P Ballesta

S Manneville

The Faraday instability in wormlike micelle solutions

Keywords: Wormlike micelles, Faraday instability, particle imaging velocimetry, birefringence

The behaviour of semi-dilute wormlike micelle solutions under vertical vibrations is investigated using classical measurements of the Faraday instability threshold along with two new experiments based on particle imaging velocimetry and birefringence. We provide evidence for the presence of oscillations of the critical acceleration and wave number with the vibration frequency, which are linked to oscillations of the velocity and birefringence fields along the vertical direction. These observations are interpreted in terms of standing elastic waves across the fluid layer. Such an interpretation is confirmed numerically by using the model proposed by S. Kumar [Phys. Rev. E, 65, 026305 (2002)] for a viscoelastic fluid to calculate the velocity and deformation fields. Finally, further birefringence experiments above the instability threshold show that the Faraday instability does not induce disentanglement and breakage of the micellar network.

Introduction

In the classical "Faraday experiment" a fluid layer is submitted to vertical vibrations at a frequency f and acceleration a [START_REF] Faraday | On the forms and states assumed by fluids in contact with vibrating elastic surfaces[END_REF]. By increasing the acceleration from zero, the system undergoes a bifurcation from a simple, well-defined state (flat interface with zero velocity in the bulk) to a more complex, destabilized state: at a given acceleration a c called the critical acceleration, the flat interface goes unstable and a surface wave pattern characterized by the critical wave number k c develops (1; 2; 3; 4). When the driving acceleration is further increased above a c , secondary instabilities take place [START_REF] Edwards | Patterns and quasi-patterns in the Faraday experiment[END_REF], leading at even higher accelerations to spatio-temporal chaos [START_REF] Kudrolli | Superlattice patterns in surface waves[END_REF]. Together with the Rayleigh-Bénard convective instability, the Faraday instability has often been described as a paradigm for the study of dynamical systems at the laboratory scale [START_REF] Manneville | Dissipative Structures and Weak Turbulence[END_REF].

Until about a decade ago, the Faraday instability had been intensively studied in Newtonian fluids (4; 5; 6; 8; 9). Only recently have vertically vibrated complex fluids been the subject of experimental (10; 11; 12; 13; 14) and theoretical research (15; 16). It was shown that the coupling between the microstructure of complex fluids and the Faraday instability either only slightly modifies the instability (e.g. by affecting the critical acceleration (10; 11) or by delaying the transition to disordered states [START_REF] Kityk | Delay of disorder by diluted polymers[END_REF]) or presents non-intuitive behaviours such as stabilized "persistent holes" in shear-thickening fluids [START_REF] Merkt | Persistent holes in a fluid[END_REF]. With the notable exception of concentrated colloidal and granular suspensions (12; 17), most of these works were performed on fluids whose viscoelasticity is only a small perturbation (10; 13; 14) to Newtonian behaviour. The case of a complex fluid with significant viscoelasticity was addressed in Ref. [START_REF] Wagner | Faraday waves on a viscoelastic liquid[END_REF] and it was shown that viscoelasticity could lead to a harmonic response of the fluid surface (i.e. at f ) instead of the classical subharmonic response (i.e. at f /2).

In a previous paper [START_REF] Ballesta | Signature of elasticity in the Faraday instability[END_REF] we reported onset measurements of the Faraday instability in a semi-dilute surfactant solution (CPyCl-NaSal at 4% wt.) known to form an entangled network of "wormlike" micelles. In this very viscoelastic fluid, the critical acceleration and wave number were shown to present oscillations as a function of the driving frequency f . We interpreted these oscillations in terms of standing elastic waves between the disturbed surface and the container bottom.

The present article is devoted to a more thorough study of this striking effect of viscoelasticity on the Faraday instability. After a brief description of our wormlike micellar system and of our experimental setup, we first present new sets of Faraday experiments in various wormlike micelle solutions and explore the dependence of the fluid response on surfactant concentration and on temperature. Then, the interpretation in terms of standing elastic waves is confirmed using two novel experiments: particle imaging velocimetry (PIV) and birefringence under vertical vibrations, which show a non monotonic behaviour of the velocity and deformation fields along the vertical direction. We also use the finite-depth model of ref. [START_REF] Kumar | Vibration-induced interfacial instabilities in viscoelastic fluids[END_REF] to predict the velocity field. This numerical calculation accounts qualitatively for all our experimental observations. Finally, reasons for quantitative discrepancies between the model and the experiments are discussed. In particular, birefringence measurements above onset show that the specific nonlinear features of the rheology of wormlike micelles, namely disentanglement and breakage of the micellar network and shear-induced alignment, do not come into play in our experiments.

1 Viscoelastic fluid: wormlike micelle solution Wormlike micelles are long (typically hundreds of nanometers to tens of microns), cylindrical (a few nanometers in diameter) and flexible aggregates of surfactant molecules in aqueous solution [START_REF] Larson | The Structure and Rheology of Complex Fluids[END_REF]. Depending on the concentration and on the surfactant, wormlike micelle solutions present various characteristic behaviours: shear-thickenning at low concentration [START_REF] Liu | Shear-induced gelation and fracture in micellar solutions[END_REF], shear-induced alignment in the semi-dilute regime [START_REF] Berret | Inhomogeneous shear flows of wormlike micelles: a master dynamic phase diagram[END_REF], and equilibrium isotropic to nematic phase transition at even larger concentrations [START_REF] Berret | Isotropic-to-nematic transition in wormlike micelles under shear[END_REF]. In the following we focus on semi-dilute solutions [START_REF] Berret | Linear rheology of entangled wormlike micelles[END_REF]. As shown by Cates [START_REF] Cates | Reptation of living polymers: dynamics of entangled polymers in the presence of reversible chain-scission reactions[END_REF], the unique viscoelastic behaviour, namely almost a perfect Maxwell fluid [START_REF] Rehage | Rheological properties of viscoelastic surfactant systems[END_REF], of these solutions is due to the interplay of two processes: reptation motion as in conventional polymers [START_REF] Bird | Dynamics of Polymeric Liquids[END_REF] and breaking and recombination of the micelles under thermal agitation. Therefore, the complex viscosity of these solutions can be written:

η * (ω) = η 0 1 + iωτ 2 1 + iωτ 1 , (1)
where η 0 is the fluid zero-shear viscosity, τ 1 the relaxation time, ω the pulsation, and τ 2 a second characteristic time.

For a pure Maxwell fluid, τ 2 = 0. However, due to the presence of the solvent, we shall rather use τ 2 = τ 1 η S /η 0 , where η S ≪ η 0 is the solvent viscosity (in our experiments, η S = η brine = 10 -3 Pa.s and η 0 ≃ 10 Pa.s). Thus, in the case of a Maxwell fluid, there are really only two free parameters η 0 and τ 1 in eq. ( 1) since τ 2 simply derives from the solvent viscosity η S . Another widely used model for the linear rheology of wormlike micelles is the Oldroyd fluid whose complex viscosity is also given by eq. ( 1) but where τ 2 is left as a free parameter (see e.g. refs. (27; 28)). We shall see below that this additional free parameter allows one to account for the behaviour of our solutions over a much larger frequency range than the Maxwell fluid. Note that such a use of the Oldroyd model assumes the validity of the Cox-Merz rule which may be violated in some wormlike micelle solutions for ωτ 1 > 1 (27; 29). We still use fits by the Oldroyd model since equation (1) yields good empirical representation of our data. Also note that more elaborate models that account not only for the solvent but also for high frequency relaxation modes based on Rouse or Zimm approaches may apply to wormlike micelle solutions [START_REF] Fischer | Rheological master curves of viscoelastic surfactant solutions by varying the solvent viscosity and temperature[END_REF]. Finally we introduce the shear modulus as the high-frequency limit of the imaginary part of -ωη

* (ω) i.e. G 0 = lim ω→∞ ℑ(-ωη * (ω)) = η 0 (τ 1 -τ 2 )/τ 2 1 ≃ η 0 /τ 1 .
In the following, we study various wormlike micelle solutions made of cetylpyridinium chloride (CPyCl, from Aldrich) and sodium salicylate (NaSal, from Acros Organics) dissolved in brine (0.5 M NaCl) with a fixed concentration ratio [NaSal]/[CPyCl]=0.5 as described in refs. (21; 25). The CPyCl-NaSal 2 Best fit parameters for an Oldroyd fluid (eq. ( 1) where both τ 1 and τ 2 are free parameters) for CPyCl-NaSal solutions of various concentrations c. concentration of our solutions varies from 2% to 8% wt. This concentration range ensures that (i) the micelles are in the semi-dilute regime, i.e. remain entangled and form a viscoelastic network ( 21) and (ii) the solutions are not too viscous (η 0 ≃ G 0 τ 1 < 100 Pa.s) so that the onset of Faraday waves can be observed.

Linear rheological measurements were performed in the cone-and-plate geometry using a shear rate controlled rheometer (ARES, TA Instruments) with small shear rate oscillations of amplitude 0.1 s -1 (the linear regime extends up to at least 1 s -1 for all concentrations under study). Such measurements yield reliable values for the viscoelastic moduli G ′ (ω) = -ωℑ[η * (ω)] and G ′′ (ω) = ωℜ[η * (ω)] as long as the oscillation frequency f = ω/2π remains smaller than about 40 Hz. Figure 1 shows the dimensionless data G ′ /G 0 and G ′′ /G 0 as a function of ωτ 1 for the various solutions used in the present work. The fact that the G ′′ /G 0 vs ωτ 1 data do not collapse on a single curve at high frequency provides a clear indication that the Maxwell model fails for ωτ 1 > 3. Indeed both G ′ and G ′′ are well accounted for by eq. ( 1) over the whole frequency range provided τ 2 is left as a free parameter (Oldroyd fluid). Tables 1 and2 gather the best fit parameters using eq. ( 1) as a function of surfactant concentration. As shown by the dashed lines in Fig. 1 (see also Fig. 5(a) for raw data measured on a 3% wt. CPyCl-NaSal solution and the corresponding Maxwell and Oldroyd fits), the Maxwell model (where τ 2 = τ 1 η S /η 0 is fixed) leads to a good description of the G ′′ /G 0 data only for ωτ 1 smaller than about 3. This means that higher frequency modes, which are captured by the Oldroyd model, come into play for ωτ 1 > 3, a feature that is classically observed in wormlike micelles [START_REF] Fischer | Rheological master curves of viscoelastic surfactant solutions by varying the solvent viscosity and temperature[END_REF]. Finally a power-law fit of our η 0 vs c data yields an exponent of 2.7±0.3 in rough agreement with the value of 3.3 reported in refs. (23; 22) and consistently with the "fast-breaking" limit for micellar

dynamics (24)

2 Critical acceleration and wave number measurements

Experimental setup

Our experimental setup is similar to the one used in refs. (5; 18). It consists of a cylindrical Plexiglas container of diameter d = 60 mm filled to a height h = 10 mm under the brimful boundary condition [START_REF] Edwards | Patterns and quasi-patterns in the Faraday experiment[END_REF]. This cell is sealed by a Plexiglas cover that prevents evaporation and surface contamination, and vertically vibrated by an electromagnetic shaker (Ling Dynamic Systems V406). Except when stated otherwise, the fluid temperature is controlled to 21±0.5 • C by water circulation beneath the container. The acceleration is measured by a small piezoelectric accelerometer (Endevco 2224C) attached to the cell. The wave number is inferred from images of the surface taken by a CCD camera (Cohu). The illumination technique allows us to detect surface deformation down to an amplitude of about 30 µm [START_REF] Ballesta | Instabilité de Faraday dans les fluides complexes[END_REF]. We checked that our container is large enough for the system to be considered as laterally unbounded by performing experiments in various Newtonian fluids (silicon oils and waterglycerol mixtures) and by comparing the experimental measurements of a c and k c to the numerical calculation of ref. [START_REF] Kumar | Parametric instability of the interface between two fluids[END_REF] for an unbounded viscous fluid. As already mentioned in ref. [START_REF] Ballesta | Signature of elasticity in the Faraday instability[END_REF], the agreement between experiment and theory is almost perfect over the whole range of frequencies f = 20-120 Hz, so that finite-size effects due to lateral boundaries are negligible.

Measurements of a c and k c

At a given driving frequency f , the instability threshold is determined as follows: (i) the acceleration is quickly increased until the surface is fully destabilized then quickly decreased until the instability completely disappears; (ii) from this last value, the acceleration is slowly increased again (by approx. 1%/min) until the interface goes unstable, which yields an "upper limit" a cmax for the critical acceleration; (iii) once the instability appears, we wait for the whole surface to be fully destabilized (which takes about 1 min); (iv) a picture of the surface is taken from which the critical wave number k c is estimated; (v) the acceleration is finally slowly decreased (by approx. 1%/min) until the instability completely disappears, which yields a "lower limit" a cmin for the critical acceleration. In all cases, the upper and lower limits of a c differ by less than 2%: no significant hysteresis is observed and we define a c as the average a c = (a cmin + a cmax )/2. For all our solutions and over the whole range of investigated frequencies f = 20-120 Hz, the surface response was found to be subharmonic i.e. the standing wave pattern oscillates at f /2.

As reported in ref. [START_REF] Ballesta | Signature of elasticity in the Faraday instability[END_REF], the critical acceleration and wave number in wormlike micelles strikingly differ from their equivalent in Newtonian fluids. Indeed both curves a c and k c present pronounced oscillations as a function of the driving frequency f . Figure 2 shows a c and k c vs f for a 3% wt. CPyCl-NaSal solution (see also fig. 1 of ref. [START_REF] Ballesta | Signature of elasticity in the Faraday instability[END_REF] for similar measurements on a 4% wt. solution). A consequence of this peculiar behaviour is that k c does not follow the usual dispersion relation for gravito-capillary surface waves:

ω 2 = (gk c + σk 3 c /ρ) tanh(k c h),
where ω = πf for subharmonic response, g is the acceleration due to gravity, σ the surface tension and ρ the fluid density (3). Instead we find a non-monotone dispersion relation. Since the same experiments performed on Newtonian fluids with similar viscosity and surface tension do not display such behaviour, the oscillations observed in fig. 2 must be linked to the viscoelastic properties of our micellar solutions.

Simple physical interpretation of the oscillations

In ref. [START_REF] Ballesta | Signature of elasticity in the Faraday instability[END_REF] we proposed to interpret the oscillations in a c and k c as the signature of "standing elastic waves" across the cell. Indeed in a strongly viscoelastic fluid, the disturbed surface is likely to generate shear waves that propagate downward across the fluid layer, are reflected at the bottom of the cell, backpropagate toward the surface where they may interact with the surface wave provided the attenuation over a distance 2h is not too large. The velocity of shear waves is given by c(ω) = G ′ (ω)/ρ ≃ G 0 /ρ for a Maxwell or an Oldroyd fluid in the ωτ 1 ≫ 1 limit, which is always verified in our experiments (from table 2, ωτ 1 = 2πf τ 1 is in the range . Using this simplification one can calculate the phase of the reflected shear wave as it reaches the surface. Indeed, the total travel time from the surface to the bottom of the cell and back to the surface is 2h/c, which corresponds to a phase shift (2h/c)(ω/2) since the response of the surface is subharmonic. Moreover the no slip boundary condition at the bottom of the cell leads to an additional phase shift of π so that the phase of the reflected shear wave at the surface is ϕ = ωh ρ/G 0 + π.

If ϕ = 2πn, with n an integer, the reflected shear wave and the surface wave are in phase leading to constructive interference, amplification of the surface wave, and a local minimum of a c . On the other hand, when ϕ = (2n + 1)π, the interference is destructive and surface waves are hindered, leading to a local maximum of a c , hence the oscillations of Fig. 2(a). This simple interpretation, which only focuses on a single reflection and neglects attenuation of the shear wave, leads to the following expression for the distance δf between two maxima of a c :

δf th = 1 h G 0 ρ . (2)
The experiments presented in ref. [START_REF] Ballesta | Signature of elasticity in the Faraday instability[END_REF] were restricted to only one sample at a single surfactant concentration of 4% wt. The main objective of the present work is to further check the prediction of equation ( 2) over the whole semidilute regime. More experiments were thus performed by varying the surfactant concentration, i.e. for different elastic moduli G 0 . Figure 3(a) shows the critical acceleration versus the reduced frequency f /δf th for 3% and 4% wt. CPyCl-NaSal solutions, where the value of G 0 in eq. ( 2) is given by linear rheological measurements. In this representation, critical accelerations oscillate roughly with the same period as a function of reduced frequency. However the maxima of a c do not exactly correspond to integer multiples of f /δf th , which points to a first limitation of the simple physical picture proposed above. We shall come back to this discrepancy in sect. 3.2.

Equation (2) also predicts that δf does not depend on the fluid relaxation time τ 1 . For CPyCl-NaSal wormlike micelle solutions, it is well known that τ 1 greatly depends on the temperature while G 0 is nearly temperature independent [START_REF] Fischer | Rheological master curves of viscoelastic surfactant solutions by varying the solvent viscosity and temperature[END_REF]. Indeed on our 4% wt. Finally, more experiments at different fluid heights, surfactant concentrations, and temperatures were performed. Figure 4 gathers experimental values of δf inferred from these measurements as a function of δf th predicted by eq. ( 2) where G 0 was extracted from linear rheology. This plot clearly shows that eq. ( 2) holds for all our experiments so that our interpretation in terms of standing elastic waves is relevant.

Analytical and numerical predictions of the linear stability analysis

In this section, we compare our experimental results to the numerical approach first proposed by L. Tuckermann and K. Kumar (8) and adapted to viscoelastic fluids by S. Kumar ( 15) by taking into account a complex, frequency dependent viscosity η * (ω).

Numerical predictions for a c and k c

The numerical scheme relies on a linear stability analysis based on Floquet theory. Fourier expansions of the velocity field v and of the surface deformation ξ together with linearization of the Navier-Stokes equations lead to a N × N eigenvalue problem, where N corresponds to the cut-off in the Fourier expansions (we chose to use N = 8, but N = 4 already provides good estimates). The eigenvalues a(ω 0 , k) correspond to marginal values of the acceleration i.e. for a given pulsation ω 0 = 2πf of the excitation, surface modes with wave number k go unstable when a > a(ω 0 , k). For a fixed ω 0 , a = a(ω 0 , k) defines a set of so-called "resonance tongues" in the (a, k) plane. Each resonance tongue is associated to a different resonance frequency ω = nω 0 /2 of the surface, where n is a positive integer. Odd (resp. even) values of n correspond to a subharmonic (resp. harmonic) response. This linear stability analysis allows one to determine the critical acceleration a c as the global minimum of the resonance tongues and to define the corresponding wave number as the critical wave number k c . In all the cases investigated in the work, a c and k c were found to lie on the first resonance tongue, so that the surface response is always subharmonic at ω = ω 0 /2 = πf . The only severe constraint of the numerical method of ref. [START_REF] Kumar | Vibration-induced interfacial instabilities in viscoelastic fluids[END_REF] is that it requires the knowledge of η * (ω) over a wide range of frequencies, typically 0-N f /2 ≃ 2f for an external forcing frequency f . Since f can reach 120 Hz in our experiments, rheological measurements of η * (ω) are in principle required up to about 250 Hz, which is far above the upper limit of our rheometer (about f = 40 Hz). Thus, above f ≃ 20 Hz, extrapolations of η * (ω) were used in the numerical calculation based on the two models discussed above: the Maxwell and the Oldroyd models. To account for viscoelasticity at higher frequencies, the rheological measurements were fit by an Oldroyd fluid. As already pointed out, the Oldroyd model has two free parameters τ 1 and τ 2 and leads to a better fit of the viscoelastic moduli for f > 2 Hz (see the continuous lines in fig. 5(a)). The linear stability analysis using the corresponding Oldroyd fluid also yields a much better agreement between numerical and experimental a c and k c . In particular the amplitude of the oscillations now decreases with increasing frequency as seen in the experiment. However, for f > 60 Hz, the critical acceleration found numerically remains smaller than the experimental one by about 20%.

At this stage the lack of rheological data at frequencies larger than 40 Hz does not allow us to draw any definite conclusion on whether a better agreement between theory and experiment would be achieved by using a more elaborate rheological model at high frequencies. Experimental imperfections such as absorption of shear waves at the bottom of the container or additional dissipation at the walls, which are not taken into account in the numerical calculation, may also account for the observed discrepancies. Another possibility is the fact that surface waves may affect the microstructure of the fluid in a nonlinear way (see sect. 5 below). Therefore we did not try to look further for more ap-propriate rheological models but we rather focused on the original features of the Faraday instability in our strongly viscoelastic fluid, namely the presence of elastic waves across the sample.

Calculation of the velocity field

Interestingly the velocity field v of a vertically vibrated viscoelastic fluid at onset can be inferred from the numerical model of ref. [START_REF] Kumar | Vibration-induced interfacial instabilities in viscoelastic fluids[END_REF]. Indeed, if the velocity vector is expanded in terms of a Fourier series

v = u e x + v e y + w e z = ∞ n=0 v n (x, y, z)e i(n+α)ω 0 t , (3)
with α = 0 (resp. α = 1/2) for a harmonic (resp. subharmonic) response, then ref. [START_REF] Kumar | Vibration-induced interfacial instabilities in viscoelastic fluids[END_REF] shows that the n th Fourier mode of the vertical component w of the velocity reads

w n (x, y, z) = v n (x, y, z) • e z = f (x, y)w n (z) (4)
= f (x, y) a 1 e kcz + a 2 e -kcz + a 3 e qnz + a 4 e -qnz ,

where

∂ 2 ∂x 2 + ∂ 2 ∂y 2 f (x, y) = k 2 c f (x, y) , (6)
q 2 n = k 2 c + i (n + α)ω 0 ρ η * ((n + α)ω 0 ) , (7)
and k c is the critical wave number.

The a i 's are deduced from the boundary conditions and depend on n, α, k c , η * , ω, and h (see ref. [START_REF] Kumar | Vibration-induced interfacial instabilities in viscoelastic fluids[END_REF] for their full expressions). They are also proportional to the (n + α) th Fourier component of the surface wave amplitude ξ n . However, since eq. ( 5) derives from a linear stability analysis, the velocity field is only known up to some multiplicative constant and a weakly nonlinear approach would be required to get a quantitative estimate of the velocity amplitude as a function of the distance from instability threshold. In the present work, we shall assume that our experiments are performed close enough to onset so as to allow us to use eq. ( 5) where the absolute value of the velocity will be left as a free parameter. Finally, the horizontal velocity u e x + v e y can be calculated from w and the incompressibility equation ∇ • v = 0. Figure 6 presents the first subharmonic mode (n = 0, α = 1/2) of the velocity field calculated from eq. ( 5) for an Oldroyd fluid. The velocity field consists of an array of counterrotating rolls whose number across the fluid height increases with increasing frequency. Of course, these counterrotating rolls are not steady rolls but oscillate in time at the frequency of the surface response. As shown by eq. ( 7), oscillations along the z direction arise from the term a 3 e qnz + a 4 e -qnz in eq. ( 5). Indeed, since q n is a complex number, it can be rewritten as q n = ik n + 1/l n , where k n is the vertical wave number and l n is the attenuation length for the (n + α) th mode.

Focusing on the case (n = 0, α = 1/2) and neglecting higher-order modes in eq. ( 5), it is straightforward to understand why couterrotating rolls should be detected in experiments on wormlike micelles but should remain unseen in Newtonian (or weakly viscoelastic) fluids. Indeed, with k c = 600 m -1 and ω 0 = 380 rad.s -1 in standard experimental conditions and for an Oldroyd fluid with typical parameters η 0 = 20 Pa.s, τ 1 = 0.5 s, and τ 2 = 0.5 ms, one gets l 0 = 16 mm, which is greater than the vertical wavelength 2π/k 0 = 9 mm. On the other hand, for a Newtonian fluid of same absolute viscosity η = |η * (ω 0 /2)| = 0.43 Pa.s, the calculation yields l 0 = 1.6 mm ≪ 2π/k 0 = 36 mm. Note also that in the latter case l 0 < h = 10 mm so that elastic waves are fully damped when they reach the bottom of the cell and no stationary elastic wave pattern can form across the fluid height.

Finally eq. ( 7) also allows one to recover eq. ( 2) in the case of an Oldroyd fluid and when only the first subharmonic mode is considered. In such a case, inserting eq. ( 1), n = 0, and α = 1/2 in eq. ( 7) leads to:

q 2 0 = k 2 c + i ωρ η 0 1 + iωτ 1 1 + iωτ 2 , (8)
where ω = ω 0 /2 is the pulsation of the surface response. Since in our experiments ωτ 2 ≪ 1 ≪ ωτ 1 (see table 2) and k 2 c η 0 /ρω 2 0 τ 1 ≪ 1, the zero-order approximation of eq. ( 8) yields:

q 0 ≃ ik 0 ≃ iω ρ G 0 . ( 9
)
As explained above the oscillations of a c and k c are linked to the wave number q 0 . More precisely one period of the oscillation of a c corresponds to an increase of k 0 by π/h (addition of a local maximum of v) i.e. an increase of ω by π G 0 /ρh 2 . For a subharmonic response this leads to δf = G 0 /ρh 2 , i.e. eq. ( 2). Thus our simple interpretation in terms of standing elastic waves is consistent with the analytical expression [START_REF] Manneville | Dissipative Structures and Weak Turbulence[END_REF] at least in a first approximation.

Expanding q 0 to first order, one gets :

q 0 ≃ iω ρ G 0 - ik 2 c 2ω G 0 ρ + ρ G 0 1 + τ 1 τ 2 ω 2 2τ 1 . (10)
With q 0 = ik 0 + 1/l 0 , the previous equation leads to

k 0 = ω ρ G 0 - k 2 c 2ω G 0 ρ , (11)
l 0 = G 0 ρ 2τ 1 1 + τ 1 τ 2 ω 2 . ( 12
)
Since k c increases roughly linearly with ω, eq. ( 2) becomes less accurate as frequency increases due to the corrective term in eq. ( 11). This explains why maxima of a c do not correspond to integer values of f /δf th in fig. 3(a). Moreover eq. [START_REF] Merkt | Persistent holes in a fluid[END_REF] shows that the attenuation length l 0 decreases with ω. At high frequencies oscillations should thus disappear. This accounts qualitatively for the decrease of the amplitude of the oscillations in a c and k c observed experimentally and numerically.

So far we have shown that the linear stability analysis of ref. [START_REF] Kumar | Vibration-induced interfacial instabilities in viscoelastic fluids[END_REF] predicts an oscillating behaviour for a c and k c . The calculation of the velocity field close to instability threshold unveils the existence of counterrotating rolls. Such rolls, unseen in Newtonian fluids, are a clear signature of viscoelasticity in the Faraday experiment. Our next step is to provide direct experimental evidence for the existence of these "elastic rolls."

Experimental study of the velocity field

In this section we first describe the experimental apparatus used to access the velocity field in the Faraday experiment. Measurements are then presented and discussed in light of the above theoretical predictions.

Particle imaging velocimetry setup

In order to provide experimental evidence for the existence of large-scale counterrotating rolls in our viscoelastic fluid, particle imaging velocimetry (PIV) experiments were performed under vibrations. Since wormlike micelle solutions are transparent, the scattering properties of our system were enhanced by seeding the solutions with 0.1% wt. hollow glass spheres (Sphericel, Potters Industries) of mean radius 11.7 µm and density 1.1. Thanks to the high zero-shear viscosity of the micellar solutions, sedimentation of the spheres is negligible over the ∼8 hour duration of a Faraday experiment (a rough estimate yields a sedimentation velocity of about 0.03 µm.s -1 ). We also checked that the rheological properties of the micellar system were not significantly affected by the addition of 0.1% wt. scatterers i.e. remain very well described by an Oldroyd fluid. The best fit parameters obtained for the Oldroyd model on a 4% wt. CPyCl-NaSal solution seeded with 0.1% wt. scatterers are η 0 = 24.3 Pa.s, τ 1 = 0.87 s, and τ 2 = 0.55 ms, very close to those of the corresponding solution without scatterers (see table 2).

In order to visualize the motion of the scatterers under vibrations, we use a parallelepipedic Plexiglas cell of length L = 180 mm (in the x direction), width l = 4 mm (in the y direction), and depth h = 10 mm (in the z direction) under the brimful boundary condition. Thanks to the large aspect ratio L/l, a onedimensional surface wave pattern is forced perpendicular to the x direction [START_REF] Douady | Experimental study of the Faraday instability[END_REF]. The cell is lit in the (x, z) plane by a vertical laser sheet crossing the cell at y 0 = 2 mm. Images of the (x, z) plane are recorded by a high-speed CCD camera (Mikroton MC1310) at a frequency ∼ 10f during 40 periods of the vertical oscillations.

The first step in the image processing is to detect the vertical position of the cell by following a bright spot made by a laser at the bottom of the cell (around (x ≃ 0, z ≃ -10 mm in fig. 7, see also fig. 11c). The displacement field of the scatterers and the two-dimensional velocity vector u e x + w e z at a given point (x, z) are then computed in the moving reference frame of the vibrated cell by cross-correlating the intensity fields of two consecutive images over small square regions of size 0.25 mm 2 . Figure 7 shows a typical image of the micellar solution along with the velocity field obtained by PIV under vibrations.

For the sake of clarity and since y is fixed to y 0 in all the PIV experiments, we shall drop the dependence on y of the various variables defined in the previous section whenever such dependence is not needed. Note also that the ±1 pixel uncertainty on the vertical position of the cell may remain too large compared to the vertical displacement of the fluid and in some cases leads to artifacts in the estimation of the vertical component w of the velocity. Therefore, in the following, we focus on the velocity component u along the x direction. In order to compare experimental results to numerical calculations, we extract the mode oscillating at ω 0 /2 from u(x, z, t) i.e. u 0 (x, z) as defined by eq. ( 3) with α = 1/2.

Experimental velocity measurements compared to numerical predictions

Shown in fig. 8 are the amplitude u(x, z) and phase φ(x, z) of u 0 (x, z). It is clearly seen that the velocity does not decrease exponentially with z as would be expected for a Newtonian fluid, but rather presents local maxima consistent with the presence of counterrotating rolls (see fig. 6(b) and (c) for a qualitative comparison with numerical results).

In the case of our one-dimensional pattern with large aspect ratio, one may recover the horizontal velocity u from the numerical calculation of w by assuming < u(x, z) > averaged over x = 6-9 mm. The continuous lines are the predictions for φ and u given by equations ( 5) and ( 13) with n = 0 and α = 1/2 and using an Oldroyd fluid with η 0 = 24.3 Pa.s, τ 1 = 0.87 s, and τ 2 = 0.55 ms. Both the amplitude of u 0 and the reference phase are free parameters.

L ≫ l. Indeed under this assumption and the no slip condition at the cell border (v = 0 for y = 0, l) one gets v = 0 and w(x, y, z) = cos(k x x) cos(k y y)w(z), with k y = π/l and k 2

x = k 2 c -k 2 y . This is equivalent to having a half mode along the y direction. Next, incompressibility leads to:

u n (x, y, z) = - 1 k x sin(k x x) cos(k y y) dw n dz (z) , (13)
where w n (z) is given by eq. ( 5).

Figures 8(c) and 8(d) compare the experimental results (averaged over x = 6-9 mm) to the numerical prediction of eq. ( 13) for (n = 0, α = 1/2) corresponding to the best fit of the linear rheological measurements by an Oldroyd fluid. The number of rolls as well as their positions in the cell are very well accounted for by the numerical calculation. We believe the discrepancies near the surface, especially the large experimental values of u 0 , to be linked to the non-zero amplitude ξ 0 of the surface waves that prevents an accurate estimation of the velocity field.

Birefringence experiments

Measurements of a c and k c , experimental characterization of the velocity field close to onset, and comparison with numerical predictions have confirmed the interplay of parametrically excited surface waves and elastic waves in a strongly viscoelastic fluid under vertical vibrations. Thus the presence of a microstructure, namely an entangled network of wormlike micelles, deeply affects the Faraday instability. We now address the question raised in ref. [START_REF] Ballesta | Signature of elasticity in the Faraday instability[END_REF] of whether the instability modifies the microstructure. In particular, in the nonlinear regime, semi-dilute solutions of wormlike micelles are known to disentangle and align under shear leading to a shear-induced isotropic-nematic transition (19; 33; 34). Therefore one may wonder whether the flow induced by Faraday waves is strong enough to induce such a transition, which could provide an alternative explanation for the discrepancy between experiments and linear stability analysis observed in sect. 3.1.

Flow-induced alignment have already been much studied in semi-dilute wormlike micelle solutions using birefringence (35; 36; 37; 38; 39; 40; 41). Following these previous works and in order to investigate the deformation of the microstructure under vertical vibrations, we performed birefringence experiments close to onset and further above the instability threshold. The experimental setup is modified as follows. A parallelepipedic glass cell of length L = 72 mm and width l = 6 mm is filled with the micellar solution to a height h = 7.1 mm. This cell is set between two polarizers and lit from backward by an extended white light source. The high-speed CCD camera used previously for PIV measurements captures the transmitted light at about 1000 fps. For given vibration frequency f and acceleration a, two measurements can be performed depending on the configuration of the polarizers. When the polarizers are aligned, the contour of the surface wave is easily accessed by image processing. This yields the surface deformation ξ(x, t) from which the amplitude of the surface wave ξ 0 may be extracted. When the polarizers are crossed, the birefringence intensity I(x, z, t) is recorded and analyzed.

First observations and qualitative discussion

Figure 9 shows a typical birefringence pattern obtained at a given time t 0 between crossed polarizers on an 8% wt. CPyCl-NaSal solution slightly above the onset of Faraday waves. Evenly spaced bright spots fill the whole cell with the same horizontal periodicity as the surface wave. Such bright spots are observed below antinodes of the stationary surface wave. The birefringence intensity I(x, z, t) oscillates in phase with ξ(x, t) so that bright spots alternatively turn to black when the surface deformation goes through zero. Regions in the bulk located below nodes of the surface wave remain dark at all times.

Based on the previous analysis of the velocity field, the interpretation of the birefringence pattern is rather straightforward. Indeed, from the velocity fields of figs. 6 or 7, the two-dimensional rate of deformation tensor γ can easily be estimated using γi,j = ∂ i v j , with i and j being x, y, or z and (v x , v y , v z ) = (u, v, w). In the (x, z) plane, γ (not shown) is composed of localized compression and stretching zones that oscillate in time. Compression and stretching result in an oscillatory deformation of the micellar network, which leads to the observed birefringent spots.

Moreover, as shown in fig. 10(a), the number n spot of bright spots along the vertical direction increases with increasing frequency. Figure 10(b) proves that drops of k x are linked to the discretization of the number of spots along the z direction. This is a direct consequence of the link between the vertical wave number and k c (see eq. ( 7)): each time the number of spots increases by one (or equivalently each time a new set of counterrotating rolls fits in the cell height), the vertical wave number increases (by π/2h) and the surface wave number abruptly decreases.

Finally, if one defines an equivalent shear rate near the surface as γ = ξ 0 k c ω 0 /2 (15), a rough estimate with ξ 0 = 30 µm (the minimum amplitude that our setup can detect) yields γ ≃ 3 s -1 . Since such a shear rate is of the same order of magnitude as the critical shear rate γc = 2.6/τ 1 for micelle disentanglement and alignment [START_REF] Spenley | Nonlinear rheology of wormlike micelles[END_REF], nonlinear rheology may come into play and we should check for the possibility of an alignment transition of the micelles under vervibrations. The next sections are devoted to a detailed analysis of the birefringence intensity and to more measurements above onset.

Birefringence measurements above onset

For an optically anisotropic medium composed of elongated and oriented micelles, the transmitted birefringence intensity takes the following general expression (37):

I(x, z, t, θ) = I ′ sin 2 δ(x, z, t) 2 sin 2 2(χ(x, z, t) -θ) , (14)
where the retardation δ reads

δ(x, z, t) = 2πl λ ∆n(x, z, t), (15)
I ′ is the incident intensity, λ ∼ 500 nm is the light wavelength, and ∆n(x, z, t) is the birefringence intensity. χ(x, z, t) is defined as the angle between some reference direction and the average micelle direction. θ is the angle between the polarization of the incident beam and the reference direction.

To address the issue of whether or not the nonlinear deformation regime is reached and the micelles disentangle, we shall make use of the so-called "stressoptical rule" according to which the quantity ∆n sin 2χ is proportional to the shear stress σ (37; 38). This rule was shown to hold at low shear stresses in CTAC wormlike micelle solutions for various salt (NaSal) concentrations [START_REF] Decruppe | Flow birefringence, stress optical rule and rheology of four micellar solutions with the same low shear viscosity[END_REF]. Although to our knowledge no experimental validation of the stress-optical rule is available for our CPyCl-NaSal in brine system, we may assume that it is general enough to also hold in our experiments. As long as the network structure remains intact and the micelles remain entangled, the deformation is elastic, i.e. σ ∝ γ, so that one should have ∆n sin 2χ ∝ γ, where γ stands for the deformation. However the stress-optical rule no longer applies when the micelles disentangle and the network is broken (37; 38).

In the case δ ≪ 1 and for θ = 0, which is always verified for wormlike micelles (37), eq. ( 14) and the stress-optical rule lead to

I(x, z, t) ∝ ∆n(x, z, t) 2 sin 2 2χ(x, z, t) ∝ γ(x, z, t) 2 . ( 16
)
Focusing on the first subharmonic mode, eq. ( 16) shows that I(x, z, t) should oscillate at the forcing pulsation ω 0 with an amplitude I 0 (x, z) ∝ γ 0 (x, z) 2 , where γ 0 (x, z) denotes the amplitude of the first subharmonic mode of γ(x, z, t). Experimentally γ 0 is varied by increasing the driving acceleration a above the critical acceleration a c . Indeed the deformation at the surface γ 0 (x, z = ξ 0 (x)) is directly proportional to the surface wave amplitude through γ 0 = k c ξ 0 . Thus increasing a leads to larger surface wave amplitudes and to larger deformations, which allows one to test eq. ( 16). 19) over a = 88.3-90.1 m.s -2 and the continuous line is the best fit by eq. ( 20) over a = 88.3-97 m.s -2 .

More precisely, we first extract the position of the surface ξ(x, t) from images recorded with aligned polarizers. The subharmonic amplitude of the deformation ξ 0 (x) is then computed and fitted to ξ 0 (x) = ξ 0 | cos(k x x)| (see fig 11(a)), which yields the amplitude of the deformation γ 0 = k c ξ 0 at the surface. Since the dependence of γ 0 (x, z) on z is decoupled from the dependence on x (see also eq. ( 4)), one has γ 0 (x, z) = γ 0 | cos(k x x)|f (z), where f (z) does not need to be specified in order to test the scaling law [START_REF] Müller | Faraday instability in a linear viscoelastic fluid[END_REF].

Figure 12 shows the evolution of the surface wave amplitude ξ 0 with the driving acceleration a above onset. In the case of the Faraday instability, it is well known that ξ 0 should obey the following amplitude equation [START_REF] Douady | Experimental study of the Faraday instability[END_REF]:

τ g dξ 0 dt = ǫξ 0 - ξ 3 0 l 2 3 - ξ 5 0 l 4 5 + ... , (17)
where τ g is the characteristic growth time of the instability, ǫ = a/a c -1 is the reduced acceleration, and l 3 and l 5 are characteristic lengths linked to the dissipation process. At steady state eq. ( 17) becomes:

ǫξ 0 - ξ 3 0 l 2 3 - ξ 5 0 l 4 5 + ... = 0 . (18) 22
If the expansion is truncated to third order, one finds the usual expression:

ξ 0 = l 3 √ ǫ . (19)
As seen in fig. 12, eq. ( 19) accounts for the experimental data over a rather restricted range of accelerations and the ξ 5 0 term should be included [START_REF] Wernet | Amplitude measurements of Faraday waves[END_REF]. In this case, solving eq. ( 18) leads to:

ξ 2 0 = l 4 5 2l 2 3   1 + 4ǫ l 4 3 l 4 5 -1   , (20)
which yields a much better description of the ξ 0 measurements.

Once ξ 0 (hence γ 0 ) is known, the birefringence intensity I(x, z, t) is measured from images obtained between crossed polarizers at θ = 0 so that the intensity at the bright spots is maximized. Figure 11(b) shows I(x 0 , z 0 , t) recorded at the position of a bright spot. Contrary to the prediction of eq. ( 16), I(x 0 , z 0 , t) is not perfectly harmonic but contains a significant subharmonic component. This is most probably due to the finite amplitude of the surface waves which induce a small displacement of the position of the birefringence spots at frequency f /2. To decrease the noise in the determination of I 0 , I(x 0 , z 0 , t) is averaged over two horizontally adjacent bright spots and I 0 is defined as the harmonic component of this mean intensity.

Figure 13 presents the evolution of I 0 as a function of γ 0 for two different concentrations and driving frequencies. We find that I 0 ∝ γ 2 0 up to a = 1.1a c . Beyond this point the surface becomes chaotic. Our conclusion is that the stress-optical rule and eq. ( 16) hold and that the micellar network does not break, at least for a < 1.1a c . In particular micelles remain entangled close to the instability threshold and nonlinear rheology cannot be invoked to explain deviations from linear stability analysis. Thus, even if the amplitude of the equivalent shear rate falls into the nonlinear regime, it appears that the micellar network remains intact under vibrations. We suggest that this is linked to the short period of the forcing (1/f ≃ 0.02 s) compared to the characteristic time of our fluid (τ 1 ∼ 0.5-1 s). Rather than the amplitude of the equivalent shear rate, it would probably be more relevant to compare the deformation ξ 0 k c to the critical deformation τ 1 γc for disentanglement and alignment. A good approximation is τ 1 γc ≃ 2.6 while ξ 0 k c ≃ 0.2 for a = 1.1a c . Since ξ 0 k c ≪ τ 1 γc , the fluid deformation over one period of the forcing is too small to break the network and induce the alignment transition.

Conclusion

In this paper we have presented new results for the Faraday instability in wormlike micelles. We have shown that the strong viscoelasticity of these solutions gives rise to original features characterized by oscillations of the critical acceleration and wave number. We clearly linked these oscillations to the presence of standing elastic waves across the cell. Velocimetry experiments have provided evidence that such elastic waves generate counterrotating rolls so that the velocity field does not decrease exponentially with depth. Linear stability analysis was shown to predict and fit both these phenomena. Finally birefringence measurements have shown that the deformation generated at the interface tends to orient the micellar network but is not large enough to induce disentanglement.

We believe that the present experiments may contain worthwile implications for rheological studies of strongly viscoelastic fluids. As noted above and as already mentioned in ref. [START_REF] Raynal | Faraday instability with a polymer solution[END_REF], the Faraday instability constitutes an easy way to probe a complex fluid in an oscillating elongational flow at frequencies above 30 Hz. One may thus wonder whether rheological information can be gained from measurements under vertical vibrations and how the Faraday experiment may be used as a "high" frequency rheometer. For instance, we tried to infer the rheological properties of our micellar solutions by fitting our velocity field

Fig. 1 .

 1 Fig. 1. Linear rheology of 2, 3, 4, 6, and 8% wt. CPyCl-NaSal solutions. (a) Storage modulus G ′ /G 0 and (b) loss modulus G ′′ /G 0 versus ωτ 1 , where ω is the shear oscillation pulsation, τ 1 the main relaxation time, and G 0 = lim ω→∞ G ′ (ω) the shear modulus. The dashed line and the continuous line respectively represent a Maxwell fluid (η 0 = 49 Pa.s, τ 1 = 0.84 s) and an Oldroyd fluid (η 0 = 50 Pa.s, τ 1 = 0.85 s, τ 2 = 0.91 ms).

Fig. 2 .

 2 Fig. 2. (a) Critical acceleration a c and (b) critical wave number k c plotted against the vibration frequency f for a 3% wt. CPyCl-NaSal solution at T = 21 • C. Error bars on a c are estimated from the experimental "upper" and "lower" limits for a c by ±(a cmax -a cmin )/2 (see text). Error bars on k c correspond to the standard deviation of 3 to 6 estimations performed on various areas of the picture of the surface.

Fig. 3 .

 3 Fig. 3. (a) Critical acceleration a c versus f /δf th with δf th = G 0 /ρ/h for a 3% wt. (•) and a 4% wt. (⋄) CPyCl-NaSal solution at T = 21 • C. (b) Critical acceleration a c versus f for a 4% wt. CPyCl-NaSal solution at 23 • C (△) and 19 • C (▽).

Fig. 4 .

 4 Fig. 4. Experimental δf (•) versus predicted δf th = G 0 /ρ/h. The continuous line is the first diagonal. tures led to crystallization of the sample), and G 0 = 32 Pa and τ 1 = 0.37 s at T = 23 • C: a 4 • C change in temperature only induces a 3% change in G 0 while τ 1 is doubled.Figure 3(b) shows a c measurements performed at T = 19 • C and T = 23 • C (slightly below room temperature to prevent condensation on the Plexiglas cover). As expected from eq. (2) the experimental curves are almost undistinguishable.

Fig. 5 .

 5 Fig. 5. (a) Linear rheology of a 3% wt. CPyCl-NaSal solution: experimental viscoelastic moduli G ′ (•) and G ′′ (•) versus shear oscillation frequency f . The dotted line and the continuous line respectively represent the best fits by a Maxwell fluid over f = 0-3 Hz (see table 1), and by an Oldroyd fluid over f = 0-40 Hz (see table 2). (b) Critical acceleration a c and (c) critical wave number k c versus vibration frequency for a 3% wt. CPyCl-NaSal solution. The dotted (resp. continuous) lines are the result of the numerical calculation for the previous Maxwell (resp. Oldroyd) fluid.

Figure 5 (

 5 Figure 5(a) shows the best fit of the viscoelastic moduli of our 3% wt. CPyCl-NaSal solution by a Maxwell fluid, which yields an accurate description of the experimental G ′ and G ′′ up to f ≃ 2 Hz. Feeding the numerical calculation of ref. (15) by this Maxwell model leads to the values of a c and k c shown as dotted lines in fig. 5(b) and (c).The first noticeable result is the presence of strong oscillations in a c with the same periodicity as the experimental ones. Moreover the evolution of k c is qualitatively similar to that of the experimental data. Still experiment and theory remain quantitatively different. The critical acceleration predicted numerically is smaller than the experimental a c by about an order of magnitude and, for both a c and k c , the amplitude of the oscillations predicted from the Maxwell model increases with frequency whereas it is clearly seen to decrease in the experiment. Since the vibration frequency is always greater than 20 Hz, these discrepancies are most probably linked to the poor fit of G ′′ by the Maxwell model for f > 2 Hz.

Fig. 6 .

 6 Fig. 6. (a) Numerical velocity field v(x, z, t 0 = 0) for an Oldroyd fluid with η 0 = 20 Pa.s, τ 1 = 0.5 s, and τ 2 = 0.5 ms vibrated at f = 70 Hz. (b) Phase φ(x, z) and (c) amplitude u(x, z) of the x-component of the velocity v(x, z, t).

Fig. 7 .

 7 Fig. 7. Picture of a 4% wt. CPyCl-NaSal solution seeded with hollow glass spheres recorded under vibrations at f = 75 Hz by the high-speed CCD camera. The arrows show the velocity field u e x + w e z computed by PIV on two consecutive images. The white spots around x ≃ 0 correspond to the laser beam used for detecting the bottom of the cell.

Fig. 8 .

 8 Fig. 8. (a) Phase φ(x, z) and (b) amplitude u(x, z) of u 0 (x, z) measured by PIV in a 4% wt. CPyCl-NaSal solution vibrated at f = 75 Hz. (c) < φ(x, z) > and (d) < u(x, z) > averaged over x = 6-9 mm. The continuous lines are the predictions for φ and u given by equations (5) and (13) with n = 0 and α = 1/2 and using an Oldroyd fluid with η 0 = 24.3 Pa.s, τ 1 = 0.87 s, and τ 2 = 0.55 ms. Both the amplitude of u 0 and the reference phase are free parameters.

Fig. 9 .Fig. 10 .

 910 Fig. 9. Typical birefringence pattern I(x, z, t 0 ) of an 8% wt. CPyCl-NaSal solution vibrated at f = 80 Hz.

Fig. 11 .

 11 Fig. 11. (a) Surface wave amplitude ξ(x 0 , t), (b) birefringence intensity I(x 0 , z 0 , t), and (c) position of the bottom of the cell b(t) for an 5% wt. CPyCl-NaSal solution vibrated at f = 65 Hz. In (b) x 0 = 7.5 mm and z 0 = -2.1 mm were chosen to coincide with a birefringence spot.

Fig. 12 .

 12 Fig. 12. (a) Surface wave amplitude ξ 0 versus acceleration a for an 8% wt. CPyCl-NaSal solution vibrated at f = 80 Hz. (b) ξ 2 0 versus a. The dotted line is the best fit by eq. (19) over a = 88.3-90.1 m.s -2 and the continuous line is the best fit by eq. (20) over a = 88.3-97 m.s -2 .

Fig. 13 .

 13 Fig. 13. Square root of the transmitted intensity I 0 versus the deformation γ 0 for a 5% wt. CPyCl-NaSal solution vibrated at f = 65 Hz (⋄) and for a 8% wt. CPyCl-NaSal solution vibrated at f = 80 Hz (•). The continuous line is the first diagonal. (a) Linear scales. (b) Logarithmic scales.

Table 1

 1 Best fit parameters for a Maxwell fluid (eq. (1) where τ 2 = τ 1 η S /η 0 is fixed) for CPyCl-NaSal solutions of various concentrations c.

	c (% wt.)	2	3	4	6	8
	η 0 (Pa.s) 2.97	12.8	27.6 74.1 129
	τ 1 (s)	0.421 0.713 0.882 1.08 1.11
	c (% wt.)	2	3	4	6	8
	η 0 (Pa.s) 2.95	12.6	27.7	77.5	129
	τ 1 (s)	0.411 0.714 0.891 1.13	1.10
	τ 2 (ms)	2.00 0.754 0.611 0.344 0.849
	Table

measurements to equations (3)(4)(5)(6)(7) without assuming any rheological model (31). Although the results did not prove conclusive enough (probably due to boundary effects as mentioned at the end of sect. 3.1), this may still constitute an interesting direction for future research. Finally another perspective for this work would be to study complex fluids with shorter characteristic times, e.g. liquid crystals, in which the surface waves may induce microstructural changes.

The authors wish to thank P. Lettinga and B. Pouligny for fruitful discussions. S. Lerouge is thanked for advice on the birefringence experiments and P. Snabre for technical help on the PIV measurements.


Marion Richard

Camille Trottier

Marc Verdegem

Jérôme Hussenot

Variation sources of marine periphyton

Keywords: periphyton, fouling, artificial substrate, marine pond, aquaculture

published or not. The documents may come

Submersion time, depth, substrate type and sampling method as variation sources of marine periphyton

Introduction

Periphyton refers to the entire complex of attached aquatic biota on submerged substrates, including associated non-attached organisms and detritus [START_REF] Van Dam | The potential of fish production based on periphyton[END_REF]. This assemblage comprises bacteria, fungi, protozoa, phyto and zoo-plankton, benthic organisms and detritus (Azim et al., 2005). It can be used as additional food in aquatic production systems. Aquaculture based on periphyton was originally derived from traditional fishing methods known in Africa as Acadja [START_REF] Welcomme | An evaluation of the acadja methods of fishing as practised in the coastal lagoons of Dahomey (West Africa)[END_REF] and in Asia as Kathas and Samarahs (Van Dam et al., 2002). Artificial substrates are added into aquatic system to enhance the food availability. This semi-extensive aquaculture system is well known to increase the production of fish [START_REF] Ramesh | Comparison of three plant substrates for enhancing carp growth through bacterial biofilm[END_REF][START_REF] Umesh | Enhancing growth of common carp, rohu an Mozambique tilapia through plant substrate: the role of bacterial biofilm[END_REF]Azim et al. 2001a).

Although widely tested in freshwater fish culture (Azim et al., 2005), the use of periphyton in brackish or marine waters [START_REF] Van Dam | The potential of fish production based on periphyton[END_REF][START_REF] Huchette | Periphyton-based cage aquaculture, Periphyton: Ecology, exploitation and management[END_REF][START_REF] Khatoon | Formation of periphyton biofilm and subsequent biofouling on different substrates in nutrient enriched brackishwater shrimp ponds[END_REF] is limited to shrimp [START_REF] Bratvold | Effects of sand sediment and vertical surfaces (AquaMats TM ) on production, water quality, and microbiological ecology in an intensive Litopenaeus vannamei culture system[END_REF][START_REF] Moss | Effects of artificial substrate and stocking density on the nursery production of pacific white shrimp Litopenaeus vannamei[END_REF][START_REF] Arnold | Intensive production of juvenile tiger shrimp Penaeus monodon: An evaluation of stocking density and artificial substrates[END_REF] and abalone cultures [START_REF] Kawamura | Importance of periphyton in Abalone culture, Periphyton: Ecology, exploitation and management[END_REF].

Variation of periphyton quantity and quality depends on a range of factors such as (i) submersion time (Azim and Aseada, 2005), (ii) substrate type [START_REF] Ramesh | Comparison of three plant substrates for enhancing carp growth through bacterial biofilm[END_REF][START_REF] Keshavanath | Use of articicial substrates to enhance production of freshwater herbivorous fish in pond culture[END_REF]Azim et al., 2002a), and (iii) light intensity and quality [START_REF] Kirk | Light and photosynthesis in Aquatic ecosystems[END_REF][START_REF] Goldsborough | Periphyton in freshwater lakes and wetlands, Periphyton: Ecology, exploitation and management[END_REF]. The latter is strongly influenced by the depth of the substrates [START_REF] Asaeda | Spatial structure and populations of a periphyton community: a model and verification[END_REF]. Thus, Azim et al. (2001aAzim et al. ( , 2003b) ) waited minimum 2 weeks to allow periphyton to develop on the substrates before stocking fishes. [START_REF] Keshavanath | Use of articicial substrates to enhance production of freshwater herbivorous fish in pond culture[END_REF] observed that fish production based on periphyton depends on artificial substrate type and preferred to use bamboo rather than PVC pipes or sugarcane bagasse bundles when culturing masheer (Tor khudree) fingerlings. Azim et al. (2001bAzim et al. ( , 2002aAzim et al. ( , 2004a) ) and [START_REF] Keshavanath | Use of articicial substrates to enhance production of freshwater herbivorous fish in pond culture[END_REF] pooled several sub-samples of periphyton collected at equally spaced depths along vertical substrates to analyse the composition of periphyton. This pooled sample was considered by these authors to represent the mean composition of periphyton developed on substrate, going from the photic zone close to the surface to the aphotic zone above the bottom.

The potential contributions of semi-extensive aquaculture to environmental protection and restoration of coastal areas have been clearly recognised within EU policy. The SEACASE program (Sustainable extensive and semi-extensive coastal aquaculture system in Southern Europe) was started in 2007 to develop sustainable extensive and semi-extensive coastal aquaculture systems in Southern Europe [START_REF] Conceição | Sustainable extensive and semi-intensive coastal aquaculture in Southern Europe -the SEACASE project[END_REF]. The present SEACASE study is a preliminary assessment of the feasibility to grow periphyton on artificial substrates in temperate marine ponds. The effects of submersion time, substrate type, water depth, and total or partial sampling methods, on the quantity and quality of periphyton collected, were evaluated. The goals of this study were to identify (i) the best periphyton substrate type and (ii) a methodology of periphyton sampling for further studies on periphyton-based marine aquaculture.

Materials and methods

1. Experimental site and design

The experiment was carried out from 9 May till 6 June 2007 in a 200 m 2 marine pond in the IFREMER-L'Houmeau experimental facilities, located on the Atlantic coast of France, near La Rochelle. Four types of substrates were used for this experiment (Fig. 1): (i) 2.5 cm wide square wooden poles (fir tree: W), and 5 cm wide strips of (ii) smooth fiber-glass (S), mosquito screen (1mm-mesh; m) and (iv) garden netting (5mm-mesh; M). The mean submersion depth of the substrates (± SE) was 76.3 ± 2.7 cm after 15 days of submersion whereas it was 66.3 ± 4.6 cm after 30 submersion days. The mean submerged surface area (± S.E.) was 713 ± 9.3 cm² and equal for each substrate type. Eleven poles or strips (called units) of each substrate type were deployed in the marine pond. The units were put 20 cm apart from the closest other units in 4 parallel rows with 11 units each within a 1.0 m x 2.4 m plot, randomly assigning the different unit types to the available locations. The different strip types were suspended in the water column from iron bars fixed on a horizontal wooden frame standing slightly above the surface on poles driven in the bottom, while the pole units were standing in the sediment, under the iron bar.

Total sampling: Influence of substrate type and submersion time

On sampling days, four units of each substrate type were randomly collected. All the periphyton on each unit was collected. Sampling was done 15 (23 May 2007; T15d) and 30 days after submersion (4-5 June 2007; T30d). Collected units were not placed back. In total, 32 units were collected (4 units/type/date x 4 types x 2 dates).

Sub-sampling: Influence of substrate type and submersion depth

At the end of the experiment (T30d), the remaining 3 units of each substrate type (W, S, m, M) were sampled in a random order. The submerged area of each unit was divided in five 15-cm-segments starting from the bottom (Fig. 2a: 1: 0-15 cm, 2: 15-30 cm, 3: 30-45 cm, 4: 45-60 cm, 5: 60-75 cm). Each 15-cm sub-sample (i.e. 1 to 5; Fig. 2a) was completely cleaned.

The order of the segment cleaning was randomly assigned for each unit. Each sub-sample was next separately stored. In total, 60 samples were collected (3 units/type x 4 types x 5 subsamples/unit).

Total vs. sub-sampling: comparison of both sampling methods

Each 15-cm sub-sample was analysed separately. The average periphyton composition on each unit was calculated in two ways (Fig. 2b):

1. Per unit, the data of the five 15-cm sub-samples (1 to 5) were added together, to represent the whole surface area (S-5), and 2. Per unit, the top (1: 0-15 cm), middle (3: 30-45 cm) and bottom (5: 60-75 cm) subsamples were added together, and extrapolated to the total unit area (S-3). These data were compared with the results of the whole unit samples (T) collected on the same day (30d). 28 data were thus used for each set of comparison ((4 units/types x 4 types) + (S-3 or S-5 sampling method 3 units/type * 4 types)).

Three units (one W, m and M) were incorrectly treated and could not be included in the data set. It explains why the total degree of freedom was lower than expected (Tables 1 through 3).

Sampling and storage

At T15d and T30d, water temperature (°C), salinity, pH were measured with a multiparameter probe (HI9828 HANNA) at the water top 15 cm of three sites in the pond, at 5:00 PM. Mean water temperature, salinity and pH (± SE) were 24.4 ± 0.76 °C, 32.2 ± 0.14 ppt and 8.1 ± 0.07 at T15d vs. 26.3 ± 0.93 °C, 32.9 ± 0.14 ppt and 8.2 ± 0.03 at T30d. Mean oxygen concentration (± SE) was at 6.4 ± 0.1 mg.L -1 (92.8 ± 2.2 %) T15d and 6.9 ± 0.1 mg.L -1 (102.7 ± 2.7%) at T30d. The water samples were collected immediately after the probe recording. Means of suspended matter (± SE) and particulate organic matter were 13.7 ± 1.04 mg.L -1 and 1.9 ± 0.2 mg.L -1

, respectively, at T15d. The suspended matter was composed of 85.9 ± 0.4 % of inorganic matter. At T30d, mean Chl a was 5.3 ± 0.3 g.L -1 . Chlorophyll pigments included 15.2 ± 1.2 percent of Phaeophytin a.

Periphyton

The order and the location of collected units were randomly assigned. Each unit was sampled by putting a PVC tube (diameter of 6 cm x 110 cm of length) over it and closing it with a 100 µm-meshed stopper to avoid periphyton loss. The length of the submerged part of the collected substrate was measured in order to calculate the exact substrate area with periphyton (cm²). Each unit was carefully and completely cleaned with fingers and a toothbrush into a plastic flask with a fixed volume of 0.7 µm-filtered sea-water (200 ml for total unit samples and 40 ml for 15-cm samples). All material from the inner part of the net of meshed substrates was removed. Each sample was next sub-sampled using a Motoda boxsplitter [START_REF] Motoda | Devices of simple plankton apparauts[END_REF]: 1/8 part was stored in a dark box at -20°C for Chla analysis, 7/16 parts were stored with 4% formalin for taxonomic analysis, and 7/16 parts were used for periphyton weight analyses, putting it directly in pre-weighted box at 60°C.

Sample analyses Dry weight and Ash free dry weight

Periphyton samples were dried at 60°C for 72h, weighed (DW: dry weight), and burned for 4h at 450°C to calculate the ash-free dry weight (AFDW; [START_REF] Byers | A comparison of methods of determining organic carbon in marine sediments, with suggestions for a standard method[END_REF]. DW, AFDW and the weight of ash (ASH) were measured to the nearest 10 -5 g with an AE240 Mettler Toledo Balance. As filtered sea-water was used to clean units, "salt correction" was applied on periphyton weight. The effects of added filtered seawater (7/16 of 200ml or 40 ml according to the type of sampling) on DW, ASH and AFDW of periphyton were determined considering the salinity of the cleaning water and the corresponding calibration curves (DW (g.L -1 ) = 1.17 Salinity (R 2 =0.99), ASH (g.L -1 ) = 0.94 Salinity (R 2 =0.99), and AFDW (g.L -1 ) = 0.23 Salinity (R 2 =0.96). These equations were established using based on DW, AFDW and ASH content of three replicates of 0.7 µm-filtered water in which the salinity was either 0, 10.7, 20.4, 28, 28.7, 36.37, 36.42, 38.03, 38.12, 40.17 or 40.38 ppt. Sea-water (28 to 40) was collected in marine ponds. Water in which the salinity ranged from 10 to 20 corresponded to diluted sea water by Milli-Q water (0 ppt).

Values were reported to the total sample volume (200 or 40 ml) and to the total length of the unit. DW, ASH and AFDW were thus expressed in mg.cm -2 .

Chlorophyll a and Phaeophytin a

Chlorophyll a (Chl a) and phaeophytin a (Phaeo a) observed in periphyton were determined with a Turner TD 700 fluorometer after 12 hours of acetone extraction at 4°C in the dark without and with acidification. Nine ml of 100% acetone were added to 1 mlperiphyton as could performed Azim's team (M.C.J. Verdegem, Pers. Com.). Chl a and Phaeo a data were reported to the total sample volume (200 or 40 ml) and to the total length of the unit. Values were expressed in µg.cm -2 for periphyton. The ratio of phaeophytin vs. sum of chlorophyll pigments was also calculated as (Phaeo a).(Phaeo a + Chl a) -1 and expressed in % (% Phaeo a). The autotrophic index (AI) was calculated as: AFDW (mg.cm -2 )/ Chl a (µg.cm -2 ) * 1000 µg/mg (APHA 1992).

Statistical analyses

The assumptions of normality and homoscedasticity were evaluated using Shapiro-Wilk [START_REF] Shapiro | An analysis of variance test for normality (complete samples)[END_REF] and [START_REF] Brown | Robust tests for the equality of variances[END_REF]) tests, respectively. When required, data were transformed to satisfy both assumptions. ANOVAs were next performed to test the influence of (i) submersion time (TIME), (ii) substrate type (TYPE), (iii) submersion depth (DEPTH), (iv) sampling method (SAMPLING), and (v) their interactions on periphyton DW, AFDW, Chl a, Phaeo a, %Phaeo a and AI. Tukey's HSD (honestly significant differences) pairwise multiple comparison tests were used to identify the differences when a source of variation was significant (P < 0.05).

Results

Total sampling: Influence of submersion time and substrate type

According to the ANOVA results (Table 1), dry weight, ash free dry weight, chlorophyll a and phaeophytin a varied significantly among submersion time (TIME; Table 1). Means were greater at T30d than at T15d. Mean AFDW and Phaeo a were more than twice higher at T30d than at T15d (DW: 6.3 mg.cm -2 vs. 2.3 mg.cm -2 and Phaeo a: 0.2 µg.cm -2 vs. 0.1 µg.cm -2 ; Fig. 3a,b).

Substrate type (TYPE) significantly affected the amount of periphyton collected in terms of DW, AFDW, Chl a and Phaeo a (Table 1). Tukey HSD tests revealed that means of DW, AFDW, Chl a and Phaeo a were larger on mosquito screen (m) that on the other substrate types (M, S or W; Fig. 3c,d). Mean periphyton DW and total chlorophyll pigment varied between 3.4 and 6.4 mg.cm -2 (Fig. 3c) and between 0.5 and 1 µg.cm -2 , respectively, among substrate types (Fig. 3d).

The interaction of both factors (TYPE x TIME) was a significant variation source of % Phaeo a (Table 1). Relatively more Phaeo a was present on mosquito screen at T30d (m-T30d: 27.7%) than at T15d (m-T15d: 16.7%). In contrast, mean % Phaeo a did not significantly differ over time on wooden poles (W), fiber-glass (S) and garden netting (M). Means (± SE) were respectively 30.2 ± 0.9 %; 28.2 ± 1.1 % and 18.7 ± 1.6 %. At T15d, a higher % Phaeo a was observed on smooth substrates (W, S) than on meshed substrates (m, M). At T30d, the % Phaeo a observed on wooden poles (W) was higher than on garden netting (M).

The autotrophic index was significantly different for the factors TYPE and TIME, and showed a significant interaction (Table 1). The mean AI observed on wooden poles (W) was more than 6 times lower at T15d (1554 ± 410) than at T30d (9449 ± 1479). In contrast, the mean AI observed on the other substrates (S, m, M) did not vary over time.

Sub-sampling: Influence of substrate type and submersion depth

Chl a, % Phaeo a and the AI were significantly different among substrate type (Table 2). HSD tests showed that at T30d, the Chl a mean was greater on meshed substrates and fiberglass than on wooden poles (m, M, S: 0.6 ± 0.12 µg.cm -2 > W: 0.26 ± 0.08 µg.cm -2 ). At T30d, % Phaeo a varied such as W > S, M ≥ M, m. The AI mean was almost three times higher on wooden poles (W: 2815 ± 816) than on the other substrates (S, M, m: 939 ± 299).

Periphyton DW, Chl a and Phaeo a significantly changed with depth (DEPTH; Table 2). More periphyton was collected at 60-75 cm depth than at 0-15 cm depth (Fig. 4).

Respectively 2 and 12 times more DW and total chlorophyll a was collected in the bottom 15 cm than at the top 15 cm. Mean differences between sampling depths of DW and Phaeo a

were not statistically significant whereas means seemed to increase between 15 and 60 cm depth. In contrast, Chl a increased gradually with depth (Fig. 4b). The mean AI (± SE) was more than three times larger in the top 15 cm (3406 ± 1002) than between 15 and 60 cm (900 ± 260). The % Phaeo a did not vary with depth whatever the type substrate (Table 2) and was 25.7 ± 2.05%.

Total vs. sub-sampling: comparison of both sampling methods

Five sub-sampling

The mean DW, AFDW, Chl a and Phaeo a differed significantly between sampling methods (SAMPLING (S-5 vs. T); Table 3). Means were higher with the total sampling (T) than the S-5 sub-sampling method whatever the substrate type (Fig. 5a,b). It was particularly right for periphyton quantity rather than quality. DW and AFDW determined through S-5 sampling were 2 and 8 times, respectively lower than T means (Fig. 5a) whereas the mean of chlorophyll pigment obtained with S-5 sampling corresponded to 82.6% of means obtained with total sampling T (Fig. 5b).

Three sub-sampling

Sampling was a significant source of variation for DW and AFDW (SAMPLING (S-3 vs. T); Table 3). More DW and AFDW were measured with total sampling (T) than with the S-3 sub-sampling method (Fig. 5a). As S-5 means, S-3 means of DW and AFDW were 2 and 8 times, respectively, lower than the T means (Fig. 5a). In contrast, Chl a and Phaeo a means did not significantly differ between S-3 and T (Fig. 5b). In situ observations showed that periphyton was also composed of detritus and small-sized organisms as harpacticoid copepods (Richard et al., unpublished data).

Discussion

Marine periphyton and its variation sources

Submersion time

A significant increase of periphyton DW, AFDW and photosynthetic pigments was shown on all substrate types. According to periphyton colonization models [START_REF] Hoagland | Colonization and community structure of two periphyton assemblages with emphasis on the diatoms (Bacillariophyceae)[END_REF][START_REF] Steinman | Effect of grazers on freshwater benthic algae, Algal ecology: freshwater benthic ecosystems[END_REF], AFDW and Chl a levels increase exponentially until a biomass peak.

Organisms at the base of the biofilm become light and nutrient limited, eventually die and detach from the substrate [START_REF] Hansson | Factors regulating periphytic algal biomass[END_REF][START_REF] Asaeda | Spatial structure and populations of a periphyton community: a model and verification[END_REF]Keshavanath et al., 2001a;Azim and Aseada, 2005). In this investigation, periphyton was still in its accretion phase on day 30 on all substrate types. As noted [START_REF] Eding | Design and operation of nitrifying trickling filters in recirculating aquaculture: A review[END_REF], biofilm establishment seems to be slower in marine than in freshwater. It would be better to wait a minimum of 4 weeks rather than 2 as Azim et al. (2001aAzim et al. ( , 2003b) ) did in freshwater, before introducing fish in marine periphyton-based ponds.

Substrate type

DW, AFDW, Chl a, Phaeo a and %Phaeo a varied according to substrate type. [START_REF] Keshavanath | Use of articicial substrates to enhance production of freshwater herbivorous fish in pond culture[END_REF] showed that biodegradable substrates could be more efficient than synthetic substrates (eg. Bamboo vs. PVC tubes) because of the nutrient leaching that occurred at the substrate-water interface [START_REF] Van Dam | The potential of fish production based on periphyton[END_REF]. In the same way, Anderson and Underwood (1994) reported higher recruitment by epifauna on plywood than on fibreglass or aluminium substrates in an estuary. In contrast, periphyton biomass was not larger on natural (i.e. wooden poles) than on fiberglass strips in this study. 30 dayssubmersion time might have been too short to permit to a significant nutrient leaching at the interface of wooden poles. Nevertheless, the periphyton grown on wooden poles contained relative more phaeophytin (higher % Phaeo a) and non autotrophic matter (higher AI) than the other substrates. The observed increase in AI could have originated from uptake of decomposition products from the wood.

More dry matter and Chl a were found on meshed substrates (mosquito and garden meshes) than on smooth substrates (i.e. wood and fiber-glass). The meshes might favour the trapping of particles, in contrast to smooth surfaces. Moreover, higher circulation of water and nutrients across the meshed substrates could stimulate periphyton growth and explain this result.

The substrate type could also influence the nature of the heterotrophic associated community. [START_REF] Richard | Influence of suspended scallop cages and mussel lines on pelagic and benthic biogeochemical fluxes in Havre-aux-Maisons Lagoon, Îles-de-la-Madeleine (Quebec, Canada)[END_REF] observed that mesh substrates, as aquaculture pens, offered appropriate structures for infauna, as Corophium sp. whereas newly submerged smooth substrate, favoured epifauna recruitment. In this study, some polychaete tubes were observed on mosquito screen, but not on smooth substrates. The results of this study indicate that more and qualitatively better periphyton grew on mosquito screen than on the other substrates.

Submersion depth

The light intensity and its spectral composition change with depth, influencing the quality and type of flora [START_REF] Boston | Photosynthesis-light relations of stream periphyton communities[END_REF][START_REF] Hansson | Factors regulating periphytic algal biomass[END_REF][START_REF] Kirk | Light and photosynthesis in Aquatic ecosystems[END_REF], as periphyton [START_REF] Goldsborough | Periphyton in freshwater lakes and wetlands, Periphyton: Ecology, exploitation and management[END_REF]. In contrast to the observations of Azim et al. (2002a), periphyton DW and chlorophyll pigments increased with depth in this investigation. A decrease of the 10 cm-water level at T30d could explain why less periphyton was collected on the 0-15 cm part of substrates than on the deeper parts. Nevertheless, the lower chlorophyll pigment concentration observed on the 15-60 cm part of substrates compared to the deeper part (60-75 cm) could be due to a photo-inhibition processes, as [START_REF] Hansson | Factors regulating periphytic algal biomass[END_REF] suggested when periphyton Chl a was negatively correlated with light. Unfortunately, light incidence was not measured during this study.

Maximal periphyton biomass could be observed where the combination of light and nutrient are optimal (Hansson et al., 2002). In this way, periphyton observed on the deeper part of substrates could have the advantage over the one observed on the surface part by benefiting from nutrient released at the water-sediment interface. Moreover it could benefit from trapping suspended sediment and microphytobenthos present at the bottom of the pond.

Comparisons Sampling method

The sum of 3 samples taken between 0-15 cm, 30-45 cm and 60-75 cm (S-3 method) led to comparable Chl a and Phaeo a means with the ones obtained with total sampling (T).

That was not the case with S-5 method. Nevertheless, the DW and AFDW of the periphyton collected with both sub-sampling methods (S-5 and S-3) were significantly lower than with total sampling. The sub-sampling, especially the one of mesh substrates, necessitated extra handling for cutting before periphyton collection. Each handling event results in losses, making both the S-5 and S-3 methods less accurate than whole unit sampling. Total sampling was easier and more periphyton was collected. In further studies, the total sampling method will be preferred to sub-sampling one.

In this investigation, the use of 200 ml of filtered salt-water for unit cleaning induced over-estimation of periphyton weight. The DW, the ASH and the AFDW added when cleaning 750 cm²-periphyton substrate with 200 ml of 0.7 µm seawater of 32.55 ppt were respectively 10.1, 8.1 and 2 mg.cm -2 . These values are very important compared to the real periphyton weight (Table 4), especially for DW and ASH. Without the salt correction, ASH would be more than 6 times greater than the real values (with correction) at T15d and 3 times at T30d. Analysis of three blanks of cleaning water should be envisaged at each sampling date in subsequent studies. To avoid the salt correction, the use of milliQ water could be envisaged in case where the determination of periphyton weight would be the only analysis to carry out on the sampled unit. The periphyton fauna and flora could be analysed from other units cleaned with filtered seawater to avoid osmotic shock of the living cells.

Periphyton in other aquatic systems

Absolute values which described the quantity and the quality of periphyton developed on our substrates deployed in marine water were different with the one observed mainly in freshwater by others authors (Table 4). The mean DW observed on our substrates reached 8.8 mg.cm -2 on mosquito screen m at T30d (Table 4). This is relatively high since 10 studies out of 13 found a DW < 5mg.cm -2 . Maximal mean organic periphyton (AFDW) observed in this study (4.5 mg.cm -2 ) was greater than means observed by others authors which generally did not exceed 1 mg.cm -2 with the exception of Azim et al., 2002b (Table 4). In contrast to this investigation, in most of the cited studies, periphyton substrates are simply removed from the water causing probably a lot of loosely attached to be lost and could explain lower mean of AFDW. The use of a tube with a stopper for substrate sampling is recommended to avoid underestimation of periphyton development.

High autotrophic index of this investigation (Table 4) was induced by greater AFDW but also by very low chlorophyll a concentration observed on substrates (0.4 to 0.6 µg.cm -2 ; Fig. 3b).

Numerous studies observed Chl a levels above 10-15 µg.cm -2 (Azim et al., 2001b(Azim et al., ,c, 2002a(Azim et al., 2003a;;[START_REF] Keshavanath | Use of articicial substrates to enhance production of freshwater herbivorous fish in pond culture[END_REF]; Table 4). Low periphyton concentration could originate partly from the use of inert substrate [START_REF] Huchette | The impacts of grazing by tilapias (Oreochromis niloticus L.) on periphyton communities growing on artificial substrate in cages[END_REF]Azim et al. 2003b;[START_REF] Liboriussen | Structure, biomass, production and depth distribution of periphyton on artificial substratum in shallow lakes with contrasting nutrient concentrations[END_REF] This study: Table 4) rather than nutrient-leaching substrate (Azim et al., 2001b(Azim et al., , 2002a(Azim et al., , 2002b)). Nevertheless, others factor could influence the primary productivity, such as temperature, light and nutrient availability [START_REF] Liboriussen | Periphyton macroinvertebrate interactions in light and fish manipulated enclosures in a clear and a turbid shallow lake[END_REF][START_REF] Vermaat | Periphyton dynamics and influencing factors[END_REF]. The high densities of periphyton recorded by Azim et al. (2001bAzim et al. ( , 2002aAzim et al. ( , 2002b) ) and [START_REF] Keshavanath | Use of articicial substrates to enhance production of freshwater herbivorous fish in pond culture[END_REF] were observed in tropical ponds in Bangladesh and India with more light and higher temperatures than in temperate ponds in France, in the Netherlands (Azim et al., 2003b) or Denmark [START_REF] Liboriussen | Structure, biomass, production and depth distribution of periphyton on artificial substratum in shallow lakes with contrasting nutrient concentrations[END_REF]. It is the same in water column where mean Chl a was above 200 µg.L -1 in Bengali fresh ponds (Azim et al. 2002b), whereas it was 5 µg.L -1 in our temperate marine pond. Productivity in freshwater is generally higher than in marine water. However, the ponds were fertilized with urea, manure, food in most studies listed in Table 4 whereas our pond was not fertilized. Azim et al. (2001cAzim et al. ( , 2003a) ) showed that periphyton biomass increased with increasing fertilization rate up to a maximum.

Thus, in future studies, as part of EU policy of environmental protection and restoration of coastal areas, fertilized effluents of intensive farms could be used to maximise periphyton production and the associated production of herbivorous fishes.

The present investigation showed that (i) periphyton biomass in a marine pond increased between day 15 and day 30, (ii) more periphyton was collected on mosquito screen than on wooden poles, fiberglass strips or garden netting, (iii) periphyton biomass increased with water depth submersion, (iv) sub-sampling methods underestimated periphyton development compared to whole unit sampling, and (v) a correction of periphyton biomass must be carried out for the dissolved inorganic salts present in marine or brackish systems using blank weight of cleaning salt filtered water. The use of a tube with stopper for substrate sampling will reduce periphyton sampling losses. Finally, the autotrophic fraction in the periphyton communities was very low compared to periphyton developed on biodegradable substrates used in fish cultures in fertilized tropical ponds. Thus, pond fertilization and use of biodegraded substrates (i.e. long-time submerged wood) should be envisaged in further studies on periphyton-based marine aquaculture in temperate regions.

Tables

Table 1 Results of analyses of variance (ANOVAs) testing the effect of substrate type (TYPE: W:

wooden poles, S: fiber-glass strip, m: mosquito screen, M: garden netting), submersion time Results of ANOVAs testing the effect of periphyton sampling method (SAMPLING: T: total vs. S-5: addition of all five sub-samples and vs. S-3: addition of three sub-samples 0: 0-15 cm, 3: 30-45, 5: 60-75 cm), substrate type (TYPE: W: wooden poles, S: fiber-glass strips, m: mosquito screen, M: garden netting) and their interactions on Periphyton dry weight (DW), ash free dry weight (AFDW), Chlorophyll a (Chl a) and phaeophytin (Phaeo a) observed on collected substrates. df: degrees of freedom, MS: mean square, F: Fischer, * P < 0.05, ** P < 0.01, *** P < 0.001

  Marine periphytonA thin mat of matter was observed on all the immersed surface of the different types of substrate after 15 days of submersion. The inorganic fraction of periphyton (ASH) could originate from trapping of suspended inorganic particles. The latter would be favoured during resuspension caused by wind driven turbulence or people working around units during sampling. The organic matter (AFDW) fraction originated from the accumulation of detritus, bacteria, fungi, flora and fauna on substrates. The presence of photosynthetic pigments (Chl a and Phaeo a) could indicate flora colonization of artificial substrates. The presence of phaeophytin a indicated that the flora observed was partly degraded (15 to 30 %). The mean autotrophic index ranged between 250 (60-75 cm section of mosquito screen at T30d) and 9450 (wooden poles at T30d). These high values indicate that the periphyton contained mainly heterotrophic organisms and dead organic matter, as specified by[START_REF] Huchette | The impacts of grazing by tilapias (Oreochromis niloticus L.) on periphyton communities growing on artificial substrate in cages[END_REF] for an AI above 200.

(

  TIME: T15d, T30d) and their interactions on periphyton dry weight (DW), ash free dry weight (AFDW), Chlorophyll a (Chl a), Phaeophytin (Phaeo a), % Phaeo a (Phaeo a.(Chl a + Phaeo a) -1 ) and autotrophic index (AI: AFDW.Chl a -1 ) observed on collected substrates. df: degrees of freedom, MS: mean square, F: Fischer, * P < 0.05, ** P < 0.01, *** P < 0.001

Figures captions Fig. 1 :

 captions1 Figures captions

Fig. 5 :

 5 Fig. 5: Mean (± Standard Error) periphyton dry weight (a) and chlorophyll pigment (b)

  Figures 1 2

Table 3

 3

Table 4

 4 Ranges of mean variables characterizing quantity and quality of periphyton developed on different submerged substrates in natural and exploited aquatic systems observed by different authors over the world

	Variation source	df MS	F	P	MS	F	P	MS	F	P	MS F	P
			log DW		log AFDW		Chl a		Phaeo a
	SAMPLING (S-5 vs. T) 1 3.838 74.62 < 0.0001 *** 24.383 130.80 < 0.0001 *** 0.085 9.30 0.0069 ** 0.017 6.46 0.0205 *
	TYPE	3 0.118 2.30 0.1124	0.626 3.36 0.0419 *	0.156 17.09 < 0.0001 *** 0.008 3.09 0.0532
	SAMPLING x TYPE	3 0.109 2.11 0.1342	0.454 2.44 0.0980	0.028 3.08 0.0537	0.007 2.67 0.0786
	Error	18 0.051			0.186			0.009			0.003
	SAMPLING (S-3 vs.T) 1 3.833 65.13 < 0.0001 *** 23.978 119.66 < 0.0001 *** 0.038 3.03 0.0986	0.008 2.94 0.1034
	TYPE	3 0.129 2.19 0.1249	0.680 3.39 0.0406 *	0.194 15.61 < 0.0001 *** 0.009 3.35 0.0421 *
	SAMPLING x TYPE	3 0.105 1.79 0.1858	0.368 1.84 0.1763	0.029 2.34 0.1074	0.007 2.53 0.0895
	Error	18 0.059			0.200			0.012			0.003

*: Fortnightly cow manure, urea, Triple super phosphate TSP, nd: no data

Acknowledgements

This study has been carried out with the financial support from the Commission of the European Communities, specific RTD programme "Specific Support to Policies", SSP-2005-44483 "SEACASE -Sustainable extensive and semi-intensive coastal aquaculture in Southern Europe", and does not necessarily reflect the European Commission views and in no way anticipates the Commission's future policy in this area. This study was co-funded by the IFREMER institute. This paper was written thanks to the administrative help of P. Bustamante and L. Picard (CNRS, UMR 6250, University of La Rochelle). The authors thank M. Prineau and N. Lachaussée for their advice and help during structure and substrate construction. Thanks go to G. Colli, C. Couturier, N. Lachaussée, F. Mornet and L. Pavie for their precious help in the field. Finally, authors thank A. Bodoy, M.L Begout, M. Breret, L. Joassard, B. Lebreton and P. Richard for the loan of materials.

Table 2 Results of ANOVAs testing the effect of substrate type (TYPE: W: wooden poles, S: fiberglass strips, m: mosquito screen, M: garden netting), submersion depth (DEPTH: 1: 0-15 cm; 2: 15-30 cm; 3: 30-45 cm, 4: 45-60 cm, 5: 60-75 cm) and their interactions on Periphyton dry weight (DW), ash free dry weight (AFDW), chlorophyll (Chl a), phaeophytin (Phaeo a) and % Phaeo a (Phaeo a.(Chl a + Phaeo a) -1 ) and autotrophic index (AI: AFDW.Chl a -1 ) observed on collected substrates. df: degrees of freedom, SS: sum square, MS: mean square, F: Fischer, * P < 0.05, ** P < 0.01, *** P < 0.001