F Tournus

K Sato

T Epicier

T J Konno

V Dupuis

Multi-L1 0 domain CoPt and FePt nanoparticles revealed by electron microscopy

Keywords: Magnetic nanoparticles, nanoalloy, chemical order

Although it has recently attracted a considerable attention, the structure of bi-metallic magnetic nanoparticles such as CoPt or FePt is still debated. These nanoalloys are promising for applications, in particular due to the extremely high magnetocrystalline anisotropy (MCA) of the bulk face-centered tetragonal (fct) L1 0 phase where pure atomic Co (or Fe) and Pt planes are stacked along the [001] direction. At small sizes, these systems appear to be much more complex than single element metallic particles: in addition to the usual existence of peculiar symmetries (icosahedral and decahedral) when the size is reduced, the various possibilities of chemical ordering offer another degree of freedom [1]. Despite this increased complexity, several theoretical investigations have tackled this question, predicting for nanoparticles smaller than 3 nm the stability (or metastability) of exotic structures displaying at the same time a five-fold symmetry and a chemical order [2][3][4]. However, evidence of few nanometers chemically-ordered particles retaining a five-fold symmetry is still missing.

We discuss the atomic structure of CoPt and FePt nanoparticles (with a diameter between 2 nm and 5 nm), synthesized by the mass-selected low-energy cluster-beam deposition technique. For both alloys, we show using transmission electron microscopy (TEM) that crystalline nanocrystals coexist with multiply-twinned particles (MTP) having decahedral or icosahedral symmetries. In particular, we show that the chemical order can be preserved across twin boundaries which can give rise to spectacular chemically ordered decahedral particles made of five L1 0 domains, as theoretically predicted [2][3][4].The samples have been characterized by TEM either in a high resolution mode (HRTEM) or with a high angle annular dark field detector in a scanning TEM mode (STEM-HAADF). In addition to a JEOL 2010F microscope (operating at 200 kV and with a field emission gun), we have used FEI Titan 80-300 microscopes operating at 300 kV with a field emission gun, either with a C s -corrector for the objective lens (for HRTEM images with highly improved spatial resolution) or with an aberration corrected probe (for atomic resolution STEM-HAADF images).

The three types of particle structure (fcc crystal, decahedron, icosahedron) can be observed, both for CoPt and FePt. These experimental observations indicate that, in this size range, the energy differences between the different types of clusters are smaller or of the order of the thermal energy. Both for FePt and CoPt, we observe crystalline particles with a single chemically ordered L1 0 domain all along the nanoparticle, down to D = 2.3 nm (smallest diameter considered here). In the present case we do not observe any threshold size. Moreover, there is no indication of a preferential surface segregation of one element. Remarkably, we also observe a chemical order in crystalline particles consisting of several L1 0 domains. In some cases, the different c-orientations are due to (111) twins (see Fig. 1a-b) which can preserve a coherent chemical order. Even more surprisingly, it is also possible to find different L1 0 domains in a 2 nm CoPt particle free of twins (Zcontrast image, see Fig. 1c-d). In this case, one domain has a c axis at 90° with respect from two other domains that are joined by an antiphase boundary (APB).

Finally, as illustrated in Fig. 2, particles showing a striking pentagonal pattern can also be found. They consist in decahedral clusters made of five L1 0 domains joined by (111) twins (see Fig. 2b). Multislice images simulations have been performed, considering a particle with a perfect chemical order, and are in excellent agreement with the experimental Cs-corrected HRTEM images (Fig. 3). This remarkable particle structure is precisely the one predicted by theoretical calculations [2][3][4]. From a magnetic point of view, it is expected that the MCA of these decahedral particles will be very low, despite they are exclusively formed of L1 0 ordered domains. In the same way, for crystalline particles, twinning and coexistence of several domains having different c orientations will certainly drastically reduce the MCA as compared to mono-L1 0 domain fct particles [START_REF]This work has been supported by GDR-CNRS 3182[END_REF].

Figure 1 .

 1 Figure 1. a) C s -corrected HRTEM image of a CoPt particle displaying a continuous L1 0 chemical order over two (111) twins (indicated by arrows in b). As a guide to the eyes, in b) the atomic columns are colored according to their apparent size in the original image. c) STEM-HAADF image of another CoPt particle: the bright dots correspond to Pt-rich atomic columns. Three different L1 0 domains can be distinguished, as schematized in d).

Figure 2 .

 2 Figure 2. a) HRTEM image of a FePt particle displaying simultaneously a five-fold symmetry and L1 0 ordered domains. b) schematic view of the corresponding chemically ordered decahedral structure, predicted by theoretical calculations.

Figure 3 .

 3 Figure 3. a) C s -corrected experimental HRTEM images of a chemically ordered decahedral FePt particle, for two defocus values. b) Multislice image simulation, using the geometry shown in Fig. 2b, with Fe atoms at the center.


Seung Hwan Lee

Minju Lee

Heehong Yang

Youngtak Cho

Seunghun Hong
email: seunghun@snu.ac.kr

Tai Hyun Park
email: thpark@snu.ac.kr

Bioelectronic sensor mimicking the human neuroendocrine system for the detection of hypothalamic-pituitary-adrenal axis hormones in human blood

Keywords: bioelectronic sensor, field-effect transistor, nanodisc, hypothalamic-pituitaryadrenal axis, corticotropin-releasing hormone, adrenocorticotropic hormone

In the neuroendocrine system, corticotropin-releasing hormone (CRH) and adrenocorticotropic hormone (ACTH) play important roles in the regulation of the hypothalamic-pituitary-adrenal (HPA) system. Disorders of the HPA system lead to physiological problems, such as Addison's disease and Cushing's syndrome. Therefore, detection of CRH and ACTH is essential for diagnosing disorders related to the HPA system. Herein, receptors of the HPA axis were used to construct a bioelectronic sensor system for the detection of CRH and ACTH. The CRH receptor, corticotropin-releasing hormone receptor 1 (CRHR1), and the ACTH receptor, melanocortin 2 receptor (MC2R), were produced using an Escherichia coli expression system, and were reconstituted using nanodisc (ND) technology. The receptor-embedded NDs were immobilized on a floating electrode of a carbon nanotube field-effect transistor (CNT-FET).

The constructed sensors sensitively detected CRH and ACTH to a concentration of 1 fM with high selectivity in real time. Furthermore, the reliable detection of CRH and ACTH in human plasma by the developed sensors demonstrated their potential in clinical and practical applications. These results indicate that CRHR1 and MC2R-based bioelectronic sensors can be applied for rapid and efficient detection of CRH and ACTH.

Introduction

In the human body, the endocrine and nervous systems are used for communication between internal systems (Hiller-Sturmhofel and Bartke 1998). The neuroendocrine system enables communication between the endocrine and nervous systems, and contributes to maintaining homeostasis (Vigas and Jezova 1996). The system consists of the hypothalamus, pituitary gland, and other glands, and can be classified into the hypothalamic-pituitary-adrenal (HPA) axis, hypothalamic-pituitary-thyroid axis (HPT), and hypothalamic-pituitary-gonadal (HPG) axis, depending on the gland (Marques-Deak et al. 2005).

Among them, the HPA axis plays an important role in the regulation of homeostasis in response to stress (Herman and Cullinan 1997). In response to a stressful event, corticotropinreleasing hormone (CRH) is synthesized, and released from the hypothalamus. Subsequently, CRH stimulates corticotropin-releasing hormone receptor 1 (CRHR1) on corticotropic cells in the pituitary, and induces the secretion of adrenocorticotropic hormone (ACTH) from the anterior pituitary. ACTH travels and binds to melanocortin 2 receptor (MC2R) on adrenocortical cells of the adrenal cortex. As a result, glucocorticoid hormone is released from the adrenal cortex (Nieuwenhuizen and Rutters 2008). The secretion of glucocorticoid hormone can be controlled by a negative feedback mechanism in the hypothalamus and pituitary (Juruena et al. 2004).

The HPA axis is associated with such functions of the body as energy usage, digestion, reproduction, and the immune system (Habib et al. 2001;Sapolsky et al. 2000;Smith and Vale 2006). Dysfunction of the HPA system causes pathologic, psychological, and physiological problems, such as Addison's disease, Cushing's syndrome, general adaptation syndrome, posttraumatic stress disorder, obsessive-compulsive disorder, anxiety disorders, and diabetes (Arnett et al. 2016;Coryell et al. 1989;Faravelli et al. 2012;Fries et al. 2015;Kinlein et al. 2015;Naughton et al. 2014). Because these diseases and syndromes are caused by tumors or damage to the hypothalamus or pituitary, and are subdivided depending on the location of the tumor and impaired part, monitoring of CRH and ACTH levels is crucial for making a correct diagnosis and prescribing the correct medication (Hiroi et al. 2001;Pyter et al. 2009).

Additionally, CRH and ACTH are prohibited by the World Anti-Doping Agency (WADA), to prevent its misuse (Knoop et al. 2017;Thevis et al. 2006). Therefore, a novel detection system is needed to enforce anti-doping regulations (Duran et al. 2019;Knoop et al. 2017).

Furthermore, the CRH-and ACTH-based interaction between the hypothalamus, pituitary, and adrenal cortex is complex and dynamic (Markovic et al. 2011). Therefore, a sensitive and selective sensor system to enable the real-time monitoring of CRH and ACTH is necessary.

Various detection methods have been used to detect CRH and ACTH (Crosby et al. 1988;Håkanson and Sundler 1971;Kertesz et al. 1998;Knoop et al. 2017;Shi et al. 2019).

These include radioimmunoassay (RIAs), enzyme-linked immunosorbent assay (ELISA), high-performance liquid chromatography (HPLC), and liquid chromatography-tandem mass spectrometry (LC-MS/MS). Although these methods have shown precise and quantitative results, they require a complicated pretreatment of the sample, long analysis time, large and expensive instruments, and complicated operations (Lee et al. 2015b;Ojeda et al. 2013). To overcome these limitations, electrochemical sensors were introduced (Duran et al. 2019;Li et al. 2017;Moreno-Guzmán et al. 2012;Moreno-Guzmán et al. 2012). However, these sensors require either an antibody (Ab) or a signal amplification for the detection of CRH and ACTH.

These requirements are disadvantageous because they make the sensing process timeconsuming and costly. Therefore, a novel, sensitive, selective, and efficient system is required.

Recently, several highly sensitive and selective sensor systems have been reported for the detection of target molecules using the combination of G-protein-coupled receptors (GPCRs) with a carbon nanotube field-effect transistor (CNT-FET) (Kim et al. 2013;Kim et al. 2009;Kim et al. 2011). GPCRs contribute to the regulation of physiological functions in the hormonal, neuronal, and sensory systems by interacting with their ligands (Premont and Gainetdinov 2007;Wettschureck and Offermanns 2005). Their specific characteristics of recognizing a corresponding ligand, such as CRH and ACTH, can be utilized for the construction of a bioelectronic sensor system (Ko and Park 2006;Kwon et al. 2018;Lim et al. 2013;Park et al. 2012). CRHR1 and MC2R are members of the GPCRs, and show high affinity with CRH and ACTH, respectively (Hofmann et al. 1974;Lovejoy et al. 2014). Although the recognition characteristics of CRHR1 and MC2R, for CRH and ACTH have been studied in human physiology, the application of these receptors for the detection of CRH and ACTH through construction of bioelectronic sensor system has not yet been reported.

Herein, we report a bioelectronic sensor system that mimics the human neuroendocrine system for the detection of CRH and ACTH. The receptors that recognize CRH and ACTH, CRHR1 and MC2R, were expressed in Escherichia coli (E. coli), and embedded into nanodiscs (NDs) for reconstitution. To construct the bioelectronic sensor system, the NDs containing each receptor were immobilized on a floating electrode that was combined with a CNT-FET sensor.

This sensor system was used to detect the levels of CRH and ACTH in the buffer solution and in human blood plasma. These approaches provide an effective strategy for the detection of HPA axis-related hormones.

Materials and methods

Materials

CRHR1 and MC2R cDNA were purchased from Origene (USA), and the MSP1E3D1 bacterial

Fabrication of CNT-FETs with floating electrodes

Carbon nanotube-field effect transistors (CNT-FET) were fabricated following a previously reported method (Kim et al. 2012;Lee et al. 2015a). Briefly, a CNT dispersion with a concentration of 0.05 mg/mL in 1,2-dichlorobenzene was prepared using sonication for 5 h.

For selective CNT assembly, non-polar (octadecyltrichlorosilane, OTS) and polar (SiO2) monolayers were patterned on SiO2 substrate using a photolithography process. CNTs were selectively adsorbed on SiO2 regions by immersing the substrate in the CNT solution.

Afterwards, source, drain, and floating electrodes (Pd/Au 10 nm/15 nm) were fabricated using photolithography and thermal metal deposition. The length and width of the CNT channels were 170 µm and 5 µm, respectively. The length and width of the floating electrode were 10 µm and 200 µm, respectively. Finally, a passivation layer (photoresist, DNR) was fabricated to expose only channel regions and to eliminate leakage currents in liquid environments.

Functionalization of gold surfaces with NDs containing hormone receptor proteins

The gold surfaces of floating electrodes were functionalized with hormone receptor proteins following a previously reported method (Yang et al. 2017). V5 antibodies (Thermo scientific)

were reduced using 2-mercaptoethylamine hydrochloride (2-MEA) for 1 h at 37 °C, to prepare half-V5 antibody fragments (Karyakin et al. 2000). By treating the channel regions with solution containing half-V5 antibody fragments at 37 °C for 1 h, the half-V5 Ab fragments were immobilized selectively on the gold surface of the floating electrodes. The CRHR1-or MC2RNDs were immobilized on the floating electrodes by the interaction between the V5 epitope of the NDs, and the antigen binding site of the half-V5 antibody.

Preparation of samples

All the tested samples, including CRH, ACTH, DA, SE, and EP, were dissolved in HEPES buffer containing 20 mM HEPES-NaOH, 100 mM NaCl, and 1 mM EDTA, pH 8.0. For the preparation of blood plasma samples, blood plasma was centrifuged at 3,000 × g for 15 min at 4 °C. The supernatant was stored at -80 °C, until used. For the electrical measurements, the frozen samples were thawed and mixed with CRH and ACTH, to prepare blood plasma samples with various concentrations of CRH and ACTH.

Real-time electrical measurements

Real-time measurement using bioelectronic sensors based on CNT-FETs was performed (Lee et al. 2011;Lee et al. 2015a). A bioelectronic sensor was connected to a semiconductor analyzer (4200-SCS, Keithley, USA). A source-drain voltage of 0.1 V was constantly applied during electrical measurement, and source-drain currents were monitored. HEPES buffer solution was placed on the channel region of a bioelectronic sensor. Afterwards, various solutions were consecutively introduced into the channel, and the responses of the bioelectronic sensor were monitored.

Results and discussion

Construction of the bioelectronic sensor system

Fig. 1 shows the HPA axis of the human neuroendocrine system and the bioelectronic sensor system mimicking the human HPA axis for the detection of CRH and ACTH. In the human HPA axis, CRH is secreted from the hypothalamus, and binds to CRHR1 in the pituitary.

Subsequently, the pituitary releases ACTH, which interacts with MC2R in the adrenal cortex.

CRHR1 and MC2R are receptors in the HPA axis and recognize CRH and ACTH, respectively.

To apply these characteristics to a sensor system, CRHR1 and MC2R were produced in an E.

coli system and embedded in NDs through a reconstitution process.

The CRHR1NDs and MC2RNDs were immobilized on the floating electrode-based CNT-FET sensor system as described previously (Lee et al. 2018;Yang et al. 2017). Based on these processes, the bioelectronic sensor system mimicking HPA axis was constructed. In this system, CRHR1NDs and MC2RNDs were used as recognition elements and the floating electrode-based CNT-FET was used as a sensor transducer to detect CRH and ACTH.

Because GPCRs have a transmembrane domain and hydrophobic character with complicated charge distribution (Gafvelin and von Heijne 1994;Grisshammer and Tateu 1995), reconstitution is required to preserve the structure and functions of the GPCR. CRHR1 and MC2R are members of the GPCR family, and reconstitution processes can provide the structural and functional preservation of CRHR1 and MC2R (Bender et al. 2015;Roy et al. 2011). This means that reconstituted CRHR1 and MC2R can recognize the CRH and ACTH selectively and sensitively. NDs have been proposed as a reconstitution method to mimic the native GPCR structure (Denisov and Sligar 2016;Serebryany et al. 2012). In this method, the receptor and lipid bilayer are wrapped with membrane scaffold proteins (MSPs). The stability of NDs embedded with GPCRs and the superiority in detection by the sensor is reported (Oh et al. 2019). However, studies describing the reconstitutions of CRHR1 and MC2R using NDs or the application of the CRHR1ND and MC2RND in the sensor have not been carried out before.

The sensor transducer was also improved by the addition of floating electrodes to the CNT-FET (Kim et al. 2012;Lee et al. 2012;Lee et al. 2015a). The combination of the improved sensor and reconstituted receptors resulted in enhanced performance in terms of sensitivity and selectivity as previously described (Lee et al. 2018;Yang et al. 2017). In previous studies, floating electrode CNT-FET has shown high sensitivity in detecting analytes and high selectivity was achieved by immobilization of reconstituted receptors (Lee et al. 2018;Yang et al. 2017). Because CRHR1 and MC2R can recognize CRH and ACTH in the human HPA axis with high sensitivity and selectivity, CRHR1NDs and MC2RNDs play critical roles in enhancing the sensing performances by combining with floating electrode CNT-FET.

Characterization of CRHR1 and MC2R expressed in Human embryonic kidney (HEK-293) cells and E. coli

Membrane proteins, especially transmembrane proteins such as GPCRs, have been expressed

in HEK-293 cells, and their functions have been tested in many studies (Atwood et al. 2011;Liberles and Buck 2006). HEK-293 cells were transfected with the gene for CRHR1 or MC2R, and their ligand binding function was evaluated using a Dual-Glo luciferase assay system (see the details in the S.1 of Supplementary Materials). In Figs. 2a andc, CRHR1-and MC2Rexpressing cells showed dose-dependent responses to CRH and ACTH, respectively; however, the non-transfected cells had no meaningful responses to CRH and ACTH. These results show that both CRHR1 and MC2R were successfully expressed in HEK-293 cells, and retained their ligand-binding interactions.

To ensure the selective response of CRHR1 and MC2R to their ligand, cells expressing CRHR1 or MC2R were stimulated with 1 µM of (CRH, ACTH, DA, EP, and SE). DA, EP, and SE are neurotransmitters that contribute to the control of body functions as chemical messengers (Di Chiara et al. 1994;Hadjiconstantinou et al. 1983;Mariussen and Fonnum 2001).

When cells expressing CRHR1 were treated with CRH, a change in the luminescent signal was observed. However, ACTH, DA, EP, and SE did not elicit meaningful responses from the cells expressing CRHR1 (Fig. 2b). Similarly, cells expressing MC2R responded to only ACTH, while CRH, DA, EP, and SE did not cause change in the luminescent signal (Fig. 2d). These results indicate that CRHR1 and MC2R can discriminate their ligand.

To develop the receptor-based biosensor, CRHR1 and MC2R were produced using an E. coli expression system (Lee et al. 2018;Yang et al. 2017). Subsequently, the receptors were solubilized, and purified using Ni 2+ affinity chromatography. For the formation of NDs, the MSP1E3D1 was also successfully overexpressed in E. coli, and purified (S.1 of Supplementary Materials). Fig. 2e shows the results of the western blot analysis of the produced CRHR1 (left), and MC2R (right) (See the detailed procedures in S.1 of Supplementary Materials). The revealed bands correctly indicate the estimated size of the receptors, which implies that CRHR1 and MC2R were successfully expressed in E. coli. The receptor-embedded NDs were constructed by mixing the receptor, lipids, and MSP1E3D1. Fig. 2f presents a SEM image of CRHR1NDs, showing that the produced NDs have the appropriate discoidal shape, and that their average size was approximately 20 nm. These results indicate that the production, purification, and reconstitution of CRHR1 and MC2R were successfully achieved. Bacterial expression is a simple, convenient, inexpensive, and efficient for production of recombinant proteins (Park et al. 2016). In this study, E. coli was used as a representative host cell as commercial CRHR1 and MCR2 are expensive and their functionality is not guaranteed.

Although E. coli displays robust production of recombinant proteins, the expression and purification of GPCR is difficult because it has complicated structures with a hydrophobic character (Kiefer et al. 1996;Yang et al. 2015). In this regard, overexpression of CRHR1 and MC2R using an E. coli expression system and their purification are a remarkable aspect of this work. Furthermore, the purified CRHR1 and MC2R were reconstituted with the ND technology for the first time. These trials allowed the detection of CRH and ACTH using CRHR1 and MC2R, which had structural stability and in vivo-like functionality.

Optical and electrical characterization of the sensor system

An optical image of a CNT-FET with floating electrodes is shown in Fig. 3a. Detailed fabrication procedures of a floating electrode-based CNT-FET system are described in the Materials and Methods section. Floating electrode-based CNT-FET sensors have been reported

to exhibit better sensing performance than a conventional CNT-FET sensor system (Lee et al. 2012;Lee et al. 2015a). The size of the sensor 2 cm × 1 cm and the sensor has two CNT channel regions.

The fabrication of floating electrode CNT-FET system was verified using fieldemission scanning electron microscopy (FE-SEM). The SEM image of the channel region in the fabricated CNT-FET with floating electrode is shown in Fig. 3b. Five floating electrodes were constructed between the source and drain electrodes. The length and width of the floating electrode were about 10 µm and 200 µm, respectively. Furthermore, the right image of Fig. 3b is the enlarged SEM image of one floating electrode, which shows the CNT channel region under the floating electrode. The CNTs were well-aligned between the source and drain electrodes. The width of the CNT channel was about 5 µm. These results indicate that the CNT channel and floating electrodes of a CNT-FET system were well-fabricated. Especially, the CNTs were selectively attached on SiO2 regions. Fig. 3c shows the I-V curves of a floating electrode-based CNT-FET system before and after the immobilization of NDs. The bias voltage was varied from -0.4 V to +0.4 V and source-drain currents were monitored using a semiconductor analyzer. This result shows that the immobilization of NDs caused an increase in resistance and the slope of the I-V profile after the immobilization decreased. However, the linear behavior of the I-V curves was maintained in both before and after the immobilization of NDs on floating electrodes. This indicates a stable ohmic behavior despite the immobilization of NDs on the floating electrodes. Fig. 3d shows the liquid gate profiles of a sensor system before and after the immobilization of NDs. Source-drain currents were monitored in the gate bias range from -0.25 V to 0.25V. The currents decreased when the gate voltage increased, which indicates a p-type characteristic. Although the immobilization of NDs caused a decrease in channel conductance, the p-type behavior was maintained even after the immobilization of NDs on the floating electrodes. The decrease in channel conductance would be from the negatively-charged C-terminuses of NDs on the floating electrodes (Lee et al. 2018). These results confirm that the gate characteristic of a CNT-FET with floating electrodes was maintained after the immobilization of NDs, which indicates the constructed system could be appropriate for the detection of CRH and ACTH as a biosensor.

Electrical measurement of CRH and ACTH in a standard sample

Fig. 4a shows the real-time responses of a bioelectronic sensor functionalized with CRHR1NDs to different concentrations of CRH. The bioelectronic sensor was connected to a semiconductor analyzer, and a source-drain bias voltage of 0.1 V was constantly applied to the bioelectronic sensor during the measurement. Source-drain currents were monitored while CRH solutions were introduced to the channel region of the bioelectronic sensor. Here, ∆G⁄G0

was used as the signal of the bioelectronic sensor, where ∆G and G0 are the conductance change and original conductance of the CNT-FET, respectively (Lee et al. 2015a). The addition of the CRH solutions in the range of 1 fM to 1 pM led to increase in the conductance of the CRHR1NDs-functionalized bioelectronic sensor. However, a bare CNT-FET did not respond to various concentrations of CRH (Fig. S1a of the Supplementary Materials). This implies that the responses arose from specific binding events between CRHR1 and CRH. Presumably, the binding events induced a structural change in receptor proteins, leading to a change in the charge state of receptor proteins on the floating electrodes. This resulted in a change in the channel conductance of the bioelectronic sensor via Schottky barrier modulation (Kim et al. 2012;Lee et al. 2018). The result shows that the bioelectronic sensor functionalized with CRHR1NDs could sensitively respond to CRH in real-time. et al. 2018;Park et al. 2014;Song et al. 2014). The estimated equilibrium constant between CRHR1 and CRH obtained using bioelectronic sensors was 5.70 × 10 9 M -1 . This value is quite similar to previously reported values (Grammatopoulos et al. 1999;Hatzoglou et al. 1996;McDowell et al. 2012). These results indicate that the bioelectronic sensor responded to CRH dose-dependently and reproducibly. These results imply that the bioelectronic sensor functionalized with MC2RNDs could detect ACTH with high sensitivity in real-time.

In Figs. 4a andd of ACTH increased in a dose-dependent manner. The curve was saturated after the injection of 10 µM ACTH into the bioelectronic sensor because the binding sites of MC2R were occupied by ACTH. The equilibrium constant between MC2R and ACTH was estimated to be 5.47 × 10 10 M -1 . This value is similar to a previously reported value using a cell-based system [START_REF] Melmed | The pituitary[END_REF]. These results indicate that the bioelectronic sensor dose-dependently responded to ACTH with reproducibility.

Fig. 4f shows the real-time responses of a bioelectronic sensor functionalized with MC2RNDs to various hormones and neurotransmitters at a concentration of 1 nM. The introduction of SE, EP, and CRH solutions did not induce a conductance change in the bioelectronic sensor, and only the addition of ACTH solution led to significant increase in the channel conductance. These results show that MC2R on the bioelectronic sensor could discriminate ACTH from other hormones or neurotransmitters. As noted earlier, SE, EP, CRH, and ACTH exist in the human blood and play important roles in the regulation and homeostasis of bodily functions. The interaction of these hormones or neurotransmitters with their counterpart receptors results in the regulation of various processes in the body. Based on this, CRHR1NDs and MC2RNDs were reconstituted and combined with a floating electrode-based CNT-FET system. Hence, CRH and ACTH were selectively detected among other hormones or neurotransmitters by utilizing our sensor system (Figs. 4c andf). Therefore, the developed CRHR1ND and MC2RND-based bioelectronic sensor system had an advantage of selective detection of CRH and ACTH.

Evaluation of the sensing performance for CRH and ACTH in human blood plasma

To assess the possibility of the bioelectronic sensor for medical and clinical use, the detection of CRH and ACTH in human blood plasma was performed. The blood plasma contains many compounds, which could affect the conductance change of sensor platforms (Lim et al. 2014).

Thus, plasma sample was serially diluted with buffer solution to 1/10 6 . The non-specific influence of blood plasma was tested through injection of diluted plasma samples to a bare CNT-FET (Fig. S2 of the Supplementary Materials). The plasma diluted 1/10 to 1/10 2 had an effect on the conductance change of the sensor platform (data not shown). The diluted plasma samples in the range of 1/10 3 to 1/10 6 did not affect the conductance change of the bioelectronic sensor platform. Based on these results, a 1/10 4 diluted plasma sample was mixed with CRH or ACTH, and different concentrations of CRH or ACTH samples from 10 -18 to 10 -5 M were prepared. Fig. 5 shows the dose-dependent responses of a CRHR1NDs or MC2RNDsfunctionalized bioelectronic sensor to various concentrations of CRH (Fig. 5a) or ACTH (Fig. 5b) in the diluted plasma. The bioelectronic sensors dose-dependently responded to the CRH or ACTH in diluted plasma. The curves reached a plateau after the injection of 10 -6 M of CRH or ACTH. Typically, CRH and ACTH levels in blood plasma are in the picomolar range (Linton et al. 1987;Talbot et al. 2003). These levels are in the detection range of our bioelectronic sensor, based on the results shown in Fig. 5. When the dose-response curves of CRHR1NDs or MC2RNDs to CRH or ACTH were compared in plasma environment (Fig. 5) and in the buffer condition (Fig. 4), similar slopes were observed. The equilibrium constant between CRHR1 and CRH in diluted plasma was estimated as 2.20 × 10 9 M -1 , and the equilibrium constant between MC2R and ACTH in diluted plasma was estimated as 1.32 × 10 11 M -1 . The equilibrium constants in the plasma environment were similar to those from the experiments under buffer condition shown in Fig. 4b (CRHR1 to CRH) and Fig. 4e (MC2R to ACTH).

In order to verify the selectivity of the bioelectronic sensor functionalized with CRHR1NDs and MC2RNDs in human blood plasma, a 1/10 4 diluted plasma sample was mixed with various hormones and neurotransmitters, such as SE, EP, CRH, and ACTH. The intensities of the responses to CRH and ACTH were quantitatively compared with those of the other hormones and neurotransmitters of 1 nM concentration. Fig. 5c shows the selective responses of a CRHR1NDs functionalized bioelectronic sensor to CRH with 1 nM. The sensors showed significant changes in conductance in response to CRH. However, other hormones and neurotransmitters such as the SE, EP, and ACTH did not cause changes in the conductance of the sensor. Fig. 5d shows the selective responses of a MC2RNDs functionalized bioelectronic sensor to ACTH with 1 nM. The sensors showed a significant response to 1nM ACTH.

However, the introduction of SE, EP, and CRH induced negligible changes in the conductance of the sensor. These results indicate that the biosensors could sensitively and selectively detect CRH or ACTH in blood plasma as well as pure environment such as buffer. Our biosensor could be also effectively applied for the diagnosis of HPA axis-related diseases and anti-doping regulations. Therefore, the biosensors developed in this work could have potential for practical and clinical applications.

Although several studies have reported the detection of CRH and ACTH before, our receptor-based system has higher sensitivity and selectivity than the other systems (Table 1).

Recently, laser desorption/ionization mass spectrometry (LDI MS)-based analysis has been introduced as an advanced method of MS that features high sensitivity, accuracy, and in-second detection (Sun et al. 2018;Vedarethinam et al. 2019). Nevertheless, it is still difficult to make a conclusive comparison with the other systems because LDI MS-based analysis is yet to be applied for the detection of CRH and ACTH. Although electrochemical methods have shown good sensitivity, most of them required Ab-based detection approaches, and such approaches are time-consuming. Furthermore, several sensors additionally required signal amplification for detection due to their low sensitivity. Therefore, the sensing procedures are less efficient, more expensive, and time-consuming. However, the sensor developed in this study can detect CRH and ACTH using receptors like those of the human neuroendocrine system. This development circumvents the Ab requirement and signal amplification in the detection of CRH and ACTH. Therefore, our detection system is superior to the other reported systems in terms of cost, analysis time, and efficiency. Furthermore, the receptor-based sensing and recognition approaches developed in this study can be applicable for the detection of target molecules by other label-free sensing and recognition methods (Liu et al. 2020;Sun et al. 2019).

Conclusions

We developed a bioelectronic sensor system that mimicked the human neuroendocrine system for the detection of the HPA axis hormones, CRH and ACTH. The CRH receptor and ACTH receptor, CRHR1 and MC2R, were produced in E. coli, and reconstituted using NDs. The receptors were combined with a floating electrode-based CNT-FET. The bioelectronic sensor exhibited real-time responses to CRH and ACTH at concentrations down to 1 fM, and discriminated each hormone from other hormones and neurotransmitters. The performance was maintained in human plasma samples. These results indicate that our system can be applied to basic neuroendocrine-related research, and shows excellent potential for use in medical and practical applications. (f) Selective response of the MC2RND-based bioelectronic sensor to various hormones and neurotransmitters (1 nM). The sensor exhibited a response only when 1 nM ACTH was injected into the system, and did not respond to SE, EP, or CRH.

Fig. 4b

 4b Fig.4bshows the dose-dependent responses of bioelectronic sensors functionalized

Fig. 4c

 4c Fig.4cshows the real-time responses of a bioelectronic sensor functionalized with

Fig. 4d

 4d Fig. 4d displays the real-time responses of a bioelectronic sensor functionalized with

  , we performed the tests in the process from low to high concentrations of target molecules. Because the responses of our sensor system were based on the binding of CRH to the CRHR1 or ACTH to the MC2R, the binding sites of CRHR1 or MC2R could be occupied by CRH or ACTH. Therefore, if a sensitivity test was performed from high to low concentrations, the high concentration of CRH or ACTH would first occupy the binding sites of receptors and would disturb the binding of the low concentration of CRH or ACTH molecules. Hence, we performed the sensitivity test of sensors from low to high concentrations of target molecules.

Fig. 4e

 4e Fig.4eshows the dose-dependent responses of bioelectronic sensors functionalized

Figure captions Fig. 1 .

 captions1 Figure captions

Fig. 2 .

 2 Fig. 2. Characterization of CRHR1 and MC2R expressed in HEK-293 cells and E. coli (a)

Fig. 3 .

 3 Fig. 3. Optical and electrical characterization of a bioelectronic sensor system (a) Optical image

Fig. 5 .

 5 Fig. 5. Dose-dependent responses of the CRHR1ND-and MC2RND-based bioelectronic

  Fig. 1

Table 1 . Comparison of performance between the developed sensor with other reported sensors.

 1

	Analyte	Recognition element	Analytical method	Detection limit (M)	Assay time (min)	Reference
		-	LC-MS/MS	52.1 × 10 -12	> 20 min (Precipitation: >1 hr)	(Knoop et al. 2017)
	CRH	Antibody	EIS	0.7 × 10 -6	>100 min	(Duran et al. 2019)
		Receptor	FET	10 -15	Real-time	This study
		-	LC-MS/MS	2.0 × 10 -12	> 3 min (Precipitation: >1 hr)	(Shi et al. 2019)
	ACTH	Antibody	Voltammetry	4.0 × 10 -15	>100 min	(Moreno-Guzmán et al. 2012)
		Antibody	EIS	22.0 × 10 -15	>100 min	(Li et al. 2017)
		Receptor	FET	10 -15	Real-time	This study

Abbreviation. EIS: electrochemical impedance spectroscopy. LC/MS: liquid chromatography chromatography-tandem mass spectrometry. FET: field effect transistor

Acknowledgments

This work was supported by the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT (MSIT) of Korea (nos. 2013M3A6B2078961, 2018R1A2B3004498, 2018R1C1B5085757), the Ministry of Education (2018R1A6A1A03024231), and the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement no. 682286). S.H. also acknowledges the support from Samsung Electronics, and from the National Research Foundation of Korea (no. 2017R1A2B2006808).

This work was also supported by KIST Institutional Program (no. 2E29530).

Conflict of Interest:

The authors declare no competing financial interest.

Author Contributions

ML fabricated bioelectronic sensors and performed sensing measurements. YC characterized the sensors. SHL and HY prepared CRHR1 and MC2R and stabilized in NDs. THP and SH are responsible for the project and contributed to data analyses. The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

CNT channel and floating electrodes. The five floating electrodes were fabricated between the source and drain electrodes. The CNT channel was also observed under the floating electrodes.

(c) I-V curves of a floating electrode-based bioelectronic system before and after the immobilization of NDs. The dI/dV exhibited a linear behavior. When the NDs were immobilized on the floating electrodes, the dI/dV decreased as the resistance increased.


Trent Victor

Mickël Regan

Alexandra Fort

Christophe Jallais

Keywords: Distraction and Inattention: Engagement duringmanual, assisted, and automated driving modes

come

Editorial

Driver Distraction and Inattention: Engagement during manual, assisted, and automated driving modes.

Trent Victor, Michael Regan, Alexandra Fort, Christophe Jallais Email: trent.victor@chalmers.se; trent.victor@volvocars.com We are in an era of mobility transformation [START_REF] Ertrac | Strategic Research Agenda. Input to the 9 th EU framework programme[END_REF]. Pervasive personal communication technologies and in-vehicle digitalization already impacts daily life, while vehicle automation is on the cusp of widespread impact. Consequently, there is a critical need to understand distraction and the limits of attention in many modes of transportation -both in-vehicle modes (manual-, assisted-, and automated driving) and in new forms of transportation -as they all are affected by digitalization and limits of attention. Distracted driving has long been a contributor to motor vehicle crashes [National Center for Statistics and Analysis], and is flourishing in these new digitalized mobility environments [START_REF] Cunningham | Driver distraction and inattention in the realm of automated driving[END_REF]. This Special Issue expands on some presentations from the 6th DDI conference held in Gothenburg, Sweden (October 15-17, 2018). The main focus of the DDI conference series is on driver distraction and inattention, presentations and papers on distraction affecting other road users. During DDI conferences, recent developments and trends in the field of inattention and distraction in driving are presented. Presentations and discussions are organized covering basic and applied research, challenges in mitigating distraction and inattention, the latest policy developments, and priorities for research and countermeasures development. Therefore, research about new forms of mobility (e.g. autonomous vehicles, electric vehicles), emerging sources of distraction (e.g., smartphone, touchscreen) and driver monitoring were discussed [Proceedings of the DDI2018 conference can be downloaded here: http://ddi2018.org/?page_id=1272].

The adverse effects of distraction have been studied and reported extensively at DDI conferences. Distraction as a new health problem for pedestrians was studied by Vollrath and colleagues. Activities of students before and while crossing the street were observed near major high schools. Using the mobile phone substantially reduced safety-related behaviours before crossing the street and 10.5% of the students used their phone while crossing the street. Using data from the Australian Naturalistic Driving Study (ANDS), Young and collaborators revealed that drivers spend approximately 45 percent of their driving time engaging in potentially distracting tasks. Using different methodologies, Young and collaborators and Petzholdt and collaborators studied contextual characteristics that influence the decision to initiate a secondary task. Both research groups found that drivers are in agreement in some contexts, such as waiting until a vehicle is stationary. In other contexts, Young found drivers do not consider factors that may impact risk (e.g. weather, lighting) and Petzholdt found opinion was split. Further, Petzholdt found that drivers' explanations for their decision to text was explained by scenarios with low demand on attention or handling, and the idea that there would be some margin of error.

The theme of the 6th conference was "Driver Engagement during Assisted Driving". Within vehicle automation, some key safety issues hinge on operator responsibilities while using automation [START_REF] Seppelt | Potential solutions to human factors challenges in road vehicle automation[END_REF]. Terminology describing these responsibilities (such as Overreliance, Driver engagement, Driver-in-the-loop, Fall-back readiness, Receptiveness, and Availability) are related to Distraction and Inattention. These issues were addressed in several papers, presentations, keynotes, and a panel discussion at the 6 th conference.

Euro NCAP's strategy on assisted and automated driving, and their new Occupant Status Monitoring rating element, were presented. These and existing automation standards, regulations, and design guidance were discussed in the panel. Tivesten and collaborators developed an out-of-the-loop crash prediction algorithm suitable for eye-tracker-equipped vehicles with highly reliable but supervised automation. Three pre-conflict behavioural patterns were found to be associated with increased risk of crash involvement: low levels of visual attention to the forward path, high Percent Road Centre (i.e., gaze concentration), and long visual response times to attention reminders if visually distracted. Interestingly, these behaviours were found despite the presence of attention reminders. Beyond eye trackers for driver monitoring, Woerle and colleagues studied physiological measures to detect drivers becoming drowsy and falling asleep during highly automated driving. Using electroencephalography (EEG) as a ground truth, electrodermal activity (EDA) and electrocardiography (ECG) were found to differentiate between sleep and wakefulness. These papers address and discuss implications for the implementation of driver monitoring systems in invehicle systems, a topic of increasing relevance in ratings and regulation.

Further, on the topic of driver engagement in automation, the impact of long (1 hour) and short (10 min) durations was studied for three successive periods of automated driving by Bourrelly and collaborators. Poorer take-over performance (longer reaction time and a sharper avoidance manoeuvre) and increased drowsiness was found in a longer autonomous period compared to a shorter one. Their results indicate that sequencing the autonomous phase into several short periods is preferable to a long automated driving phase.

The 6th DDI conference saw an increase in the number of studies showing the importance of new issues associated with automation and the necessity to keep drivers in the loop to regain the control of the car in a safe way. Future studies will likely take into account what kinds of secondary tasks the drivers could do when in a highly automated mode. Therefore, new research on distraction and inattention topics should deal with the level of engagement/disengagement of the driver during the use of such automated systems.

The next, 7 th , DDI conference will be held in Lyon, France, during 12-14 October 2020 and will have a theme on the disconnected driver. The term disconnected driver can be seen as an expansion of the current focus of the DDI conference towards other states of disconnection with driving such as intoxication, sleeping (in addition to fatigue), sudden sickness, and automation overreliance related states. As this research area matures, we are currently seeing an increase in the efforts within rules and regulation (e.g. EU), introduction of on-market driver monitoring technology (driver monitoring cameras) and other countermeasures. Whilst focus will likely remain on topics related to understanding and quantifying driver distraction and inattention, as in previous conferences, upcoming conferences will likely see reporting of impact assessments of on-market countermeasures such as in-vehicle driver monitoring technologies, as well as assessment or shaping of regulations and safety ratings. Further, we will likely see an increase in focus on automationrelated topics, related driver states, other road users, and other domains of road transport.

  The bi-annual International Conference on Driver Distraction and Inattention (DDI) was initiated in 2009 by Michael Regan (University of New South Wales, Sydney) and Trent Victor (SAFER, Vehicle and Traffic Safety Center at Chalmers, and Volvo Cars). The 6th DDI conference (2018) was coorganized by SAFER, IFSTTAR (French Institute of Science and Technology for Transport, Development and Networks), and the Australian Road Research Board (ARRB).


Adrien Rohfritsch

Jean-Marc Conoir

Tony Valier-Brasier

Régis Marchiano

Impact of particles size and multiple scattering on the propagation of waves in stealthy-hyperuniform media

Propagation of waves in materials that exhibit stealthy-hyperuniform long range correlations is investigated. By using a modal decomposition of the field that takes multiple scattering into account at all orders, we study the impact of the concentration of particles on the transparency of such materials at low frequency. An upper frequency limit for transparency is defined that include both the particles size and the degree of stealthiness. We show that the independent scattering approximation is not relevant to calculate elastic mean free paths when wavelength become comparable to the size of particles. We find that transparency is very robust with regard to the degree of heterogeneity of the host random medium and the polydispersity of particles. Finally, it is shown that resonances can be used as frequency filter.

I.

INTRODUCTION

The study of wave propagation in random heterogeneous media has been in constant evolution from the last century. Understanding and predicting the propagation of waves while multiple scattering occurs is a challenge that is in constant evolution. Multiple scattering in random media is of practical interest in many fields of physics ranging from the theory of randomly disordered crystals [1], the resonant scattering of light [2], the ultrasonic monitoring of colloidal mixtures [3], the effective dynamic mass density of composites [4], metamaterials [5] to name a few.

Reviews of the best-known models and their mathematical backgrounds can be found in Refs. [START_REF] Economou | Green's Functions in Quantum Physics[END_REF][START_REF] Tsang | Scattering of Electromagnetic Waves: Advanced Topics[END_REF][START_REF] Sheng | Introduction to Wave Scattering, Localization and Mesoscopic Phenomena[END_REF][START_REF] Rytov | Principles of Statistical Radiophysics[END_REF]. Many of these models, at their best, are based on the use of paircorrelation functions in order to describe the microstructure of the random media and the Percus-Yevick approximation to take account the correlation between particles [START_REF] Tsang | [END_REF][11][12]. By nature, they are limited to describe media which are statistically uniform. An evolution has been noticeable at the turn of the two past decades where the tuning of the microstructure has been widely explored to design materials with specific properties, as the cloaking for example [13].

Recently, a new state of matter has been discovered that is characterized by cancellation of normalized density fluctuations at infinite wavelength. This state is called hyperuniformity (or superhomogeneity) [14,15] and lead to very particular and interesting properties that have already been observed at natural mesoscopic [16] or microscopic scales [17]. Self assembly of dense packings can also be hyperuniform [18,19]. Among hyperuniform media, a subclass exist for which normalized density fluctuations cancel for finite wavelengths λ > λ min . These ones are called stealthy hyperuniform (SHU) and are at the center of our study. In this context, several recent works have shown the importance of employing n-points correlation functions of order higher than two for the effective dielectric tensor of electromag-netic waves propagating in a two-phase composite random medium with isotropic components [20,21].

Drawing on this experience, it seemed interesting to use a numerical software to calculate the wave propagation through a disordered distribution of particles because there is a priori no consideration on the correlation in such approach. By correlation, we understand not only the geometric arrangement of the particles but also the constructive or destructive interferences between the waves which propagate in the medium. In the following, we will use an in-house software, called MuScat, which is able to address this problem with several tens of thousand cylinders, regardless of the type of cylinders with high contrast ratio [22] or polydispersion, the frequency range investigated, and the kind of geometry [23]. This allows us to model any microstructure made of randomly distributed cylinders in detail without approximation except the inherent numerical errors.

To illustrate the great potential of stealth hyperuniformity, one can cite the numerous studies dealing with the emergence of band gaps in SHU materials that have a bandwidth comparable to periodic structures, but are isotropic [24,25]. A recent remarkable numerical study has investigated this phenomenon for elastic waves propagating in a medium with rather large particles [28]. The band gap effect typically appears for wavelengths λ < λ min . Combining advantages of ordered and disordered media, it allows interesting applications such as freeform waveguiding [27,28] or optical cavities [29]. These phenomena -although very interesting -are out of the scope of our study. For λ > λ min , the cancellation of normalized density fluctuations induces transparency of the medium. However, this property is derived from a point pattern analysis, the question of the impact of the size of the scatterers being still open. Furthermore, strong multiple scattering effects are also neglected in general, considering that only weak interactions occur between the particles of the medium. A recent remarkable study by Kim and Torquato has already shown that wavelength can be finite compared to particles size without losing transparency [21]. In order to analyze this topic, they introduce an attenuation function that incorporates the contributions from all diffracted waves due to single (elastic) scattering events. Furthermore, Leseur et al. have proposed a transparency criterion for pointscatterers (monopoles) in the multiple scattering regime that depends both on the size L of the medium considered and on the scattering mean free path of an uncorrelated medium [30]. This theoretical result has been recently qualified by an experimental study [26].

Following on from these recent works, we tackle here at the same time the two questions of strong multiple scattering and particles size effects on transparency. To do that, we focus on the propagation of acoustic waves in two dimensional SHU media made of elastic cylinders with large radius. In this case, cylinders can no longer be considered as point-scatterers, and the influence of the multiple scattering increases with the size of radii so that interactions between particles become more complicated and might go beyond single scattering events. The software MuScat allows exactly to face this topic (see Appendix to have details on the modeling). In addition, it allows us to introduce an important acoustic parameter for realistic clustering distribution which is the concentration of particles, denoted φ. The position of the particles in SHU is usually controlled by the degree of stealthiness χ [31]. Once the position of particles/cylinders is fixed, thanks to χ, the concentration φ can be increased by growing the size of the cylinders. This is what we do in the following. This leads to characterize SHU by the parameters χ and φ.

II. SHORT AND LONG RANGE CORRELATIONS IN MEDIA WITH LARGE PARTICLES

A. Short range correlations

In heterogeneous media constituted by finite size cylinders, a minimal exclusion distance b = 2a between their centers is naturally imposed by their non-penetrability. This distance can also be larger but has to be lower than a maximum value imposed by the concentration. For random monodisperse media, the maximum concentration being φ r ≈ 83%, the upper limit for b is [22]

b max = 2a φ r φ . (1)
This type of constraints is called "short-range correlations" (SRC) because the presence of a cylinder only constraints its neighbourhood, that is to say the position of cylinders located in a circle of radius b around it. For all systems considered in this paper, the minimal exclusion distance b = 2a has to be taken into account in order to avoid overlapping. Imposing b ≥ 2a strongly and variously affect propagation of waves, both at large and small wavelength regimes [22]. Important effects around resonant frequencies can also appear and were recently studied in Refs. [22,32]. Interestingly, SRC media can be seen as an intermediate state between purely random and SHU media, as illustrated in Figure 1. Practically, to build SRC medium, we use the method known as Random Sequential Addition Process.

B. Long range correlations

The generation of SHU media is performed following an adaptation of the procedure described by Uche et al. [33]. The main quantities are shortly recalled here.

The point pattern representing the centers of the particles can be described either in real space by the pair correlation function g 2 (r) or in Fourier space by the structure factor S(k) given for a pattern of N points located at (r 1 , r 2 , ..., r N ) by

S(k) = 1 N N j=1 e ik.r j 2 . (2)
A point pattern is stealthy-hyperuniform if its structure factor S(k) vanishes around the origin

S(|k| < K) = 0. (3)
Generally speaking, the bound K > 0 can depend on the orientation of the vector k, which means that the region where the structure factor vanishes can be arbitrary. Choosing a square domain Ω of side K is convenient to get an analytical expression of S(|k| < K). K is then given as a function of the number of particles N , the size of the medium L and the degree of stealthiness χ as [30]

K = π L 4χN + 1. (4)
The parameter χ is the ratio between the number M (Ω) of independent vectors in the domain Ω in reciprocal space and the number of degrees of freedom in real space (2N in two dimensional space) [33]

χ = M (Ω) 2N . (5)
Extreme values χ → 0 and χ → 1 lead respectively to Poisson distribution and perfect crystal lattice. Both parameters M (Ω) and N are finite because only finite systems can be considered numerically. The apodization of the infinite medium by a rectangular window (in both space directions) leads to the discretization of the reciprocal space. The spectrum is hence descretized in P = 2K 2π/L points. This apodization also imposes that S(k = 0) = N rather than 0. For a square domain Ω of size 2K, the structure factor can be written analytically in a potential form which can be minimized in order to reach stealth hyperuniformity [30,33]. A recent work focuses on other very efficient strategy to build very large hyperuniform packings [34].

In a wave scattering context, the structure factor appears in the expression of the scattered intensity as S(q) = S(k i -k s ), where k i is the incident wavenumber and k s the scattered wavenumber. Noting k = ||k i || = ||k s ||, this leads to the upper limit k < K/2 for which, under monopolar and independent scattering approximations, only forward scattering appears which is called "perfect transmission" or "transparency" effect. For the sake of legibility, we introduce the reduce wavenumber k = 2k/K.

The word "transparency" has here to be understood as a propagation without loss (effective attenuation equal 0) rather than no perturbation of the total field in transmission [21]. Interesting properties such as complete photonic/phononic band gaps in SHU media with circular stealthy regions are known to be isotropic up to χ ≈ 0.57 [31,35]. For higher values, anistropric effects appear for wavenumbers k > 1.

The main objective of this work is to tackle the question of the conservation of transparency in dense material with large particles. As already mentioned, it is worth noting that both short and long range correlations have to be imposed in the same time, due to the non-interpenetrability of the particles.

For each case presented in this paper, except in Section VI, particles are infinite steel cylinders with properties c L = 5700 m/s, c T = 3000 m/s, ρ = 7850 kg/m 3 . The host medium is water (c 0 = 1500 m/s and ρ = 1000 kg/m 3 ) with wavenumber k = ω/c 0 . These properties are interesting because on one hand they lead to a strong impedance variation between particles and host medium (which is the case in many situations in wave physics) and on the other hand it allows a quantitative comparison with previous work that deals with SRC [22]. The source is a plane wave propagating in x-direction (p inc (r) = Ae ikx = Ae ikr cos(θ) ). Next section is dedicated to the derivation of an expression for the theoretical upper limit for transparency in SHU media.

C. Upper theoretical a/λ limit for transparency Before going further, it is worth pointing out that it is impossible to build SHU materials transparent for arbitrary high ratio a/λ. To explain that, let us recall that the maximum concentration in 2D periodic distribution is φ p = π/(2 √ 3) ≈ 90.7% and approximately φ r ≈ 83% for random packings. Having that in mind, let us consider an arbitrary surface of size L×L with concentration φ, noting N = φL 2 /πa 2 the number of particles and a the radius of the particles. As said in Section II, SHU materials are transparent for k < 1, K being given by Eq. ( 4). It gives in the limit 4χN 1, that

ka < Ka 2 = πφχ ≤ πφ p χ. (6)
Eq. [START_REF] Economou | Green's Functions in Quantum Physics[END_REF] shows that it is impossible to arbitrarily enlarge the domain Ω when considering media composed of finite-size particles. Secondly, it imposed an upper limit a/λ for transparency that is reachable for a given concentration φ. The limit case χ = 1 and φ = φ p leads to (ka) max = 1.68, which leads to λ ≈ 3.74a. In this situation, both long range and short range correlations are maximized and the medium is a perfect crystal. This extreme situation is not reached in the following. We chose to impose at most χ max = 0.6 and φ max = 50%, which gives (ka) max = 0.97 (λ = 6.47a), which already corresponds to large particles, and which is an "intermediate" frequency regime very common in acoustics.

It is worth noting that Eq. ( 6) mixes together the concentration φ, related to SRC for large particles, and the parameter χ which is linked with long range correlation. As long as parameter χ is only a function of the N points coordinates, for low values (typically χ < 0.4), two points can be very close to one another. This leads to the idea already known [START_REF] Torquato | Random Heterogeneous Materials: Microstructure and Macroscopic Properties[END_REF] that for a given particles size (and thus a given concentration), a minimum value of parameter χ is imposed. This explains why χ stays relatively high in this work (χ min = 0.4).

III. TRANSMISSION IN SHU MEDIA WITH FINITE-SIZE PARTICLES

This section is dedicated to the study of the transmission through dense media with large particles at low and high concentrations.

A. Transmission in systems with large particles at low concentration

In real space, SHU media are not easily recognizable. In Figure 1 An illustration of perfect transmission is given in Figure 2, representing the energy field scattered by a SHU medium composed of 6400 steel rods immersed in water, for two different wavenumbers : in the case k = 0.929, only forward scattering is observed, whereas for k = 1.047, the backscattering effects become stronger. These two frequencies have been chosen close to k = 1 to show that the transition appears to be very sharp. These two fields being calculated over only one distribution, no averaging over disorder is performed. This illustrates the surprising impact of correlations, leading to results very different from pure random distributions for which the incoherent part of the field can be much stronger.

In order to illustrate further the effect of hyperuniformity, reflection/transmission coefficients R/T are defined such as

       R = |p R | 2 |p R | 2 + |p T | 2 , T = |p T | 2 |p R | 2 + |p T | 2 , (7)
with p R and p T the scattered acoustic field at positions x = -L/2 and x = L/2, averaged on the transverse direction y. The source being a plane wave, the average process is only performed over |y| < 0.2L in order to avoid diffraction effects. Expressions of Eq. ( 7) are chosen because they impose in a trivial way R + T = 1.

In the single scattering regime, the transmission coefficient T through a slab can be totally determined with the help of the structure factor. S(q) → 0 implies T → 1, and at the opposite S(q) 0 implies an increase of the coefficient R. To illustrate that in several configurations of disorder, R and T coefficients for b/2a = 1, b/2a = 2.24 and SHU distributions are shown in Figure 3 (same types of distributions than the ones presented in Figure 1). The more the structure factor is close to 0, the more T gets close to 1. Here, the particles radii are a = 5 mm and the concentration is φ = 10%, leading to the upper limit for transparency Ka/2 = 0.35. This is a regime for which the monopolar mode is already insufficient: N m = 3 is necessary to reach convergence (for elastic particles, scattering modes T 0 and T 1 are of same amplitudes for large wavelength, T 1 can never be neglected).

Before continuing, let us look at both Figures 1 and3 together. It is interesting to note that SRC materials appear as a transition between purely random and SHU materials. This suggests another reading of the SRC materials. In fact, the local constraint imposed by the exclusion radius b forces the medium to organize itself over a long distance as soon as the concentration is blocked. We then can understand why their behaviors tend to behave like those of SHU materials, at least at low frequency, even if they stay of course very different.

The perfect transmission property of SHU materials with non-ponctual particles has been demonstrated in the case of low concentrations. The comparison between other types of correlation (b/2a = 1 and b/2a > 1) have illustrated the specific behaviour of SHU materials.

B. Transmission in systems with large particles at high concentration

In this section, the question of the reflection/transmission ratio in SHU media is discussed through the evolution of parameter T with regard to the couple (a, L). To this end, it is interesting to make an average of T, either on N config realisations of the disorder < T >= N -1 config N config p=1 T p or, to have a more synthetic point of view, an average over the frequency range 0 < k < 1,

T = 2 P P/2 p=1 T k = 2πp L (8)
where the value p ≤ P/2 introduced in Section II.B.

corresponds to k ≤ 1. It is clear that < T > =< T >.
Even if in a sense Eq. ( 8) seems to cover up information, this corresponds to a more realistic case, for which the source is not harmonic.

Even if this transmission coefficient is strongly frequency dependent, a first result is presented here at frequency k = 0.5, in order to see how < T > in the middle of the transparency window is impacted by strong variation of couple (a, L). For each couple of parameters (a, L), the quantities |p T | 2 and < T > are presented in Figure 4. On each curve presented in the two lower subplots in this Figure, each dot represents a transmission coefficient T calculated for a given couple. For the largest radii (a ≥ 0.06λ), a maximum appears after which < T > is exponentially decreasing. In order to enhance the analysis, simulations under the Born approximation (without particles interactions) are displayed on figures 4(b) and (d). In this case, the quantity |p T | 2 never stops increasing with medium size L, showing the cumulative effect of the simple scattering in the direction of propagation of the source. It also increases with particles size a, because the total scattering cross section is increasing with a at a given wavelength.

Having all that in mind, one can deduce from cartographies in Figure 5 what is the limit for transmission performances considering a given particles size and a given medium size. The four examples of < T > with respect to the frequency plotted in the same figure also show that the evolution of < T > with frequency can be very complicated when increasing a, even if an average process is performed over 20 different SHU configurations. This justifies the introduction of average transmission coefficient < T > to study the transparency in a global manner.

As an important result of our study, we also connect decrease of transparency to the emergence of an incoherent part in the acoustic field. This incoherent part is visible on the cartography of 5(b). This quantity is the normalized standard deviation to the average field. For a SHU medium composed by point scatterers (a λ), it has to be 0, because the scattered field is totally deduced from the structure factor, which is equally zero for all the disorder configurations. In other words, for this case, < T >= T p , ∀p. When strong multiple scattering effects appear due to particles size or long propagation distance, an incoherent part appears (< T > = T p ). From Figure 5, we see a very good agreement between decrease of < T > and increase of δ(< T >). If we compile these results, we first observe that transparency is more and more affected when the size of particles and the size of the medium increase, i.e. when the multiple scattering is reinforced and the distance on which waves propagate increases. In the latter case, there is a "conflict of interest" between the size of the medium and the elastic mean free path that is the distance on which the field is still ballistic. This last point is discussed in terms of effective attenuation in the next section.

δ(< T >) = < (T -< T >) 2 >/< T > in Figure

IV. EFFECTIVE ATTENUATION IN SHU DENSE MEDIA WITH LARGE PARTICLES

Now that transmission performances have been shown to strongly vary with parameters (a, L, λ), we explore the question of the losses inside the medium. To do that, the effective attenuation is computed numerically and compared to different models from the literature.

A. Criterion for transparency at large wavelength regime in finite medium

Let us recall the effective wavenumber of the Independant Scattering Approximation (ISA) k ISA [11,22], also called Boltzmann approximation [30] that takes into account single scattering events, given by where

k 2 ISA = k 2 -4in 0 f (0) (9)
f (θ) = +∞ n=-∞
T n e inθ is the far field form function, n 0 = φL 2 /πa 2 and T n the scattering coefficients of the particles. The scattering cross-section γ L is defined with the help of these coefficients as

γ L = 2 ka +∞ n=-∞ |T n | 2 . ( 10
)
The ISA model, which is of first order in concentration, is not adapted to take into account geometrical constraints but considers the medium as purely uncorrelated. The corresponding scattering mean free path is given by l e,ISA = 1/(2 (k ISA )).

The question of evaluating the scattering mean free path l e in SHU media is not new. Leseur et al. used a diagrammatic approach based on the use of the selfenergy to derive the following order of magnitude (see Section 4 of Ref. [30])

l e ≈ kl 2
e,ISA (11) in the frequency range k < 1. It is noteworthy that Eq. ( 11) is only applicable for k < 1. Furthermore, it is derived in under long-wavelength and independent scattering approximations. Considering the ballistic regime to be valid for L < l e , with l e the real scattering mean free path of the SHU medium, they state that

L < kl 2 e,ISA (12)
is the criterion for staying in the ballistic regime and ensuring transparency. The first question is whether the relation ( 11) is still correct for large particles ? Recent experimental work of Aubry et al. [26] showed that transparency seems to be conserved up to k → 1, even for their dense system made of high index large particles for which relation (12) was not satisfied. Reacting to these results, we have investigated this problem in more details.

In the following, MuScat effective attenuation α eff , elastic mean free path l e and phase velocity c eff (useful for section V.A.) are calculated numerically. The medium being located between x = 0 and x = L, one has two different pressure fields p 0 and p 1 [22] p 0 = Ae ikL in homogeneous medium, p 1 = Ae ik eff L in heterogeneous medium, (13) with k eff = ω/c eff + iα eff . Effective coefficients α eff = 1/2l e and c eff are calculated using the expression

         α eff = - 1 L ln p 1 p 0 , c eff = ωL ωL c 0 + arg p 1 p 0 -1 . (14)

B. Effective attenuation in thin SHU materials with large particles at low concentration

We focus now on the effective attenuation for SHU materials built with different values of χ and different concentrations φ.

In this part, Ka/2 stays below 0.4 (a = 2.5 mm). The size L of the medium is chosen such that Eq. ( 12) is satisfied. Imposing these parameters, we focus on the stealthiness of the medium χ and its impact on the effective attenuation. In Figure 6, MuScat simulations as well as Leseur predictions [30] clearly indicate that attenuation stays very low compared to ISA for k < 1 (the vertical dotted lines being the limit k = 1 for each value of χ). It is worth noting that, for φ = 10% and χ = 0.55, Leseur and MuScat predictions are in good qualitative agreement up to ka > 0.38 for which attenuation starts increasing significantly. Very similar results are presented by Kim and Torquato for electromagnetic waves in Ref. [21], Our results can be seen as a numerical confirmation regardless the size of the scatterers of the analytical formulation developed by these authors. The original and key point of their model is to take into account n-point correlation functions of order higher than two for the effective dielectric tensor of electromagnetic waves propagating in random media. That way, they approximately accounts for spatial correlations at all orders between scatterers via the spectral density that appears in their so-called "attenuation function". In fact, the good agreement between the theory of Ref. [21] and the results of the present paper shows that this theory correctly captures multiple scattering effects. In their work, they point out numerous applications. In particular, this shows that isotropic lowpassfilters can be built by introducing correlation into a purely random heterogeneous medium. To illustrate the link between SRC and SHU, we also present results similar to those of our earlier work [22] which focused on the impact of the increase of SRC parameter b on the propagation. Results in Figure 6(c) demonstrates that at low frequency the introduction of SRC also makes the medium more transparent but in a smoother way compared to SHU. From these results, we demonstrate that wavelength approximation of Eq. ( 11) seems well appropriate to this case for which concentration is low but with large particles and strong multiple scattering.

C. Effective attenuation in thin SHU materials with large particles at high concentration

To go further into this investigation, we discuss now about how the behavior of SHU materials changes when the particles size increases at constant wavelength. To achieve this, 20 point patterns are built with stealthiness χ = 0.6. Imposing φ ≤ 50%, Eq. ( 6) imposes (ka) max = 0.97. Furthermore, we impose a max = 11.5 mm. For each point pattern, we performed simulations increasing particles radii, bearing in mind that multiple scattering effects become stronger. Results in Figure 7 show the evolution of the parameter α eff . Four different wavelengths k = 0.5, k = 0.75, k = 0.9 and k = 1.1 have been chosen in order to have a large overview on what happens when increasing the ratio a/λ. For a = a max , values of k leads respectively to (ka) max = 0.48, (ka) max = 0.72, (ka) max = 0.864, (ka) max = 1.067.

For the lowest frequency ranges k = 0.5 and k = 0.75, Leseur predictions (l e ≈ kl 2 e,ISA ) and MuScat simulations quantitatively agree for all radii, proving the validity of Eq. ( 11) for large wavelength regime, even at very high concentration. In these cases, multiple scattering effects are weak enough to be quantitatively described by the ISA coupled with the long range correlation analysis. The third case k = 0.9 ((ka) max = 0.864) is still below the upper bound k = 1. For this case, the concentration effects start playing an important role for a = 9.2 mm which gives ka ≈ 0.7. This value is also in agreement with the extreme value a = 11.5 mm of case (b) and appears to be the limit of validity of the large-wavelength independent scattering regime of Eq. ( 11). For such high concentrations values, it is not surprising that the ISA analysis fails to describe properly the propagation. One should consider adding second order terms in the effective wavenumber. This is also visible on the last case ( k = 1.1), for which MuScat simulations and ISA predictions are in almost perfect agreement for a ≤ 9.2 mm. After this value, the multiple scattering effects are too strong to be well approximated by the ISA.

This analysis leads to the conclusion that the model proposed by Leseur [30] is in quantitative agreement with MuScat simulations up to a ratio ka ≈ 0.7. This limit appears to be related to the independent scattering approximation. Hence, we proved that this order of magnitude is particularly appropriate to evaluate the effective attenuation of a SHU material up to critical values φ = 30% and ka ≈ 0.7. MuScat simulations presented here have been performed for a medium size of L = 0.5 m. Other values have been tested that gave the same results.

V. MEDIA WITH VARIOUS PARTICLES SIZES: EXCEEDING THE LIMITS OF POINT PATTERNS ANALYSIS

In this part, the purpose is to complete the analysis of the transparency and enrich the structure factor information by considering new characteristics of the particle as the polydispersity, particles with subwavelength resonances, and by inserting the SHU material in a dispersive random medium.

A. Heterogenous host medium

What happens if the host medium that surrounds the large particles building the SHU material is heterogeneous? This question has already been discussed by Kim and Torquato [START_REF] Kim | [END_REF] for point particles. They show that increasing the concentration of defects implies the loss of hyperuniformity. Here, to study this problem staying within the framework of multiple scattering, the host medium is made heterogeneous and dispersive by inserting small particles randomly distributed around the large ones (see Figure 8). More specifically, we built a SHU point pattern with χ = 0.6, N = 2700 and a = 3.8 mm. Then, we randomly added N s small particles inside this medium. To ensure that our study remains in the framework of SHU materials, we assume that multiple scattering effects associated to the host random medium are weaker than those related to the SHU material. This lead us to impose two limits, namely, the radius of the N s particles is equal to a/2 (in order to reduce their scattering cross sections, the ratio of the T 0 scattering coef-ficients being approximately of 0.25 for the whole frequency range) and N s ≤ 4N so that the concentration φ of small particles stays below the concentration φ of the SHU material (both concentrations are equal when Ns = 4N). Before going further, using the numerical procedure described in previous part, the effective wavenumber k eff = ω/c eff + iα eff that characterizes the heterogeneous host medium is shown in Figure 9. As expected at low frequencies, one can note that the dispersion of the velocity is rather weak but the attenuation is increasing.

In order to study the propagation through the medium, the transmission coefficient T is introduced as follows

T = |p T | 2 |p R,h | 2 + |p T,h | 2 ≈ |p T | 2 |p T,h | 2 (15)
where p R,h and p T,h are respectively the back-scattered and forward-scattered pressure for the case N s = 0, consisting of a purely homogeneous host medium. The quantity |p R,h | 2 is small compared to |p T,h | 2 . This allows us to write approximation of Eq. ( 15). T and T give different types of information. As previously, T gives the ratio between reflection and transmission.

On the other hand T allows to estimate the losses due to the dispersion for different degrees of short-scale heterogeneity (number of small particles N s ) in the host medium.

It is worth noting in Figure 8 that T remains close to unity and is significantly modified only for large values of N s . Transmission performances stay very high, which endorses the fact that this phenomenon is resistant to the dispersion even if the transmitted field intensity, represented by T , undergoes a significant decrease.

One must keep in mind that results presented here are only valid if the two populations (namely the N and the N s particles) can be studied separately. Here, we show that the whole medium can be understood as the superposition of a SHU material immersed in a dispersive medium. This unexpected result has a limit of course, especially in term of concentration, but this limit is out of the scope of this paper. However, this result is significant because it shows how a particles size study can reveal SHU properties where the simple point pattern analysis fails.

To support this result and in order to go further, we will now examine whether the SHU medium surrounded by the dispersive random medium preserves its main characteristic, namely the cancellation of the structure factor. From a first point of view, based on homogenization approaches, assuming the SHU immersed in the effective medium amounts to consider that the quantity

S(k) is computed taking k = (k x , k y ) t = ( c c eff 2π L + iα eff ) (n x , n y )
t , with -P/2 < (n x , n y ) < P/2. This leads to the structure factors plotted in Figure 10 (b) (d), where φ stands for the concentration of the SHU N -points pattern and φ for the effective host medium composed by the N s particles. The structure factor almost perfectly cancels out and the transmission behavior can be explained in simple terms, as a direct consequence of the vanishing of the structure factor which depends on the effective wavenumber.

To explore the subject further, it is also interesting to examine what happens when considering the whole point pattern in the structure factor calculation, taking into account the particles size through the scattering coefficient T 0 , in the manner of Romero-García et al. [38] S(k, T 0 ) = 1

N +Ns j=1 T (j)2 0 N +Ns j=1 N +Ns k=1 T (j) 0 T (k) 0 e ik.(r k -r j ) , (16)
with the idea that taking into account the type of particles gives a different weight for the large compared to the small ones (note that Eq. ( 16) is equivalent to Eq. ( 2) if all the scatterers are identical). We chose here to keep only the monopolar mode because the definition was derived for point scatterers [38]. The resulting structure factor is plotted for two values of N s in Figures 10 (a),(c). We clearly see in these Figures that the domain Ω is identifiable, even if the structure factor is not equal to zero. However, the structure factor appears to be affected homogeneously in the entire domain Ω, which is in less quantitative agreement with the observed weakening of the transmission coefficient T that is most affected for k close to 1.

As strong and important result of this study, we show that a complex medium made of two distinct populations of different sizes can be successfully understood as the superposition of a dispersive random medium (that can be homogenized) and a SHU medium that keeps its transparency properties. These properties are only affected by the dispersion of the host medium, if the concentration stays rather low.

B. Influence of polydispercity

This part deals with the question of the influence of polydispersity. Nature never reaches perfect monodispercity, and laboratory conditions for experiment also integrates error bars on the size of the particles. The proof of the great importance of accounting for polydispercity in many cases is already established [39]. Here, the introduction of such effects is possible thanks to the T-matrix formalism that allows to change one by one the properties of each particle. The mean radius is chosen to be a 0 = 1 mm, and the variations are random, following a Gaussian distribution of standard deviation σ, with imposed upper limit a max = 2a 0 to ensure no interpenetrability between particles and lower limit a min = 1 µm. Here, parameter χ = 0.5. Figure 12 shows that transmission is weakened by polydispersity. However, considering experimental uncertainty, for which 10% of variation is quite important, it is worth noting that transmission stays very close to 1 up to k = 1.

Gathering this conclusion with previous ones, it is meaningful to remember that all the particles do not play the same role in the propagation inside the structure. With the use of Eq. ( 16), the structure factor is calculated for the case σ = 0.3a 0 and presented in Figure 11(b). Comparing with the monodisperse case (a), one can observe that the resulting structure is no more stealthy, mainly for the highest wavenumbers in the square, as shown in Figure 11(b). This is in agreement with the fact that T starts decreasing before k = 1. However, having in mind experimental constraints, it is worth noting that transmission stays very close to 1 for relatively large polydispersity.

VI. RESONANT STEALTHY HYPERUNIFORM MEDIA

In the past few years, many studies have been devoted to artificial mesoscopic materials. Among these, particular attention has been paid to locally resonant metamaterials for which the strategy for obtaining extraordinary phenomena is to exploit strong subwavelength resonances of the particles embedded in the propagation medium. The coupling between resonances and propagating waves in the matrix surrounding the particles can lead to the formation of band gaps, often referred to as hybridization gaps [40]. However, the wave attenuation in the matrix is often a limiting factor and in this context, SHU appears as a promising candidate due to their transparency property. That leads then to the question of the influence of a resonance on the propagation of waves through a SHU material. To achieve this, we consider steel shells filled by water, with outer radius a out = 5 mm and various inner radius a in that drive the resonance frequency of the mode of vibration n = 1.

Figure 13 shows the impact of the resonance on the coefficient of transmission T , reducing it by a factor of 0.65 close to resonance frequencies. This effect can be enhanced, by using more complicated resonators, but this is out of the scope of our study. The important thing is not this point but the fact that resonances perturb the transparency of the SHU material. It is not easy to explain this phenomenon. One hypothesis is as follow; periodic and SHU media have in common to involve interference phenomena. Bragg interferences for periodic media and cancellation of the structure factor for SHU materials.

For periodic media, a resonance results into the opening of a localized pass-band inside stop-bands [41,42]. In other words, transmission is created in a frequency range where total reflection should be observed. This can be understood by remembering that stop-bands are due to Bragg interferences. When there is a resonance, waves scattered by the particles undergo a phase shift which varies very rapidly on the whole width of the resonance [42]. Such a phase shift locally destroys the interference phenomena that were responsible of the stop-bands. This explains why a localized pass-band is opened just around the resonance frequencies. The phenomenon is similar for SHU material. If we think that transparency is due to interferences, transparency must also be perturbed by the phase shift which varies very rapidly around the same resonance frequencies. In this case we observe in Figure 13 that the transparency is perturbed on the whole width of the resonance. Periodic and SHU materials are opposite from each other. In one case the resonance creates transmission, in the other it creates reflection. This suggests that SHU materials, like periodic ones, could be used for the construction of frequency filters.

VII. CONCLUSION

This paper deals with the propagation of acoustic waves in two dimensional SHU medium made of elastic cylinders with large radii, with a particular focus on transparency. To achieve this, we have used the software MuScat [23] based on the modal decomposition of the fields that takes multiple scattering into account at all orders. In this context, we introduce an important acoustic parameter for realistic clustering distribution which is the concentration of particles. Once the position of cylinders is fixed, thanks to the degree of stealthiness χ [31], the concentration φ can be expanded by increasing the size of the cylinders. The perfect transmission property of SHU materials with non-ponctual particles has been demonstrated. However, it has been established that SHU materials are transparent only on condition that ka ≤ πφχ. Furthermore, it is observed that transmission performances strongly depend on three parameters: the radius of cylinders a, the size of the medium L and the wavelength. In particular, it is shown that transparency disappears for higher values of a and L, i.e. when the multiple scattering effect between particles increases with the onset of an incoherent field. From another analysis based on diagrammatic approach, Leseur et al. have proposed a order of magnitude for the scattering mean free path that depends on the scattering mean free path l e,ISA of uncorrelated random media [30]. We show that this analysis is still valid for large particles even for high concentrations but our investigation points out that the use of the independent scattering approximation in SHU must be put into question above and beyond ka ≈ 0.7.

The question of the impact of polydispercity is also tackled and shows that, under common experimental uncertainties, transparency behavior is conserved. Further-more, it is shown that a SHU media also keeps its properties when immersed in an heterogeneous scattering background, in the limit of rather low concentration, Lastly, by changing the elastic properties of our particles, we show how the introduction of a sub-wavelength resonance in the frequency range k < 1 can be used as filter that stops the propagation of particular frequencies.

VIII. ACKNOWLEDGMENTS

We wish to thank Romain Pierrat (ESPCI ParisTech, PSL Research University, CNRS, Institut Langevin, Paris, France) for fruitful discussions.

IX. APPENDIX : CLUSTER EXPANSION IN TWO DIMENSIONS AND MUSCAT NUMERICAL PROCEDURE

The total field resulting in the interaction of a source p inc and N infinite cylinders located at (r 1 , r 2 , ..., r N ) in an homogeneous matrix can be written as p(r) = p inc (r) + N j=1 p (j) s (r j ), (17) where the acoustic pressure scattered by each cylinder is expanded on the cylindrical harmonics p (j) s (r j ) = +∞ n=-∞ A (j) n H (1) n (kr j )e inθj ,

with H

(1)

n the Hankel function of the first kind and order n and (r j , θ j ) the cylindrical coordinates of the point r in the coordinates system of the j th cylinder. For a given set of cylinders, the unknown amplitudes A (j) n (truncated at order N m -1 so that 2N m -1 modes are taken into account) forming the vector A are determined by solving the following linear system

[I -T M] A = T E, (19)
with T the block matrix containing all the T (j) (diagonal) matrices that contain the scattering coefficients of every scatterer j. The vector E is the modal amplitude of the incident wave and the matrix M, built thanks to the Addition Theorem, models all the interactions between scatterers (neglecting it leads to the Born approximation).

The resolution of Eq. ( 19) is challenging for great numbers of cylinders or high frequencies. To achieve this, we use the in-house software MuScat [23] that is adapted for solving problems for sizes of several thousands of cylinders. It is worth pointing out that Eq. ( 19) allows the calculations of any kind of distributions without restriction concerning the number, the elastic properties, the source type nor the source frequency. A key point is that not only the monopolar mode is calculated here, but also the higher ones. In the following, N m -1 will be the highest mode taken into account in the modal sum, and is computed for each case such as |T Nm | and |T Nm+1 | < 10 -4 .

  , three distributions are shown : SRC, b/2a = 1 (a), SRC, b/2a = 2.24 (b) and SHU (c), with their structure factors. Properties of SRC are not enough to reach stealth hyperuniformity since the structure factor does not fully vanishes around the origin.

FIG. 1 .

 1 FIG. 1. (Color online) Random packings of cylinders (SRC with b/2a = 1) (a), short-range correlated medium (b/2a = 0.5/φ = 2.24) (b) and stealthy hyperuniform medium (c) (χ = 0.4) with their structure factors (respectively (d), (e) and (f)).

FIG. 2 .

 2 FIG. 2. (Color online) Scattered energy field with a incidence plane wave propagating from the left to the right for k = 0.929 (a) and k = 1.047 (b), for distributions made of N = 6400 particles of radius a = 5 mm, with χ = 0.4, with concentration φ = 3.1% and N/L 2 = 400.

FIG. 3 .

 3 FIG. 3. (Color online) Reflection and transmission coefficients through short-range correlated medium with b/2a = 1 (a), b/2a = 2.24 (b) and stealthy-hyperuniform (c). All distributions are built with N/L 2 = 1270.

FIG. 4 .

 4 FIG. 4. (Color online) Scattered energy |pT | 2 and transmission coefficient T for different size of medium L and different particles radii a, for k = 0.5. Left panel shows results for Multiple Scattering simulation (MS) and right panel shows single scattering simulation (Born approximantion). The continuous (green) line represents the highest level of the reflection coefficient R. Every curve is calculated with MuScat and the point density N/L 2 = 1200 is kept constant.

FIG. 5 .

 5 FIG. 5. (Color online) Mean of transmission coefficient < T > over the frequency range k ∈ [0, 1], for different medium size L and particles size a (a). Standard deviation δ(< T >)) is presented (b), as well as four examples of transmission coefficients < T > for four couples of parameter (a, L) with respect to the frequency (below). Number of cylinders is fixed such that N/L 2 = 1200.

→ 1 b

 1 FIG. 6. (Color online) Effective attenuation for a medium with concentration φ = 5% (N = 1908, N/L 2 = 2550) (a) and φ = 10% (N = 3816, N/L 2 = 5100) (b), a = 2.5 mm and different values of parameter χ. The vertical dotted lines correspond to the frequency for which k = 1. (c) presents effects of SRC and is of the same type than results from [22], Figure 8(a).

1 FIG. 7 .

 17 FIG. 7. (Color online) Effective attenuation at four wavenumber k = 0.5 (a), k = 0.75 (b), k = 0.9 (c) and k = 1.1 (d) as a function of particles size for a SHU medium with parameter χ = 0.6 and N/L 2 = 600. Vertical dotted lines indicate the radii used to build distributions on the right, which are illustrative examples of SHU distributions smaller but similar to the ones used to perform the simulation with MuScat.

FIG. 8 .FIG. 9 .

 89 FIG. 8. (Color online) Transmission coefficient T and T for media composed of a stealthy distribution of N particles with radius a (black circles), N/L 2 = 1200 and a uncorrelated distribution of Ns particles of radius a/2 (red/light grey circles). (Green) dotted line represents the stronger level of reflection coefficient R, whose definition is analogous to T .

5 FIG. 10 FIG. 11

 51011 FIG. 10. (Color online) Structure factors as defined by Eq. (16) for Ns = N (a) and Ns = 4N (c) for which the medium is made of N + Ns particles. Structure factors calculated thanks to Eq. (3) for the SHU medium of N particles (concentration φ) immersed in an effective medium with concentration φ = 0.25φ (corresponding to Ns = N ) (a) and φ = φ (corresponding to Ns = 4N ) (d) (see text).

  FIG. 12. (Color online) Transmission coefficient through polydisperse media with various radii distribution functions, χ = 0.5 and N/L 2 = 26460.

FIG. 13

 13 FIG. 13. (Color online) Transmission coefficient T through resonant stealthy medium (χ = 0.5 and N/L 2 = 26460) made of tubes of inner radius ain and outer radius aout for ratio a in a out = 0.31, 0.33, 0.35 (the arrow indicates the direction for which the ratio increases). The scattering coefficients T0 and T1 are also represented (right y-label, the resonance appears on the second scattering coefficient T1).