Zhenyu Zhao

Koenraad Muylaert

Anthony Szymczyk

Ivo F J Vankelecom

Harvesting microalgal biomass using

Keywords: Patterned membrane, Computational fluid dynamics, Cross-flow filtration, XDLVO, Interaction energy, Membrane technology, Membrane development

come

Introduction

Microalgae have gained increasing attention due to their fast growth rate, high CO 2 fixation rate and content of bio-active compounds [START_REF] Chen | Optimization of ultrasonic-assisted extraction, refinement and characterization of water-soluble polysaccharide from Dictyosphaerium sp. and evaluation of antioxidant activity in vitro[END_REF][START_REF] Suparmaniam | Insights into the microalgae cultivation technology and harvesting process for biofuel production: A review[END_REF][START_REF] Zhao | Combustion characteristics and kinetics of five tropic oilgal strains using thermogravimetric analysis[END_REF]Zhao et al., 2020e). Moreover, microalgae can use a variety of waste components as nutrients to produce high-value products and treat wastewater [START_REF] Li | Prescreening, identification and harvesting of microalgae with antibacterial activity[END_REF][START_REF] Shahid | Cultivating microalgae in wastewater for biomass production, pollutant removal, and atmospheric carbon mitigation; a review[END_REF]Zhao et al., 2016b). For further commercialization purposes, technologies are required to reach high microalgal biomass concentrations at low cost and high efficiency. However, harvesting microalgae still remains a problem, because of the small size of the microalgal cells, the high dilution of cells in the broth and the high energy required for conventional harvesting via centrifugation (Zhao et al., 2020c;Zhao et al., 2020d). Membrane filtration processes have been seen as an alternative to harvest microalgae, due to their high microalgae retention, high harvesting efficiency, ease of upscaling and low energy input (Bilad et al., 2012b;Zhao et al., 2020d). A membrane with high flux, high microalgal cell retention and low fouling propensity is the prerequisite to harvest microalgae. However, membrane fouling is always inevitable, which directly lowers membrane flux and requires frequent cleaning cycles. The latter lowers membrane life-time and increases operational costs while creating waste streams (Bilad et al., 2012a;Zhao et al., 2020d).

Increasing the hydrophilicity of the membranes and introducing surface charge are commonly used approaches to prevent fouling (Bilad et al., 2012a;Marbelia et al., 2016a). The adhesion forces between the membrane surface and water increase with increasing membrane hydrophilicity, resulting in a strong adsorbed layer of water molecules on the membrane surface. As a result, less attachment is found between the foulant and membrane surface [START_REF] Zhu | Layer-by-layer assemblies for antibacterial applications[END_REF]. Introducing surface charge is an efficient approach to eliminate charged foulants. When the membrane surface and the foulants have similar charge, electrostatic repulsion can prevent the foulants from attaching to the surface, therefore mitigating fouling and enhancing membrane fluxes [START_REF] Zhong | Positively charged nanofiltration (NF) membranes via UV grafting on sulfonated polyphenylenesulfone (sPPSU) for effective removal of textile dyes from wastewater[END_REF]. The foulants in a microalgal broth mainly come from microalgal cells and extracellular organic matter (EOM). Their charge is normally negative, mainly due to sulphated extracellular polysaccharides, carboxylic acids, uronic acids or acidic sugars on the cell surface and suspended in the broth (Marbelia et al., 2016b;[START_REF] Zaouk | Filterability of exopolysaccharides solutions from the red microalga Porphyridium cruentum by tangential filtration on a polymeric membrane[END_REF]. A membrane with higher hydrophilicity and negative charge may thus present a better anti-fouling performance and higher membrane fluxes. A negatively charged polycarbonate (PC) membrane has thus been used to harvest Chlorella vulgris (C. vulgris), showing higher hydrophilicity, microalgal retention and membrane flux [START_REF] Huang | Layer-by-layer assembly of high negatively charged polycarbonate membranes with robust antifouling property for microalgae harvesting[END_REF]. A negatively charged polyacrylonitrile membrane was used to harvest eight types of microalgae, also showing higher water permeances and less fouling (Marbelia et al., 2016b). Sulfonated polysulfone (sPSf) has been incorporated into membranes to successfully create negative charges and a higher hydrophilicity [START_REF] Song | Polysulfone/sulfonated polysulfone alloy membranes with an improved performance in processing mariculture wastewater[END_REF]Zhao et al., 2020a;[START_REF] Zhao | Harvesting microalgae using vibrating, negatively charged, patterned polysulfone membranes[END_REF].

The physico-chemical interaction between membrane surface and foulants in an aqueous solution can normally be revealed using the extended Derjaguin, Landau, Verwey, Overbeek (XDLVO) theory which contains the attractive Lifshitz-van der Waals (LW), Lewis acid-base (AB) and electrostatic repulsive double layer (EL) interactions [START_REF] Hoek | Extended DLVO interactions between spherical particles and rough surfaces[END_REF][START_REF] Van Oss | Acid-base interfacial interactions in aqueous media[END_REF]. However, this theory can only be used to quantitatively calculate the interaction energy between two infinitely flat surfaces. Many microalgal shapes are spherical or elongated, and many membrane surfaces are rough and/or patterned [START_REF] Lin | A novel approach for quantitative evaluation of the physicochemical interactions between rough membrane surface and sludge foulants in a submerged membrane bioreactor[END_REF]Marbelia et al., 2016b). An improved theory, the Derjaguin approximation, has been developed to quantitatively deal with the physico-chemical interactions between a spherical microalgal cell and a flat membrane surface [START_REF] Wang | Effects of ionic strength on membrane fouling in a membrane bioreactor[END_REF]. The interactions between polyvinylidene fluoride (PVDF)/polyethersulfone membranes and microalgae/extracellular organic matter (EOM) have been successfully characterized (Zhao et al., 2016a), but the interaction between sPSf blend membranes and microalgal cells is still unknown.

In addition to the hydrophilicity and surface charge modifications, surface patterning is considered an emerging method to alleviate fouling due to the unique hydrodynamic flow behavior near the membrane surface and the extended active area [START_REF] Ilyas | Synthesis of patterned PVDF ultrafiltration membranes: Spray-modified non-solvent induced phase separation[END_REF][START_REF] Jung | Transport and deposition of colloidal particles on a patterned membrane surface: Effect of cross-flow velocity and the size ratio of particle to surface pattern[END_REF][START_REF] Marbelia | Preparation of patterned flat-sheet membranes using a modified phase inversion process and advanced casting knife construction techniques[END_REF]Zhao et al., 2020b).

Patterned membranes have been used for waste water treatment and microalgae harvesting, showing higher membrane fluxes than corresponding flat membranes [START_REF] Lee | Flow analysis and fouling on the patterned membrane surface[END_REF]Zhao et al., 2020b;[START_REF] Zhao | Harvesting microalgae using vibrating, negatively charged, patterned polysulfone membranes[END_REF]. The traditional ways for modifying the membrane surface are template-based molding, direct printing (e.g. 3-D printing and etching) and chemical modification [START_REF] Heinz | Surface-patterning of polymeric membranes: fabrication and performance[END_REF], which are always limited by equipment/mold cost, complex parameter control, compressed porosity and/or low pattern fidelity [START_REF] Ma | Controlling biofouling of reverse osmosis membranes through surface modification via grafting patterned polymer brushes[END_REF][START_REF] Marbelia | Preparation of patterned flat-sheet membranes using a modified phase inversion process and advanced casting knife construction techniques[END_REF]. A recently reported one-step method, phase inversion [START_REF] Hołda | Understanding and guiding the phase inversion process for synthesis of solvent resistant nanofiltration membranes[END_REF] by spraying using corrugated casting knives, may solve these problems [START_REF] Marbelia | Preparation of patterned flat-sheet membranes using a modified phase inversion process and advanced casting knife construction techniques[END_REF]. The polymer solution is cast using a modified patterned knife, and non-solvent is simultaneously sprayed on the cast film to rapidly solidify the surface pattern. The membrane prepared using this modified phase inversion method showed a significantly higher clean water permeance (CWP) than when using the traditional phase inversion method [START_REF] Ilyas | Synthesis of patterned PVDF ultrafiltration membranes: Spray-modified non-solvent induced phase separation[END_REF][START_REF] Marbelia | Preparation of patterned flat-sheet membranes using a modified phase inversion process and advanced casting knife construction techniques[END_REF]Zhao et al., 2020b). The anti-fouling property of the patterned membrane is highly related to the pattern shape [START_REF] Choi | Effect of Pattern Shape on the Initial Deposition of Particles in the Aqueous Phase on Patterned Membranes during Crossflow Filtration[END_REF]. However, current studies only focus on one kind of pattern shape, the comparison between different pattern shapes still needs to be further investigated.

In this study, sPSf was used to create negative charges on PSf membranes. The comparison of membranes with different pattern shapes for microalgae harvesting was investigated. The one-step method, phase inversion by spraying, was used to prepare membranes with patterns at their active side. The interaction energies between microalgal cells and the membrane surface were calculated to unravel the mechanism governing the foulant deposition and release in the cross-flow system. The fluid behavior near the membrane surface in a cross-flow system was analyzed using computational fluid dynamics (CFD).

Materials and methods

Microalgae cultivation

Desmodesmus sp. (Zhao et al., 2020e) was cultivated in a 30 L plexiglas column photobioreactor with 25-L BG-11 medium (Zhao et al., 2016b). The cultivation was under continuous illumination with a light intensity of 100 μmol photons m -2 s -1 .

Continuous aeration was kept constant at a rate of 15 L/min. After 10 days of cultivation, 10 mL microalgae broth was filtrated through a 0.1 μm filter to remove the cultivation medium and salts. The filter with microalgal paste was put in an oven under 105 °C until constant weight. The final dry weight of the microalgae was 0.88±0.02 g/L.

sPSf synthesis and characterization

sPSf was synthesized using direct sulfonation according to a previous study [START_REF] Manea | Characterization of polymer blends of polyethersulfone/sulfonated polysulfone and polyethersulfone/sulfonated polyetheretherketone for direct methanol fuel cell applications[END_REF]. Briefly, PSf (25 g) (Solvay, Udel P-1700, Belgium) was completely dissolved in 300 mL of chloroform (Sigma-Aldrich, Belgium). A 10 mL 6M chlorosulfonic acid (99%, Sigma-Aldrich, Belgium) solution was prepared in chloroform and was then added dropwise to the PSf solution while stirring vigorously for 1 h at 0 °C until a viscous precipitate was formed. The supernatant was then removed. The precipitate was washed first using chloroform and then with deionized water until neutral pH. After drying in a vacuum oven for 4 days at 40 °C, sPSf powder was obtained.

The degree of sulfonation (DS) and the ion-exchange capacity (IEC) of the obtained sPSf powder were determined using the back-titration method [START_REF] Manea | Characterization of polymer blends of polyethersulfone/sulfonated polysulfone and polyethersulfone/sulfonated polyetheretherketone for direct methanol fuel cell applications[END_REF][START_REF] Song | Polysulfone/sulfonated polysulfone alloy membranes with an improved performance in processing mariculture wastewater[END_REF]. sPSf powder (0.1 g) was soaked in a 1 M H 2 SO 4 solution for 24 h, then washed using deionized water until neutral pH, and soaked into a 3 M NaCl solution for another 24 h. The NaCl solution was then titrated using a 0.01 M NaOH solution, and using phenolphthalein (5%, Sigma-Aldrich, Belgium) as indicator. The DS and IEC were calculated as follows:

𝐷𝑆 (%) = 0.442𝑐𝑣 𝑀-0.08𝑐𝑣 × 100%
(1)

𝐼𝐸𝐶 (𝑚𝑒𝑞 𝑔 ⁄ ) = 1000𝐷𝑆 442+81𝐷𝑆 ( 2
)
where c is the concentration of NaOH solution (M), v the reacted volume of NaOH solution (mL), M the mass of sPSf (g).

Membrane preparation

Patterned membranes were prepared using four different patterned knives (with a wave, triangle, rectangle or trapezoid pattern) and a spray-modified non-solvent induced phase separation (s-NIPS) method according to a previous study [START_REF] Marbelia | Preparation of patterned flat-sheet membranes using a modified phase inversion process and advanced casting knife construction techniques[END_REF]. Patterned knives were 3D printed using VeroWhitePlusTM RGD835

(Stratasys Ltd, Eden Prairie, USA) as resin material with constant pattern height (h) of 1000 μm and inter-pattern distance (d) of 2000 μm (Fig. 1a). PSf (13.5w%), sPSf (4.5w%) and deionized water (1w%) were dissolved in N-methyl-2-pyrrolidone (99w%, Acros organics, Belgium) at 60 °C with 25w% polyethylene glycol (PEG, Mw~1000 Da, Fluka, Belgium) as pore-forming agent (Zhao et al., 2020b;[START_REF] Zhao | Harvesting microalgae using vibrating, negatively charged, patterned polysulfone membranes[END_REF]. After degassing overnight, the solution was cast on a smooth glass plate with a wet thickness of 200 μm (measured from the valley point of the corrugation to the bottom of the polymer film, below referred to as base-layer) at a casting speed of 2.3 cm/s. Deionized water as non-solvent was sprayed on the cast film simultaneous with the move of the casting knife in order to rapidly initiate the phase inversion process and fix the patterns [START_REF] Hołda | Understanding and guiding the phase inversion process for synthesis of solvent resistant nanofiltration membranes[END_REF][START_REF] Marbelia | Preparation of patterned flat-sheet membranes using a modified phase inversion process and advanced casting knife construction techniques[END_REF]. The cast film was immediately immersed into deionized water (20 °C) to complete the phase inversion [START_REF] Hołda | Understanding and guiding the phase inversion process for synthesis of solvent resistant nanofiltration membranes[END_REF]. Four different patterned membranes were referred to as Wave, Tri, Rec and Trap, respectively (Fig. 1). To investigate the effect of sPSf and s-NIPS on membrane performance and properties, three different non-patterned membranes were prepared with the same wet thickness using a conventional casting knife: (1) one prepared using the s-NIPS method and a casting solution with 13.5w% PSf, 4.5% sPSf (total polymer concentration of 18w%), 25w%

PEG and 1w% deionized water;

(2) one prepared using the conventional non-solvent induced phase separation (NIPS, i.e. by directly immersing the cast film into the coagulation bath without any spraying) using a casting solution with 13.5w% PSf, 4.5% sPSf, 25w% PEG and 1w% deionized water;

(3) one uncharged reference membrane prepared using the conventional NIPS method and a casting solution with 18w% PSf, 25w% PEG and 1w% deionized water. These three flat (F) membranes were referred as F-sNIPS, F-NIPS and F-PSf, respectively.

The casting solutions of all patterned membranes, F-sNIPS and F-NIPS membranes are the same. All patterned membranes and F-sNIPS membrane were prepared using the s-NIPS method, while F-NIPS and F-PSf membranes were prepared using the conventional NIPS method.

Analytical methods

The actual pattern height (H) and inter-pattern distance (D), the base-layer thickness (B) (Fig. 1b), and the cross-sections of patterned membranes were determined and observed using scanning electron microscopy (SEM, JEOL JFC-1300, Tokyo, Japan)..

The contact angles of three probe liquids (MiliQ water, glycerol and diiodomethane)

were determined using a contact angle goniometer (Kruss, Drop Shape Analysis System, Germany). The hydrophilicity of the membrane was characterized using the contact angle measured with MiliQ water. A higher surface tension (lower MiliQ water contact angle) corresponds to a more hydrophilic surface [START_REF] Dobrak-Van Berlo | Parameters determining transport mechanisms through unfilled and silicalite filled PDMS-based membranes and dense PI membranes in solvent resistant nanofiltration: Comparison with pervaporation[END_REF].

The membrane surface chemical composition was characterized using attenuated total reflectance Fourier-transform infrared spectrometer (ATR-FTIR, Bruker Alpha, Germany) with 4000-400 cm -1 scanning range.

The zeta potential (ZP) of the membrane surface was determined by streaming current measurements using an adjustable-gap measuring cell (SurPASS Electrokinetic Analyzer, AntonPaar), following the procedure described before (Marbelia et al., 2016b). Visiolab software was used for data analysis.

Because the surface geometry may influence the measured results, three different flat membranes (F-sPIS, F-NIPS and F-PSf) were used. The property of F-sPIS can reflect the properties of patterned membranes.

The size and morphology of Desmodesmus sp. were determined and observed using an inverted microscope operated through CellSens (IX83, Olympus Corporation, Japan).

The ZP of Desmodesmus sp. was determined using a Nanobrook Omni Zeta Potential Analyzer (Brookhaven Corporate Park, Holtsville, USA), following the procedure described before (Marbelia et al., 2016b). Microalgal ZP was calculated using the Smoluchowski equation.

To determine the contact angle of Desmodesmus sp., fresh mciroalgal broth was filtrated through an F-PSf membrane. The retained microalgae paste on the membrane surface was pressed to form a flat film. The resulted microalgal flat film was then dried at ambient temperature for 24 h to remove surplus water. The microalgal contact angle was also determined using contact angle goniometer and three probe liquids.

Filtration experiment

Dead-end filtration

Membrane fluxes (J m ) of different membranes were determined using a high-throughput dead-end filtration system (HTML, Belgium) [START_REF] Vandezande | Compositional Optimization of Polyimide-Based SEPPI Membranes Using a Genetic Algorithm and High-Throughput Techniques[END_REF] containing 16 membrane coupons with a 1.77 cm 2 filtration active area each.

Before filtrating the microalgal broth, the membrane was first stabilized using deionized water at 1 bar for 20 min to stabilize the membrane. Membrane flux was measured using microalgal broth as the feed under 0.5 bar pressure for 10 min, and was calculated using Eq. ( 1). The CWP was measured using deionized water as feed.

The optical density (OD) was determined to calculate microalgae harvesting efficiency (E) using a spectrophotometer (UV-1800, SHIMADZU, Belgium) at 680 nm and was calculated using Eq. ( 2).

𝐽 𝑚 (𝐿 𝑚 2 ⁄ ℎ 𝑏𝑎𝑟) = 𝑉 𝐴•𝑡•𝑇𝑀𝑃
(1)

𝐸 (%) = 𝑂𝐷 𝑓 -𝑂𝐷 𝑝 𝑂𝐷 𝑓 (2)
where A is the membrane active area (m 2 ); t the filtration time (h); TMP the trans-membrane pressure (bar); OD f the absorbance of the feed; OD p the absorbance of the permeate.

Cross-flow filtration

The filtration performances of different membranes were determined using a pressure-driven HT cross-flow filtration system (HTML, Belgium) which contains 4 membrane samples with 10 cm 2 filtration active area each, and microalgal broth was used as feed. Before filtrating the microalgal broth, the membrane was first stabilized using deionized water with a maximum pump speed of 0.015 m/s at 2.5 bar for 6 h until the CWP leveled off. The patterned membranes were fixed with the pattern lines perpendicular to the flow direction. The critical pressure (CP, bar) was determined using a stepwise method (Bilad et al., 2012a;Zhao et al., 2020b). Briefly, the filtration was carried out at a cross-flow velocity of 0.015 m/s and initialized under a pressure of 1 bar. After every 2 h, the pressure was increased by 0.5 bar until the maximum pressure of 2.5 bar.

The Reynolds number (Re) of the feed flow in the cross-flow system was calculated using Eq. ( 3) with a known broth density of around 1024 kg/m 3 and an apparent viscosity of 0.0018 Pa/s [START_REF] Michels | Effects of shear stress on the microalgae Chaetoceros muelleri[END_REF].

𝑅𝑒 = 𝜌𝑣𝑟 𝜇 ( 3
)
where  is the broth density (kg/m 3 ); v the cross-flow velocity (m/s); r the inner diameter of the tube (m);  the apparent viscosity (Pa/s).

CFD modeling

Flow behavior near the surface of the different membranes was simulated using a commercial fluid dynamics software (Ansys 19.0, Ansys Inc., USA). The colored areas and path lines represent the distribution of velocity and flow direction. Wall shear on the membrane surface was expressed using single colored lines.

Statistic analysis

All experiments were carried out three times, and the results were expressed as mean values±standard deviation. The significance of the results was analyzed using statistical product and service solutions (SPSS) 17.0 and analysis of variance (ANOVA) with least significant digit (LSD) analysis (P<0.05).

Interaction energy calculation

Considering that the surface energy and interfacial forces between the membrane surface and microalgae are related to the membrane and microalgae intrinsic properties, only F-sNIPS, F-NIPS and F-PSf membranes were used for calculation.

F-sNIPS membrane was prepared using the same casting solution and phase inversion method as for the patterned membranes, which can be used to reflect the patterned membranes surface properties.

The total interaction energy (ΔG Tot , mJ/m 2 ) between the membrane surface and microalgae cells in the medium, including LW (ΔG LW , mJ/m 2 ), EL (ΔG EL , mJ/m 2 ) and AB (ΔG AB , mJ/m 2 ) interactions [START_REF] Hoek | Extended DLVO interactions between spherical particles and rough surfaces[END_REF], can be expressed as the XDLVO interaction energy per unit area between two infinite planar surfaces using

Eqs. ( 4)-( 7) [START_REF] Wang | Effects of ionic strength on membrane fouling in a membrane bioreactor[END_REF].

𝛥𝐺 𝑇𝑜𝑡 = 𝛥𝐺 𝐿𝑊 + 𝛥𝐺 𝐸𝐿 + 𝛥𝐺 𝐴𝐵 (4)

𝛥𝐺 𝐿𝑊 = - 𝐴 𝐿 12𝜋𝑙 2
(5)

𝛥𝐺 𝐸𝐿 = 𝜀𝜀 0 𝜅𝜁 𝐴 𝜁 𝑀 [ 𝜁 𝐴 2 +𝜁 𝑀 2 2𝜁 𝐴 𝜁 𝑀 [1 -coth(𝜅𝑙)] + 1 sinh(𝜅𝑙)
] ( 6)

𝛥𝐺 𝐴𝐵 = 𝛥𝐺 𝑙 0 𝐴𝐵 exp( 𝑙 0 -𝑙 𝜆 ) (7)
where l is the separation distance between two infinite planar surfaces (l 0 is the minimum separation distance); A L (= 12π𝑙 0 2 𝛥𝐺 𝑙 0 𝐿𝑊 ) the Hamaker constant; ε (78.5) and ε 0 (8.854×10 -2 CV -1 /m) the dielectric constant of water and the dielectric permittivity of the vacuum, respectively (Zhao et al., 2016a); κ, the inverse Debye length, calculated using Eq. ( 8) [START_REF] Ahmad | Harvesting of microalgal biomass using MF membrane: Kinetic model, CDE model and extended DLVO theory[END_REF][START_REF] Gojkovic | Cryogenic X-ray photoelectron spectroscopy determines surface composition of algal cells and gives insights into their spontaneous sedimentation[END_REF]Wang et al., 2014); ζ A and ζ M are the microalgae and membrane surface ZPs, respectively; λ (0.6 nm) the decay length of AB interaction in aqueous solution [START_REF] Lin | A novel approach for quantitative evaluation of the physicochemical interactions between rough membrane surface and sludge foulants in a submerged membrane bioreactor[END_REF];

𝛥𝐺 𝑙 0 𝐿𝑊 , 𝛥𝐺 𝑙 0 𝐸𝐿 and 𝛥𝐺 𝑙 0 𝐴𝐵 are three interaction components at the minimum separation distance for two planar surfaces, which can be calculated using Eqs.

(9)-( 11).

𝜅 = √ 𝑒 2 ∑ 𝑛 𝑖 𝑧 𝑖 2 𝜀𝜀 0 𝐾𝑇 (8) 𝛥𝐺 𝑙 0 𝐿𝑊 = -2(√𝛾 𝑀 𝐿𝑊 -√𝛾 𝑊 𝐿𝑊 )(√𝛾 𝐴 𝐿𝑊 -√𝛾 𝑊 𝐿𝑊 ) (9) 𝛥𝐺 𝑙 0 𝐸𝐿 = 𝜀𝜀 0 𝜅 2 (𝜁 𝐴 2 + 𝜁 𝑀 2 )[1 -coth(𝜅𝑙 0 ) + 2𝜁 𝐴 𝜁 𝑀 𝜁 𝐴 2 +𝜁 𝑀 2 csch(𝜅𝑙 0 )] (10)
𝛥𝐺 𝑙 0 𝐴𝐵 = 2[√𝛾 𝑊 + ( √ 𝛾 𝐴 -+ √ 𝛾 𝑀 --√ 𝛾 𝑊 -) + √ 𝛾 𝑊 -(√𝛾 𝐴 + + √𝛾 𝑀 + -√𝛾 𝑊 + ) -√𝛾 𝐴 -𝛾 𝑀 + - √𝛾 𝐴 + 𝛾 𝑀 -] (11)
where the subscripts A, W and M represent microalgae, water and membrane, respectively; γ the surface tension parameter; γ LW , γ + and γ -the LW component, electron-acceptor and electron-donor parameters, respectively; T the absolute temperature; e the electron charge; n i the number concentration of ion i in microalgal broth; z i the valence of ion i; K the Boltzmann's constant.

To calculate three energy components (LW, EL and AB), the surface tension parameters/components (γ) should be determined. The total tension parameter (γ Tot ), including LW (γ LW ) and AB (γ AB ) components [START_REF] Van Oss | The role of van der Waals forces and hydrogen bonds in "hydrophobic interactions" between biopolymers and low energy surfaces[END_REF], can be calculated using Eq. ( 12), which can also be calculated using the extended Young equation (EYE) by testing the contact angle using three probe liquids [START_REF] Li | Comparison of the Effects of Extracellular and Intracellular Organic Matter Extracted From Microcystis aeruginosa on Ultrafiltration Membrane Fouling: Dynamics and Mechanisms[END_REF].

The EYE can be expressed as Eq. ( 13) [START_REF] Huang | Understanding the fouling of algogenic organic matter in microfiltration using membrane-foulant interaction energy analysis: Effects of organic hydrophobicity[END_REF]:

𝛾 𝑇𝑜𝑡 = 𝛾 𝐿𝑊 + 𝛾 𝐴𝐵 (12)
(1 + cos 𝜃)𝛾 𝐿 𝑇𝑜𝑡 = 2(√𝛾 𝑆 𝐿𝑊 𝛾 𝐿 𝐿𝑊 + √𝛾 𝑆 + 𝛾 𝐿 -+ √𝛾 𝑆 -𝛾 𝐿 + ) (13)

𝛾 𝐴𝐵 = 2√𝛾 -𝛾 + (14)
where the subscripts S and L represent the solid (i.e. membrane and microalgae)

surface and the liquid, respectively.

Eqs. ( 9)-( 11) can only allow for two planar surfaces, while Desmodesmus sp. is spherical according to the microscope observation, and the cell size ranges from 5.1-7.0 μm. The Derjaguin approximation (DA) was thus used to calculate the interaction energy components (𝑈 𝑠𝑝ℎ 𝐿𝑊 (𝑙), 𝑈 𝑠𝑝ℎ 𝐸𝐿 (𝑙) and 𝑈 𝑠𝑝ℎ 𝐴𝐵 (𝑙)) between a spherical surface (mciroalgal cell) and planar surface (membrane) [START_REF] Derjaguin | Untersuchungen über die Reibung und Adhäsion, IV[END_REF]. The calculation can be expressed using Eqs. ( 15)-( 18). 𝑈 𝑠𝑝ℎ 𝐿𝑊 (𝑙 0 ), 𝑈 𝑠𝑝ℎ 𝐸𝐿 (𝑙 0 ) and 𝑈 𝑠𝑝ℎ 𝐴𝐵 (𝑙 0 )

represent three interaction energy components at the minimum separation distance for spherical and planar surfaces.

𝑈 𝑠𝑝ℎ 𝑇𝑜𝑡 (𝑙) = 𝑈 𝑠𝑝ℎ 𝐿𝑊 (𝑙) + 𝑈 𝑠𝑝ℎ 𝐸𝐿 (𝑙) + 𝑈 𝑠𝑝ℎ 𝐴𝐵 (𝑙) (15) 𝑈 𝑠𝑝ℎ 𝐿𝑊 (𝑙) = 2𝜋𝛥𝐺 𝑙 0 𝐿𝑊 𝑙 0 2 𝑎 𝐴 𝑙 (16) 𝑈 𝑠𝑝ℎ 𝐸𝐿 (𝑙) = 𝜋𝜀𝜀 0 𝑎 𝐴 [2𝜁 𝐴 𝜁 𝑀 ln ( 1+𝑒 -𝜅𝑙 1-𝑒 -𝜅𝑙 ) + (𝜁 𝐴 2 + 𝜁 𝑀 2 ) ln(1 -𝑒 -2𝜅𝑙 )] (17)
𝑈 𝑠𝑝ℎ 𝐴𝐵 (𝑙) = 2𝜋𝑎 𝐴 𝜆𝛥𝐺 𝑙 0 𝐴𝐵 exp ( 𝑙 0 -𝑙 𝜆 ) (18)
where 𝑈 𝑠𝑝ℎ 𝑇𝑜𝑡 (𝑙) is the modified total interaction energy; a A the mciroalgal cell radius.

Results and discussion

Characterization of membranes and microalgae

The DS and IEC of the resulting sPSf in current study are 71% and 1.4 meq/g, respectively, indicating a high degree of sulfonation and a relatively high hydrophilicity [START_REF] Song | Polysulfone/sulfonated polysulfone alloy membranes with an improved performance in processing mariculture wastewater[END_REF]. sPSf with 71% DS is not water-soluble, preventing sPSf from leaching out from the membrane during phase inversion and filtration. To verify the successful incorporation of sPSf in membranes, FTIR was used. The spectrum (Fig. 2) shows a small characteristic peak at ~1041 cm -1 , which corresponds to the symmetric stretch vibration of the S=O bond in the sulfonic acid group, indicating the successful incorporation of sPSf in the blend membrane.

The cross-sectional morphologies of the patterned membranes are shown in Fig. 3, showing a typical asymmetric structure with finger-like macrovoids with a remarkable triple-layered structure. The dense skin layer is thicker in the valley than at the apex (Fig. 3), indicating that the fluxes at the apex may be higher than in the valley. The possible reason is that the phase inversion process in the valleys is different from that at the apex as out-diffusing solvent mixes with a smaller amount of non-solvent, which is present and stagnant in the valley than on top of the patterns (Fig. 4). As the spraying nozzle covers a width of around 6 cm, and the volume of sprayed water is 3.6 mL/s at a casting speed of 2.5 cm/s. The theoretical corresponding sprayed non-solvent thickness is then about 660 μm (assuming a flat surface). This amount of water is enough for the initial solidification. However, as the volume of water in the valley is less than on the apex, the skin layer properties change along the exact position on the waves [START_REF] Hołda | Understanding and guiding the phase inversion process for synthesis of solvent resistant nanofiltration membranes[END_REF]. A similar result was found in a previous study (Zhao et al., 2020b).

After introducing sPSf into the membrane, the membrane surface hydrophilicity and negative charge increased significantly (Table 1) (i.e. the water contact angles and ZPs of the F-sNIPS and F-NIPS membranes were lower than those of the F-PSf membrane), indicating the antifouling potential for microalgae harvesting.

Interfacial Gibbs free energies at a minimum separation distance between membrane surface and microalgal cells were calculated based on the surface tension data from Table1 and are shown in Table 2. The contribution of EL free energy is normally a magnitude lower than that of LW and AB free energies and can be considered as negligible [START_REF] Cao | Understanding particle deposition kinetics on NF membranes: A focus on micro-beads and membrane interactions at different environmental conditions[END_REF]Zhao et al., 2016a). The total interfacial Gibbs free energy (𝛥𝐺 𝑙 0 𝑇𝑜𝑡 ) was therefore calculated by the sum of 𝛥𝐺 𝑙 0 𝐿𝑊 and 𝛥𝐺 𝑙 0 𝐴𝐵 .

A negative value of 𝛥𝐺 𝑙 0 𝑇𝑜𝑡 represents an attractive interaction, on the contrary a repulsive interaction [START_REF] Li | Comparison of the Effects of Extracellular and Intracellular Organic Matter Extracted From Microcystis aeruginosa on Ultrafiltration Membrane Fouling: Dynamics and Mechanisms[END_REF]. In addition, a higher absolute value of 𝛥𝐺 𝑙 0 𝑇𝑜𝑡 means a higher attractive strength between two surface [START_REF] Li | Comparison of the Effects of Extracellular and Intracellular Organic Matter Extracted From Microcystis aeruginosa on Ultrafiltration Membrane Fouling: Dynamics and Mechanisms[END_REF]. From Table 2, negative values of 𝛥𝐺 𝑙 0 𝑇𝑜𝑡 indicate an attractive interaction between membrane and microalgal cells as well as microalgal cells themselves. The most negative value is found between microalgal cells (-67.7), indicating that microalgal cells tend to aggregate more and create fouling. By adding sPSf and using the s-NIPS method, the lowest negative value of 𝛥𝐺 𝑙 0 𝑇𝑜𝑡 is found between F-sNIPS membrane and microalgal cells (-33.6), indicating an alleviated membrane fouling. The surface property of F-sPIS can be approximately considered as the same as for patterned membranes made from the same casting solutions and the s-NIPS method, indicating reduced fouling even though without unique flow behavior on the patterned membrane surface.

Effect of the pattern shape on membrane performance

To investigate the effects of pattern shape, the membranes were prepared using four different patterned knives (wave, triangle, rectangle and trapezoid) with fixed PSf, sPSf, PEG and deionized water concentrations of 13.5w%, 4.5w%, 25w% and 1w% in the membrane casting solutions, respectively. Three flat membranes (F-sNIPS, F-NIPS and F-PSf) with the same wet casting thickness were prepared for comparison.

Membranes were first potted in a dead-end filtration system to determine the CWP and the membrane flux in a microalgal broth (Fig. 5). The patterned membranes were more water-permeable compared with the flat membranes (Fig. 5a), in line with previous studies [START_REF] Marbelia | Preparation of patterned flat-sheet membranes using a modified phase inversion process and advanced casting knife construction techniques[END_REF]Zhao et al., 2020b;[START_REF] Zhao | Harvesting microalgae using vibrating, negatively charged, patterned polysulfone membranes[END_REF]. The flat membranes prepared by adding 4.5w% sPSF showed a higher CWP. Incorporation of sPSf improves the affinity between water and the membrane surface (i.e. higher hydrophilicity, Table 1) by creating more hydrogen bonds and enhancing polar interactions [START_REF] Luo | Molecular interaction between acidic sPPSU and basic HPEI polymers and its effects on membrane formation for ultrafiltration[END_REF]. In addition, the porosity, mean pore size and hydrophilicity also increase, therefore facilitating transport of water [START_REF] Song | Polysulfone/sulfonated polysulfone alloy membranes with an improved performance in processing mariculture wastewater[END_REF]Zhao et al., 2020a;[START_REF] Zhao | Harvesting microalgae using vibrating, negatively charged, patterned polysulfone membranes[END_REF]. The CWP of the F-sNIPS membrane was significantly (P<0.05) higher than that of F-NIPS, indicating that mere use of the s-NIPS method did increase the membrane permeability. Similar results were explained in a previous study [START_REF] Marbelia | Preparation of patterned flat-sheet membranes using a modified phase inversion process and advanced casting knife construction techniques[END_REF]. All these results showed that the high CWP of the patterned membrane was not only caused by the extended surface area and the introduction of charges, but can also be partly intrinsically ascribed to the s-NIPS method.

A similar trend was found when harvesting microalgae (Fig. 5b). The harvesting efficiency of each membrane was higher than 99%, validating the feasibility of harvesting microalgal biomass. Interestingly, while the CWPs of F-NIPS and F-PSf had no significant difference (P>0.05), the membrane fluxes of these two membranes in the microalgal broth showed significant differences (P<0.05). This can be linked to the fact that most of the microalgal cells are surrounded by carboxylic acids, uronic acids or acidic sugars, hence carrying negative charges that are rejected by the membrane surface charge (Marbelia et al., 2016b;Zhao et al., 2020a). Membrane flux can thus effectively be enhanced when less microalgal cells and/or EOM deposit on the membrane surface. For the patterned membranes, the Wave-membrane showed the highest CWP and membrane flux in the microalgal broth, followed by the Tri,

Trap and Rec membranes. The patterned and flat membranes were potted in a cross-flow filtration system to investigate the effect of pattern shape on filtration performance (Fig. 6). The anti-fouling property of each membrane was evaluated using CP. With step-wise increasing the pressure, the fouling on the membrane surface became more significant. A higher CP therefore represents a higher anti-fouling capacity. The experimental limit was at 2.5 bar. Compared with the F-PSf membrane (CP=2.5 bar), all other membranes showed a CP higher than 2.5 bar, thus proving the efficiency of incorporating sPSf as anti-fouling strategy. The filtration curves of Tri and Trap membranes almost overlap, which is in line with the results in the dead-end filtration system. These two shapes thus rendered similar filtration performance. The filtration curves of all patterned membranes are situated above those of the flat membranes. Even after 8-h filtration, they still can give a relatively high membrane flux. For the patterned membranes, more microalgal cells were deposited in the valley regions rather than on the ridges, probably due to the enhanced wall shear on the ridge regions or the different local fluxes [START_REF] Marbelia | Preparation of patterned flat-sheet membranes using a modified phase inversion process and advanced casting knife construction techniques[END_REF][START_REF] Maruf | Critical flux of surface-patterned ultrafiltration membranes during cross-flow filtration of colloidal particles[END_REF]Zhao et al., 2020b). For the flat membranes, the cake layer can easily cover the whole membrane surface due to the constant flow behavior on the membrane (Zhao et al., 2020b;[START_REF] Zhao | Harvesting microalgae using vibrating, negatively charged, patterned polysulfone membranes[END_REF]. The filtration curve of the Wave-membrane is situated above that of the other membranes, indicating that the wave shape does significantly improve the filtration performance.

These numbers are all much higher than in literature, even though broth conditions will surely be slightly different. A flat PSf membrane used in the cross-flow system for C. pyrenoidosa harvesting, showed a maximum CWP of 270 L/m 2 h bar and a broth flux of 92 L/m 2 h bar [START_REF] Sun | Microalgae filtration by UF membranes: influence of three membrane materials[END_REF]. A negatively charged PC membrane was used for C. vulgaris harvesting, showing a maximum membrane flux in a microalgal broth of 133 L/m 2 h bar [START_REF] Huang | Layer-by-layer assembly of high negatively charged polycarbonate membranes with robust antifouling property for microalgae harvesting[END_REF]. A recent study using flat PVDF membranes without surface pattern in tilted panel and axial vibration systems to harvest Euglena sp. and C. pyrenoidosa showed membrane permeances of 724 and 640 L/m 2 h bar, respectively [START_REF] Lau | Effect of membrane properties on tilted panel performance of microalgae biomass filtration for biofuel feedstock[END_REF]Zhao et al., 2016a). Even with enhanced shear near the membrane surface (i.e. 14512 s -1 ) and lower microalgal concentration (i.e. 0.6 g/L of Euglena sp. and 0.55 g/L of C. pyrenoidosa), the membrane permeances were still much lower than those reported here using negatively charged membranes with a wave pattern. All negatively charged patterned membranes in current study showed a significantly higher membrane permeance in both microalgal broth (up to 1000 L/m 2 h bar) and CWP (up to 2250 L/m 2 h bar),

indicating an important advantage of the synergy between surface charge and pattern.

A previous study using negatively charged PSf patterned membrane to harvest C.

vulgaris also showed a huge improvement in membrane flux upon charge creation [START_REF] Zhao | Harvesting microalgae using vibrating, negatively charged, patterned polysulfone membranes[END_REF].

Interaction energy analysis

The profiles of interaction energies between microalgal cells and three flat membranes as a function of the separation distance are shown in Fig. 7. A positive value represents a repulsive energy; a negative value an attractive energy. The interaction energy curves of F-sNIPS and F-NIPS almost overlap, indicating a similar interaction energy with microagal cells for these two membranes, as could be expected. This confirms that phase inversion does not change the membrane surface property. With decreasing the distance between microalgal cells and membrane surface, repulsive electrostatic and attractive Van der Waals energies all increase (Figs. 7b and7d). The total interaction energy profile shows an actual microalgal adhesion energy barrier on the membrane surface that the microalgal cells must overcome an energy barrier [START_REF] Wang | Effects of ionic strength on membrane fouling in a membrane bioreactor[END_REF]. Therefore, a higher repulsive energy barrier represents that only a small part of microalgal cells can eventually adhere on the membrane surface even though the interaction energy at minimum separation distance is highly negative. The energy barriers significantly increase after incorporating sPSf in the membranes, while s-NIPS method does not give significant improvement (Fig. 7a). Similar interaction free energies trends were also found in a previous study [START_REF] Wang | Effects of ionic strength on membrane fouling in a membrane bioreactor[END_REF], where the interaction energy between a PVDF membrane and sludge flocs (both surface charges are negative) also showed attractive LW and AB energies, a repulsive EL energy and a high energy barrier. Analogously, the interaction energy properties of the F-sNIPS membrane can approximately represent these of the patterned membranes as well.

CFD analysis of the flow regime near the membrane surface

A velocity path line profile near the membrane surface was obtained using CFD modeling (Fig. 8). The red color represents a higher velocity, and the blue color a lower. For patterned membranes, the velocity at the ridge region is higher than in the valley region, confirming the results in 3.2 that two different flows occurred near the membrane surface leading to a high-velocity on the ridge and a low-velocity in the valley. Small vortices were found in the bottom of the valley regions, which can alleviate microalgal cell deposition by mixing and scouring effects [START_REF] Ma | Controlling biofouling of reverse osmosis membranes through surface modification via grafting patterned polymer brushes[END_REF].

A linear streamline is found near the flat membrane surface with a relatively low velocity, confirming a constant flow behavior. The results are in line with previous studies in which activated sludge particle and microalgal cells depositions were mitigated by using patterned membranes compared with the corresponding flat membranes under low Re number conditions [START_REF] Lee | Flow analysis and fouling on the patterned membrane surface[END_REF]Zhao et al., 2020b).

A contour line profile was used to present the wall shear on the membrane surface (Fig. 9). The red color represents a higher wall shear, and the blue color a lower. With a higher wall shear, the fouling can be better alleviated [START_REF] Du | Shear stress in a pressure-driven membrane system and its impact on membrane fouling from a hydrodynamic condition perspective: a review[END_REF]. Like the velocity profile, the wall shear at the ridge region is higher than in the valley region, confirming the results in 3.2 that an enhanced wall shear at the ridge region can give patterned membranes more channels to pass water. The wall shear on the flat membrane surface is too low to show, indicating that the wall shear on the flat membrane surface may be not high enough to remove the fouling, resulting in a serious cake layer on the flat membrane surface and a lower membrane flux. Similar results were found in previous studies, in which sludge particles and microalgal cells were detached from the ridge of the patterned membranes with enhanced shear, while a flat membrane surface was fully covered by sludge and microalgal cells [START_REF] Lee | Flow analysis and fouling on the patterned membrane surface[END_REF]Zhao et al., 2020b).

The CFD modeling clearly shows the advantage of using patterned membranes for harvesting microalgae. However, no significant difference is found between different pattern shapes, indicating that flow behavior may not be the main reason which influences the filtration performance.

Combing the results from 3.1 and 3.2, the patterns in the flow channel act as turbulence promotors, which can generate high velocity and wall shear on the ridge, and vortices in the valley. This can hinder microalgal cell attachment on the membrane surface, and in turn reduce fouling [START_REF] Du | Shear stress in a pressure-driven membrane system and its impact on membrane fouling from a hydrodynamic condition perspective: a review[END_REF]. In addition, the wave patterns give a relatively larger un-fouled contact area in microalgal broth, which allowed more clean water to pass through, in turn resulting in a higher membrane flux and CP.

Conclusions

The effects of pattern shape on the morphology and filtration performance of negatively charged PSf membranes was investigated to harvest microalgal biomass.

By adding sPSf to the PSf membrane, the hydrophilicity and surface charge were enhanced. In cross-flow filtration, all patterned membranes showed a higher CP, and wave-patterned membranes showed the highest CWP and flux in microalgal broths with 100% harvesting efficiency.

The mechanism of alleviating fouling for patterned membranes to harvest microalgae was unraveled based on the membrane-cell interaction energy using the improved XDLVO theory, showing a higher repulsive energy when incorporating sPSf, thereby proving the fouling alleviating property of sPSf containing membranes.

CFD modeling showed a higher velocity and shear stress on the apexes, indicating that the patterns can provide areas with less-fouling to allow more water passage. Tables Table 1 Contact

Figures

  Figures

Fig. 1 Fig. 4

 14 Fig. 1 Schematic diagram (a) of four different 3D-printed patterned knives cast

Fig. 6 Fig. 7

 67 Fig. 6 Microalgal filtration using the stepwise method in a cross-flow system.

Fig. 8 Fig. 9

 89 Fig. 8 Velocity streamline profiles simulated by CFD modeling for the (a) Wave,

angle, zeta potential and surface tension data of membranes and Desmodesmus sp..

  θ wat , θ gly , θ dii are the contact angles determined using Milli-Q water, glycerol and diiodomethane, respectively.

	Sample	Contact angle (°)		Surface tension (mJ/m 2 )	ZP (mV)
		θ wat a	θ gly	a	θ dii a	γ LW	γ -	γ +	γ AB γ TOL
	F-sNIPS 57.3±1.7 60.3±0.7 29.4±0.9	44.5 25.8 0.02 1.4 45.9	-95.8
	F-NIPS 66.0±0.8 68.7±0.3 34.9±0.4	42.1 20.9 0.2	4.1 46.2	-95.8
	F-PSf	86.0±0.8 88.7±0.3 44.9±0.4	37.1 11.0 1.5	8.1 45.2	-52.6
	Algae	83.2±2.5 64.7±1.1 43.8±1.8	37.6 0.04 4.8	0.9 38.5	-18.0
	Graphical abstract

a

Table 2 Interfacial Gibbs free energies (mJ/m 2 ) at a minimum separation distance.

 2

	Sample	𝛥𝐺 𝑙 0 𝐿𝑊	𝛥𝐺 𝑙 0 𝐴𝐵	𝛥𝐺 𝑙 0 𝑇𝑜𝑡
	F-sNIPS	-5.8	-27.8	-33.6
	F-NIPS	-5.3	-31.0	-36.3
	F-PSf	-4.1	-38.7	-42.8
	Algae	-4.3	-63.4	-67.7

Acknowledgements

Zhenyu Zhao acknowledges the China Scholarship Council for financial support.

Declaration of interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


Richard Dusséaux
email: richard.dusseaux@latmos.ipsl.fr

Saddek Afifi
email: saddekafifi@yahoo.fr

Keywords: Multilook intensity ratio, layered rough surfaces, scattering amplitudes, small perturbation method. I

 

investigated under the hypothesis of multivariate Gaussian and K-distribution models for the underlying complex scattering amplitudes [START_REF] Kong | Identification of terrain cover using the optimum polarimetric classifier[END_REF]- [START_REF] Lee | Polarimetric Radar Imaging: From Basics to Applications[END_REF]. The derivations presented in [START_REF] Kong | Identification of terrain cover using the optimum polarimetric classifier[END_REF]- [START_REF] Lee | Polarimetric Radar Imaging: From Basics to Applications[END_REF] are relevant but the connection with an electromagnetic model is not made [START_REF] Gardashov | Determination of the distribution of the number of specular points of a random cylindrical homogeneous Gaussian surface[END_REF]- [START_REF] Chen | Polarimetric simulations of SAR at L-band over bare soil using scattering matrices of random rough surfaces from numerical threedimensional solutions of Maxwell equations[END_REF].

In [START_REF] Afifi | On the co-polarized scattered intensity ratio of rough layered surfaces: The probability law derived from the small perturbation method[END_REF], we derive the closed-form expression for the probability distribution for the scattered intensity ratio for a stratified medium under a monochromatic plane wave illumination. The scattered intensities are derived from the first-order small perturbation method and we assumed slightly rough interfaces of infinite extent and centered Gaussian height distributions. For single-look signatures, the co-and cross-polarized intensity ratios follow heavy-tailed distributions which don't have finite mean and variance. In [START_REF] Afifi | The coand cross-polarized scattered intensity ratios for 3D layered structures with randomly rough interfaces[END_REF], we showed that the statistics of the co-and crosspolarized intensity ratios allow differentiating a stratification air/clayey soil/rock with or without snow cover. In [START_REF] Afifi | Statistical distribution of the Normalized Difference Polarization Index for 3-D layered structures with slightly rough interfaces[END_REF], we derive the statistics for the Normalized Difference Polarization Index (NDPI) which is defined as the ratio of the difference between two intensities under different polarizations on the sum of these two intensities. The NDPI therefore takes values between -1 and 1. Contrary to the intensity ratio, the first and second order statistical moment values of this discriminator are finite.

In the present paper, we are interested in the statistical properties of the co-and cross-polarized intensity ratios but for multilook configurations. We derive the theoretical expressions for the probability density function (PDF) and the cumulative distribution function (CDF) for a stratified medium bounded by random slightly rough interfaces under a plane wave illumination. The scattered intensities are given by the first-order SPM [START_REF] Afifi | Scattering by anisotropic rough layered 2D interfaces[END_REF]- [START_REF] Afifi | Electromagnetic scattering from 3D layered structures with randomly rough interfaces: Analysis with the small perturbation method and the small slope approximation[END_REF]. For electromagnetic signatures based on more than two looks, we show that the probability distribution has finite mean and variance. To our knowledge, it is the first time that the analytical expressions for the mean and variance and the CDF are found.

This paper is organized as follows. Section II is dedicated to the statistical properties of the stratified medium. Section III is devoted to the analytical expressions derived from the firstorder SPM for the scattering amplitudes and the scattered intensities. In section IV and appendix, we derive closed-form expressions for the PDF and CDF and for the mean and variance. In Section V, the theory is verified by comparison with Monte-Carlo results. ˆ( , ) () () exp 88

x x y y yy xx R q l l l l ll ll                 (2)
The coefficient q is a mixing parameter with 1 q  [3], [START_REF] Afifi | The coand cross-polarized scattered intensity ratios for 3D layered structures with randomly rough interfaces[END_REF]. If 0 q  , both rough interfaces are uncorrelated, otherwise, they are fully or partially correlated.

A monochromatic h-or v-polarized plane wave of wavelength  impinges on the structure. The incident direction is defined by the wave vector 0 k and the angles 0  and 0  . The structure is composed of three different regions characterized by an isotropic and homogeneous permittivity. The top and bottom regions are half-spaces. The permeability of all regions is assumed to be 0  .

III. SCATTERED INTENSITY AS A RANDOM PROCESS

For an observation direction defined by the angles  and  (see Fig. 1), the first-order amplitudes of the co-and crosspolarized contributions of the field scattered in the upper region are derived from the SPM as follows [START_REF] Afifi | Scattering by anisotropic rough layered 2D interfaces[END_REF]:

(1) 1,( ) 1,( ) 1 0 0 2,( ) 2 0 0 ( , ) ( , ) ( , ) ˆ ( , ) ( , ) ba ba ba A K a Ka                     (3) with sin cos k     , sin sin k     , 0 0 0 sin cos k     , 0 0 0 sin sin k     and 2/ k   .
The subscript (a) gives the impinging plane wave polarization (h or v) and the subscript (b), the scattered wave polarization (h or v), respectively. The function ˆ( , )

IA        (4) where (1) (1) ( ) ( ) ( , ) ( , )
I K K R             (5)
where the brackets denote a statistical average over the ensemble of realizations of () ( , ) ba I  .

IV. MULTILOOK INTENSITY RATIO STATISTICS

A. Scattering amplitudes as complex random variables Let () ( , )  , respectively. We assume that the height distributions of both interfaces are centered and Gaussian. Since any linear operator transforms a Gaussian process into another Gaussian process, we deduce that the first-order scattering amplitudes given by [START_REF] Afifi | Scattering by anisotropic rough layered 2D interfaces[END_REF]

2 ba R i ba j ba ij ij K K R           (6) '' 22 * ,( ) ,( ' '
) 0 0 11 1 Re[ ( , )] 2 ba b a R R i ba j b a ij ij K K R           (7) '' 22 * ,( ) ,( ' '
) 0 0 11 1 Îm[ ( , )] 2 ba b a R J i ba j b a ij ij K K R           (8)
We find from ( 5) and ( 6) that:

2 2 0 () 2 cos ( , ) 2 cos ba ba R I        (9)

B. Probability density function of the multilook intensity ratio

By using polar coordinates, we derive from the joint PDF of

                 (10)
where

'' 2 () 0 2 ( ' ') ( , ) ( , ) ba ba R ba R b a I p I         (11)
and

2 0 0 22 cos 2 cos ba R     (12)
The function 0 I denotes the zeroth-order modified Bessel function. The quantity r is the modulus of the correlation coefficient between the two complex random variables

The n-look averaged intensity is defined as follows:

() ( ), 1 1 n ba ba i i II n    (14)
We assume that the n-look intensities are means of n independent identically distributed random variables. Because the n-look intensities are identically distributed, all the random variables ( ), /

    0 ( ), 0 ( ' ') ( ) ( ' ') ( ) ( ' ') '' 0 0 ' ' , 2 , ' ' ,,
                           (16)
where

1 n I  is the ( 1 n  )-th order modified Bessel function. Let ( ) ( ' ') ,( , ' ') / ba b a n ba b a V I I 
be the n-look intensity ratio.

Finally, we obtain the PDF for the ratio ,( , ' ')

n ba b a V as follows: ,( ' ') ( ) ( ' ') , , 0 ( ) ( , ) n ba b a ba b a V II p v w p vw w dw       (17)
By substituting ( 16) into [START_REF] Mishra | A statistical-measure-based adaptive land cover classification algorithm by efficient utilization of polarimetric SAR observables[END_REF], we find:

,( ' ') (1

                                (18)
) ( ) (2 ) () ( ) ( ) 2 (1 2 ) n ba b a n n n V n r p v v p n pv nn v vp r p            (20)
The PDF only depends on the three parameters n, r and 0 p . For 1 n  , we get the PDF for the single-look intensity ratio. It's a heavy-tailed distribution with infinite mean and variance.

The closed-form expression [START_REF] Lee | Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery[END_REF] has been established in [START_REF] Joughin | Probability density functions for multilook polarimetric signatures[END_REF] and [START_REF] Lee | Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery[END_REF]. The derivation begins with a multivariate Gaussian model for the underlying complex scattering amplitudes. But, our approach is not based on the a priori assumption that the complex scattering amplitudes follows a multivariate Gaussian distribution. For slightly rough interfaces with Gaussian height distributions, we establish this property in the context of the first-order SPM.

C. Cumulative density function of the multilook intensity ratio

We find from [START_REF] Lee | Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery[END_REF] the recurrence relation between

,( ' ') , (
)

n ba b a V pv and 1,( ' ') , (
)

n ba b a V pv  : 1,( ' ') ,( ' ') 2 2 0 0, 2 0, 2 (1 2 ) ( ) 21 2(1 ) ( ) n ba b a n ba b a V V p v p r p v v n r p p v n           (21)
By integration, we find from (21):

1,( ' ') 1,( ' ') 1,( ' ')
,( ' ') 2 0 , , 0

, 22 0 0 , 0 2 (1 2 ) ( ) ( ) () 21 2(1 ) ( ) n ba b a n ba b a n ba b a n ba b a v VV v V V p r F v xp x dx px n p dx r p F v xn          (22)
where the function ,( ' ') , ()

n ba b a V
Fv designates the CDF for the n- look intensity ratio. We write the (n+1)-look PDF as follows:

1,( ' ')

0 ,1 1 1/ 2 () () () n ba b a n Vn n v v p p v q Rv      (23) where 2
() R v cv bv a    with, 2 0 2 0 2 (1 2 ) 1 ap b p r c         (24)
and,

2 1 1 10 (2 2) (1 ) ( 1) ( 1) nn n n q r p nn         (25)
In accordance with the following recursion formula [30, Eq.

(2.263), p.95] with 2 mn  ,

1 , 1, 1/2 2, (2 2 1) ( 2 ) 2( 2 ) ( 1) ( 2 ) m m n m n n mn x m n b KK m n cR m n c ma K m n c              (26)
where

, 1/2 m mn n x K dx R    (27)
we find from ( 23),

1 1 1, 1 1 0 , 1 (
)

n n n n n n n F v q K q p K        (28)
and we also establish that,

  1 1 00 1 2, 1 0 1, 1 , 1 0 1, 1 () () vv n n n n n n n n n n n px xp x dx a dx x q K p K aK ap K                (29)
By substituting ( 29) into ( 22) and using the relation [START_REF] Afifi | The coand cross-polarized scattered intensity ratios for 3D layered structures with randomly rough interfaces[END_REF], after some algebraic manipulations, we obtain a mathematical recurrence formula for the CDF:

1,( ' ') ,( ' ') 0 ,, ( ) ( )
n ba b a n ba b a n V V n vp v F v F v q R R       (30)
Finally, we obtain from (30) the analytical expression of ,( ' ') , ()

n ba b a V Fv : ,( ' ') 1,( ' ') 0 ,, 2 2 2 00 2 1 0 2 2 2 1 00 ( ) ( ) 2 (1 2 ) (1 ) (2 ) ( ) ( ) 2 (1 2 ) n ba b a ba b a VV m n m vp F v F v v vp r p r p v m m m v vp r p                    (31)
where the function Fv is the CDF for a single-look intensity ratio. We established in [START_REF] Afifi | On the co-polarized scattered intensity ratio of rough layered surfaces: The probability law derived from the small perturbation method[END_REF]

that 1,( ' ') 0 , 2 2 2 00 1 () 2 2 2 (1 2 ) ba b a V vp Fv v vp r p      (32)
This is new compared to previous works on the n-look intensity ratio statistics. We find from (31) that for any n,

D. Mean and variance of the multilook intensity ratio

Using the recurrence formula (26), we show that the mean ,( , ' ') n ba b a V  of ,( , ' ')

n ba b a V exists if 1 n  with, 2 0 ,( , ' ') () ( 1)
n ba b a n r p V n     (33)
and that the variance ,( , ' ') increase linearly with 0 p . The limit of the mean is 0 p as n tends to infinity and, the limit of the variance is zero. When n goes to infinity, the intensity ratio value is no longer random.

For uncorrelated scattered amplitudes

(1) () ba A and (1) ( ' ') ba A , 0 r  . For this case, the multilook intensity ratio follows a Fisher distribution for which the closed-form expressions for the mean and variance are well known [START_REF] Kendall | Kendull's Avdanced Theory of Statistics[END_REF]. To the best of our knowledge, it is the first time that the analytical expressions for the mean and variance for correlated scattered amplitudes are found.

V. NUMERICAL RESULTS

We consider a two-layer rough ground. The wavelength  is equal to 24 cm. The relative dielectric constants of the ground layers are 2 4.66-0.29

 

for the upper and lower surfaces, respectively. A 5cm-thickness ( 0 u in Fig. 1) is assumed for the middle layer [START_REF] Boisvert | Effect of surface soil moisture gradients on modelling radar backscattering from bare fields[END_REF]. The Gaussian spectrum of the upper interface is anisotropic with the correlation lengths in the backscattering direction. The theoretical probability law obtained for surfaces of infinite extent is given by [START_REF] Lee | Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery[END_REF]. The histograms are estimated over results associated with 2 13 surface realizations where 20 L   [START_REF] Afifi | The coand cross-polarized scattered intensity ratios for 3D layered structures with randomly rough interfaces[END_REF]. The probability law is representative of the values which the intensity ratio can take and does not lose some details as for the statistical averages. The agreement between the theoretical probability law with the histogram obtained from Monte-Carlo simulations is very good for each value of n and validates the theory for the co-polarized intensity ratio in the backscattering direction.

Given the simulation parameter values, the correlation coefficient r is equal to 0.994 and therefore, the co-polarized scattering amplitudes (1) () hh In the plane perpendicular to the incidence plane, there is a depolarization and we established in the context of the firstorder perturbation theory that the cross-polarized intensity ratio is only defined for a (v)-polarized impinging plane wave [START_REF] Dusséaux | Statistical distributions of the coand crosspolarized phase differences of stratified media[END_REF]. Fig. 5 shows the theoretical PDF of ,( , ) ( 30 , 90 ) n hv vv V       and the associated histogram for four values of n. Given simulation parameter values, the correlation coefficient r is equal to 0.872 and therefore, the co-polarized scattering amplitude (1) () vv A and the cross-polarized amplitude (1) () hv

A are strongly correlated. The parameter 0 p is equal to 11. vp  so that the ratio mean approaches 0 p and the maximum value increases. As previously mentioned, the histograms are estimated from 2 13 realizations over areas of 2 400 . We find a very good agreement between the histograms and the corresponding theoretical probability laws for the four values of the number of looks. Fig. 6 shows the theoretical CDF and the CDF estimated from Monte-Carlo simulation data. For each value of n, the two curves are superimposed and this comparison validates the analytical formulation for the CDF for the cross-polarized ratio. At 1, n  the distribution has a heavy tail and as a result, the CDF converges very slowly towards 1 as the ratio tends to positive infinity. As n increases, the convergence is faster. Fig. 7 gives the values of the theoretical mean and standard deviation for the variable ,( , ) ( 30 , 90 ) n hv vv V       and the values estimated over Monte-Carlo simulations versus the number of looks. As previously in the backscattered the mean ,( , ) n hv vv V  quickly converges towards the median 0 p and for large values of n , the standard deviation ,( , ) n hv vv V  decreases in 1/ n . The comparison is satisfactory and validates the closed-form expressions for the mean and variance for the multilook cross-polarized intensity ratio. For 2  , there is a slight difference between the estimated and theoretical mean values and for 3 n  , a weak difference for the variance values, respectively. This difference decreases when increasing the area LL  .

VI. CONCLUSION

We have derived the theoretical statistics for multilook intensity ratios for 3-D layered structures under a monochromatic plane wave illumination. The derivation begins with a multivariate Gaussian model for the underlying complex scattering amplitudes. In contrast to previous works [START_REF] Joughin | Probability density functions for multilook polarimetric signatures[END_REF]- [START_REF] Lee | Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery[END_REF], our approach is not based on the a priori assumption that the complex scattering amplitudes follow a multivariate Gaussian distribution. For slightly rough interfaces with an infinite extent and centered Gaussian height distributions, we establish this property in the context of the first-order small perturbation method.

Assuming that the n-look intensities are means of n independent identically distributed single look intensities, we obtain a 3-parameter probability distribution and we show that the PDF and CDF only depend on the number of looks n, the correlation coefficient r between the complex scattering amplitudes under study and the ratio 0 p between the associated average single-look intensities. We have shown that the parameter 0 p is the median of the n-look intensity ratio for any value of n and that the mean exists for 1 n  and the variance for 2 n  , respectively. The mean and the standard deviation of the n-look intensity ratio increase linearly with 0 p . The limit of the mean is 0 p as n tends to infinity and for large values of n , the standard deviation decreases in 1/ n . The obtained formulae are derived from the first-order SPM and they are valid for slightly rough interfaces only. For a two-layer rough ground, the analytical results were compared with those derived from Monte Carlo simulations. We have assumed an upper interface with an anisotropic Gaussian spectrum and a lower interface with an isotropic one. Both random surfaces are partially correlated. The stratified medium is studied in the backscattering direction and in the transverse plane. The analytical formulas assume rough interfaces with infinite extent. Nevertheless, we have shown by Monte-Carlo simulations that these analytical expressions can be used for surfaces of a few hundred wavelength squared. The agreement between simulated data and theory is observed to be very good for the probability density function, the cumulative density function, the mean and the standard deviation for the co-and cross-polar ratios. We have considered random processes with Gaussian spectra and crossspectra but the analytical results established in the paper can be used for all random processes with finite memory and for an arbitrary number of interfaces.

APPENDIX

                       (A1)
The determination of 0 ( ), 0 ( ' ') ( , )

                      (A3)
and after some mathematical calculations, we find:

                             (A5)
The joint probability law of LT  ) as follows:

( ) ( ' ')

baI

   and ( ' ') ( , ) ba I  be the (b)-and (b')-polarized contributions of the scattered intensity under the (a)-and (a')polarized incident waves, respectively. In a first stage, we determine the probability law for the multilook intensities and finally, the probability law for the multilook intensity ratios. Let () ( , ) ba R  and () ( , ) ba J  be the real and imaginary parts of the scattering amplitude (1) () ( , ) ba A  and let ( ' ') ( , ) ba R  and ( ' ') ( , ) ba J  be the real and imaginary ones of (1) ( ' ') ( , ) ba A

Fig. 1 .

 1 Fig. 1. Structure with two nonparallel interfaces

(

  the n-look intensities are independent random variables, the joint probability law ( times[28, chap. 5]. For determining this joint PDF, we also use, as in[START_REF] Joughin | Probability density functions for multilook polarimetric signatures[END_REF], the properties of the two-dimensional Laplace transform of 0 ( ), 0 ( ' ') derivation is included in the appendix and gives the following analytical expression, ( ) ( ' ')

p

  for any value of n.

2 n

 2

  surfaces are partially correlated with a correlation coefficient[START_REF] Déchamps | Electromagnetic scattering from a rough layer: Propagation-inside-layer expansion method combined to the forward-backward novel spectral acceleration[END_REF]  equal to 0.2. The incident wave vector angles 0  and 0  are equal to 30° and 0°. Fig.2shows the PDFs of the co-polarized intensity ratio ,( , ) n hh vv V

  shown by Fig.2, as the number of looks increases, the PDF curve becomes narrower and concentrated near 0 p because multilook processing reduces the statistical variation and this causes an increase of the maximum value of the PDF.

Fig. 3

 3 Fig.3shows the theoretical CDF and the CDF estimated from the set of realizations of the co-polarized scattered intensities. For each value of n, the two curves are superimposed. As the number of looks increases, the CDF becomes a staircase function about the value 0 p because the reduction of the statistical variation by the multilook processing.

Fig. 2 .

 2 Fig.2. Theoretical PDF of random variable ,( , )( 30 , 0 ) n hh vv V

Fig. 3 .Fig. 4 .

 34 Fig. 3. Theoretical CDF of random variable ,( , ) ( 30 , 0 ) n hh vv V

  a result, for this medium under consideration, the depolarization is strong within the transverse plane. At 1, n  the random variable 1,( , ) hv vv V follows a heavytailed law and has a large amount of skewness. As n increases, the PDF starts to become symmetrical about 0

Fig. 5 .Fig. 6 .

 56 Fig.5. Theoretical PDF of random variable ,( , )( 30 , 90 ) n hv vv V

Fig. 7 .

 7 Fig. 7. Values of theoretical means and standard deviation of ,( , ) ( 30 , 90 ) n hv vv V

  The two-dimensional Laplace Transform (LT2) of

  As shown in Fig.1, the structure we consider is a stack of two surfaces separating three media. The rough surfaces are randomly deformed over an area LL  . The quantity 0

	Multilook Intensity Ratio Distribution for 3D II. ANALYZED STRUCTURE
	u Layered Structures with Slightly Rough designates the layer average thickness. The two boundaries are located at the height 1 ( , ) z a x y  and 20 ( , ) z a x y u  . They Interfaces are realizations of second order stationary, centered Gaussian
	stochastic processes with Gaussian spectrum ˆ( , ) R  :
					ii
	2 2 xi ll 2 2 4 yi        2 exp i xi yi l l    ˆ( , ) ii R 	Richard Dusséaux and Saddek Afifi (1)
	There are two distinct correlation lengths,	l and	l . The i-th
					xi	yi
	interface is isotropic if	ll  and anisotropic if	ll  .
					xi	yi	xi	yi
	The rms-heights	 are small numbers compared to the
				i
	incident wavelength and the gradients of the surface height
	functions	a	( , ) x y are much smaller than 1 so that the
		i
	perturbation method may be used to study wave scattering [6]-
	[8].
	The cross-spectrum 12 ˆ( , ) R  is given in the following form:
	12		1 2		1 2 1 2
					22	22
				22 12	12
	S

  /

		A	 		A	 	L	. The scattered intensity
		ba				ba
	ba () ( , ) I  depends on rough boundary height profile
	realizations and for a given direction, it is a random variable.
	ba When L , the mean of () ( , ) I  is given by,
	() ba	( , )
	2 cos cos 2	0	22 11 ij	Re[	* ,( ) i ba	j	,( ) ba	ˆ	ij	(	0	,	0	)]

  By using the following relationship found in the Laplace transform table[29, pp. 245, Eq. 9],

		p	, II ba	b	a	( , ) w w  	1  2 LT
	22 00 2 2 1 0 np r s r   n 1   2 2 2 0 0 0 0 0 0 2 2 2 2 1 1 (1 ) n n n p n p r n p s s r r                                     r The two-dimensional inverse Laplace transform (A6) becomes (A6)
	with two consecutive one-dimensional inverse Laplace
	transforms,
										
		p	( ) ( ' ') , ba b a II	( , ) w w  	22 00 2 1 np r     	n	1  1 TL	00 2 np 1 1 r   s           	n	(A7)
	2 2 2 0 0 22 ) ( ' 1 r n p np 10 2 p rs  0 2 (1 n 1 r    1 / ) n s                        r Knowing that 1 1 TL  
	 the relationship (A7) becomes:  1 1 ( 1)! exp( ) () n n n TL w aw     sa	(A8)
	p	( ) ( ' ') , ba b a II	22 0 0 22 1 0 exp 1 ( 1)! 1 n n n p n w r n r     w    ( , ) w w      
		2 2 2 0 0 22 ) ( ' 1 r n p w np 00 2 rs r  / ' 1 ) n p 0 0 2 s r    exp (1 n                  1 1 TL   	(A9)
	1 11 ( 1)/ 2 exp( / ) ( / ) n n n as TL w a I s      	(2	aw	)	(A10)

We find the closed-form formula

[START_REF] Ulaby | Radar Polarimetry for Geoscience Applications[END_REF]

giving the joint PDF for the multilook intensities () ba I and ( ' ') ba I .


Quentin Palomar

Chantal Gondran
email: chantal.gondran@univ-grenoble-alpes.fr

Jean-Paul Lellouche

Serge Cosnier

Michael Holzinger
email: michael.holzinger@univ-grenoble-alpes.fr

Functionalized tungsten disulfide nanotubes for dopamine and catechol detection in a tyrosinase-based amperometric biosensor design

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Nanomaterials are nowadays essential tools in biosensor research not only due to the enhanced surface area but also to a variety of particular phenomena related to the nanosized structure [1][2][3] . Within the vast availability of different nanomaterials, carbon nanotubes 4, 5 and later graphene 6, 7 became prominent candidates for biosensing applications because of their beneficial electronic and optical properties for enhanced signal capture of the biological recognition event. More recently, 2D transition metal dichalcogenides such as WS 2 and MoS 2 showed promising properties in biosensor designs 8 which are principally based on the layered form of these metal dichalcogenides. The detection of the target is commonly transduced via optical detection 9 or in a field effect transistor design 10 but such dichalcogenides have also shown their appropriateness in electrochemical biosensor setups 11 . The synthesis of WS 2 and MoS 2 can also form differently shaped materials like nanotubes or nanorods 12,13 with improved surface activities and might be more appropriate for biosensor applications 14 . Raichman et al. reported an efficient functionalization method for tungsten disulfide inorganic nanotubes using a highly electrophilic acidic Vilsmeier-Haack reagent for homogeneous covalent carboxylation of the WS 2 walls. The carboxylic acid functions lead to improved dispersions in aqueous media and represent efficient anchor groups for the immobilization of bioreceptor units 15,16 . Furthermore, the clearly reduced cytotoxicity of such transition-metal dichalcogenides compared to carbon nanomaterial represents an advantage when combined with biological entities 17,18 . In this work, we used such carboxylated WS 2 (WS 2 -COOH) nanotubes for the immobilization of the tyrosinase enzyme as protein model and evaluated this sensor for the detection of catechol and dopamine.

Catechol (CA) is a toxic compound 19 but serves as starting materials for the synthesis of many different chemicals and therein a wide variety of pesticides 20 . This compound therefore contributes to the pollution of soils and groundwater. It further has a low degradation rate which worsens its environmental impact. This journal is © The Royal Society of Chemistry 20xx

Please do not adjust margins Please do not adjust margins . An efficient filtration and transfer technique 23 to form identic WS 2 -COOH nanotube deposits was adjusted for the electro-enzymatic detection of these two compounds. As illustrated in Scheme 1, the carboxylic acid groups served as anchor for the immobilization of tyrosinase, an enzyme that oxidises specifically catechol and dopamine to its respective quinones. These quinones are then monitored at cathodic potentials via electrochemical reduction to the initial hydroquinones. The WS 2 -COOH nanotubes serve here principally as porous nanostructures on the working electrodes to immobilize and enhanced amount of tyrosinase enzymes.

Results and discussion

Characterization of the WS 2 -COOH nanotubes

Before the construction of the biosensor, the morphology and electrochemical behaviour of the WS 2 -COOH nanotubes were characterized. Figure 1 shows the scanning electron microscopy images of these nanotubes as synthesized (Figure 1A, B) and after film formation and transfer on GCE (Figure1 C, D). At the same magnification (Figures 1A andC), the as synthesized WS 2 -COOH nanotubes and the formed film do not show significant differences and demonstrate that the stirring dispersion, filtration and transfer procedure does not affect these nanotubes.

The length of these nanotubes is comprised between 2 and 10 microns for a diameter of about 100 to 200 nm. It also seems that the WS 2 -COOH nanotubes are formed of rolled-up WS 2 sheets. This lamellar structure is particular visible at higher magnification as shown in Figure 1B. Figure 1D shows the deposits at lower magnification and reveals an assembly of randomly oriented nanotubes forming a highly porous structure. To determine the thickness and roughness of the formed WS 2 -COOH films, a confocal laser microscope was used. This technique allows mapping the height and the morphology of the films by profilometry. Figure 2 shows images taken at the boundary between the gold substrate and the WS2-COOH films.

A thickness of 6.2 μm was determined for a roughness of ± 0.6 μm. This relatively high roughness, 10% of the total thickness, can be explained by the highly porous nature of the formed film. Finally, it appears that thickness and roughness values for 3 different films, exhibit a relative standard deviation of 6% and 3.6% respectively highlighting the reproducibility of the procedure. The electrochemical behaviour of these films on GCEs was studied using cyclic voltammetry in organic media (Figure 3). Within a scan range between -2.0 and +2.0 V vs Ag+/Ag, the first cycle shows a strong and large irreversible anodic peak at 1.7 V that starts at 0.6V. This peak almost disappears during the second scan. According to the studies about the inherent electrochemistry of transition-metal dichalcogenides by Bonde et al. , measured using a four-point probe) and can therefore not be considered as extension of the electrode. .

Detection of catechol

The detection of catechol using tyrosinase is based on the enzymatic oxidation of phenols and o-bisphenols like catechol to oquinones. The product of the enzymatic reaction is then reduced at the electrode at -0.2 V vs SCE. The resulting chronoamperometric current density is proportional to the analyte concentration. In addition, the target molecule is regenerated during the reduction at the electrode which allows the amplification of the electrochemical signal and leads to increased sensitivity 26,27 . The measurements were done by immersing the modified electrode in an electrochemical cell containing 20 mL of PBS (0.1 mol.L -1

, pH = 7.4).

Figure 4A shows the increase of steady-state current response of the bioelectrode upon successive additions of catechol and illustrates the fast response time (3s). Figure 4B shows the amperometric response of the bioelectrode as a function of catechol concentration in the range of 0.3-220 µmol L -1 resulting in a calibration curve which follows a Michaelis-Menten kinetic. The maximum current density, j max , is related to the amount of immobilized enzymes on the electrode surface and to the affinity of the enzyme for its target and thus to the apparent Michaelis-Menten constant (K M app ). The linear fit equation (R 2 = 0.99897) obtained with the Lineweaver-Burk representation leads to 36 µA cm -2 and 183 μmol L -1 for j max and K M app , respectively (Figure 4C):

The K M app used here reflects the kinetics of the immobilized enzyme and not that of the free enzyme in solution. Surprisingly, the determined K M app is slightly lower than the K M of free enzymes (300 μmol L -1 with catechol

28

) in spite of the constraints to the mass transport at immobilized states. Nevertheless, this behaviour is quite common and was already observed with tyrosinase biosensors 29,30 . To explain this phenomenon, it is necessary to consider the formula for K M : This journal is © The Royal Society of Chemistry 20xx

Please do not adjust margins

Please do not adjust margins

Where k 1 represents the kinetic of the enzyme-substrate complex formation and k 2 the transformation of enzyme-substrate complex into the product of the enzymatic reaction and the regenerated

( k 2 = k 2 °×[O 2 ]
), K M app can be calculated:

Thus, the value of K M app is directly related to the oxygen concentration. However, the use of nanomaterials considerably increases the quantity of immobilized enzymes and the co-substrate amount is rapidly consumed. Since the oxygen supply is limited by diffusion, a concentration gradient of oxygen can thus be established between the solution and the enzyme. The co-substrate concentration in the enzyme's environment is then lower than in solution and results in a decrease of K M app since the value of k 2 is lower than expected. Assuming that the enzyme is here in a favorable environment, it may then be assumed that the oxygen concentration decreases to 40% close to the enzymes compared to the rest of the solution.

The linear part of the calibration curve within the concentration range between 0.6 μmol.L -1 and 70 μmol.L -1 reveals a sensitivity value of 152.5 mA L cm -2 mol -1 (R 2 = 0.9986).It should be noted that the linearity range and the sensitivity are identic for the two electrodes. The electrochemical characteristics in terms of molecular architecture, K M app , maximum current density, linearity, and sensitivity are summarized in Table 1.

The linearity range of our system are similar to those found in literature 35 . This can be explained by the presence of WS 2 -COOH nanotubes which allows a high amount of immobilized enzymes on the electrode surface as other nanomaterials. However, the sensitivity remains relatively low compared to other reports

34

. This can be due either to the rapid oxygen consumption and/or to reduced permeability of the WS 2 -COOH film but finally results most likely from the almost inexistent conductivity at this potential.

Detection of dopamine

As for catechol, tyrosinase also oxidizes dopamine to dopamine-oquinone which can be detected at the same potential (-0.2 V vs SCE). The detection of dopamine was thus performed under identic conditions as described above. Figure 5A illustrates the chronoamperometric response of the modified GC/WS 2 -COOH/tyrosinase electrodes as a function of the dopamine concentrations. It can be seen that the current stabilizes rapidly after dopamine injection confirming the fast electronic transfer and the permeation of the substrate through the WS 2 -COOH nanotube deposit. The range of concentrations tested varies from 0.5 µmol L -1 to 137 µmol L -1

. From the shape of the curve, it seems that the dopamine oxidation via tyrosinase is also governed by Michaelis-Menten kinetics. Figure 5B shows the evolution of the current density as a function of the dopamine concentration. The observed maximum current density is clearly higher than for catechol detection.

This may be due to the fact that the amine function of dopamine is partially positively charged at pH 7.4 and hence can display a faster and deeper penetration inside the WS2-COOH matrix which is negatively charged. As a consequence, the calibration curve, which is obtained by plotting the current density as a function of the dopamine concentration in solution, does not necessarily reflect the dopamine concentration in the WS2-COOH matrix.

Please do not adjust margins

Please do not adjust margins From the Lineweaver-Burk representation (not shown), a K M app of 41 μmol L -1 and j max of 57 μA cm -2 could be determined following the linear fit equation (R 2 = 0.99344):

The maximum current density, j max , is higher for the detection of dopamine (57 μA cm -2

) than for the detection of catechol (36 µA cm -2

) at the same amounts of enzymes. It seems that the WS 2 -COOH nanotubes provide a better accessibility to the enzyme for the dopamine rather than for catechol.

The experimental K M app value is, as for the catechol experiments, lower than the K M of the free enzyme (K M = 2.2 mmol L -1 for the tyrosinase towards dopamine) 28 related to the limited oxygen concentration. It should be noted that the experimentally determined K M app is lower for dopamine (41 μmol L -1

) than for catechol (183 μmol L -1

). In solution, the K M value of tyrosinase for dopamine is higher (2.2 mmol L -1

) compared to catechol (300 μmol.L -1

). This inversion may be related to an accumulation of dopamine in the WS 2 -COOH matrix related to electrostatic interactions. This can be deduced by the arc shaped (and not linear) calibration curve at low dopamine concentrations. In order to estimate this accumulation effect, we fitted this part of the calibration curve in Figure 5C . It can therefore be concluded that below the value of 35 µmol L -1 , the dopamine concentration is greater around the enzymes than in solution. At higher values, the dopamine concentration is the same in the WS 2 -COOH matrix and in solution.

Figure 5D shows that at low dopamine concentration, the calibration curve presents two successive linear parts as other examples reported in literature for PPO biosensors [36][37][38] . For a concentration range varying from 0.5 to 10 μmol L -1 , the biosensor sensitivity for dopamine is 88.0 mA L cm -2 mol -1 (or 6.2 mA L mol -1

) and 48.8 mA L cm -2 mol -1 (or 3.4 mA L mol -1

) between 10 and 40 μmol L -1 of dopamine. This setup and phenomena are quite good reproducible with a relative standard deviation of 4.8% for three different electrodes. This journal is © The Royal Society of Chemistry 20xx

Please do not adjust margins Please do not adjust margins . The system presented here has slightly lower performances in terms of linearity range, sensitivity, and limit of detection compared to carbon nanotubes (CNTs), widely used for biosensing application, despite good maximum current densities. We have shown that our WS 2 -COOH nanotube deposits are not conductive at appropriate potentials and can therefore contrary to CNTs, not contribute to the electrochemical signal capture. However, the increased surface area of the nanotubes for improved enzyme loading and the particular interaction with dopamine almost compensates this lack of conductivity.

Experimental Chemicals

Dopamine hydrochloride 98%, 1,2-Dihydroxybenzene (catechol), Phosphate buffered saline (PBS) tablets, N-Hydroxysuccinimide (NHS) 98%, 4-(Dimethylamino)pyridine (DMAP) and Tyrosinase from mushroom were purchased from Sigma-Aldrich. WS 2 nanotubes were provided by NanoMaterials Ltd (Yavne, Israel). N-(3-Dimethylaminopropyl)-N′-ethylcarbodiimide hydrochloride (EDC) was purchased from Fluka. All chemicals and solvents were of analytical grade and were used as received, without further purification. WS 2 -COOH nanotubes were synthesized and characterized as described in 15 . The obtained material was characterized using FT-IR spectroscopy (Figure 6) where the appearance of characteristic peaks, summarized in table 3, indicate the successful grafting of carboxylic acid functions.

Apparatus

Wavenumber (cm - tetrabutylammonium perchlorate, TBAP), whereas a saturated calomel electrode (SCE) was used as a reference electrode in aqueous media.

Preparation of the WS2-COOH nanotube films

Prior to any modification, the surface of the glassy carbon electrodes (GCE,  = 3 mm) was polished with a 2 μm diamond paste purchased from Presi (France) and rinsed successively with water, acetone, and ethanol under sonication (5 min). Afterwards, the GCEs were modified with the WS 2 -COOH nanotubes. These nanotubes were dispersed by mechanical stirring in distilled water (15 mg of WS 2 -COOH in 1L of H 2 O). The solution is then allowed to settle for 60 min in order to sediment the larger nanotube clusters.

The supernatant was then filtered over cellulose nitrate filter (Sartorius, 0.45 µm,  3.5 cm) The obtained film was finally transferred to a GCE surface by dissolution of the cellulose nitrate filter according to a procedure described in 23 The same deposition technique was used during laser microscopy characterization, this time using a gold substrate.

Enzyme immobilization on the WS2-COOH nanotube films

The WS 2 -COOH nanotube films with a controlled thickness of 6.2 μm, were then used as a support for the immobilization of tyrosinase. This reaction involves the activation of the carboxylic acid functional groups of the film in order to allow the coupling of the NHS function. ), NHS (10 mmol L -1

) and DMAP (10 mmol L -1

) for 12 h at 4°C. EDC and DMAP serve as activation agents. Eventually, the GCE is incubated with a solution containing the concentration of 0.3 mg mL -1 of tyrosinase in PBS 0.1 mol L -1 pH 7.4 for 12h at 4°C in order to immobilize the enzyme. This enzyme concentration was chosen after a short optimization procedure using 0.1, 0.3, and 0.5 mg mL -1 tyrosinase concentration in a cyclic voltammetric setup.

The reduction peak currents at -0.1 V vs SCE were recorded in 0.1 molL -1 PBS; pH 7.4, containing 5 µmol L -1 catechol. Figure 7 shows the obtained values revealing an optimal concentration of 0.3 mgmL -1

. . This journal is © The Royal Society of Chemistry 20xx

Please do not adjust margins

Please do not adjust margins

The resulting electrodes were then washed carefully several times with PBS solution.

Conclusions

COOH functionalized WS 2 nanotubes were used as platform for enhanced immobilization of the tyrosinase enzyme leading to a catechol and a dopamine biosensor with very satisfying performances. Despite the lack of conductivity of this semiconductor material at appropriate redox potentials for amperometric biosensor applications, these tubes form highly porous deposits with improved enzyme wiring capacities. The determined K M app values of immobilized tyrosinase are close to the values for the enzyme in solution which represents optimal accessibility of the target molecules to the active site of the enzyme. For dopamine detection, we observed an accumulation of the enzymatically produced dopamine oquinone leading to enhanced signals at low analyte concentration. The possibility to functionalize these WS 2 nanotubes with carboxylic acid functions make this material easier to process and facilitate post functionalization. At the current state of research, these WS 2 -COOH nanotubes provide not superior properties, compared to other described nanomaterials, for electrochemical biosensing applications but they can become promising components in electrochemical biosensor setups as additives due to the sown beneficial properties.

  Dopamine (DA) is a neurotransmitter and plays an important role as an extracellular chemical messenger in the cardiovascular, renal, hormonal and nervous systems. It is at the centre of cell signal transmission and has therefore a strong influence on human behaviour. Abnormal concentrations of DA in biological fluids are an indicator of several diseases such as schizophrenia, Huntington's disease or Parkinson 21, 22

Scheme 1 :

 1 Scheme 1: Sketch of the functionalization of WS2 modified glassy carbon electrodes with the enzyme tyrosinase via a standard EDC/NHS coupling reaction. These modified bioelectrodes served for the detection of catechol (bottom left) and dopamine (bottom right) at -O.2 V vs Ag + /Ag.

Figure 1 :

 1 Figure 1: SEM images of A-B) as synthesized WS2-COOH powder at a magnification of A) ×20000, and B) 50000. SEM images of C-D) thin films of WS2-COOH nanotubes at a magnification of C) ×20000, and D ×5000.

Figure 2 :

 2 Figure 2: Height mapping images and 3D views realized with a confocal laser microscope at a magnification of × 50 of WS2-COOH film at the cut edge (A and B) with

24 ,

 24 this peak can be assigned to the irreversible oxidation of WS 2 on surface forming WO 4 2-. The potential shift in the second scan is related to the oxidation of remaining WS 2 in a WO 4 2-environment. At the reverse scan towards negative potentials, an irreversible cathodic peak was observed at -0.7 V which also disappeared at the second scan. Here, it is more likely that loosely adsorbed species are released during the first scan or a reduction of the protons since WS 2 can be used as electrocatalysts for hydrogen evolution reaction (HER) 25 Most important, the WS 2 -COOH nanotubes are electrochemically silent in the potential range around -0.2 V where should occur the amperometric detection of dopamine and catechol with the tyrosinase based biosensor. Unfortunately, films of WS 2 -COOH are almost not conductive at this potential (1.6 ± 0.1 × 10 -07 S cm -1

Figure 3 :

 3 Figure 3: Voltammograms obtained in CH3CN + 0.1 mol L -1 TBAP; first cycle (a) and second cycle (b) with the GCE/WS2-COOH modified electrode between -2.0 and +2.0 V vs Ag + /Ag; Scan rate: 100 mV s -1

Figure 4 : 1 .

 41 Figure 4: A) Chronoamperometric response of the GC/WS2-COOH/tyrosinase electrode (E = -0.2 V vs. SCE; 0.1 mol L -1 PBS; pH 7.4) for successive increase of catechol concentration from 0.3 µmol L -1 to 220 µmol L -1 . B) Plot of the current density towards catechol concentration on Tyrosinase/WS2-COOH/GCE. C) Lineweaver-Burk plot of the corresponding data. D) Plot of the linear part of the current density towards increasing catechol concentrations.

Figure 5 : 1 ,

 51 Figure 5: A) Chronoamperometric response of the GC/WS2-COOH/tyrosinase electrodes (E= -0.2 V vs. SCE; 0.1 mol L -1 PBS; pH 7.4) for increasing dopamine concentrations from 0.5 µmol L -1 to 137 µmol L -1 , B) Plot of the current density towards dopamine concentrations. C) Fit of the calibration curve with the Michaelis-Menten equation taking into account the accumulation of dopamine in the biosensor matrix. D) Plot of linear parts of the current density towards increasing dopamine concentrations measured with GC/WS2-COOH/tyrosinase electrode.

  using an equation that takes the concentration change of dopamine into account: at the equilibrium: with K d as the equilibrium dissociation constant. [DA], [NC] and [DA] acc are the dopamine concentrations in solution, negative charges on WS 2 -COOH nanotubes, and accumulated dopamine in the deposit, respectively. The unknown amount of negative charges on WS 2 -COOH can be substituted by the difference between the maximum accumulating dopamine concentration, and the actually accumulated dopamine concentration, [DA] acc .: Leading for [DA] ads to: The integration of this equation inside the Michaelis-Menten equation gives: By fitting the first part of the calibration curve with this equation (figure 5C, R 2 = 0.99728), the dissociation constant, K d , is estimated at 5×10 -5 and the maximum accumulating dopamine concentration, , at 35 µmol L -1

Figure 6 :

 6 Figure 6: IR spectra of WS2 nanotubes before (top) and after (below) carboxylation. Moreover and as additional proof for COOH functionalization, characteristic fluorescence-based KAISER tests 41 have been performed after EDC-based derivatization/activation of the polyCOOH shell by 1,3-diaminopropane and ninhydrin reactivity. Fluorescence measurements helped to obtain the optimized carboxylation degree of 0.5 mmol COOH groups g -1 of functional WS 2 nanotubes.

Figure 7 :

 7 Figure 7: Plot of the peak currents at -0.1 V versus different tyrosinase concentration from 0.1 to 0.5 mg mL -1

Table 1 :

 1 Comparison of the characteristics (system used, KMapp, imax, linearity and sensibility) of different nanostructured electrochemical catechol biosensors.

		System used	KMapp / µmol L -1	imax / µA	Linearity/ µmol L -1	Sensitivity / mA L mol -1
	This work	WS2-COOH nanotube + GCE	183 ± 20	2.5 ± 0.6	0.6 -70	10.7 ± 0.2
		Gold nanoparticule +
	Karim et al. 31	screen printed carbon	-	-	0.01 -80	13720
		electrode
	Sethuraman et al. 32	PEDOT-rGO+ GCE	30.48	92	0.04 -62	1600
	Zhou et al. 33	Mesoporous carbon nitride + GCE	11.07	11.07	0.05 -12.5	593
	Bujduveanu et al. 34	CNT + CaCO3 nanoparticules + GCE	7	39.2	0 -8	2500
	Mei et al. 35	rGO-PdCu Nanocage + GCE (laccase)	-	-	500 -1155 1655 -5155	12.65 5.51

enzyme. This formula for K M does not take into account the cosubstrate (oxygen) concentration of the enzymatic reaction but this is actually the case since tyrosinase reduces oxygen to water to regenerate itself. By integrating the co-substrate concentration in the equation for k 2

Table 2

 2 compares our findings with representative examples found in literature about dopamine detection using tyrosinase combined with a nanomaterial focusing various criteria such as

Table 2 :

 2 Comparison of the obtained characteristics (system used, KM app , imax, linearity and sensibility) of different nanostructured electrochemical dopamine biosensors.

					System used	K M	app / µmol L	-1	i max / µA	Linearity/ µmol L	-1	Sensitivity / mA L mol -1
	This work				WS 2 -COOH nanotube + GCE	41 ± 2		4 ± 0.5	0.5 -10 10 -40	6.2 ± 0.7 3.4 ± 0.4
	Bujduveanu et al.	34	CNT + CaCO 3 nanoparticules + GCE	30			-	0.015 -30	250
	Wang et al.	39		SiO 2 nanoparticule+ acide phytic + GCE (laccase)	-			-	0.99 -103.1	1.9
	Lete et al.	40		CNT + glutaraldehyde + PEDOT + gold IDE	11.1		-	100 -500	14.1
	Zhuang et al. 36		Overoxidized Polypyrrole/Graphene + GCE	-			-	0.5 -10 25 -1000	94 ± 15
	sensitivity, linearity, the maximum current densities and the
	calculated K M app

Table 3 :

 3 selected characteristic FT-IR peaks with vibrational modes

Table 2 :

 2 Comparison of the obtained characteristics (system used, KM app , imax, linearity and sensibility) of different nanostructured electrochemical dopamine biosensors.

					System used	K M	app / µmol L	-1	i max / µA	Linearity/ µmol L	-1	Sensitivity / mA L mol -1
	This work				WS 2 -COOH nanotube + GCE	41 ± 2		4 ± 0.5	0.5 -10 10 -40	6.2 ± 0.7 3.4 ± 0.4
	Bujduveanu et al.	34	CNT + CaCO 3 nanoparticules + GCE	30			-	0.015 -30	250
	Wang et al.	39		SiO 2 nanoparticule+ acide phytic + GCE (laccase)	-			-	0.99 -103.1	1.9
	Lete et al.	40		CNT + glutaraldehyde + PEDOT + gold IDE	11.1		-	100 -500	14.1
	Zhuang et al. 36		Overoxidized Polypyrrole/Graphene + GCE	-			-	0.5 -10 25 -1000	94 ± 15

For this, the electrode is incubated with PBS (75 µL) containing EDC (20 mmol L -1

This journal is © The Royal Society of Chemistry 20xx J. Name., 2013, 00, 1-3 | 3

This journal is © The Royal Society of Chemistry 20xx J. Name., 2013, 00, 1-3 | 5

Acknowledgements

We gratefully acknowledge the University Grenoble Alpes (Contrats Doctoraux thématiques UJF 2014) for the PhD fellowship funding of Quentin Palomar, the CNRS PICS program 6344 France Israel for further support. The authors wish also to acknowledge the support from the platform Chimie NanoBio ICMG FR 2607 (PCN-ICMG), from the LabEx ARCANE (ANR-11-LABX-0003-01 and CBH-EUR-GS, ANR-17-EURE-0003), and from the Institut Carnot PolyNat (CARN 0007-01).

Conflicts of interest

There are no conflicts to declare.