Yuhan Zhong

Vishnu Vijayakumar

Viktoriia Untilova

Mounib Bahri

Laurent Herrmann

Laure Biniek
email: laure.biniek@ics-cnrs.unistra.fr

Nicolas Leclerc
email: leclercn@unistra.fr

Martin Brinkmann
email: martin.brinkmann@ics-cnrs.unistra.fr

A Heeger

A G Mcdiarmid

H Shirakawa

Bringing Conducting Polymers to High Order: Toward Conductivities beyond 10 5 S

Keywords: Organic Thermoelectrics, Conducting Polymers, Polymer Crystallization, Thin films. I. Introduction

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Figure for Title of Content

An effective design strategy of polymer thermoelectric materials based on structural control in doped polymer semiconductors leads to a substantial improvement of charge conductivity (>10 5 S/cm) and thermoelectric power factors (2 mW . m -1. K -2 ). The highly anisotropic conducting polymer films are ideal model systems to investigate the correlations between thermopower S and charge conductivity . poly(ethylenedioxythiophene) (PEDOT) doped with tosylate (Tos). [10,11] Conducting polymers such as PEDOT-Tos have a layered structure of positively charged poly(ethylenedioxythiophene) chains alternating with layers of tosylate anions. (11) PEDOT-Tos shows remarkable thermoelectric properties due to the combination of high charge conductivity, reasonable thermopowers and low thermal conductivity. [10,11] Despite promising TE properties, the crystallization of PEDOT-Tos in thin films is difficult to control. So far, no simple method is available to either orient in-plane the PEDOT chains or to select the preferential contact plane of crystals on a given substrate. For a large part the synthetic method is to blame: it requires the use of in-situ polymerization techniques with a low control over the polymerization process and finally the structure of the polymer thin films. The only improvements in charge transport were obtained via control of secondary crystallization e.g. by using co-solvents such as ethylene glycol that helps extract the non-conducting polystyrene sulfonate from the bulk of the PEDOT-PSS films. [12,13] In strong contrast, advanced control of crystallization, crystal orientation and alignment have all been demonstrated for polymer semiconductors (PSCs) such as regioregular poly(3-hexylthiophene) (P3HT) or poly(2,5-bis(3-dodecyl-2-thienyl)thieno [3,2-b]thiophene) (C 12 -PBTTT) (See Figure 1a). [18,19] Numerous methods including self-seeded growth, blade-coating, epitaxy or high temperature rubbing help control precisely the crystal dimensions and their orientation in thin films, resulting in diverse morphologies: single crystals, spherulites, aligned crystalline lamellae. [14][15][16][17][18][19][20] It is therefore natural to take profit of the high structural control gained on polymer semiconductors such as P3HT or PBTTT to further improve their charge transport and thermoelectric properties after doping. Doping of thin films of P3HT or PBTTT with fluoroalkylsilanes (FTS) or F 4 TCNQ has led to high conductivities typically of the order of a few hundreds to one thousand S/cm. [21][22][23] Vaporphase doping of P3HT and PBTTT with F 4 TCNQ leads to conductivities of 12.7 S/cm and 200 S/cm, respectively. [22,23] Controlled doping of well-crystallized and oriented conjugated polymer can further lead to anisotropic conducting polymer films. [24] Solution-doping of rubbed P3HT films with F 4 TCNQ afforded aligned conducting polymers with conductivities of the order of 22 S/cm whereas vapor phase doping of highly crystalline P3HT afforded 12.7 S/cm conductivities. [22,24] More recently, the impact of the alkyl side chains on the efficiency of doping, its kinetics and the ultimate TE properties, was demonstrated for the family of PBTTT with alkyl side chains from C 8 to C 18 . [25] However, those values remain well below the best conductivities reported for doped polyacetylene (PA) or even PEDOT-Tos.

Is it possible to reach the conductivities of doped PA close to 10 5 S/cm for oriented films of P3HT or PBTTT (see Figure 1.a) after doping and what limits the ultimate conductivities? To answer these questions, we have investigated the possibility to reach high doping levels in films of P3HT and PBTTT oriented by high temperature rubbing. [18,24,26,27] The choice of the two polymers is guided by their structural differences in terms of nanomorphology in rubbed thin films: P3HT has a periodic semi-crystalline lamellar morphology whereas C 12 -PBTTT shows a liquid-crystalline-like structure. [18,19] The choice of FeCl 3 is motivated by the fact that it is a stronger oxidant than F 4 TCNQ. [28] Accordingly, this manuscript is divided in three parts dealing with: i) the structural and spectroscopic features induced by FeCl 3 -doping on aligned P3HT and C 12 -PBTTT films, ii) the dependence of charge transport and thermopower on [FeCl 3 ] concentration and iii) the analysis of the S- correlations in the oriented films of P3HT and C 12 -PBTTT.

II. Results.

1) Structural signatures of FeCl 3 doping.

As demonstrated previously for P3HT and F 4 TCNQ, the combination of an orientation method such as mechanical rubbing and sequential doping in an orthogonal solvent (See As seen in Figure 2.c and 2.f, doping modifies the unit cell parameters of C 12 -PBTTT and P3HT but to different extents (see also figure S1). For both polymers, the unit cell expands along the alkyl side chain direction (d 100 ) and the -stacking repeat (020) is reduced, demonstrating that dopant molecules penetrate inside the layers of alkyl side chains of the crystalline phase and that the change in lattice parameters correlate with the amount of incorporated dopants. [24,25] For both polymers, the interlayer spacing d 100 increases with For both polymers, doping results in a small decrease of the -stacking periodicity similar to that observed with F 4 TCNQ doping. [24,25] The main difference between C 12 -PBTTT and P3HT resides in the evolution of the side chain spacing. In C 12 -PBTTT, the maximum change in d 100 is equivalent to 3 CH 2 -CH 2 bonds i.e. 3.9 Å whereas in P3HT it is only 1.7 Å. In both cases, the increase of d 100 is smaller than the 7.4 Å van der Waals diameter of FeCl 4 -. Even more telling is the variation in unit cell volume ∆V in both cases. Assuming orthorhombic unit cells with two chains per unit cell for both P3HT (a=16.5 Å, b=7.8 Å and c=7.77 Å) and C 12 -PBTTT (a=19.7 Å, b=7.4 Å, c=13.5 Å), ∆V=20-30 Å 3 for P3HT versus ∆V=230-250 Å 3 for C 12 -PBTTT. Intercalation of small molecules such as phenyl-C 71 -butyric acid methyl ester (PC 71 BM) into the structure of PBTTT has been extensively analyzed and modelled. [29][30][31] It has been shown that PC 71 BM molecules intercalate in the structure of PBTTT and form bimolecular crystals of PC 71 BM and C 14 -PBTTT. The bimolecular crystal phase is characterized by an expanded unit cell parameter along the alkyl side chain direction. Since the van der Waals diameter of FeCl stoichiometry is further consistent with that observed in the heavily doped phase of PEDOT:Tos. [32,33] To confirm this stoichiometry, we further analyzed our samples using EDX -STEM. PBTTT are particularly loose and offer the possibility to harbor dopant molecules. Indeed, the areas per al l stem are 2 . 2 for C 14 -and 2 2 for P3HT (form I). Both values are substantially lower than the 18.4 Å 2 for the classical polymethylene subcell, indicating that the layers of side chains can indeed host small molecules such as F 4 TCNQ, FeCl 4 -or FeCl 2 . [24,25,34,35] 2) Nanomorphology of doped thin films. 50-100 nm for C 12 -PBTTT and 10-15 nm for P3HT. These differences in nano-morphologies are related to different persistence lengths of the two polymer chains: 3 nm for P3HT versus 9 nm for C 12 -PBTTT. [36,37] Accordingly, in strong contrast to the periodic semi-crystalline lamellar structure of P3HT, C 12 -PBTTT displays rather a liquid-crystal-like morphology with extended planarized chain segments along the chain direction (see illustration in 3) Spectroscopic signatures of doping.

TEM has shown that the amount of FeCl 3 incorporated in the crystals of C 12 -PBTTT and P3HT depends on the concentration [FeCl 3 ]. This trend is further supported by Polarized UV-vis-NIR spectroscopy that helps visualize the level of chain oxidation versus [FeCl 3 ] (see Figure 4). Increasing the doping concentration results in a corresponding increase of the polaronic features P1 and P2 and a decrease of the neutral polymer absorption (N). For both polymers, the NIR absorption bands P1 and P2 are polarized parallel to the alignment direction R and the films are strongly birefringent under the polarizing microscope. Most remarkably, the P3HT and PBTTT films doped at 10 mM exhibit a spectrum without significant contribution of the original neutral polymer (1-2%), suggesting an oxidation level of the polymer backbones similar to that of PEDOT-PSS (30-40%). [10] As a result, the films doped with 10 mM FeCl 3 are almost optically transparent with a light blue color similar to PEDOT:PSS (excess FeCl 3 gives rise to a small band at 374 nm). [38] . The presence of the P2 band indicates also that the regime of pure bipolaron formation is not attained for [FeCl 3 ]= 10 mM (for a nondegenerate ground-state polymer, the P2 transition is forbidden for the bipolaron. [39] ) P1 and P2 are more red-shifted in C 12 -PBTTT than in P3HT:  max (P2) = 809 nm for P3HT and 854 nm for C 12 -PBTTT. Although the position of the P1 peak cannot be observed directly and lies beyond 2500 nm, the slope of the absorbance curve at 2500 nm indicates that the P1 peak is also substantially more red-shifted for C 12 -PBTTT than for P3HT. Positions of polaronic features are determined by the extent of delocalization and/or crystalline perfection. [40,41] Hence the red-shifted polaron bands in C 12 -PBTTT suggest a larger polaron delocalization length relatively to P3HT. This is consistent with the larger extension of aligned and planarized chain segments observed by HR-TEM in C 12 -PBTTT (see Figure 3).

Typically, C 12 -PBTTT films doped with 10 mM FeCl 3 have a dichroic ratio of the P2 and P1 band of 10.6, which is very close to 10.8 of undoped C 12 -PBTTT. These similar values confirm that FeCl 3 doping does not alter substantially the orientation of the C 12 -PBTTT chains. For oriented P3HT films, the situation is different: the dichroic ratios of the P1 and P2 bands decreases with increasing [FeCl 3 ] (see Figure S6). This loss of orientation is possibly related to the loss of both alignment and crystallinity upon intercalation of dopant molecules. This is consistent with the decreased intensity of ED reflections and the loss of vibronic structure in the absorption of the non-doped phase (N). Interestingly, UV-vis spectra for POL  R (see Figure 4) indicate that the amorphous fraction of P3HT and C 12 -PBTTT is also doped with FeCl 3 . Indeed, upon doping with increasing [FeCl 3 ], the absorption band of the amorphous phase (at 526 nm in C 12 -PBTTT and 506 nm in P3HT, see Figure S7)

progressively disappears. This result is at variance with the case of F 4 TCNQ-doped films for which no evidence for doping of the amorphous domains was seen by polarized UV-vis spectroscopy (See Figure S8). [24] The fact that both crystalline and amorphous phases are doped with FeCl 3 explains, at least in part, the higher overall conductivity relatively to F 4 TCNQ. [24] Comparison with F 4 TCNQ-doped films indicates that the doping level achieved with FeCl 3 is substantially larger. In particular, the remaining proportion of neutral C 12 -PBTTT and P3HT is much lower (only a few percent) for FeCl 3 with respect to F 4 TCNQ. [16] Following the work by Miller et al., the high oxidation levels at saturation of C 12 -PBTTT and P3HT upon doping with FeCl 3 could reflect a higher dopant miscibility as compared to F 4 TCNQ. [29] Moreover, the spherical shape of the FeCl 4 -anions may also favor a closer proximity of the dopant to the polymer backbone as compared to the rod-like F 4 TCNQ whose long axis tends to align parallel to the alkyl side chains i.e. in a plane perpendicular to the polymer backbone. [24,25] As seen hereafter, the higher overall oxidation of P3HT and C 12 -PBTTT chains achieved upon FeCl 3 doping accounts for the improved charge conductivities.

4) Anisotropy of charge transport.

As demonstrated in the previous section by TEM and UV-vis-NIR spectroscopy, the proportion of oxidized versus neutral chains in the oriented P3HT and C 12 -PBTTT films can be tuned by adjusting [FeCl 3 ] while maintaining the high level of in-plane orientation. This gives a handle to control the charge carrier density in the oriented films and to determine its impact on anisotropic charge transport and TE properties. Therefore, a sequential doping protocol was used to probe the changes in charge conductivity and thermopower on a given oriented film sample: the same oriented film was doped in successive steps with a solution of FeCl 3 /nitromethane of increasing concentration. This alleviates all issues related to differences in morphology and structure of the rubbed samples that would result in a scattering of the charge conductivity values  and Seebeck coefficients S. Figure 5 shows the evolution of  and S versus FeCl 3 concentration for P3HT and C 12 -PBTTT oriented films that were sequentially doped with FeCl 3 solutions of increasing concentration. Table 1 collects the highest values of conductivity and power factors for both doped polymers and corresponding values from the literature for iodine-doped PA.

Upon doping, both P3HT and C 12 -PBTTT show a strong increase of conductivity with a saturation at higher FeCl 3 concentration (for both // and  directions). In the aligned C 12 -PBTTT films, the conductivity increases by several orders of magnitude from 150 S/cm at 0.1 mM to 2.2 10 5 S/cm at 5 mM. The highest conductivities of C 12 -PBTTT surpass the 10 5 S/cm reported for stretch-oriented bulk films of polyacetylene doped with iodine. [4] However, it is worth to note that it is necessary to realize a progressive doping with increasing dopant concentration to reach such large conductivities. Indeed, C 12 -PBTTT films doped directly with a 10 mM solution reach only a 10 4 S/cm conductivity. As a matter of fact, PBTTTs with shorter or longer alkyl side chains showed also very high conductivities (see Table SI1) albeit below those of C 12 -PBTTT. This specificity of C 12 -PBTTT was previously noted in the case of F 4 TCNQ-doped PBTTTs and reflected a higher doping level for this polymer. (25) For the highly conducting C 12 -PBTTT films, one can estimate the charge carrier density based on the structural data indicating that the unit cell contains two FeCl 4 -anions per unit cell, which translates to a maximum of two charge carriers per unit cell i.e. a carrier density of approx. 9 . 10 20 cm -3 . [42] This value is larger than the 0.3-0.95 . 10 20 cm -3 reported for FTS-doped PBTTT. [43] However, the present result appears consistent in view of the UV-vis-NIR results indicating full oxidation of the C 12 -PBTTT backbone contrary to the FTS-doped PBTTT that showed a substantial amount of neutral polymer chains and an oxidation level of only 3.5-11%. [43] As for P3HT, alignment improves substantially the conductivity from 63 S/cm in nonoriented FeCl 3 -doped P3HT films to 570 S/cm along the rubbing direction in oriented films (for [FeCl 3 ] = 5mM). [38] The anisotropies of charge conductivity are much larger in C 12 -PBTTT than in P3HT ( // /  =70 for C 12 -PBTTT and 12 for P3HT). This difference is partly due to different film structures. As shown by TEM, the rubbed C 12 -PBTTT films have a dominant face-on orientation whereas P3HT has a mixture of edge-on and face-on oriented domains. As a consequence,   is essentially probed along the insulating C 12 side chains of C 12 -PBTTT whereas transport along -stacking and side chains coexist in P3HT films. As an alternative explanation, the reduced anisotropy of the charge conductivity in P3HT could be due to the loss of orientation/crystallinity and/or to the doping of amorphous zones for increasing [FeCl 3 ] as suggested by polarized UV-vis spectroscopy. As a rule, the maximum conductivity of doped P3HT films is always below that of C 12 -PBTTT. It does not reflect a difference in the degree of in-plane orientation since both polymers have similar high dichroic ratios prior to doping. It is also not due to different doping levels since the spectroscopic features for 10 mM-doped films of P3HT and C 12 -PBTTT are almost identical. The lower conductivity of doped P3HT must thus reflect intrinsic charge transport limitations related to its semi-crystalline periodic lamellar structure.

In doped P3HT films, charges must necessarily channel through the less conductive amorphous zones that harbor chain ends and chain folds, hence, lowering the macroscopic conductivity  // . In the liquid crystal-like structure of C 12 -PBTTT, more effective percolation between ordered domains of highly aligned chains in the rubbing direction (see HRTEM image in Figure 2) results in a larger conductivity  // . Even in the direction perpendicular to the rubbing, C 12 -PBTTT films display particularly high conductivities of  2100 S/cm that are close to those observed in non-oriented FTS-doped C 14 -PBTTT. [21] This indicates that the conductivity measured in non-oriented PBTTT samples is essentially limited by the charge transport perpendicular to the chains. Achieving such high conductivities in doped C 12 -PBTTT on a macroscopic scale implies also that charges must be transported quite effectively between the well doped and ordered domains seen in HRTEM. Charge tunneling between highly conducting domains is the mechanism invoked in a granular metal model of Sheng to account for comparably high conductivities in heavily-doped PA. [44] In strong contrast, in P3HT, the large extension of amorphous interlamellar zones will hamper tunneling between doped crystalline regions and limit the macroscopic conductivity.

5) Anisotropy of thermopower and power factor.

As seen in Figure 5.b and 5.e, the thermopower variation with dopant concentration is inverse to that of charge conductivity: both Seebeck coefficients S // and S  decrease with [FeCl 3 ]. In all samples, the thermopower probed along the chain direction is larger than that measured perpendicular: S // > S  . [23] The anisotropy factor of the Seebeck coefficient S // /S  tends to decrease with increasing FeCl 3 concentration for both polymers. The anisotropies are higher for C 12 -PBTTT (4.6) than for P3HT (2.4). Overall, as observed previously for F 4 TCNQ-doped P3HT, the anisotropy in thermopower is always smaller than that of the charge conductivity. [24,25] The samples with highest conductivities have very small thermopowers (see Table 1). For instance, the rubbed C 12 -PBTTT sequentially doped with 5 mM FeCl 3 /nitromethane have  // =2.2 10 5 S/cm and S // =9.4±0.5 V/K. Their Seebeck coefficients measured perpendicular to the alignment are even smaller with S  =1.5 ± 0.5 V/K. Such low thermopowers are characteristic of metals. [START_REF] Blatt | Thermopower of Metals[END_REF] For comparison, oriented films of PA doped with iodine have S // =15 V/K. [3,4] At this point we focus on the origin of the anisotropy of the thermopower S in oriented doped films of P3HT and C 12 -PBTTT. The present situation bears similarity with the case of doped oriented PA fibers that show S // /S  >1. [START_REF] Kaiser | [END_REF]47] Kaiser proposed a heterogeneous model of metallic fibrils separated by electrical barriers to explain this observation. [START_REF] Kaiser | [END_REF] In doped PA, semi-conducting barriers alternate with metallic domains and modulate the thermopower and its anisotropy. The situation is similar for doped oriented P3HT and C 12 -PBTTT. Indeed, HRTEM evidences the semi-crystalline morphology of doped P3HT and C 12 -PBTTT with a clear coexistence of highly ordered domains and amorphous interlamellar zones. Moreover, it is known that the charge conductivity is higher in crystalline versus amorphous domains of doped P3HT (typically a few S/cm are obtained in F 4 TCNQ-doped regio-regular P3HT versus 0.01 S/cm in regio-random amorphous P3HT). (22) This difference in charge transport between amorphous and crystalline domains must translate into a corresponding difference of the thermopower S a and S  for amorphous and crystalline domains, respectively, with S a > S  .

Accordingly, the semi-crystalline morphology of oriented conducting polymer films can be analyzed by using a heterogeneous material model similar to that proposed for doped PA.

When the temperature gradient ∆ is oriented parallel to the rubbing, it is distributed over a succession of crystalline and disordered (amorphous) domains connected in series. We consider that the sample consists of a fraction  of crystalline domains. The crystalline domains are characterized b an average temperature gradient ∆  (along the chain direction)

and disordered/amorphous domains b an average temperature gradient ∆ a . We also assume that the heat current flows along the same path as the charges, so that the temperature gradients across crystalline and amorphous domains are proportional to their thermal resistances (∆  W  and ∆ a  W a with W  and W a the thermal resistances of crystalline and amorphous domains, respectively). Hence, the Seebeck coefficient in the rubbing direction S // can be written as: [START_REF] Kaiser | [END_REF] S // =S  +(1-)S a (1) Where  is given by:

(

In the case of P3HT, the crystallinity  is determined by the rubbing temperature and is close to 50% for T R = 200°C. [26] In semi-crystalline polymers such as polyolefins, the thermal resistance of crystalline lamellae along the chain direction W  is small whereas the one perpendicular to the chains is significantly larger and comparable to that of the amorphous phase i.e. W  W a . [48] This implies S // S a i.e. the Seebeck coefficient S // is dominated by the contribution from the doped amorphous and less conducting phase.

Let us now consider what happens when the temperature gradient is oriented perpendicular to the chain direction. In this situation, crystalline and disordered zones are put in parallel along the temperature gradient and charges are short-circuited through the most conductive domains i.e. along the crystalline domains. As seen in TEM of the rubbed 180°C

P3HT sample, the crystalline lamellae make a quite continuous conduction path given their lateral extension perpendicular to the rubbing direction (several hundreds of nm) and the way such lamellae are interconnected. Accordingly, charges will be tunneled out of the sample within the crystalline domains in the direction perpendicular to the rubbing and this implies that the observed Seebeck coefficient S  is dominated by that of the most conductive lamellae and not by the disordered (amorphous) domains. This implies that S  S  . As a consequence, the Seebeck coefficient anisotropy becomes S // /S  S a /S  >1.

The present analysis indicates that a nanostructured morphology in conducting polymers such as doped P3HT or PBTTT that results in a modulation of electronic properties can be beneficial in oriented films since both the thermopower S // and the conductivity  // are increased in the direction of polymer chains. It is important to recall that one major strategy to improve TE properties in inorganic systems is also based on material nano-structuring.

Indeed, the objective of nano-structuring inorganic materials such as Bi 2 Te 3 is to break the heat transport in the materials through enhanced scattering of phonons at multiple interfaces between domains without breaking charge transport. [49] The resulting power factors for the most conductive C 12 -PBTTT films are very high and can reach values close to 1.94 mW . m -1. K -2 , a value slightly larger than for I 2 -doped PA (1.35 mW . m -1. K -2 ). [3] Such power factors are quite large compared to those of non-oriented films of C 12 -PBTTT doped with F 4 TCNQ from the vapor phase (0.1 mW . m -1. K -2 ). The combination of orientation and a higher oxidation of the C 12 -PBTTT polymer after FeCl 3 doping accounts for the remarkable power factors obtained in rubbed thin films. Previous UV-vis studies indicate that a high proportion of neutral polymer chains is present in oriented P3HT and C 12 -PBTTT films after doping with F 4 TCNQ. This implies a high proportion of semi-conducting chain segments and domains that prevents reaching similarly high conductivities as for FeCl 3 doping. and S  -ln() (see text).

6) Correlations between thermopower and charge conductivity in oriented conducting

polymer films.

Kang and Snyder have recently demonstrated that details of microscopic charge transport can appear through the correlations between S and  [50] A general theoretical frame relating to the S=f( scaling law was recently proposed by these authors. They predicted a scaling behavior with an exponent -1/s where s is a function of the charge transport mechanism. [50] In PEDOT:TOS the S- curve is well fit with s=1 pointing at a transport mechanism dominated by acoustic phonon scattering. The exponent -1/3 with s=3 is obtained for a transport mechanism involving scattering of carriers by ionized impurities. From an experimental point of view, Chabinyc et al. showed that S and  are related via the relation S  . [51,52] The set of (S,) data obtained over a large range of conductivities (1.0-10 5 S/cm) in this study helps further probe the S=f() correlations and its anisotropy especially in a regime of high conductivities (> 10 3 S/cm). This methodology alleviates the experimental difficulties associated with charge carrier determinations in doped films e.g. using AC Hall measurements, especially for such aligned films.

The S- plots for directions parallel and perpendicular to the chains are shown in Fig. 6.a and 6.b for C 12 -PBTTT and P3HT, respectively. For both polymers, the S = f( plots show different behaviors parallel and normal to the chain directions. In the direction parallel to the chains (R), the correlation between S // and  // is well described by a power law with an exponent -1/4. This is consistent with previous results by Chabinyc and coworkers for F 4 TCNQ-doped C 14 -PBTTT. [51,52] In the direction perpendicular to R, both polymers show a deviation from the power law, especially for high values of the conductivity. The conductivity   is observed to saturate close to 200 S/cm in P3HT and 2000 S/cm in C 12 -PBTTT. The departure from the scaling law in the case of S  -  can have different origins. First, it could be related to some intrinsic structural change in the films when the doping concentration becomes important. We observe that the dichroic ratio for the P1 band as well as the intensity of the reflections in the ED pattern of doped P3HT films tend to decrease when [FeCl 3 ] approaches 10 mM. A disordering of the layers of -stacked backbones may be induced by the introduction of increasing amounts of FeCl 3 in the layers of side chains for both P3HT and PBTTT (Figure S9), indicating that it is apparently independent of the chemical nature of the dopant. The similarity of the behavior for FeCl 3 and F 4 TCNQ further supports the fact that the high conductivity in FeCl 3 -doped films is not related to a contribution from ionic conductivity that should result in a departure from the S  scaling law. To the best of our knowledge, it is the first time that such a difference between the S- correlations is evidenced along and normal to the chain direction. As a matter of fact, different S- correlations along and normal to the chain direction would further explain, at least in part, the large spread in experimental data in reference 49 and 50 as the measurements concern isotropic films for which both contributions parallel and normal to the chain direction can combine to different extents.

III. Conclusion.

Controlling crystallization and orientation of polymer semiconductors prior to doping gives a unique handle over the charge transport and thermoelectric properties of the obtained conducting polymer films. This strategy is versatile and can be applied to numerous doped polymer semiconductors including n-type polymers [54] or low-bandgap polymers [55,56] that can be oriented by different methods such as blade-coating, high temperature rubbing or epitaxy and subsequently doped using different types of dopants and doping methods. Central to this approach is the preservation of high order present in the pristine polymer semiconductors upon doping. This is possible when the dopants intercalate progressively into the layers of side chains by incremental increase of the dopant concentration in such a way that dopant intercalation preserves orientation and -stacking of the conjugated polymer backbones.

Three conditions are identified to obtain conductivities of 10 5 S/cm: i) achieve high in-plane orientation of conjugated polymers with a high persistence length, ii) ensure a regular intercalation of dopant molecules in the polymer structure to form bimolecular polymer-dopant crystals and iii) maintain a percolating nano-morphology along the chain direction e.g. via a liquid-crystal-like morphology (semi-crystalline lamellar morphologies are detrimental).

Moreover, using a heterogeneous model approach based on the semi-crystalline structure of the doped and aligned polymer films, we are able to account for the anisotropy of the Seebeck coefficient that is enhanced in the direction of alignment.

Two different charge transport behaviors are evidenced in the directions parallel and normal to the polymer chains: S //   -1/4 prevails in the direction parallel and S  -ln(   in the direction normal to the polymer chains. Regarding TE applications, the combination of effective chain alignment and controlled oxidation by sequential doping helps to reach a record power factor of 2 mW . m -1. K -2 that could result in high ZT factors provided the thermal conductivity is well contained along the chain direction. Such a result is important as it sets a new upper limit for the ultimate TE power factor of such conducting polymers, but, it will be essential to further address the important issue of thermal conductivity and its correlation to the charge conductivity to obtain TE polymers with outstanding performances.

IV. Experimental Section.

1) Material and characterization.

P3HT was purchased from Merck whereas C 12 -PBTTT was synthesized following the detailed procedure described in ref. 19. The macromolecular parameters are collected in Table 2.

Sodium polystyrenesulfonate (NaPSS), anhydrous FeCl 3 and solvents (anhydrous nitromethane and ortho-dichlorobenzene) were obtained from Aldrich. All the chemicals were used as received. 2) Orientation of thin films.

The preparation of oriented P3HT and C 12 -PBTTT films by high-temperature rubbing followed the protocol described in previous publications. [19,20] P3HT and C 12 -PBTTT films were prepared by doctor blading a hot polymer solution (10 mg/ml) in ortho-dichlorobenzene (oDCB) on NaPSS-coated glass substrates at 160°C. The NaPSS films are spin coated (3000 rpm, 10 mg/ml aq) on pre-cleaned glass substrates and serve as sacrificial substrates to float the rubbed films on the pre-patterned devices for Conductivity/Seebeck measurements or on copper grids for TEM analysis. High temperature rubbing was performed by using a homemade set up consisting of a translating hot plate on which the sample is fixed and a rotating cylinder covered with a microfiber cloth. The films were rubbed at different rubbing temperatures depending on the polymer (125°C for C 12 -PBTTT and 180°C for P3HT). To determine the film thickness after rubbing, the films were melt-annealed at 350°C under nitrogen in a linkam hot stage (HS-95 controller and LTS420 stage) to randomize the in-plane chain direction and the thickness was extracted from the UV-vis absorbance (See Figure S10).

3) Doping protocol.

All doping experiments were carried out in a glove box (Jacomex) with P O2 < 3 ppm and P H2O <1 ppm. 10 mM FeCl 3 solutions were prepared by dissolving 8 mg of FeCl 3 in 5 ml anhydrous nitromethane in a glass vial and subsequent dilution yielded the solutions with concentration in the range 0.1 -5 mM. Doping was done by dipping the oriented polymer film in the dopant solution for 30 s -1 min until no color change was any more visible. Aging of the FeCl 3 -doped films is observed when the samples are placed in ambient (dark). Figure S11exemplifies the evolution of the UV-vis-NIR absorption spectrum and the sheath resistance when the films are transferred from the glovebox to ambient atmosphere. The clear decrease in polaronic absorption correlates with the increase of sample resistance. Typically, the sample's resistance increases b a factor after one hour in ambient. Therefore, all electrical characterizations were performed in a glovebox whereas UV-vis-NIR measurements were performed within 1-2 min after transfer of the sample to the spectrometer in ambient to avoid substantial dedoping.

3) Thin film characterization.

Structural analysis: TEM: The preparation of the samples for TEM analysis involved three steps: i) floating the oriented polymer films on distilled water and recover them on TEM copper grids, ii) coat the films with a thin amorphous carbon layer using an auto 306 Edwards appropriate shadow masks. The contact geometry used for the electrical conductivity and thermopower measurements is described and illustrated in Figure S2

  10 5 S/cm PF=2 mW . m -1. K -2

Figure 1 )

 1 Figure 1) yields highly oriented and crystalline conducting polymer films. The same strategy

Figure 1 .

 1 Figure 1. a) Chemical structure of P3HT and C 12 -PBTTT. b) Schematic illustration of the two step process to prepare oriented conducting thin films combining high temperature rubbing and subsequent doping in a solution of FeCl 3 in nitromethane. c) Illustration of the color change in rubbed C 12 -PBTTT films (left) upon doping with 1 mM and 10 mM FeCl 3 /nitromethane solutions. d) Polarized Optical Microscope images under crossed polarizers showing the high birefringence in the as-rubbed and doped C 12 -PBTTT films.

Figure 2 .

 2 Figure 2. Structure evolution in doped P3HT and C 12 -PBTTT upon doping with FeCl 3 as obtained by low dose TEM. Electron diffraction patterns of C 12 -PBTTT before (a) and after (b) doping with 10 mM FeCl 3 . c) Interlayer spacing d 100 and -stacking distance versus FeCl 3 concentration in nitromethane for C 12 -PBTTT films. TEM. Electron diffraction patterns of

[FeCl 3 ]

 3 and saturates for [FeCl 3 ] approaching 10 mM. A maximum of doping has been reached. d 100 increases from 19.7 Å (pristine) to 23.6 Å (10 mM) in C 12 -PBTTT and from 16.4 Å (pristine) to 18.1 Å (10 mM) for P3HT.

4 -and 2

 42 FigureS2and S3 (see ESI) shows the elemental analysis in terms of EDX spectra and 2D maps for 5 mM FeCl 3 -doped C 12 -PBTTT and P3HT. First, one can see that there is a uniform distribution of both S and Fe atoms in the doped layers, indicating that doping is uniform in the samples. Second, the EDX spectra help determine the atomic ratio of S and Fe elements. For C 12 -PBTTT, one obtains a value close of 1.9±0.2 whereas for P3HT it is approximately 2.5±0.2. The S/Fe ratio for C 12 -PBTTT is in full agreement with the value deduced from the unit cell parameter variation assuming it is related to the uptake of 2 FeCl FeCl 2 . For P3HT, the situation is different as the unit cell expansion is moderate and does not reflect the uptake of dopants. The absence of important unit cell variation in P3HT suggests that the alkyl side chain layers of P3HT can accommodate more readily the intercalation of FeCl 4 -and FeCl 2 molecules possibly because the side chains of P3HT are not interdigitated. However, for both polymers, the incorporation of dopants in the layers of alkyl side chains requires a structural reorganization especially when side chains are interdigitated (see the schematic illustration in Figure2.g). Lattice expansion along the side chains, partial removal of interdigitation as well as tilting of the backbones and side chains are the potential mechanisms by which the structure of the doped phase can adapt to a given concentration of dopants. It is worth to mention that both the alkyl side chain sublattices of P3HT and of C 12 -

Both C 12 -

 12 PBTTT and P3HT show a coexistence of crystalline and amorphous domains but their nano-morphologies are substantially different as revealed by bright field and HR-TEM. Firstly, bright field TEM shows that only the P3HT films have a characteristic periodic lamellar morphology (see Figure S4 and S5) whereas no such morphology is observed for rubbed C 12 -PBTTT. This difference is further visible in the HR-TEM images of figure 3. Highly ordered domains of-stacked polythiophene backbones periodically separated by layers of alkyl side chains containing the dopants appear in the form of fringed patterns with a 17.6 Å periodicity for doped P3HT and 22.5 Å for doped C 12 -PBTTT ([FeCl 3 ] = 4 mM). The ordered domains have very different lengths of planarized chain segments for both polymers:

  Figure 3.b and 3.e). As a result, the connectivity between ordered domains is far better along the chain direction in C 12 -PBTTT films (see illustration in figure 3.b and 3.d). As seen hereafter, these different nano-morphologies impact the ultimate conductivities of doped oriented films.

Figure 3 .

 3 Figure 3. Low dose HRTEM image of FeCl 3 -doped C 12 -PBTTT (a) and P3HT (c) films aligned by high temperature rubbing ([FeCl 3 ] = 4 mM). The insets correspond to the Fast Fourier Transforms. Schematic illustration of the nanomorphology in the LC phase of C 12 -PBTTT (b) and the semi-crystalline lamellar morphology of P3HT (d).

Figure 4 .

 4 Figure 4. Evolution of the polarized UV-vis-NIR spectrum in oriented P3HT and C 12 -PBTTT thin films prepared by high temperature rubbing and subsequently doped in solution of FeCl 3 /nitromethane of increasing concentration. The light polarization is parallel (//) to the rubbing direction or perpendicular to (). The asterisk highlights the absorption peak of the excess FeCl 3 present on the film surface.

Figure 5 .

 5 Figure 5. Variation of the charge conductivity  (a and d), Seebeck coefficient S (b and e) and power factor PF (c and f) as a function of increasing FeCl 3 concentration for doped P3HT and C 12 -PBTTT oriented films.  S and PF are measured along the rubbing direction (//) and perpendicular to it (). The lines are guide to the eye. The blue dotted lines in d) and e) show the typical values of conductivity and power factors reached for PA doped with iodine, respectively.

Figure 6 .

 6 Figure 6. Correlation between thermopower S and charge conductivity s in highly oriented thin films of a) C 12 -PBTTT (T R =125°C) and b) P3HT (T R =180°C) doped with FeCl 3 . // and  refer to the direction of the measurements respectively parallel and perpendicular to the chain direction. The full lines correspond to the results of the fit using a scaling law S //   -1/4

  of reference 24. To measure the anisotropy of conductivity and TE properties, the devices consist of two devices oriented along the rubbing direction and two oriented normal to the rubbing direction. The oriented P3HT and C 12 -PBTTT films prepared by rubbing on NaPSS substrates were floated on distilled water and subsequently recovered on the glass-substrates covered with the gold electrodes. Doping with FeCl 3 was performed on these devices by dipping in the solutions of given concentration.Four-point-probe measurements of electrical conductivity were performed following the procedure described in detail in reference 24 and 25. A Keithley 2634B sourcemeter and a Lab Assistant Semiprobe station in a Jacomex glovebox under N 2 atmosphere were used to measure all conductivity and Seebeck coefficients. The resistivity  was derived from the sheet resistance R following the equation =1.81Rt (consider reference 24 for the determination of the geometrical correction factor). The obtained conductivity values are an average of four devices. For the Seebeck coefficients, a differential temperature method was used i.e. a temperature gradient was established across the sample either along the rubbing direction or normal to it. Details of the experimental setup used to measure the thermopower are given in reference 24.

Table 1 .

 1 Thermoelectric characteristics of the FeCl 3 -doped oriented films of C 12 -PBTTT and P3HT and comparison with stretch-oriented iodine-doped PA (3).

	Sample	P3HT	C 12 -PBTTT	I 2 -doped PA (ref. 3)
		[FeCl 3 ]=5 mM	[FeCl 3 ]=5 mM
	 // (S/cm)	570±100	(2.2±0.5) . 10 5	6 . 10 4
	  (S/cm)	48±20	2100±300	-
	S// (V/K)	5.4±0.5	9.4±0.5	15
	S  (V/K)	2.3±0.5	1.5±0.5	-
	Power factor PF // (W.m -1. K -2 )	21±6	1944±626	1350

Table 2 .

 2 Macromolecular parameters of the semiconducting polymers used in this study. (Gel permeation chromatography was performed in hot trichlorobenzene versus PS standard)

	Polymer	M n (g/mol)	M w (g/mol)	PDI
	P3HT	24200	43600	1.8
	C 12 -PBTTT	26000	45000	1.73

Acknowledgments.

Bernard Lotz is gratefully acknowledged for fruitful discussions and careful reading of the manuscript and C. Blanck and M. Schmutz for technical support in TEM. P. Allgayer is acknowledged for technical support with the rubbing machine and N. Zimmermann preparation of pre-patterned device. We thank Ovidiu Ersen for giving us access to the EDX spectroscopy facility on the TEM platform from IPCMS. Financial supports from ANR grant Anisotherm and CNRS grant PEPS Thermobody are acknowledged. VV and VU are grateful for financial support from Idex Attractivité (University of Strasbourg) and IRTG Softmatter / Région Grand'Est, respectivel .

C 12 -

. his disordering ma account for a "saturation" of the conductivit along the chain direction for high FeCl 3 concentration.

Alternatively, the departure from the power law may indicate a different transport mechanism in the direction perpendicular to the polymer chains. As a matter of fact, attempts to fit the S  -  plot indicate that S  scales rather well with ln(  ). A similar dependence was reported by Mateeva et al. for oriented polyaniline films for which the S- relation was fitted with the equation: [53] (3)

with k B the Boltzmann constant, e the charge of the electron and  a dimensionless coefficient. i refers to the direction either parallel (//) or perpendicular () to the chain direction. [53] When fitted with equation (1), one obtains =3809 S/cm and =26. To avoid sample ageing upon transfer in ambient atmosphere, the TEM grids were mounted on the TEM sample holder inside the glove box and the holder was then transferred to the TEM in an air-tight container. The samples were only swiftly exposed to air (1-2s) just before introducing the sample holder into the TEM. TEM was performed in the bright field and diffraction modes using a CM12 Philips microscope equipped with an MVIII (Soft Imaging System) Charge Coupled Device camera. Beam exposure was set to a minimum using the low dose system to avoid de-doping under the electron beam that is observed when the same zone is exposed for a prolonged period of time.

STEM-EDX experiments were performed using a Cs-corrected JEOL JEM-2100F microscope operated at 100 keV to avoid dedoping of the films under the beam. Energy Dispersive X-Ray spectroscopy (EDX) mapping was performed using JEOL Silicon Drift Detector (DrySD60GV: sensor size 60 mm 2 ) with a solid angle of approximately 0.5 srad".

Polarized UV-Vis-NIR absorption: The orientation of the polymer films was probed by UV-Vis-NIR absorption (350-2500 nm) using a Cary 5000 spectrometer with polarized incident light and spectral resolution of 1 nm. The UV-Vis-Nir spectra of the doped polymer film were measured parallel and perpendicular to the rubbing direction R.

4) Electrical conductivity and Seebeck Coefficient.

All devices were fabricated on glass substrates following the conditions described in references 24 and 25. 40 nm-thick gold contacts were prepared by thermal evaporation using


Numerical study of a Wave Energy Converter located in front of a vertical breakwater.

Tournant a P, Perret a G., Neuvéglise a S., Marin a F., Smaoui b H. and Sergent b P.

Abstract-We consider the numerical modelling of a quayside WEC (Wave Energy Converter) using OpenFOAM. The numerical model uses sliding interfaces with moving mesh and waveFoam's method to generate and absorb wave reflection. Water overtoppings over the wall and RAO (Ratio Amplitude Response) results are investigated. The influence of clearence distance and the buoy draft is studied. The numerical model is validated by comparing the results with experimental data and analytical results using a linear potential model. We observed a good correlation between experimental and numerical results. Overtoppings decrease and RAO values increase when the clearance distance is decreased. Index Terms-CFD, OpenFOAM, vertical wall, Wave Energy Converter, I. INTRODUCTION D UE to global warming, renewable energy tech- nologies present a very active research sector. Unlike wind turbine, wave energy converters are dependent on the desired location. A system at deep water, will be very different from a nearshore system.

Nearshore systems present the advantages to reduce OPEX (Operational Expenditure) by reducing maintenance and grid connection costs. In addition, quayside systems benefit from waves reflection on the vertical breakwater.

The influence of a vertical wall on floating systems was first studied analytically in different configurations using a linear potential model. Such model was first developed by Mc Iver and Evans [START_REF] Mciver | An approximate theory for the performance of a number of wave-energy devices set into a reflecting wall[END_REF] to study the impact of additional vertical walls on energy devices performance. Hsu and Wu [START_REF] Hsu | The hydrodynamic coefficients for an oscillating rectangular structure on a free surface with sidewall[END_REF] showed that hydrodynamics coefficients of a heaving buoy were strongly modified by the presence of the wall. His model was completed by Zheng et. al. [START_REF] Yh Zheng | On the radiation and diffraction of water waves by a rectangular buoy[END_REF]. Still using linear potential model, Elchahal et. al. [START_REF] Elchahal | The effects of reflection coefficient of the harbour sidewall on the performance of floating breakwaters[END_REF] studied the clearance distance influence on the RAO (Response Amplitude Ratio) of a rectangular floating breakwater in front of a harbour. Bhattacharjee and Guedes Soares [START_REF] Bhattacharjee | Wave interaction with a floating rectangular box near a vertical wall with step type bottom topography[END_REF] added a bottom topography. WEC efficiency and the CWR (Capture Width Ratio) increase were studied for circular floating buoy [START_REF] Schay | Numerical modelling of a heaving point absorber in front of a vertical wall[END_REF]. A detailed study of a rectangular P. Tournant, S. Neuvéglise, G. Perret and F. Marin was at the LOMC, UMR CNRS 6294, Université du Havre, 53 rue de Prony, CS 80 540, 76058 Le Havre, France (e-mail: paul.tournant@univlehavre.fr,gaele.perret@univ-lehavre.fr,francois.marin@univlehavre.fr).

H. Samoui and P. Sergent was at the Centre d'Etudes et d'expertise sur les Risques, l'Environnement, la-DTechEMF, 134 rue de Beauvais-CS 60039, Margny-Lès-Compiègne 60280, France (e-mail: Hassan.Smaoui@cerema.fr,Philippe.Sergent@cerema.fr).

H. Samoui was also at Laboratoire Roberval/LHN, Sorbonne Université, Université de Technologies de Compiègne, FRE CNRS 2012, CS 60319, Compiègne 60203, France.

floater in front of a vertical wall has been recently performed analytically by Zhang et. al. [START_REF] Zhang | The effect of the coastal reflection on the performance of a floating breakwater-wec system[END_REF].

Those results were compared with experimental and numerical studies. The ability to study WEC performance using OpenFOAM is investigated in many research [START_REF] Windt | Highfidelity numerical modelling of ocean wave energy systems: A review of computational fluid dynamics-based numerical wave tanks[END_REF]. There are some dynamic mesh methods to move the buoy [START_REF] Davidson | Dynamic mesh motion in openfoam for wave energy converter simulation[END_REF]. Numerically modeling of a rectangular buoy in front of a vertical wall with sloping bottom topography has been performed using Open-Foam by Gao et. al. [START_REF] Gao | On hydrodynamic characteristics of gap resonance between two fixed bodies in close proximity[END_REF]. They have shown that for all frequencies tested, maximal vertical wave force in the gap decreases when the topography slope increases. Neuvéglise et. al. showed that analytical model were not valid for large draft or short clearance distances [START_REF] Neuvéglise | Two-dimensional modelling of a quayside floating system[END_REF]. Moreover, quayside heaving buoy were experimentally showed to reduce overtopping for regular waves [START_REF] Neuvéglise | Impact of a quayside floating system on overtopping[END_REF]. The ability of OpenFOAM to characterize overtoppings has been studied and validated by Chen et al. [START_REF] Chen | Numerical modelling of wave overtopping at dikes using open-foam®[END_REF].

In the present study, we consider the behaviour of a rectangular heaving buoy in front of a rectangular wall. RAO results are compared with the analytical potential model developed by [START_REF] Yh Zheng | On the radiation and diffraction of water waves by a rectangular buoy[END_REF]. The impact of the floater on overtopping is highlighted for regular and irregular waves. Experimental and numerical modelling are performed. Numerical results are carried out by OpenFOAM.

II. EXPERIMENTAL SET-UP

The numerical model has been validated by comparing the results with experimental data for one configuration described below. Experimental measurements have been performed in LOMC wave flume (Laboratory of waves and complex media) of Le Havre Normandie University. The wave flume dimensions are 35m long, 1.2m high and 0.9 wide (Fig. 1). The floater oscillates along two vertical spikes to en- sure only a heave motion. Draft, total height and width of the buoy are respectively Γ = 0.25m, H f = 0.4m, L = 0.1m and the clearance distance, i.e. the distance between the floater and the vertical breakwater, is D = 0.01m. The water depth is fixed at h = 0.63m. The distance between the free surface and the top of the vertical breakwater is Rc = 0.07m. Overtoppings are collected in a tank behind the vertical wall. They are measured by two pumps with volume counters. The measured water volume are rejected in the flume, to ensure a constant mean water level. Irregular and regular waves have been tested. A JONSWAP spectrum is considered with a peak enhancement factor γ = 3.3 for irregular waves. The wave height is fixed to H i = 0.1m for regular wave and the significant wave height to H s = 0.1m for irregular waves. The periods are varied between 1.15s and 3.5s for regular waves, like peak periods for irregular waves. Reflected waves are absorbed by the paddle. ore details on the experimental set-up can be found in Neuvéglise et. al. [START_REF] Neuvéglise | Two-dimensional modelling of a quayside floating system[END_REF].

III. NUMERICAL MODEL

Governing equations

Numerical modelling is performed using OpenFoam software. InterFoam solver is used to solve two-phase fluid, incompressible and inviscid. Mass and momentum equations are formulated as:

∂ρu i ∂x i = 0 (1)
∂ρu i ∂t + ∂ρu i u j ∂x i = - ∂p ∂x i -g i x i ∂ρ ∂x j + µ ∂ ∂x j ( ∂u i ∂x j + ∂u j ∂x i )
(2) Where p, u i and g i represent the dynamic pressure, the velocity, and the gravity acceleration, in air and water phases. These equations are solved in both phases. To solve them in the whole domain, we use VoF method (Volume of fluid) which consists in introducing a scalar quantity α characterizing the mass fraction, α = 1 corresponding to water and α = 0 to air phase. Density ρ and dynamic viscosity µ are defined by 3 and 4.

ρ = αρ water + (1 -α)ρ air (3)
µ = αµ water + (1 -α)µ air (4)
An other equation is added to characterize scalar α advection in 5.

∂α ∂t + ∂αu i ∂x i + ∂α(1 -α)u ir ∂x i = 0 (5)
The ∂α(1-α)uir ∂xi term allows to artificially compress the interface. The interface is defined as α = 0.5. Turbulence model k -SST is used [START_REF] Menter | Ten years of industrial experience with the sst turbulence model[END_REF].

B. Numerical configuration

Waves are generated in waveFoam relaxation zone which is added to the domain. This zone enables to properly absorb wave reflection and avoid re-reflection [START_REF] Conde | Comparison of different methods for generation and absorption of water waves[END_REF]. The numerical domain is bi-dimensional (Fig. 2).

The buoy is rectangular, moving only in heave motion. The relaxation zone lengths are set to 2λ and the flume length to 3λ, where λ is the wave length.

Regular and irregular waves are modeled. Stokes model at first order is used for regular waves. A JON-SWAP spectrum with γ = 3.3 is chosen for irregular waves. The incident and reflected wave amplitude is measured according to Mansard & Funke method [START_REF] Mansard | The measurement of incident and reflected spectra using a least squares method[END_REF]. Sampling is defined so as to respect the Shannon condition. Overtoppings are determined with the temporal evolution of the mass flux through the outlet face.

C. Mesh

The mesh is realized with blockMesh and snappy-HexMesh, implemented in OpenFOAM to generate and refine mesh.

The sliding interfaces method was chosen to model buoy motion. This method has the advantage of being less expensive in time than mesh morphing or overset mesh. Moreover, when the gap is small (D ≤ 0.01m), the mesh morphing strongly deforms the cells in the gap and does not allow to model important buoy displacements. Use of overset mesh is also difficult for small gaps. Sliding interfaces is thus a good option for one degree of freedom systems in front of structure.

However there are currently parallelisation problems with this method in OpenFOAM. Cells addition and suppression do not work well in several cases. The mesh definition is one of the biggest challenge.

D. Convergence

The size of the optimal cells is studied according to the GCI (Grid Convergence Method) method presented by Celik [START_REF] Celik | Procedure for Estimation and Reporting of Uncertainty Due to Discretization in CFD Applications[END_REF]. This method permits to estimate discretization error by comparing results of three different meshes, if the ratio between the characteristic cell size is more than 1.3. According to Fig. 3, the optimal mesh obtained with this convergence method is mesh # 4 corresponding to 100k cells. Region around buoy contains the smallest cells with a size of 1.6 mm.

IV. NUMERICAL VALIDATION

In order to validate the numerical model, the RAO is compared with experimental results. The buoy amplitude is normalized by the incident wave amplitude in Fig. 4. Both numerical and experimental results are in good agreement. Relative maximum deviation is 23%. Results are also compared with linear potential model based on Haskind decomposition [START_REF] Yh Zheng | On the radiation and diffraction of water waves by a rectangular buoy[END_REF]. However, as previously observed by Neuvéglise [START_REF] Neuvéglise | Two-dimensional modelling of a quayside floating system[END_REF], physical and numerical results are far from the linear potential model. Indeed, damping and nonlinear phenomena are not taken into account in such model.

V. OVERTOPPINGS

Numerical overtopping V N are obtained using Eq. 6. We calculated the mass flow Q m that crosses outlet face with Eq. 7. Mass flux is directly determined by waveFoam solver. The time step is ∆t = 10ms. S outlet is the outlet face surface.

V N = 1 ρ t f t=0 Q m ∆t (6)
Q m = S outlet ρφ dS (7)
For each tested condition, numerical simulations are performed with and without the floater in order to calculate the overtopping ratio ∆V N = V N withbuoy /V N withoutbuoy .

In addition to numerical overtopping, overtopping is estimated using the EuroTop manual [START_REF] Van Der Meer | Eurotop : Manual on wave overtopping of sea defences and related structures, an overtopping manual largely based on european research, but for worldwide application[END_REF] equation for a vertical sea wall without buoy (Eq. 9). The same equation is used with the buoy where the incident wave height H i is replaced by the wave height in the gap H (1) /2. The correction performed by S. Neuvéglise et al. [START_REF] Neuvéglise | Impact of a quayside floating system on overtopping[END_REF] to take into account the dissipation in the gap between the floater and the vertical breakwater, is used.

V A with buoy = 0.054•exp   -2.12 R C H (1) 2 1.3   • g H (1) 2 3 (8)
V A without buoy = 0.054 • exp -2.12 R C H i 1.3 • gH 3 i (9) Where H (1)
2 represents wave height in the gap and H i the incident wave height. The overtopping ratio is then defined as:

∆V A = V A withbuoy V A withoutbuoy • F corr (10)
Where F corr is a correcting factor defined by the buoy parameters to take into consideration the pressure drop due to viscous effects in the gap, characterised by the friction coefficient Λ.

F corr = 1 -Λ R C + Γ D (11)
Thus, overtopping ratio is equal to one without buoy.

Overtopping results are presented in Fig. 5 and 6 for regular and irregular waves. For regular waves, Neuveglise et. al. [START_REF] Neuvéglise | Impact of a quayside floating system on overtopping[END_REF] showed that overtoppings strongly increase close to the resonance. This result is confirmed by numerical simulations for large clearance distance, i.e. R c /D = 0.7 although overtopping increase is smaller than experimental ones. For a small one, R c /D = 7, overtopping ratio measured numerically is decreased for all sea-states. The same result is observed for irregular waves: overtopping ratio is reduced for a small clearance distance, i.e. when the floater is close to the wall. Indeed, dissipation in the gap increases as the clearance distance decreases leading to a decrease of overtopping in these cases.

VI. EXTREME CONDITION

To study the clearance distance influence on the RAO, two different buoys are tested experimentally in regular and irregular waves for one extreme condition. Buoys have two different drafts, Γ = 0.08m and Γ = 0.125m. The buoys' width remains constant at L = 0.1m. The sea states parameters are h = 0.65m and H s = 0.175m. For regular waves, the wave period is T = 1.79 s. A JONSWAP spectrum is used for irregular wave with T p = 1.79s and γ = 3.3.

The floater amplitude for irregular waves is determined similarly to the significant wave height using the 0th order moment m 0 defined in Eq. 12 where S f (ω) represent the amplitude spectrum of the buoy and a f the buoy amplitude.

m 0 = ∞ 0 S f (ω)dω (12)
a f = 2 √ m 0 (13)
RAO results are presented in Fig. 7. The RAO values are more important in irregular waves than regular waves. However, when the clearance distance D decreases, RAO increases in regular and irregular waves for both tested buoys.

VII. CONCLUSION

The present model carried out with OpenFOAM allows us to study the behavior of a buoy located in front of a vertical wall. Overtopping ratio and RAO obtained numerically have been compared with experimental results. Both approaches show a good agreement. It is shown that a small clearance distance increases RAO and minimizes overtopping in regular and irregular waves. A quayside device may have two main functions, recovering energy and protecting the vertical breakwater by reducing overtopping. Reducing the clearance distance provides better results for both device functions. In a future study, we will study the effort applied on the structures and estimate the amount of water volume crossing for different buoy shapes. A modeled PTO will be added to the experimental set-up and numerical model. The objective is to optimize the shape of the float to increase the production of electrical energy with a PTO, minimize overtopping and reduce the manufacturing cost, mainly depending on the shape of the main body.

Fig. 1 .

 1 Fig. 1. Scheme of the wave flume used in this study.

Fig. 2 .

 2 Fig. 2. Domain dimensions and configuration.

Fig. 4 .

 4 Fig.[START_REF] Elchahal | The effects of reflection coefficient of the harbour sidewall on the performance of floating breakwaters[END_REF]. Ratio Amplitude Operator : comparison between experimental, analytical, and numerical results

Fig. 5 . 7 Fig. 6 .

 576 Fig. 5. Overtoppings for regular waves. Full line represents the analytical model. Red markers represent results for Rc/D = 0.7 and blue for Rc/D = 7. The horizontal black line indicates the value of the overtopping ratio without buoy

Fig. 7 .

 7 Fig. 7. Influence of the clearance distance on RAO for regular waves (•) and irregular waves ( ).

ACKNOWLEDGEMENT

The authors would like to acknowledge Normandy Regional Council and ADEME for the funding of this study.