows the design of curved PAHs such as bowls, saddles, coiled ribbons, propellers and circular belts with defined molecular chirality. [2] The latter property is of a great interest in life, chemical, and physical sciences as well as technology, because it allows an additional tuning of molecular properties and functions for many applications. [3] Large chiral PAHs have thus become molecules of utmost interest as three-dimensional chiral nanographenes, and small curved PAHs are now often sought as building blocks for their construction in the racemic series. [4] In that direction, multihelicenes have emerged as a class of promising well-defined large chiral PAHs. [5] Multi-helicenes can exist as several diastereomers, each existing as a pair of enantiomers (except in the cases of some achiral meso diastereomers), that can more or less rapidly interconvert depending on the torsional energy and flexibility of the molecules. The proper accumulation of helical strain in multi-helicenes can produce exceptionally distorted molecules showing some benzene units suffering from an extreme torsion angle. For instance, record torsions of 35.3º and 35.7º were recently measured in a D2-symmetric quadruple helicene [6] and a C2-symmetric hextuple helicene, [7] respectively. A fruitful approach to racemic triple helicenes and higher order congeners was based on cyclotrimerization reactions to forge a distorted triphenylene core. [7,8] Using this approach, we report herein the diastereoselective and, when applicable, enantiospecific syntheses of two large and extremely distorted diastereomeric hextuple helicenes of formula C90H48. The conformational behavior, the structure, the chiroptical properties and the torsion-induced deficient local aromaticity in these molecules are discussed in detail, together with the exploration of the complexation ability toward silver(I) ions for one of them. 6-31G(d) in the gas phase], energies are free Gibbs energies and are expressed in kJ×mol -1 , see the Supporting Information for details. (d) Yamamoto-type cyclotrimerization of racemic 9,10dibromo [7]helicene (rac-1) leading to rac-C2-2. (e) Three-dimensional representations of rac-C2-2 obtained from single-crystal X-ray diffraction analysis; remarkable interplanar angles are highlighted in red.

new chiral PAH embeds three homochiral [7]helicene units on its outer shell and three homochiral [5]helicene units of opposite configuration on its inner edges. [7]Helicene itself has a high barrier to enantiomerization (178.8 kJ×mol -1 ) [9] precluding its inversion of configuration at a significant rate under the projected reaction conditions (ca. 130 ºC). It was assumed that the barriers to inversion of configuration of the various [7]helicene-containing species involved in the planned nickel(0)-mediated cyclotrimerization would be too high for the synthesis to rely on a thermodynamic control as in previous studies. [7,8] Thus, it should be possible to perform the synthesis from enantiopure (P)or (M)-1 to obtain the corresponding enantiopure hextuple helicene D3-2 in an enantiospecific manner. Indeed, the nickel(0)-mediated cyclotrimerization of (M)-1 afforded (-)-(M,M,M,P,P,P)-D3-2 in a 38% yield (Figure 1a). The same reaction performed with (P)-1 gave (+)-(P,P,P,M,M,M)-D3-2 in a 39% yield (not depicted). Both enantiomers of D3-2 were thus directly obtained as enantiopure hextuple helicenes, as confirmed by analytical chromatography on a chiral stationary phase. The structure and absolute configuration of (-)-(M,M,M,P,P,P)-D3-2 were unambiguously confirmed by single-crystal X-ray diffraction methods (Figure 1b) and circular dichroism spectroscopies (see Supporting Information). On a stereochemical point of view, ten diastereomers of 2 are possible: two of C1 symmetry, six of C2 symmetry and two of D3 symmetry. Their relative free Gibbs energies were calculated by DFT methods from 0.0 kJ×mol -1 for D3-2, the thermodynamic diastereomer, up to 108.3 kJ×mol -1 for a C2-symmetric metastable diastereomer. A plausible enantiomerization pathway for D3-2 was computed with a barrier to enantiomerization calculated at 160.6 kJ×mol -1 (Figure 1c), in agreement with a value experimentally determined at 162.0 kJ×mol -1 corresponding to a racemization half-life t1/2 = 43 million years at 25 ºC or 1.3 years at 130 ºC. Notably, the barrier to enantiomerization of D3-2 is lower than for the parent [7]helicene, an oddity for Dn symmetric multi-helicenes in general. [5] Amusingly, hextuple helicene D3-2 is the second triskelion-shaped nanographene reported within a short period of time. [4e] The enantiospecific synthesis of both enantiomers of D3-2 from either (P)and (M)-1 confirmed our hypothesis that no significant inversion of configuration of [7]helicene units can occur at a significant rate in the various synthetic intermediates involved in its synthesis. This translated in a unique opportunity to synthesize diastereoselectively a diastereomer of D3-2 from racemic 1, this time mixing helicities of the [7]helicene units at the outer shell of the molecule.

Accordingly, the nickel(0)-mediated cyclotrimerization of rac-1 allowed the formation of both the racemic (P,M,P,P,P,M)-C2-2 and D3-2 diastereomers in a 13.8:1 ratio and the isolation of (P,M,P,P,P,M)-C2-2 in a 16% yield (Figure 1d); note that hextuple helicene rac-(P,M,P,P,P,M)-C2-2 can also be denominated rac-(P,P,M,M,P,P)-C2-2 or rac-(M,P,P,P,M,P)-C2-2 according to the proposed nomenclature. The structure and relative configurations of rac-C2-2 were ascertained by single-crystal X-ray diffraction analysis (Figure 1e). The free Gibbs energy of diastereomer C2-2 was calculated at +32.2 kJ×mol -1 relative to D3-2, and diastereomer C2-2 is an intermediate in the computed enantiomerization pathway for D3-2 (Figure 1c). The enantiomers of diastereomer C2-2 were separated by chiral HPLC techniques and were fully characterized by chiroptical methods, including the attribution of their absolute configurations (see the Supporting Information).

Structural and chiroptical studies. The structural features of hextuple helicenes D3-2 and C2-2 were carefully examined (Figure 2). The four benzene units from the triphenylene core of D3-2 show a marked alternation of bond lengths from 1.404 Å to 1.465 Å (mean values because of the crystallographic C2symmetry of D3-2), to be compared to 1.393 Å in benzene,1.420 Å in graphene, and 1.338 Å and 1.454 Å in 1,3-butadiene, [10a,b] and may be regarded as their Kekulé structures, i.e. 1,3,5cyclohexatrienes. The central ring in D3-2 exists in a chair conformation and the three surrounding rings adopt severely twisted conformations with a mean torsion angle of 32.4º (Figure 2a). Evaluation of the photophysical properties of D3-2 indicated a maximal absorption at 417 nm (ε = 90000 M -1 cm -1 ) and maximal emission at 538 nm (quantum yield = 5.7%). The optical rotation of (M,M,M,P,P,P)-D3-2 was measured at [a]D 25 = -300 (c = 0.0325, CHCl3), which is an unexpectedly small absolute value when compared to the large optical rotation values of isolated (P)- [5]helicene ([a]D 25 = +1670) and (M)- [7]helicene ([a]D 25 = -5900). Similarly, the electronic and vibrational circular dichroism spectra of D3-2 were found of low intensity (see the Supporting Information). It was reasoned that the strong contribution to the chiroptical properties of the three moderately stretched homochiral (M)- [7]helicene units at the outer shell in (M,M,M,P,P,P)-D3-2 (mean interplanar angle of the two terminal rings = 39.7º vs 32.3º in [7]helicene itself [11a,b] ) is somehow compensated by a strong contribution of opposite intensity of the three severely stretched homochiral (P)- [5]helicene units at its edges (mean interplanar angle of the two terminal rings = 64.0º vs 46.0º in [5]helicene itself [11c] ). The structural analysis of C2-2 also revealed a pronounced bond lengths alternation of the four benzene units from its triphenylene core, with bond lengths generally comparable with those in D3-2. However, the longest interatomic distance in C2-2 is the C-C bond crossed by the C2 axis in the periphery of the triphenylene core has a length of 1.474 Å (Figure 2b), comparable with the one of the C(sp 2 )-C(sp 2 ) single bond in some biphenyls. [10c] The C2-2 diastereomer was found significantly more distorted than isomer D3-2, with a maximal torsion angle culminating at 36.9º (!) for the peripheral ring of the triphenylene core on the C2 axis, establishing the current record of torsion for a "benzene" ring (Figures 2b,d). Also, the naphthalene unit aligned with the C2 axis in C2-2 exhibits a very high end-to-end torsion of 66.7º (Figure 2c, current record = 69.5º [6] ). The optical rotation of (P,M,P,P,P,M)-C2-2 was measured at [a]D 25 = +705 (c = 0.0337, CH2Cl2) and electronic circular dichroism spectrum showed a relatively low intensity (see Supporting Information), chiroptical features comparable with those of diastereomer D3-2. The highly distorted structures of D3-2 and C2-2 limit the formation of intermolecular p-p interactions, thus allowing enhanced solubility in organic solvents. This is well illustrated by the crystal packing of (M,M,M,P,P,P)-D3-2 that shows few interactions between the individual molecules that stack to form supra-helices of (P) configuration within the crystal, creating large chiral CDCl3 solvent channels organized along the a and b crystallographic axes (Figure 3).

Evaluation of aromaticity.

The hextuple helicenes D3-2 and C2-2 sustain unprecedented distortion for stable PAHs, pointing out for unusual deficiencies in local aromaticity. Aromaticity is an extremely useful and popular concept in chemistry although there is no single quantitative definition of it, and it is not directly measurable experimentally. [START_REF]For the definition of aromaticity by IUPAC[END_REF] Application of the qualitative Clar's rule [13] to D3-2 (or C2-2, stereochemistry is not accounted for in Clar's structures) predicts an alternation of local aromaticity with 13 separated p-sextets in the Clar's structure containing the largest number of disjoint aromatic p-sextets (Figure 4d), and nine disjoint p-sextets in the complementary Clar's structure. However, the Clar's rule is strictly valid for planar polycyclic aromatic systems. Structural- [14] energetic- [15] electron density-based, [16] and magnetic-based indices [17] have been developed to characterize and to quantify aromaticity as a property of molecules. However, these indices were also developed for planar systems and their transposition to three-dimensional systems can lead to erratic results. For instance, it was earlier recognized that the popular structural-based indices as HOMA (Harmonic Oscillator Model of Aromaticity) and energetic-based indices as TRE (Topological Resonance Energy), as well as some electron density-based methods, do not perform well for coiled systems as [n]helicenes and can lead to discrepancies. [18] Indeed, we experienced difficulties with the HOMA during our early work (see Supporting Information). Actually, only magnetic-based methods appear applicable to severely twisted aromatics. When an external magnetic field is applied to a molecule, strong induced electrons currents are created at aromatic (diatropic currents) and antiaromatic (paratropic currents) rings. The ACID (Anisotropy of the Induced Current Density) method [19] allows the visualization of an isosurface on which the induced current density vectors are plotted, showing diatropic (clockwise, aromatic) and paratropic (counterclockwise, antiaromatic) currents. However, the method is sensitive to the orientation of the applied external magnetic field, normally perpendicular to the molecular plane, which can be problematic with highly distorted PAHs such as D3-2 and C2-2. The NICS (Nucleus Independent Chemical Shifts) index relies on the computation of the local magnetic environment at any desired point in space, often referred to as a ghost atom, and is expressed in ppm. [START_REF] Chen | [END_REF] NICS(0) values refer to ghost atoms placed at the center of each cycle, with large negative NICS(0) probing induced diatropic ring currents attributed to aromaticity. For instance, benzene has a NICS(0) calculated at -8.6 ppm at the level of theory employed herein. Although the combination of both methods allows for a qualitative and quantitative description of the induced electron currents, it should be kept in mind that the connection between induced diatropic ring currents and aromaticity is not straightforward. [12b,17] The NICS analysis of D3-2 (Figure 4a) allowed identifying three groups of rings with distinct local aromaticity: i) the A and B rings of the triphenylene core have NICS(0) values comprised between -0.6 and +0.8 ppm accounting for poorly directional induced ring currents indicating a non-aromatic character of these two rings; ii) the C and D rings have deficient induced diatropic ring currents with NICS(0) values calculated around -5.3 ppm pointing out for marked deficient local aromaticity; and iii) the terminal E rings with NICS(0) values of -9.6 ppm denoting full aromaticity. The over-estimation of NICS in the terminal rings of [7]helicene is documented and is due to the magnetic coupling with the neighboring ring placed below or above. [17a] Notably, for each [5]helicene and [7]helicene subunit in D3-2 an overall lower aromaticity than for the parent isolated [5]helicene and [7]helicene was computed (see graphs in Figure 4a). Also, the distribution of local aromaticity in the [7]helicene units in D3-2 differs significantly from the one of isolated [7]helicene with the central ring B being non-aromatic in D3-2. In order to differentiate torsional effects 0) values for all non-equivalent rings in D3-2. The graphs show the NICS(0) values for the [7]helicene and [5]helicene units in D3-2 (solid red and blue lines, respectively) and isolated [7]helicene and [5]helicene (dashed lines). ( b) NICS(0) values for all non-equivalent rings in C2-2. The graphs show the NICS(0) values for the [7]helicene and [5]helicene units in C2-2 (solid red and blue lines, respectively) and isolated and [5]helicene (dashed lines). (c) ACID plot of D3-2, the magnetic field is applied along the +z axis perpendicular to the view plane pointing to the observer, diatropic currents are highlighted with blue and green arrows and paratropic currents are highlighted with red arrows. (d) Clar's perspective of D3-2 showing 13 separated psextets; the Clar's perspective of C2-2 would be identical. (e) ACID plot of C2-2, the magnetic field is applied along the +z axis perpendicular to the view plane pointing to the observer, diatropic currents are highlighted with blue arrows and paratropic currents are highlighted with red arrows.

from through-space and conjugation effects in the NICS(0) calculations of the triphenylene cores of both D3-2 and C2-2, the NICS(0) were also computed for the virtual triphenylene molecules having the distorted geometries D3-2 and C2-2 (see the Supporting Information). Comparison with the NICS(0) of triphenylene itself showed an augmentation of ca. +2 ppm for the distorted rings when compared to their planar counterparts, showing that torsional effects have a significant contribution to the unusual NICS(0) value computed for D3-2 and C2-2. The ACID plot of D3-2 (Figure 4c) confirmed the NICS analysis: the external E rings display clear and strong diatropic ring currents consistent with full aromaticity, the C and D rings exhibit dominant diatropic ring currents but of lower intensity, while the A and B rings show competing small diatropic and paratropic currents indicating weak electronic delocalization and a non-aromatic character. The triphenylene core in D3-2 is probably the least aromatic known triphenylene unit, which correlates with its unprecedented torsion. More globally, a strong diatropic macrocyclic ring current is clearly visible at the periphery of the whole molecular architecture showing global electronic delocalization, which is certainly compensating for the severe distortions of the p-systems and contributing to the thermodynamic stability. The analysis of local aromaticity in diastereomer C2-2 revealed comparable features to the one of D3-2 with however a more complex situation due to the lower symmetry of the molecule (Figures 4b,e). Notably, the two homochiral [7]helicene units (those not crossed by the C2 axis) and the triphenylene core have different NICS(0) values. The ACID plot of C2-2 confirmed qualitatively its overall distribution of local aromaticity. Altogether, the multi-helicenes D3-2 and C2-2 are PAHs whose local aromaticity is subjected to variations of large amplitudes from entirely aromatic at the edges to nonaromatic at the cores, correlating the distribution of distortion in the molecules.

Complexation properties of multi-helicene D3-2. [7]Helicene itself was earlier demonstrated to behave as a chiral molecular tweezer for silver(I) ions, the metallic cation being sandwiched between the two outermost bonds of the helicene (C3-C4). [21] It was hypothesized that hextuple helicene D3-2 embedding three identical and moderately stretched [7]helicene units on its outer shell could behave as a mono, bis or triple tweezer capable of complexing up to three silver(I) ions to form some original chiral cationic metal-nanographene hybrids (Figure 5). The binding energies for the iterative complexation of one, two and three silver(I) ions inside the bay region of three [7]helicene units in D3-2 were evaluated by DFT calculations and compared to the one of Ag + with [7]helicene [21] (see the Supporting Information). This model study indicated a slightly better stabilization for [AgÌD3-2] + than for [AgÌ [7]helicene] + , and also that triscation [3AgÌD3-2] 3+ should be reasonably stable. Experimentally, a methanol/dichloromethane solution of D3-2 and AgNO3 (1:1) was analyzed by electrospray ionization mass spectrometry (ESI-MS) using very soft conditions, drastically limiting the generation of ions upon collision in the interface of the mass spectrometer (i.e. in the gas phase), so that most, if not all, observables result from the electrospray of cationic species initially present in solution. The formation of both the cation [AgÌD3-2] + and the biscation [2AgÌD3-2] 2+ in the solution could be clearly evidenced. However, performing ion mobility separation (IMS) prior to mass analysis to further enhance the dynamic range of detection permitted to evidence that [3AgÌD3-2] 3+ is also a stable complex. The relative stability of the three silver adducts was evaluated by the 11.2:100:3.8 abundance ratio measured for [AgÌD3-2] + / [2AgÌD3-2] 2+ / [3AgÌD3-2] 3+ by IMS-MS, clearly showing the biscationic [2AgÌD3-2] 2+ complex as the most stable one when observed as naked gas phase ions. The actual reasons for this preference remain unclear at this stage and may include the existence of a more compact and better adjusted conformation for biscationic [2AgÌD3-2] 2+ than for its mono-and triscationic analogues as seemingly indicated by the experimental determination of collision cross sections (see Supporting Information).

Conclusions

The molecular design and concept using p-system elongation and steric effects, based on helical strains surrounding a triphenylene core with configurationally stable [7]helicenes, has led to extreme distortion in some chiral PAHs. This idea was put into practice through a nickel(0)-mediated cyclotrimerization of enantiopure and racemic 9,10-dibromo [7]helicene, which diastereoselectively afforded large chiral hextuple helicenes of formula C90H48 with D3 and C2 symmetry, respectively. The diastereomer with D3 symmetry is a three-blade propeller-shaped chiral nanographene that embeds three homochiral [7]helicene units on its outer shell and three homochiral [5]helicene units of opposite configuration on its inner edges. Both D3-symmetric enantiomers were directly obtained as enantiopure materials by an enantiospecific Yamamoto-type cyclotrimerization, a premiere for large chiral PAHs, and a first example for this type of cyclotrimerization. The structural analysis of this molecule revealed a highly distorted triphenylene core with a marked bond lengths alternation of the six-membered rings. The diastereomer with C2-symmetry embeds three [7]helicene units of mixed helicity on its outer shell, and three [5]helicene units of mixed helicity on its inner edges so that four (P)-configured and two (M)-configured helicenes are present in the molecule. Its structural analysis also enlightened a pronounced alternation of bond lengths in its triphenylene core, and more remarkably extreme torsions establishing new limits for benzene (twist = 36.9º). The chiroptical properties of the two hextuple helicenes were found of relatively small magnitude, possibly due to a phenomenon of compensation between the intertwined (P)and (M)-configured helicenes. Aromaticity was computationally analyzed in these fascinating molecules using several methods, which was complicated not only because the quantification of the concept of aromaticity is intrinsically difficult, but also because existing models of aromaticity were developed for planar PAHs. Nevertheless, magnetic-based methods known as NICS and ACID proved to be useful, though not ideal, and they revealed a non-aromatic character of the triphenylene cores in both diastereomeric hextuple helicenes, in correlation to the severe distortion of the rings. An early exploration of the metal ions avidity and supramolecular properties of the D3-symmetric diastereomer revealed its ability to bind up to three silver(I) ions in the bay regions (cavities) of three [7]helicene units at its outer shell, as evaluated by the combination of DFT simulations with advanced mass spectrometry methods, opening the door to the field of chiral cationic metal-nanographene hybrids. To sum it up, the present study pushes the limits on the design and the synthesis of highly distorted chiral PAHs, where chirality provides additional control on molecules for modulating their properties for applications in chemical and materials science. It also deepens the knowledge on the conformational, chiroptical and supramolecular properties (solid and solution states) of chiral PAHs and nanographenes, and raises questions about aromaticity in these large distorted molecules.

Experimental Section

Detailed experimental procedures for the syntheses of (-)-D3-2, (+)-D3-2 and rac-C2-2, the resolution of rac-C2-2 by HPLC methods, the spectroscopic, structural, chiroptical, and photophysical characterization of all compounds, the experimental and computational enantiomerization study of D3-2, the full-detail analysis of aromaticity in D3-2,C2-2 and related molecules, the full-detail complexation study of silver(I) cations with D3-2, and the crystallographic data for (-)-(M,M,M,P,P,P)-D3-2 (CCDC 1835903) and rac-(P,M,P,P,P,M)-C2-2 (CCDC 1902495) are included as supporting information for this article.

Figure 1 .

 1 Figure1. Syntheses and some properties of the hextuple helicenes (-)-D3-2 and rac-C2-2. (a) Yamamoto-type cyclotrimerization of (M)-9,10-dibromo[7]helicene [(M)-1] leading to (-)-D3-2. (b) Three-dimensional representations of (-)-D3-2 obtained from single-crystal X-ray diffraction analysis; remarkable interplanar angles are highlighted in red. (c) Simplified plausible enantiomerization pathway of D3-2 obtained by DFT calculations [B3LYP/6-31G(d) in the gas phase], energies are free Gibbs energies and are expressed in kJ×mol -1 , see the Supporting Information for details.(d) Yamamoto-type cyclotrimerization of racemic 9,10dibromo[7]helicene (rac-1) leading to rac-C2-2. (e) Three-dimensional representations of rac-C2-2 obtained from single-crystal X-ray diffraction analysis; remarkable interplanar angles are highlighted in red.

Figure 2 .

 2 Figure 2. Some structural features of D3-2 and C2-2. (a) The triphenylene core in D3-2 including remarkable torsion angles and bond lengths (mean values). The solid-state conformation of D3-2 has crystallographic C2-symmetry, justifying for the non-equivalent torsions in all the three blue-colored rings. (b) The triphenylene core in C2-2 including remarkable torsion angles and bond length. The solid-state conformation of C2-2 has no crystallographic symmetry, justifying for the non-equivalent torsions in the two white-colored rings. (c) The naphthalene unit along the molecular C2 axis in C2-2 showing a very high endto-end torsion. (d) Profile view of the most twisted benzene ring in C2-2.

Figure 3 .

 3 Figure 3. (a) Crystal packing of (M,M,M,P,P,P)-D3-2 and representation of the solvent channels along the b axis. The content of the channels is a mix of explicitly determined d1-chloroform molecules and the calculated mask of solvent (yellow voids). (b) Representation of the (P)-configured supra-helix (pitch = crystallographic c vector = 27.2785 Å) in the crystal packing of (M,M,M,P,P,P)-D3-2: side view along the c axis, the four-fold 43 screw axis is in red, the centroids of the central rings of each molecule are represented as red dots and the distances between them (Å) are noted in green.

Figure 4 .

 4 Figure 4. Aromaticity analysis in D3-2 and C2-2. (a) NICS(0) values for all non-equivalent rings in D3-2. The graphs show the NICS(0) values for the[7]helicene and[5]helicene units in D3-2 (solid red and blue lines, respectively) and isolated[7]helicene and[5]helicene (dashed lines). (b) NICS(0) values for all non-equivalent rings in C2-2. The graphs show the NICS(0) values for the[7]helicene and[5]helicene units in C2-2 (solid red and blue lines, respectively) and isolated and[5]helicene (dashed lines). (c) ACID plot of D3-2, the magnetic field is applied along the +z axis perpendicular to the view plane pointing to the observer, diatropic currents are highlighted with blue and green arrows and paratropic currents are highlighted with red arrows. (d) Clar's perspective of D3-2 showing 13 separated psextets; the Clar's perspective of C2-2 would be identical. (e) ACID plot of C2-2, the magnetic field is applied along the +z axis perpendicular to the view plane pointing to the observer, diatropic currents are highlighted with blue arrows and paratropic currents are highlighted with red arrows.

Figure 5 .

 5 Figure 5. Computationally optimized structures of the most stable mono, bis and triscationic Ag + complexes of D3-2 (DFT, wB97XD/Def2TZVP//wB97XD/Def2SVP, gas phase). Left : [AgÌD3-2] + , middle : [2AgÌD3-2] 2+ , right : [3Ag + ÌD3-2] 3+ .

Acknowledgements

We thank Mrs. R. Rosas (Aix-Marseille Univ.) and Mrs. C. Troufflard (Sorbonne Univ.) for assistance with NMR spectroscopy, Prof. Dr. P. Ceroni (Univ. of Bologna) for assistance with photophysics, Mrs. M. Jean (CNRS) for assistance with HPLC, and the Centre Régional de Compétences