J B Vergnes

V Gernigon

P Guiraud

C Formosa-Dague

Bicarbonate concentration induces production of exopolysaccharides by Arthrospira platensis that mediate bio-flocculation and enhance flotation harvesting efficiency

Keywords: Arthrospira platensis, Exopolysaccharides, Bio-flocculation, Flotation, Atomic force microscopy

Arthrospira platensis, or Spirulina, is a cyanobacterial species mainly exploited for its nutritional qualities and able to produce exopolysaccharides (EPS) under specific conditions. In this work we identify that the bicarbonate concentration is a key parameter that induces EPS production by the cells. Using flocculation/flotation harvesting experiments, in addition to atomic force microscopy to characterize the biophysical properties of EPS, we show that EPS produced by A. platensis cells form a soft, adhesive gel in the medium that forms aggregates in which cells are entrapped and thus bioflocculated. This results in the creation of microalgae flocs in which ascending microbubbles produced in flotation process get trapped, thus enhancing the efficiency of the microalgae separation from water. This interdisciplinary study provides new insights into culture conditions that positively influence the EPS production, while allowing to reach both a high final biomass concentration and a harvesting efficiency superior to 90%.

Introduction

Arthrospira platensis, commonly known as Spirulina, is an edible filamentous cyanobacterium originally isolated from alkaline lakes in Africa, and presently cultivated in tropic, sub-tropic and tempered regions of the world. Because of its important nutritive qualities, this microalgae is now one of the most commercialized microalgae species [START_REF] Kim | Factors Indicating Culture Status during Cultivation of Spirulina (Arthrospira) Platensis[END_REF][START_REF] Raja | Applications of Microalgal Paste and Powder as Food and Feed: An Update Using Text Mining Tool[END_REF] . Indeed, A. platensis is considered as the most valuable source of protein, as up to 70% of its biomass can be composed of this fraction, with a profile that includes the full range of essential amino acids [START_REF] Dillon | Nutritional Value of the Alga Spirulina[END_REF][START_REF] Rodrigues | Influence of Ammonium Sulphate Feeding Time on Fed-Batch Arthrospira (Spirulina) Platensis Cultivation and Biomass Composition with and without PH Control[END_REF][START_REF] De La Jara | Impact of Dietary Arthrospira (Spirulina) Biomass Consumption on Human Health: Main Health Targets and Systematic Review[END_REF] . The most valuable protein this cyanobacterium is able to produce is phycocyanin, which is used as a coloring agent in food and cosmetics [START_REF] Spolaore | Commercial Applications of Microalgae[END_REF] . Moreover, A. platensis is also a source of polyunsaturated fatty acids, vitamins (B12) and other pigments (chlorophyll A and carotene), and has been evaluated for therapeutic use. For instance, A. platensis features include hypolipidemic, hypoglycemic and anti-hypertensive properties, but also has antioxidant and microbial-modulating activities [START_REF] Finamore | Antioxidant, Immunomodulating, and Microbial-Modulating Activities of the Sustainable and Ecofriendly Spirulina[END_REF] .

Another important aspect of A. platensis is that this microalgae species is capable of producing extra-cellular exopolysaccharides (EPS). A wide range of microalgae (eukaryotic and cyanobacteria) produce EPS, which act as a protective layer around the cells to preserve them from unfavorable environmental conditions [START_REF] Pereira | Complexity of Cyanobacterial Exopolysaccharides: Composition, Structures, Inducing Factors and Putative Genes Involved in Their Biosynthesis and Assembly[END_REF] . EPS from microalgae also have potential applications in industry and health [START_REF] Markou | Microalgae for High-Value Compounds and Biofuels Production: A Review with Focus on Cultivation under Stress Conditions[END_REF] . For example, EPS can be used as thickening agents to alter water rheological properties, as heavy-metal removal agents for wastewater treatments, or even as food additives in the food industry 10-12 . They also display several biological properties such as anti-inflammatory, anti-tumor, anti-viral, anti-parasitic and immunomodulatory activities [START_REF] Markou | Microalgae for High-Value Compounds and Biofuels Production: A Review with Focus on Cultivation under Stress Conditions[END_REF][START_REF] Delattre | Extraction and Characterization of Microalgal and Cyanobacterial Exopolysaccharides[END_REF] . For these reasons they have been the subject of many studies, with much effort focused on the determination of their composition, or the optimization of culture conditions that favor their production. In the case of the EPS produced by A. platensis, studies have shown that they are rich polymers composed of complex heteropolysaccharides with protein moieties [START_REF] Trabelsi | Partial Characterization of Extracellular Polysaccharides Produced by Cyanobacterium Arthrospira Platensis[END_REF] , and that temperature, light intensity and the cell's trophic mode could influence their production [START_REF] Trabelsi | Combined Effect of Temperature and Light Intensity on Growth and Extracellular Polymeric Substance Production by the Cyanobacterium Arthrospira Platensis[END_REF][START_REF] Trabelsi | Evaluation of Arthrospira Platensis Extracellular Polymeric Substances Production in Photoautotrophic, Heterotrophic and Mixotrophic Conditions[END_REF] . Another study also showed that their presence in water inhibits cell growth, which has implications in water recycling [START_REF] Depraetere | Influence of Culture Medium Recycling on the Performance of Arthrospira Platensis Cultures[END_REF] .

While A. platensis rich biomass and EPS are sought after for various applications, a limiting factor is their harvesting. Harvesting is usually a challenge in the case of microalgae because they grow at low concentrations, thereby resulting in a need to treat large volumes of water. This crucial step accounts for up to one third of the entire biomass production cost [START_REF] Molina Grima | Recovery of Microalgal Biomass and Metabolites: Process Options and Economics[END_REF] . In the case of A. platensis, the most commonly used technique to harvest the cells is by filtration given their large size. Indeed, filtration tends to work better with larger microalgae rather than with smaller microalgae that can clog the membranes used in filtration [START_REF] Rossi | Harvesting of Cyanobacterium Arthrospira Platensis Using Inorganic Filtration Membranes[END_REF][START_REF] Rossi | Harvesting of Cyanobacterium Arthrospira Platensis Using Organic Filtration Membranes[END_REF][START_REF] Drexler | Membrane Applications for Microalgae Cultivation and Harvesting: A Review[END_REF] . While this technique is usually efficient for A. platensis cultures that produce no EPS, it has been shown that for A. platensis cultures that produce the EPS, these often clog the membranes , thus resulting in low efficiency with high operating costs [START_REF] Rossi | Arthrospira Platensis Harvesting with Membranes: Fouling Phenomenon with Limiting and Critical Flux[END_REF][START_REF] Barros | Harvesting Techniques Applied to Microalgae: A Review[END_REF] . Centrifugation can also be used in the case of A. platensis [START_REF] Morist | Recovery and Treatment of Spirulina Platensis Cells Cultured in a Continuous Photobioreactor to Be Used as Food[END_REF] , yet the operating costs of centrifugation are high due to the high energy input of this technique, but also the cell damage and loss resulting from the high-shear forces applied [START_REF] Pragya | A Review on Harvesting, Oil Extraction and Biofuels Production Technologies from Microalgae[END_REF][START_REF] Garg | Flotation Separation of Marine Microalgae from Aqueous Medium[END_REF] . Flotation is a promising technique that has the advantages of having a low energyinput, reduced operating-costs and low space and equipment required. This technique is efficient to harvest cells cultivated in high-nutritive conditions if the cells are left to flocculate before the introduction of the bubbles [START_REF] Kim | Harvesting of Spirulina Platensis by Cellular Flotation and Growth Stage Determination[END_REF] . In this case the efficiency of the process could also be enhanced by adding cationic floculants to the cell suspension (CaCl2, MgCl2, FeCl3) [START_REF] Kim | Harvesting of Spirulina Platensis by Cellular Flotation and Growth Stage Determination[END_REF] . However such added flocculants can then end up in the harvested biomass and interfere with the final product quality [START_REF] Vandamme | Flocculation as a Low-Cost Method for Harvesting Microalgae for Bulk Biomass Production[END_REF] . An interesting and sustainable alternative to this problem is natural auto-or bio-flocculation. For A. platensis, three cases of bio-flocculation have been reported. In these three cases, the same mechanism is described where bio-flocculation is directly induced by the accumulation of carbohydrates by the cells under nutrient stress conditions (phosphorous and nitrogen) or at high salinities, leading to an increase in their specific density [START_REF] Markou | Carbohydrates Production and Bio-Flocculation Characteristics in Cultures of Arthrospira (Spirulina) Platensis: Improvements Through Phosphorus Limitation Process[END_REF][START_REF] Depraetere | Harvesting Carbohydrate-Rich Arthrospira Platensis by Spontaneous Settling[END_REF][START_REF] Toyoshima | A Pilot-Scale Floating Closed Culture System for the Multicellular Cyanobacterium Arthrospira Platensis NIES-39[END_REF] . However in these conditions, the nutrient deprivation results in reduced cell growth and biomass production.

The present study addressed the two important aspects in A. platensis, EPS production and harvesting. We show the direct influence of EPS on the recovery efficiency of the biomass using flocculation/flotation technique. For this we developed a culture strategy of A. platensis to assess the role of culture medium bicarbonate concentration on the production of EPS by the cells. Their implication in flocculation/flotation harvesting process were then evaluated in experiments at laboratory-scale. Finally, using an advanced atomic force microscopy [START_REF] Binnig | Atomic Force Microscope[END_REF] imaging technique we characterized the biophysical properties of EPS to understand their role in flocculation. Altogether we describe an original bioflocculation mechanism and we provide culture conditions that allow for optimized biomass production and efficient harvesting.

Material and Methods

Strain and culture conditions

An Arthrospira platensis strain isolated for the Paracas lake in Peru was used in this study and cultivated in tap water containing the following: Ca (0.025 g/L); Cl (0.045 g/L); K2O (0.297 g/L); NO3 (1.779 g/L); Mg (0.012 g/L); Mn (0.230 g/L); Zn (0.461 mg/L); Cu (0.229 mg/L); B (0.229 mg/L); SO4 (0.583 g/L); complexed iron (micronutriFe, Prayon, Belgium, 0.003 g/L), P2O5 (0.178 g/L) and a variable concentration of NaHCO3 and NaOH to adjust the initial pH to 9.5. Cultures were achieved in 6 L poly(methyl methacrylate) photobioreactors and were continuously agitated by the gentle bubbling of atmospheric air (2 L/min). Illumination was provided by three submerged tubes containing warm white LEDs (BDL, vs-f600ww-12V-3528-10mm-2700K) delivering an irradiance of 280 or 350 µmolυ/m²/s for a total of 916 cm² of illuminated walls. The temperature of the culture was regulated at 30 or 35°C using an immersion heater (Scientax). The protocol used to cultivate the cells is described in Figure 1.

Cultures were maintained in an inoculum culturing PBR (sub-cultures at an OD of 0.3 every week) at moderate cell growth conditions (30°C, 280 µmolυ/m²/s). For inoculating the experiments PBRs, 6 L from the inoculum culturing PBR were filtered on a 56 µm sieve (Retsch, 60.131.000056); approximately 2 g of the retentate was then diluted in 6 L of new culture medium at different NaHCO3 concentrations. The culture in the experiments PBRs lasted 4 days in high cells growth conditions (35°C, NaHCO3 concentration was maintained constant during the four days of the culture by measuring the pH twice a day and adding NaHCO3. The pH measure was used in the following equation ( 1):

(1)

[Bic]i + ∆[Bic] =

[Ct]i + 0.5 × ∆[Bic] 1 + 10 pKa1 -pHm + 10 pHm -pKa2

Where [𝐵𝑖𝑐] and [𝐶𝑡]

Flocculation/flotation experiments

A detailed presentation of the laboratory scale flotation experiments can be found in [START_REF] Besson | High-PH-Induced Flocculation-Flotation of the Hypersaline Microalga Dunaliella Salina[END_REF][START_REF] Besson | Flocculation-Flotation Harvesting Mechanism of Dunaliella Salina: From Nanoscale Interpretation to Industrial Optimization[END_REF] . DAF experiments were performed in a Multiplace Orchidis™ FTH3 Flottatest. Three flotation-test beakers were run simultaneously, in which 600 mL samples were collected from the algal cultures and added to each beaker. The depressurization at atmospheric pressure of 200 mL culture medium free of algae and saturated by air at 6 bars for 15 min induced the formation of microbubbles. The recycle ratio (pressurized culture medium volume/initial sample volume) was of 33%. In each case, prior to depressurization, the cells were left to flocculate during 20 min under gentle mechanical mixing (100 rpm). Harvesting efficiency quantification is based on optical density measurements. The harvesting efficiency represents the quantity of algae floated compared to the quantity in the initial suspension: it was evaluated using the following equation:

(2)

E(%) = (1 - OD𝑎V𝑎 OD𝑖V𝑖 ) × 100
Where ODi and Vi are the initial optical density at 680 nm and the volume of algal suspension before flotation, ODa and Va are the optical density at 680 nm of the aqueous phase and the volume of the aqueous phase after injection of pressurized water.

Filtration experiments

Filtration experiments were performed in a 200 mL Amicon cell (Merck, 10402511) using an inox filter with a 10 µm pore size (Negofiltres, stainless steel 316L). 1 L samples were collected from the algal cultures and added in a pressurized tank; filtration was then performed under 0.5 bars. The filtrate was collected in a beaker on a monitored scale and the weight of the filtrate was measured every 5 s for 900 s. The filtrate density was of 1000 g/L and the variation of the flow rate was calculated over time with the measure of the mass over time.

Optical and AFM imaging

The optical images of A. platensis cells before flocculation/flotation experiments were directly obtained by sampling 1 mL of the cell suspension. These samples were then deposited on glass surfaces and allowed to stand for 30 min at room temperature. An Alcian blue staining was then used to color the EPS [START_REF] Passow | Distribution, Size and Bacterial-Colonization of Transparent Exopolymer Particles (TEP) in the Ocean[END_REF] . To this end, 1 mL of a solution made of 0.06% acetic acid and 0.03% of Alcian blue was deposited on the glass surfaces and allowed to stand for 15 min. Glass surfaces were then immersed in distilled water for 15 min in order to remove the non-fixed staining. Finally images (resolution of 1024 × 768 pixels) were recorded at high magnification (×50) using an inverted Axio Observer Z1 microscope (Zeiss). In order to evaluate the relative quantity of EPS from the optical images, images were analyzed using ImageJ software in order to determine the area of the image covered by EPS. For that, the color optical images were first split into the three color channels red, blue and green. As only the EPS are stained by the Alcian blue, both red and green channels were subtracted to the blue channel, to obtain an image of only the blue signal. This image was then binarized: on the binary image, a white pixel corresponds to blue and a dark pixel corresponds to the other colors on the initial image.

The percentage of the area covered by EPS was calculated by extracting the number of white pixels on the binary image compared to the total number of pixels of the image. For AFM imaging, 1 mL of cell suspension was sampled and directly immobilized on polyethylenimine (PEI, Sigma P3143) coated glass slide prepared as previously described [START_REF] Francius | Nanostructure and Nanomechanics of Live Phaeodactylum Tricornutum Morphotypes[END_REF] . Briefly, freshly oxygen activated glass slides were covered by a 0.2% PEI solution in deionized water and left for incubation overnight. Then the glass slides were rinsed with deionized water and dried under nitrogen. A total of 1 mL of cell suspension was then deposited on the PEI slides and allowed to stand for 30 min at room temperature. Images were recorded using the Quantitative Imaging™ mode available on the Nanowizard III AFM (Bruker).

Multiparametric images were recorded with MLCT-BIO cantilevers (Bruker, nominal spring constant of 0.01 N/m) using an applied force of 1.0 nN, a constant approach/retract speed of 100 μm/s, and a zrange of 3 µm.

Results and Discussion

Bicarbonate concentration in the growth medium influences EPS production

Prior to initiating experimental conditions, all cells were cultured and maintained in an inoculum culturing PBR at low light intensity and moderate temperature (30°C), with a bicarbonate concentration of 0.25 mol/L (18.9 g/L), similar to Zarrouck's [START_REF] Zarrouk | Contribution à l'étude d'une cyanophycée: Influence de divers facteurs physiques et chimiques sur la croissance et la photosynthèse de Spiruline maxima (Setch et Gardner) Geitler[END_REF] 1). These concentrations of bicarbonate are lower than the ones found in the literature [START_REF] Trabelsi | Partial Characterization of Extracellular Polysaccharides Produced by Cyanobacterium Arthrospira Platensis[END_REF][START_REF] Trabelsi | Combined Effect of Temperature and Light Intensity on Growth and Extracellular Polymeric Substance Production by the Cyanobacterium Arthrospira Platensis[END_REF][START_REF] Depraetere | Influence of Culture Medium Recycling on the Performance of Arthrospira Platensis Cultures[END_REF][START_REF] Depraetere | Harvesting Carbohydrate-Rich Arthrospira Platensis by Spontaneous Settling[END_REF] . Because A. platensis grows in alkaline waters, the amount of dissolved CO2 is low, and so the bicarbonate is added to be used as a source of carbon, and assimilated due to the energy they capture from light. As bicarbonate ions are consumed by the cells, hydroxide ions are released into the medium, thereby increasing the pH over the course of the culture [START_REF] Miller | Evidence for HCO3-Transport by the Blue-Green Alga (Cyanobacterium) Coccochloris Peniocystis1[END_REF] . To avoid this, and to assess only the role of the bicarbonate concentration on the EPS production, we chose to perform our cultures at a pH of 9, and maintain the bicarbonate concentration constant over 4 days of culture in the experiments PBRs. Towards this end, the pH of the culture medium was measured twice a day, which indicated the quantity of bicarbonate consumed by the cells, which could then be compensated by adding bicarbonate into the medium (the detailed procedure is found in the materials and methods).

Characterization of the EPS produced by A. platensis have been the subject of considerable study [START_REF] Trabelsi | Partial Characterization of Extracellular Polysaccharides Produced by Cyanobacterium Arthrospira Platensis[END_REF][START_REF] Filali Mouhim | Isolation and Preliminary Characterization of the Exopolysaccharide of the Cyanobacterium Spirulina Platensis[END_REF] . Previous studies have shown that the EPS consist of heteropolysaccharides with protein moieties, and that the polysaccharidic fraction is composed of neutral sugars and uronic acids, in different proportions depending on the strains and the culture conditions used. EPS were also shown to form aggregates that could be detected using histological staining with dyes such as Alcian blue that reacts with acidic polysaccharides [START_REF] Reddy | Phenotypic Variation in Exopolysaccharide Production in the Marine, Aerobic Nitrogen-Fixing Unicellular Cyanobacterium Cyanothece Sp[END_REF][START_REF] Shiraishi | Association of Heterotrophic Bacteria with Aggregated Arthrospira Platensis Exopolysaccharides: Implications in the Induction of Axenic Cultures[END_REF] . We selected this technique to qualitatively evaluate the effects of bicarbonate concentration on the production of EPS in samples taken from the three experiments PBRs of culture 1 (1 week in inoculating PBR). Our results, presented in Figure 2, show that no EPS could be detected in cells grown at the lowest bicarbonate concentration (Fig. 2a). However, in the case of cells grown with bicarbonate concentrations of 0.05 and 0.1 mol/L (Fig. 2c andd), aggregated EPS could be detected around cells and the quantity was positively correlated to the concentration of bicarbonate. To relatively quantify the observed differences in EPS concentrations, we developed an original image treatment strategy allowing us to measure on these images the area covered by the EPS. Toward this end, the blue signal only was first extracted from each image: the blue images were then converted into binary images (Fig. 2b, d andf), where white pixels correspond to blue and dark pixels correspond to the other colors on the initial images. Then, knowing the pixel resolution of our images, we could extract the number of white pixels and calculate the percentage of the area covered by the EPS. The results of these analyses, performed in each case on 7 different images for each conditions, are presented in Fig. 2g. At low NaHCO3 concentration, the mean area covered by EPS in the recorded images is of 5.5 ± 2.3%, whereas it increases to 8.2 ± 1.0% at a concentration of 0.05 mol/L, and to 14.1 ± 0.7% at a concentration of 0.1 mol/L (differences are significant at a p-value < 0.05, Mann-Whitney test). These analyses then allow us to quantitatively show the increase in EPS production with the bicarbonate concentration, and to show that their relative quantity depends on the bicarbonate concentration. This observation proves that the bicarbonate concentration in the culture medium influences the production of EPS in our culture conditions, and that the more bicarbonate was present, the more EPS could be produced. This is the first time to our knowledge that this factor is showed to influence EPS production in A. platensis, and more generally in cyanobacteria.

Previous studies on this particular species have showed that factors influencing their production were light, temperature and the cell's trophic mode [START_REF] Trabelsi | Combined Effect of Temperature and Light Intensity on Growth and Extracellular Polymeric Substance Production by the Cyanobacterium Arthrospira Platensis[END_REF][START_REF] Trabelsi | Evaluation of Arthrospira Platensis Extracellular Polymeric Substances Production in Photoautotrophic, Heterotrophic and Mixotrophic Conditions[END_REF] . More globally for cyanobacteria, other factors have been identified such as phosphate, nitrogen and sulfate deprivation, or pH and aeration [START_REF] Pereira | Complexity of Cyanobacterial Exopolysaccharides: Composition, Structures, Inducing Factors and Putative Genes Involved in Their Biosynthesis and Assembly[END_REF] . The positive influence of carbon availability on the EPS production that we observe here could be explained by the fact that cells use the carbon needed for their growth, and, in favorable conditions (high temperature and high light intensity), they use the remaining available carbon to produce EPS. We next wanted to understand the impact of bicarbonate concentration on cell growth.

Indeed, in phosphate or nitrogen deprived conditions that also favor EPS production, cells do not grow and thus a balance must be met between EPS production and biomass. To evaluate if a low bicarbonate concentration also has impacts on the biomass production, we followed the cultures over 4 days using different concentrations, in 3 different cultures. The results, presented in figure 3, show the changes in biomass concentration over time for the 3 different cultures (Fig. 3a, b andc). The qualitative observation of these 3 cultures suggests that the NaHCO3 concentration does not have a significant impact on the growth, even if a small effect can be observed in the case of the first culture. To quantitatively assess this point, the average growth factors were calculated in each case and the histograms are presented in Fig. 3d. We found that the average growth factor obtained for each bicarbonate concentrations over the 3 cultures was similar, thus indicating that even at the lowest bicarbonate concentration, the growth rate of the cells is not affected. Moreover higher NaHCO3 concentrations were also used (up to 0.25 mol/L) which also has no effects on the cell growth rate (data not showed). One interesting note however is that while the growth rate does not vary with the bicarbonate concentration, the final biomass concentration reached decreases in cultures 2 and 3 compared to culture 1. Indeed, in culture 1, the average final biomass concentration is of approximately 1400 mg/L, while in culture 2 it is of 1230 mg/L and of 925 mg/L in culture 3, even if the initial biomass concentrations are almost identical. This can be explained by the time cells spent in the inoculum culturing PBR before being used for experiments (Figure 1). In this PBR, bicarbonate concentration is high (18.9 g/L), but temperature and light intensity are moderate. Previous work by Trabelsi et al. indicated that 30°C with low light intensity still induces the production of EPS by the cells, but at low quantity [START_REF] Trabelsi | Combined Effect of Temperature and Light Intensity on Growth and Extracellular Polymeric Substance Production by the Cyanobacterium Arthrospira Platensis[END_REF] . Thus we can suppose that after one week in this PBR (culture 1), no or low quantity of EPS were produced, but that after 4 (culture 2) and 7 weeks (culture 3), cells could have produced more EPS. Thus, when filtrating this biomass to inoculate the experiments PBRs, EPS are also retained and inoculated. The biomass in fact inoculated is lower, which leads to lower biomass production in the end. Moreover, the presence of EPS in the medium have been shown to inhibit the cell growth [START_REF] Depraetere | Influence of Culture Medium Recycling on the Performance of Arthrospira Platensis Cultures[END_REF] . These two reasons may explain why the final biomass concentration reached in the experiments PBRs decreases with the time the inoculated biomass spent in the inoculum culturing PBR.

Therefore, in the range of concentrations tested here, we show that the more bicarbonate present in the medium, the more EPS are produced by the cells in optimal conditions of light and temperature.

Another important point is that even at low bicarbonate concentration (2.1 g/L), cell growth is unaffected, if this concentration is maintained over the culture. Using higher NaHCO3 concentrations thus only increases the quantity of EPS produced by the cells and has no impact on their growth. But what about the effects of EPS production on harvesting? However in this case, a beige layer can be observed on the filter, which could correspond to the retention of EPS on the filter.

Finally, the efficiencies of harvesting were calculated using equation ( 2) by measuring the optical density in the treated phases; the results are presented in Fig. 4c. We found that in the case of the first culture (1 week in the inoculum preparing PBR), the efficiency at low bicarbonate concentration reaches only 7.4 %, while for the higher concentrations it is of 85.1 % and 97.4 %. Further increasing the NaHCO3 concentration, up to 0.25 mol/L, did not further increase the harvesting efficiency (data not showed). In the case of the first culture, where no EPS coming from the inoculum are present, only the EPS produced during the 4 days culture are present, and thus only their effect on the flocculation/flotation is assessed. It seems thus that the EPS production is directly related to the flocculation/flotation efficiency, as the more EPS produced, the more efficient the harvesting. For

EPS induce A. platensis bio-flocculation

EPS, and more globally all the algal organic matter (AOM) excreted by microalgae cells during their growth that include proteins, polysaccharides, nucleic acids, lipids and small molecules, can be involved in microalgae flocculation. Several studies have showed the influence of AOM on induced flocculation (alkaline flocculation or flocculation induced by chitosan or cationic polymers) in a positive or negative way, as for example demonstrated for Chlorella vulgaris cells [START_REF] Vandamme | Influence of Organic Matter Generated by Chlorella Vulgaris on Five Different Modes of Flocculation[END_REF][START_REF] Beuckels | Influence of Organic Matter on Flocculation of Chlorella Vulgaris by Calcium Phosphate Precipitation[END_REF] . However, the direct role of EPS in bio-flocculation has never been described for any microalgae species. For A. platensis, the only studies reporting cases of bio-flocculation describe that nutrient deprived conditions or high salinity can trigger an accumulation of carbohydrates in cells, which changes the density of the cells and allows their flocculation and settling. It is clear in these studies that no EPS are responsible for the flocculation observed [START_REF] Kim | Harvesting of Spirulina Platensis by Cellular Flotation and Growth Stage Determination[END_REF][START_REF] Markou | Carbohydrates Production and Bio-Flocculation Characteristics in Cultures of Arthrospira (Spirulina) Platensis: Improvements Through Phosphorus Limitation Process[END_REF][START_REF] Toyoshima | A Pilot-Scale Floating Closed Culture System for the Multicellular Cyanobacterium Arthrospira Platensis NIES-39[END_REF][START_REF] Depraetere | Harvesting Carbohydrate-Rich Arthrospira Platensis by Spontaneous Settling[END_REF] . In our study, the cells are not nutrient-deprived or grown at high salinities. As such, cells do not accumulate carbohydrates, therefore the mechanism previously described cannot apply to our case. Thus our hypothesis, given our harvesting results showing the direct link between EPS and flocculation/flotation efficiency, is that the EPS induces bio-flocculation of the cells in our culture conditions, thus enhancing the efficiency of cell separation by flotation.

To verify this hypothesis we performed atomic force microscopy (AFM) imaging experiments on the EPS produced by the cells. We used an advanced imaging technique, called quantitative imaging, in which a bare tip is pushed toward the sample and retracted at high-speed over a matrix of points on the sample. Recorded force curves are then analyzed in different ways, so that correlated images of the structure, adhesion, and mechanics of the samples can be obtained simultaneously [START_REF] Chopinet | Imaging Living Cells Surface and Quantifying Its Properties at High Resolution Using AFM in QI TM Mode[END_REF] with Ym values reported for other types of extra-cellular organic matter from bacterial cells [START_REF] Formosa-Dague | Sticky Matrix: Adhesion Mechanism of the Staphylococcal Polysaccharide Intercellular Adhesin[END_REF] , and demonstrate that EPS from A. platensis form a soft gel-like structure surrounding the cells. Figure 5e is the adhesion image and represents the adhesive behavior of the EPS, which is assessed by measuring the interactions between the tip and the sample for each force curves; bright pixels on the image correspond to a high force interaction meaning that the sample is adhesive. It is then clear on this image that EPS, forming a soft gel layer around the cells, are also adhesive (pixels corresponding to EPS are bright on the image), based on the adhesion values obtained: 66.5 ± 34.4 pN (n = 8059 force curves on 3 isolated EPS aggregates) (Fig. 5f). These values are comparable with other type of extra-cellular polymers [START_REF] Formosa-Dague | Sticky Matrix: Adhesion Mechanism of the Staphylococcal Polysaccharide Intercellular Adhesin[END_REF] . Thus these results validate our hypothesis that EPS are responsible of the bio-flocculation of the cells. Indeed, our nanoscale observations show that EPS form a soft and sticky gel in which the cells are enmeshed, resulting in their flocculation. This is also consistent with the observations from Alcian blue staining that showed that cells are indeed entrapped in EPS aggregates (Fig. 2b andc). We thus describe here an original bio-flocculation mechanism in A. platensis, where the EPS produced by the cells under specific culture conditions stick to the cells and aggregate them in flocs. As a result, microbubbles produced in flotation process trap microalgae flocs, thereby enhancing the harvesting efficiency.

Conclusions

The work presented here identified a new factor influencing the EPS production in A. platensis.

While temperatures superior to 30°C and intense light are needed, the concentration in bicarbonate in the culture medium and thus the availability of carbon, induced the production of EPS by the cells.

In view of our observations within the concentration range tested, the more carbon available, the more EPS could be produced. We then showed that the induced production of EPS had a key role in harvesting. Indeed, they form a soft and adhesive gel that form aggregates in which cells are entrapped and thus bio-flocculated, without requiring any addition of external flocculants. As a result, microbubbles produced in flotation process were trapped in these microalgae flocs, which then allowed for their efficient separation from the water. We provide here culture conditions in which high biomass and EPS production are possible, and for which a sustainable harvesting by flocculation/flotation results in separation efficiencies superior to 90%. These culture conditions can also be adapted for filtration harvesting. If filtration harvesting is preferred, we show that at a low bicarbonate concentration a high biomass production can be reached without production of EPS, which can clog filtration membranes. Further study is now required to characterize the EPS production for A. platensis in the conditions we describe and determine the structure-function relationship of the different fractions of the EPS. More work should also be performed to investigate if this original EPS production and bio-flocculation mechanism can also apply to different species of microalgae that are of industrial interest.

Graphical Abstract

Synopsis

Providing culture conditions that allow both production of an EPS-rich microalgae biomass and efficient flocculation/flotation harvesting to make the production of microalgae-based biopolymers cost-effective

  are HCO3 -concentration and NaHCO3 concentration at the beginning of the culture. ∆[𝐵𝑖𝑐] is the variation of the bicarbonate ions concentration and is the unknown parameter of the equation. 𝑝𝐾𝑎1 = 6.3 and 𝑝𝐾𝑎2 = 10.3 are the two acid dissociation constants of the sodium bicarbonate in a water solution. 𝑝𝐻𝑚 is the measured value of the pH. This equation was solved with the Microsoft Excel solver after every measure and a quantity of NaHCO3 corresponding to the value obtained for ∆[𝐵𝑖𝑐] was added to the culture medium.

Figure 1 .

 1 Figure 1. Schematic representation of the culturing protocol and conditions. A. platensis cells extracted from a maintaining tank where cell growth is low (outdoor lighting and temperature, in a 200 L tank) are cultured in a 6 L PBR in moderate growth conditions. This PBR is diluted every 5 days to an OD (680 nm) of 0.3, before stationary phase is reached. To perform the experiments, cells cultured in the inoculum PBR during 1, 4 or 7 weeks (culture 1, 2 and 3 respectively) are harvested by filtration: 2 g of the cells are then inoculated into the experiments PBR at high growth conditions in culture medium containing different concentrations of NaHCO3.

Figure 2 .

 2 Figure 2. Optical imaging of EPS produced by A. platensis grown at different NaHCO3 concentrations. Optical image of A. platensis cells from culture 1, grown at a NaHCO3 concentration of (a) 0.025 mol/L, (c) 0.05 mol/L and (e) 0.1 mol/L. The blue aggregates correspond to Alcian blue stained polysaccharides. Images in (b), (d) and (f) are the binary images of the blue signals from the images respectively in (a), (c) and (e). Scale-bar corresponds to 50 µm. (g) Histogram showing the average percentage of the area covered by EPS for A. platensis cells cultured at the different NaHCO3 concentrations. The stars indicate that the differences observed are significant at a p-value < 0.05 (Mann-Whitney test).

Figure 3 .

 3 Figure 3. Growth rate of A. platensis under different NaHCO3 concentrations. Biomass concentration as a function of time for (a) Culture 1 (1 week of culture in the inoculum PBR), (b) Culture 2 (4 weeks of culture in the inoculum PBR) and (c) Culture 3 (7 weeks of culture in the inoculum PBR) at a NaHCO3 concentration of 0.025 mol/L (blue line), 0.05 mol/L (red line) and 0.1 mol/L (green line). (d) Histogram showing the growth factor µ (mg/L/h) of the 3 different cultures at the different NaHCO3 concentrations.

cultures 2 and 3 ,

 3 an important point is that at low bicarbonate concentration, the harvesting efficiency is increased in culture 2 to 41.3 % and to 86.6 % in culture 3, whereas the cells in this condition do not produce EPS. As discussed previously, this can be explained by the fact that EPS produced in the inoculum culturing PBR are transferred into the experiments PBRs; their presence enhances flocculation/flotation efficiency, even if they are not produced by the cells directly during the culture in the experiments PBRs. This further shows the direct correlation between the EPS and the flocculation/flotation efficiency, as EPS have a positive influence on the harvesting efficiency. To further show this correlation, we performed filtration experiments in the case of culture 1 (Supp. Data 2). We show that for low-bicarbonate concentrated cells, both the filtered biomass and the filtration flow rate are high. By increasing the bicarbonate concentration in the medium, these two measured parameters decreased. This finding is in accordance with the literature, where the EPS present can clog the membranes used , which results in low filtration efficiency[START_REF] Depraetere | Influence of Culture Medium Recycling on the Performance of Arthrospira Platensis Cultures[END_REF][START_REF] Rossi | Arthrospira Platensis Harvesting with Membranes: Fouling Phenomenon with Limiting and Critical Flux[END_REF] . So while the positive influence of EPS on flocculation/flotation harvesting is here clearly demonstrated, we next wanted to identify the basis of this positive influence.

Figure 4 .

 4 Figure 4. Flocculation/flotation harvesting of A. platensis at different NaHCO3 concentrations. (a) Images extracted from a video recording of the flocculation/flotation of the cells from Culture 1 in a laboratory-scale DAF device. The left flotation-test beaker contains cells grown at a NaHCO3 concentration of 0.025 mol/L, the middle flotation-test beaker contains cells grown at a NaHCO3 concentration of 0.05 mol/L, and the right flotation-test beaker contains cells grown at a NaHCO3

  . Such correlated images of isolated aggregates of EPS produced by A. platensis cells from culture 1, and grown at a bicarbonate concentration of 0.1 mol/L, are presented in figure5. These images show that the EPS produced form a thick layer that reaches 4 µm in height, as shown by the cross-section (Fig 5b)taken along the grey line on the height image presented in Fig.5a. The image in Fig 5c.shows the same EPS structure but from the nanomechanical point of view. To obtain this image, the force curves obtained were analyzed using a theoretical model, the Hertz model[START_REF] Hertz | Ueber Die Beruhrung Fester Elastischer Korper[END_REF] , that allows extracting the Young modulus (Ym) of the sample. Ym refers to the resistance of the sample to the compression exerted by the tip at a given applied force. The lower the Ym value, the softer the sample. The nanomechanical image (Fig.5c) shows that EPS is a rather soft structure (pixels corresponding to EPS are dark on the image), which is further confirmed by the quantification of the Ym values plotted in Fig.5d(n = 4976 force curves on 3 isolated EPS aggregates), yielding values of 19.1 ± 9.8 kPa. These values are consistent

Figure 5 .

 5 Figure 5. AFM multiparametric imaging of A. platensis cells before harvesting. (a) AFM height image of an isolated aggregate of EPS and (b) cross-section taken along the grey line in (a). (c) Corresponding nanomechanical image of the isolated EPS aggregate presented in (a) and (d) histogram presenting the distribution of the Young modulus values obtained on 3 different isolated aggregates of EPS. (e) Corresponding adhesion image of the isolated EPS aggregate presented in (a) and (e) histogram presenting the distribution of the adhesion values obtained on 3 different isolated aggregates EPS. Scale-bars correspond to 5 µm.

Figure 1 .

 1 Figure 1. Schematic representation of the culturing protocol and conditions. A. platensis cells

Figure 2 .

 2 Figure 2. Optical imaging of EPS produced by A. platensis grown at different NaHCO3 concentrations.

Figure 3 .

 3 Figure 3. Growth rate of A. platensis under different NaHCO3 concentrations. Biomass concentration

Figure 4 .

 4 Figure 4. Flocculation/flotation harvesting of A. platensis at different NaHCO3 concentrations. (a)

Figure 5 .

 5 Figure 5. AFM multiparametric imaging of A. platensis cells before harvesting. (a) AFM height image

  medium commonly used to cultivate A.

	platensis. Cells were then filtered and inoculated in three experiments PBR containing culture medium
	with bicarbonate concentrations of 0.025 mol/L (2.1 g/L, PBR 1), 0.05 mol/L (4.2 g/L, PBR2) or 0.1 mol/L
	(8.4 g/L, PBR3) (Figure

Acknowledgements C. F.-D. is a postdoctoral researcher supported by the AgreenSkills fellowship programme, which has received funding from the EU's Seventh Framework Programme under grant agreement No. FP7-609398 (AgreenSkills+contract). This work has been supported by the ANR JCJC project FLOTALG to C. F.-D. (ANR-18-CE43-0001-01) and by the project INSPIRO, co-funded by the EU-FEDER (Fonds Européen de Développement Régional), the Région Occitanie, France and INSA-LISBP.


Dr Yifan Zhu

Dr Jrk Nikhil

Xiaoxing Gerard

Grant Xia

Liyun Stevenson

Shiwang Cao

Christopher Fan

Yun Spadaccini

B Jing

Assouar

Prof Gerard

Dr Xiaoxing Xia

Grant C Stevenson

Liyun Cao

Prof Shiwang Fan

Dr Christopher M Spadaccini

Yun Jing

Badreddine Assouar
email: badreddine.assouar@univ-lorraine.fr

Badreddine Assouar

Grant C Gerard

Prof Stevenson

Jing Yun

Systematic Design and Experimental Demonstration of Transmission-Type Multiplexed Acoustic Meta-holograms

Keywords: multiplexing, acoustic metamaterials, acoustic metasurfaces, acoustic hologram, amplitude and phase modulations, transmitted waves

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

Introduction

Drawing inspiration from their optical counterparts, acoustic holograms have been recently reported [1][2][START_REF] Marzo | Proc. Natl. Acad. Sci[END_REF][START_REF] Hirayama | [END_REF][5][6][7][8][9], and now offer fresh perspectives on a wide-range of applications like advanced sound-field reconstruction, ultrasonic therapy, haptics, and particle manipulation. In this regard, transducer arrays [2][START_REF] Marzo | Proc. Natl. Acad. Sci[END_REF][START_REF] Hirayama | [END_REF], monolithic phase plates [5], and acoustic metamaterials/metasurfaces [6][7][8][9] are often frequent candidates for the generation of holographic images that possess unique features and functionalities. Conventional approaches [1][2][START_REF] Marzo | Proc. Natl. Acad. Sci[END_REF][START_REF] Hirayama | [END_REF][5][6] are largely based on pure phase surface and therefore necessitate iterative computational optimization, such as the Gerchberg-Saxton algorithm [6]. Despite this, such approaches are still accompanied by sub-optimal image quality since the simultaneous modulation of both amplitude and phase is crucial for the reconstruction of high-precision wave fields.

To address this limitation, some recent studies have demonstrated that metasurfaces equipped with precise control over both phase and amplitude could offer a promising route for high-fidelity acoustic holography [7][8][9][10][11]. Metasurfaces [11][12][13][14][15][16][17][18][19][20] are two-dimensional materials of sub-wavelength thickness that enable unconventional wavefront shaping functionalities due to their carefully engineered metamaterial [21][22][23][24][25][26] structural units. In a prior study, it was illustrated that reflection-type lossy acoustic metamaterials (LAM) [8] could facilitate decoupled amplitude and phase modulation (APM) and therefore allow for the realization of high-quality images that were previously unattainable. Further, a major advantage is that a straightforward, deterministic time-reversal approach could be employed for the calculation of the required APM profile, and hence ensures high accuracy and bypasses the need for complex computational optimization.

Meanwhile, the past few years have also witnessed the advent of multiplexed optical metasurface-based holograms which are encoded with several predesigned patterns or functionalities, that can be selectively projected on to image planes based on the helicity [27,28], angle [29], frequency [30,31] or the direction [32] of the incident wave. Similarly, our recent paper [33] put forward the concept for a reflection-type multiplexed acoustic metamaterial (RMAM) that enables high-fidelity frequency dependent holograms for audible sound. Each unit cell of the RMAM comprises multiple Helmholtz resonators (HRs) [34][35][36][START_REF] Kinsler | Fundamentals of Acoustics[END_REF] that are tuned to distinct operating frequencies. These HRs hence possess the geometrical entities that can be utilized to precisely control the amplitude and phase responses at each of these frequencies independently. Although both LAM and RMAM give rise to advanced routes for acoustic holography, these concepts have remained limited to reflective metamaterials in the audible regime. Multiplexing for transmitted wave meta-hologram, on the contrary, holds great potential for a wider range of acoustic wave-based applications as the interference between the hologram and the incident wave is substantially reduced.

Extending prior designs to transmission-based systems, however, is a challenging problem for both optics and acoustics and has hence rendered all prior frequency multiplexed optical [30,31] and acoustical [35] meta-holograms to be of reflection-type. In acoustics, a single HR is insufficient to induce and tailor the required non-trivial local phase shift for transmitted waves and therefore implies more complicated geometries [38]. These are particularly cumbersome to fabricate for higher operating frequencies, where the feature sizes of the micro-structure are much smaller. Additionally, the amplitude modulation in the LAM and RMAM designs is made possible by artificially introducing loss in a controlled manner via leaky slits and absorbing boundaries, that are unfeasible in the case of transmission. New strategies for transmission-type multiplexed metamaterials are thus highly desirable, in order to advance acoustic holography across a broad range of frequencies.

In this paper, we put forward transmission-type multiplexed acoustic meta-holograms (MAMH) and experimentally illustrate their working for both audible sound and ultrasound. The physical mechanism hinges on unit cells that consist of multiple Fabry-Pérot (FP) resonant channels [39,40], whose effective heights correspond to the distinct operating frequencies of interest. Each of these FP resonant channels in turn consists of amplitude and phase modulating sub-structures that allow for precise and simultaneous APM. Furthermore, the amplitude modulation in this case, is made possible by elegantly engineering the inherent dissipation that exists in the sub-wavelength FP channels owing to the thermal and viscous boundary layer effects [41][42][43]. This is highly beneficial for ultrasonic applications, where thermo-viscous losses are very high and are at present deemed extremely undesirable [44,45]. We employ a two-channel version of our unit cell to systematically design and illustrate frequency dependent functionalities that leverage simultaneous APM and significantly increase the degrees of freedom for conventional acoustic wave manipulation. To this end, we demonstrate multiplexed high-quality audio holography at 14 kHz and 17 kHz, and two-channel ultrasonic focusing and frequency dependent ultrasonic annular beam generation at 35 kHz and 42.5 kHz. We choose these two groups of frequencies, not farther apart (viz. f2 ≈ 1.2f1), in order to demonstrate that the strategy that we employed is efficient in decoupling the responses of the two frequencies, even though they are close together. These demonstrations could pave the way for a host of multiplexed and switchable compact devices that could facilitate various applications in room acoustics, audio engineering, virtual reality, and biomedical ultrasound.

Results

Working Principle of the MAMH

Figures 1a,b illustrate the working principle of the MAMH, where the incident wavefront consists of two frequencies, viz., f1 and f2, marked by the green and blue arrows, respectively. For multiplexed transmission holography, when the wave propagates through the MAMH, it must be reshaped such that it projects different predesigned images for f1 and f2, on to an image plane that exists at a given distance in front of it. The desired holographic patterns can be arbitrarily chosen and captured by the acoustic pressure field, denoted here by, Pn(x, y, z), where n corresponds to the operating frequency (n = 1 or 2 in this study). In order to reconstruct this image via a meta-hologram, we first employ the time-reversal (TR) method [8] to calculate the pressure profile, pn(xj, yj), that must exist at the hologram plane. We then design the meta-hologram such that it facilitates an amplitude and phase distribution, (An(TR), φn(TR)), that is consistent with pn(xj, yj) on the hologram plane and therefore satisfies the desired Pn(x, y, z) at the image plane for the n th frequency.

Foremost, Pn(x, y, z) is decomposed into a collection of pixels that are represented in Figure 1a, by pn (xl, yl, zl), where l denotes the pixel number. The pressure at each of the l th pixels can in turn be defined as, pn(xl, yl, zl) ≡ pl ≡ A0lexp(iφ0l), where A0l and φ0l are the predesigned amplitude and phase of a pixel that is located at a point, (xl, yl, zl), on the image plane. In hologram design, we usually set φ0l = 0, which means that the phase distribution of the predesigned image is uniform. Likewise, the hologram plane can be described by a collection of pixels, and the pressure at every j th pixel can be written as, pn(xj, yj) ≡ pj ≡ Ajexp(iφj) with amplitude Aj and phase φj. Since the system obeys time-reversal symmetry, Aj and φj, can be calculated by simply back projecting the information on the image plane as follows, (1) where kn = 2πfn/c0 is the wave number, c0 = 343 m/s is the sound speed in air. N is the total number of image pixels, and the distance, rl , between the image pixel and hologram pixel can be calculated as, rl 2 = (xj-xl) 2 +(yj-yl) 2 +(zj-zl) 2 . Similarly, by employing the values of pj, we can reaffirm its accuracy by analytically calculating the pressure distribution at the image plane through the following formula

(2)

0 0 1 exp[ ( )], N l j n l l l l A p ikr r f = = + å 1 ( , , ) exp[ ( )], M j n n j j j j A P x y z i k r r f = = - - å
where M is the total number of hologram pixels and the distance, rj ,between the spatial point at (x, y, z) and the hologram pixel, (xj, yj, zj), satisfies, rj 2 = (x-x) 2 +(y-yj) 2 +(z-zj) 2 . The calculated values of Aj and φj, obtained from Eq. ( 1), would hence serve as the required values of (An(TR), φn(TR)) and can be arranged as a function of x and y in order to realize multiplexed holography.

It is quite evident from the above, that an MAMH must possess structural units that are endowed with complete control over the amplitude and phase of the transmitted wave, at each of these frequencies independently. Therefore, in what follows, we first delineate the design of our unit cell for the target audio frequencies, f1 = 14 kHz, and f2 = 17 kHz, and then couple this with the aforementioned technique to demonstrate high-precision frequency dependent holography at fn. The latter part of the paper will then put forward the extension of this design to the ultrasonic regime and the subsequent realization of two-channel focusing and annular beam generation.

Unit Cell Design

The MAMH is made up of 3D unit cells with multiple resonant sub-units, as shown in Figure 1c.

Each unit cell has two FP resonant channels [39,40], which are denoted here by C1 and C2 for channels 1 and 2, respectively. The structure of the channels is made by a solid material and the background medium is air. The size of the unit cell is 10 mm × 10 mm × 45 mm and the transverse dimensions are smaller than half the wavelength for both cases (i.e., λ1 = 24.5 mm, λ2 = 20.2 mm). It ensures that only the fundamental mode propagates, independent of the angle of incidence. Further, as illustrated in Figure 1c, the unit cell comprises frequency, amplitude, and phase modulators, viz., the FM, AM, and PM, that are marked in blue, black, and red, respectively. Figure 1d shows the 2D cross-section of C1 and C2, along with the relevant geometrical parameters. C1 is composed of AM and PM regions with heights H1 = 12.5 mm and H2 = 32.5 mm, respectively. The total height of the two regions, H, is therefore the height of the entire unit cell (H = H1 + H2 = 45 mm). The AM is an air-channel on the top that has a width, w1, which can be tailored to be between 3 mm and 8 mm to predominantly modulate the amplitude of the transmitted wave. The PM is a zig-zag channel whose total effective length can be controlled by tuning the parameter l1 between 3 mm and 8 mm and modulates the transmitted phase. The width of this zig-zag channel, w0 = 2.5 mm and its top and bottom are connecting channels with a fixed width of l0 = 7.5 mm. The two parameters (w0, l0) were optimized to make the tunable range of the PM large enough, so as to facilitate a wide span of local phase shifts.

To start with, we first design C1 by matching its resonance frequency to the first desired working frequency, f1 =14 kHz. This is done by choosing the values of H1 and H2 such that its sum satisfies the FP resonance condition for f1. Here, the total height of the unit cell, H, is chosen to be approximately twice the first work working wavelength, i.e., H ≈ 2c0/f1. The C2 is then designed by including a wider air channel, on top of the AM, that matches well with the impedance of the free space. This wider channel is defined as FM, because its height h0 = (1-β)H is changed to choose the second targeted frequency. Here, the β is a factor of reduction that is applied to the heights of the AM and PM via h1 = βH1 and h2 = βH2 respectively and allows us to tailor the frequency at which the FP resonance of C2 occurs. In this case, we search the value of β and set β = 0.8, so that the heights of the FM, AM and PM are h0 = 9 mm, h1 = 10 mm, and h2 = 26 mm, respectively, and the FP resonance for the C2 (which corresponds to h1 + h2) occurs at the second desired working frequency, f2 = 17 kHz. It is also useful to note here that, β ≈ f1/f2, due to the fact that the resonant wavelength of the acoustic meta-materials can be approximately scaled up with the structural size [21][22][23]. For more elaborate analytical and descriptive details about the unit cell, the reader is referenced to Supporting Information Note 1.

The properties of the conceived sub-units were then assessed by numerically calculating the transmission curves for C1 and C2, in the frequency range between 12 and 20 kHz, with w1 = w2 = 7.5 mm, l1 = l2 = 7.5 mm. These results are shown in Figure 1e and illustrate that at f1, the transmission through C1 (0.74) is much higher than that of C2 (0.24). Likewise, at f2, the transmission of C2 (0.73) is much higher than C1 (0.31). Further, Figure 1f shows the acoustic pressure amplitude field through a unit cell made up of these channels, at f1 and f2, respectively.

It can be clearly seen here that when f1 (f2) is incident, the response from C1 (C2) is dominant and that of C2 (C1) results in a very low amplitude background noise. This verifies our theory that the two-channel FP resonance mechanism allows for frequency selective wave transmission with a negligible background noise. Additionally, the transmitted amplitude (A) and phase (φ) for each frequency can be calculated as a function of the different geometrical parameters of C1 and C2, respectively. Figures 2a,b show the various (A1, φ1) and (A2, φ2), that can be achieved by modulating the geometrical parameters (w1, l1) and (w2, l2), for f1 and f2, respectively. It is interesting to note here, that the numerical calculations employed take into consideration the thermo-viscous dissipation that exists in these channels and contribute to the amplitude modulation that is achieved. A design library was then built for these working frequencies, with 4 discrete values of amplitude and 8 of phase. The 4 values of amplitude that were chosen are 0.32, 0.48, 0.64, and 0.8 and were normalized to 0.4, 0.6, 0.8, and 1, respectively. The values of phase were discretized between π/4 and 2π, with a step size of π/4. These 4 × 8 combinations are shown in Figures 2c,d and have (w1, l1) and (w2, l2) dimensions that both lie between 3 mm and 8 mm, with a step size of 0.5 mm between them. Therefore, a MAMH that comprises these unit cells can be conveniently realized via a commercial extrusion-based 3D printer. (For additional details about the amplitude and phase library, the reader is referenced to Supporting Information

High-Fidelity Multiplexed Audio Holography

The aforementioned methods were then employed to design a meta-hologram that could selectively project the letters N and S for f1 = 14 kHz and f2 = 17 kHz, respectively, on to an image plane that sits 5 cm away (zl = 5 cm) from the hologram plane (zj = 0 cm), as shown in Figure 3a. The source employed in our experiments was a loudspeaker of diameter 10 cm that was placed 10 cm behind the sample. This was hence regarded as a point source in the full-wave numerical simulations, and the associated amplitude and phase compensation, (Asource, φsource)

were incorporated in our design to improve its experimental accuracy. This was done by adding a compensation to An(TR) and φn(TR), to obtain the final amplitude, An = An(TR)/Asource and phase, φn = φn(TR)-φsource. Figure 3b shows the amplitude, An(TR), and phase, φn(TR), distributions that were calculated by time-reversal for the holographic rendering of the letters, N and S. The plots adjacent to them, An and φn, represent the corrected values of the same, upon including the source compensation. The final amplitude and phase profiles that were used for the final MAMH, are shown in Figure 3c as a function of x and y. One might note here that the designed images are at two different locations (two adjacent regions). This was done to minimize the coupling between the two working frequencies, and to ensure images with highest fidelity and lowest background noise.

The design library put forward in the previous section was then revisited to obtain the dimensions for a set of 17 × 17 unit cells that could build the desired MAMH sample of size 18 cm × 18 cm × 4.5 cm (including wall thickness). Figure 3d shows the 3D printed prototype of the MAMH that was experimentally characterized by mapping the field over a 20 cm × 20 cm region, at z = 5 cm. Figures 4a, b present the simulated and measured acoustic intensity distributions on the transmissive side of the metamaterial for both frequencies of incidence. The arrows in Figure 4a, indicate the direction of propagation and show the numerically calculated intensity field at z = 4 cm, 5 cm, and 6 cm away from the sample. The letters 'N' and 'S' are distinctly visible for the two cases and are unimpaired when z = 5 cm, proving that the performance of the sample is consistent with our conceived design. Figure 4b also shows that the measured acoustic intensity field at z = 5 cm, agrees very well with the simulation results and validates the ability of the MAMH to generate high-quality frequency dependent holograms, owing to its simultaneous APM and multiple FP resonance-based mechanism. This unique capability could enable new audio devices and also greatly help renovate existing applications like stereo-sound field reconstruction [46]. It is possible that some undesirable effects could arise due to the coupling between the different unit cells and the interaction between the transmitted acoustic waves and the sufficient number of channels. Although the theoretical calculations here did not take this coupling into account, the fact that the numerical results agree well with the theoretical predications is indicative of these interactions being rather trivial in this case.

Frequency Dependent Ultrasonic Holography

The principle and design of our meta-hologram can also be readily extended to operate at ultrasonic frequencies. This is of interest, since multiplexing could greatly benefit ultrasonic applications like therapy and particle manipulation, by circumventing the need for bulky and expensive phased array systems that currently serve as the state of the art for ultrasound beam forming. Besides, the strong intrinsic thermo-viscous loss in ultrasonic metamaterials would facilitate the amplitude modulation that is highly desirable for the MAMH design that is Figure 6a shows the schematic for an MUMH that acts as a two-channel ultrasonic lens [47] which exhibits frequency dependent focusing on an image plane that is 1 cm away (zl = 1 cm).

The predesigned holographic images in this case are that of focal points that exist at two different corners of the image plane i.e. (-1.1 cm, 1.1 cm, 1 cm) and (1.1 cm, -1.1 cm, 1 cm), for f1 and f2, respectively. Figure 6b here shows the required amplitude and phase distributions that were calculated by the method described earlier. To enable this frequency dependent pressure profile, the MUMH sample was designed with an appropriate set of 10 × 10-unit cells and was fabricated by a commercial high-resolution stereolithography based 3D printer. The resultant prototype of size 4.8 cm × 4.8 cm × 1.8 cm (including wall-thickness) is shown in Figure 6c. To experimentally illustrate its working, an air-coupled ultrasonic transducer of diameter 1 cm (center frequency 40 kHz), was placed well-behind the sample (i.e. in the far-field, approx. 5 cm away) such that the wavefront that impinges on the MUMH has a uniform phase distribution and could be captured by a plane wave incidence in the full-wave simulations. The left panel of Figure 6d shows the numerically calculated pressure amplitude distribution on an image plane of size, 4.8 cm × 4.8 cm, situated in front of the sample at z = 1 cm. It can be clearly seen here that the designed MUMH enables frequency dependent wave focusing that is consistent with the predesigned holographic images show in Figure 6a. This is also evident in our experimentally measured results, shown in the right panel of Figure 6d, which agree well with the numerically calculated ones and thereby validates the functionality of our prototype. To further verify the accuracy of the frequency dependent focusing and to compare the simulations with experiments, as shown in Figure 6e, the pressure amplitude along the diagonal (described by x + y = 0, z = +1, shown in the inset) of the image plane was calculated. Here, the x axis denotes the distance between the targeted coordinate and the ordinate (0 cm, 0 cm, 1 cm). The curve calculated from the measurement (squares) has a very reasonable agreement with the numerical results (solid line). Our realization of frequency dependent ultrasonic focusing via a meta-hologram, could greatly aid biomedical applications, since high intensity focused ultrasound is now an emerging candidate for treating different diseases and conditions [48]. Additionally, to demonstrate highly precise wave reconstruction at the small wavelengths associated with 35 kHz and 42.5 kHz, we designed an ultrasonic MUMH that could generate distinctive annular beam patterns [48], that are dependent on the frequency of the incident wave.

The schematic on Figure 7a (left) illustrates the desired MUMH and shows that it would generate a single annular beam which has a center coordinate at (0 cm, 0 cm, 1 cm) when f1 is incident, but when f2 is incident, it would generate two annular beams that have two center coordinates at (-0.6 cm, 0 cm, 1 cm) and (0.6 cm, 0 cm, 1 cm). Figure 7a (right) shows the predesigned images that are taken into consideration in order to project these functionalities on to an image plane at zl = 1 cm. For this case, the amplitude and phase distributions that were calculated via time-reversal (An(TR), φn(TR)), are shown in Figure 7b for f1 (left) and f2 (right),

respectively. The single air-coupled transducer was employed here as the source and was placed 3.5 cm behind the sample to ensure a good signal-to-noise ratio. However, since the incident wavefront here is not planar, amplitude and phase compensations were incorporated to the design, as done in the case of the audio meta-hologram, and the resultant (An, φn), for both frequencies, are shown adjacent to the initially calculated plots in Figure 7b. Figure 7c shows that the numerically predicted acoustic pressure amplitude distribution at the image plane, efficiently captures the details of the predesigned images for both f1 and f2. Figure 7d presents the experimentally measured pressure amplitude fields, that agree well with the numerical results and hence confirms that our ultrasonic meta-hologram generates distinctive channel/frequency dependent annular beams, whose centers lie at the desired coordinates. Such frequency dependent annular beam generation via a MUMH is highly desirable for acoustical tweezer-based particle manipulation [50].

Conclusion

In summary, we designed and experimentally illustrated the working of multiplexed metamaterials for high fidelity acoustic holographic rendering. A new class of unit cells that could simultaneously modulate the amplitude and phase at two different frequencies were put forward and leveraged to build multiplexed meta-holograms that are encoded to project different predesigned images at different frequencies. The resultant meta-holograms not only introduce

Figure 1 .

 1 Figure 1. The schematic diagram of MAMH. (a) The schematic diagram of the hologram plane and image plane. (b) The schematic diagram of the MAMH that can generate two different images at two frequencies f1 and f2. (c) The unit cells consist of two channels (1 and 2) corresponding to two Fabry-Pérot resonant frequencies, f1 and f2. The figures show the 3D structure of the unit cell and the background medium of the air channel. The schematic diagram of the unit cell consisting of three modulators, viz., the frequency modulator (FM), amplitude modulator (AM) and phase modulator (PM) marked by blue, black and red, respectively. (d) The 2D structure of C1 and C2. The parameters are marked in the figure. (e) The transmission curves of C1 and C2 from 12 kHz to 20 kHz with w1 = w2 = 7.5 mm, l1 = l2 = 7.5 mm. At f1=14 kHz (green line), the transmission of C1 is much higher than C2. At f2 = 17 kHz (blue line), the transmission of C2 is much higher than C1. (f) The acoustic pressure amplitude fields in the unit cells at f1 = 14 kHz and f2 = 17 kHz, respectively. The arrows indicate the incident and transmitted directions.

Note 1 )Figure 2 .

 12 Figure 2. Amplitude and phase responses of the unit cell for 14 kHz and 17 kHz. (a) The amplitude and phase responses as a function of the parameters w1 and l1 for f1 = 14 kHz. (b) The 4×8 combinations of (A1, φ1) varying with w1 and l1. (c) The amplitude and phase responses varying with parameters w2 and l2 for f2 = 17 kHz. (d) The 4 × 8 combinations of (A2, φ2) varying with w2 and l2.

Figure 3 .

 3 Figure 3. The holographic designs for letters "N" and "S" at 14 kHz and 17 kHz. (a) The predesigned holographic image at 14 kHz is letter "N". The image plane is 5 cm away (z = 5 cm) from the hologram plane. (b) The amplitude and phase distributions at hologram plane calculated by time reversal are (An(TR), φn(TR)). The final amplitude and phase distributions of the sample are calculated as (An, φn). (c) The final distributions of (A1, φ1) and (A2, φ2). (d) The photograph of the sample with a size of 18 cm × 18 cm × 4.5 cm with 17 × 17 unit cells. Scale bar, 4 cm.

Figure 4 .

 4 Figure 4. Simulated and experimental results of holographic images (letter "N") at 14 kHz and (letter "S") at 17 kHz. (a) The simulated acoustic intensity distributions at the distance of z = 4 cm, z = 5 cm, z = 6 cm, respectively. The arrows indicated the wave is propagating along the z-direction. (b) The measured acoustic intensity distributions at the predesigned image plane of z = 5 cm.

  proposed here. The experimental realization of such an ultrasonic meta-hologram would thus greatly bolster the interest for high precision miniaturized acoustic wave-based devices. Figures5a, bshow the amplitude and phase responses for our unit cell when it is scaled down by a factor of 0.4. The unit cell now has the dimensions 4 mm × 4 mm× 1.8 mm, and the frequencies of incidence are modified to f1 = 35 kHz and f2 = 42.5 kHz. The design library was built as done previously, with 4 × 8 combinations of A1 and φ1 (A2 and φ2) with w1 (w2) and l1 (l2 ) now varying between 1.2 mm and 3.2 mm, with a step size of 0.2 mm, as shown in Figures5c, d. (Additional details in this regard can be found in Supporting Information Notes 1 and 2) This unit cell library was then leveraged to design two multiplexed ultrasonic meta-holograms (MUMH) for the functionalities of frequency dependent focusing and annular beam generation, respectively.

Figure 5 .

 5 Figure 5. Amplitude and phase responses of the unit cell for 35 kHz and 42.5 kHz. (a) The amplitude and phase responses changing with parameters w1 and l1 for f1 = 35 kHz. (b) The 4 × 8 combinations of (A1, φ1) varying with w1 and l1. (c) The amplitude and phase responses varying with parameters w2 and l2 for f2 = 42.5 kHz. (d) The 4 × 8 combinations of (A2, φ2) changing with w2 and l2.

Figure 6 .

 6 Figure 6. The design and demonstration of two-channel ultrasound lens. (a) The schematic diagram of the MUMH for two-channel ultrasound focusing. The image plane is 1 cm away (zl = 1 cm) from the hologram plane. The predesigned holography patterns are two focal points at the coordinate (-1.1 cm, 1.1 cm, 1 cm), and (1.1 cm, -1.1 cm, 1 cm), respectively. (b) Calculated amplitude and phase distributions for 35 kHz and 42.5 kHz, respectively. (c) The photograph of the 3-D printing sample, consisting of 10 × 10 unit cells with a size of 4.8 cm × 4.8 cm × 1.8 cm. Scale bar, 1 cm. (d) The simulated (left panel) and the measured (right panel) acoustic pressure amplitude distributions for 35 kHz and 42.5 kHz, respectively. Scale bar, 1 cm. (e) The simulated (sim., green and blue lines) and experimental (exp., green and blue squares) acoustic pressure amplitude distributions along the diagonal x + y = 0, z = +1 (shown in the inset). The horizontal axis is the distance between the targeted coordinate and the ordinate (0 cm, 0 cm, 1 cm).

Figure 7 .

 7 Figure 7. The design and demonstration of two-channel annular beams. (a) The schematic diagram of the MUMH for annular beams. The image plane is 1 cm away (zl = 1 cm) from the hologram plane. The predesigned holography pattern for f1 = 35 kHz is one annular beam with the center coordinate (0 cm, 0 cm, 1 cm). The predesigned holography pattern for f1 = 42.5 kHz is a pair annular beams with the center coordinates (-0.6 cm, 0 cm, 1 cm) and (0.6 cm, 0 cm, 1 cm). (b) The amplitude and phase distributions at hologram plane calculated by time reversal is (An(TR), φn(TR)). The final amplitude and phase distributions of the sample are calculated as (An, φn). (c) The simulated acoustic pressure amplitude distributions of annular beam in x-y plane for 35 kHz and 42.5 kHz, respectively. Scale bar, 1 cm. (d) The measured acoustic pressure amplitude distributions of annular beam in x-y plane.

Figure 4

 4 Figure 4 are obtained by scanning the output side of the sample with a distance of 5 cm (viz.,

Acknowledgements

This work is supported by the Air Force Office of Scientific Research under award number FA9550-18-1-7021, and by la Région Grand Est and Institut Carnot ICEEL. Y. J. would like to thank NSF for the support through CMMI 1951221. Work at LLNL was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

frequency as a new degree of freedom in metamaterial design, but also enable the generation of high-quality audio and ultrasonic wavefront reconstruction without the need for complex computational optimization. As a proof of concept, we numerically and experimentally demonstrated high-grade multiplexed audio holography, frequency dependent ultrasonic focusing, and two-channel annular beam generation. These demonstrated functionalities are benevolent for a wide-range of acoustic applications and could open avenues to next-generation devices for stereo-sound field reconstruction [46], ultrasonic therapy [48], and particle manipulation [50] based on switchable audio and ultrasonic devices.

Experimental Section

Numerical Simulations: The simulations were performed using the commercial finite element

Conflict of Interest

All authors declare that they have no conflict of interest.

Supporting Information

Supporting Information is available from the Wiley Online Library or from the author.