M E Negretti

A Martin

Simultaneous velocity-density measurements of downslope density clouds

Keywords: Gravity currents, mixing, experiments, sloping bottom, internal structure

; Maxworthy (2010) and the theoretical model of Dai (2013a). The collected data allowed also for the determination of the entrainment rates based on the turbulent fluxes instead using the variation of the volume flux, and the regions of entrainement/detrainment have been identified. The results support previous observations of the mechanism responsible for the buoyancy loss of the cloud and its consequent deceleration down the slope, as a large scale recirculation vortex at the back of the cloud. A very complex interior structure of the cloud is observed, with a large variety of turbulent processes taking place, such as the large-scale convectively unstable recirculation vortex at the scale of the current itself, the small-scale convectively unstable motions inside the head of the cloud and close to the bottom boundary and shear (Kelvin-Helmholtz) instabilities at the boundary between the current and the ambient water. The synoptic velocity and density measurements allowed also to test existent parametrizations of turbulent fluxes, that have been quantified in 2D fields with a high spatial resolution. Results confirm that parametrization laws based on the assumption of a constant turbulent diffusivity or mixing length do not apply for buoyancy clouds due to their high spatial heterogeneity. Hence, the parametrization of the turbulent dif-

Introduction

Variations in temperature, salinity and/or sediment concentration cause variations of fluid density in the vertical direction. The resulting flow stratification, which typically occurs in environmental and geophysical flows, leads to qualitative and quantitative modifications of the flow patterns by buoyancy. When buoyancy driven flows encounter topography, a downslope dense current, i.e. gravity current, is created. Its motion is then sustained by buoyancy and deviated by the topographic slope. Gravity currents are key processes that affect ocean, atmospheric and coastal circulation.

Gravity currents can take on different forms depending on their source of supply of buoyancy being continuous [START_REF] Baines | Two-dimensional plumes in stratified environments[END_REF][START_REF] Britter | The motion of the front of a gravity current travelling down an incline[END_REF][START_REF] Martin | Development of gravity currents on slopes under different interfacial instability conditions[END_REF][START_REF] Negretti | Development of gravity currents on rapidly changing slopes[END_REF][START_REF] Odier | Entrainment and mixing in a laboratory model of oceanic overflow[END_REF] or resulting from a finite volume release [START_REF] Beghin | Gravitational convection from instantaneous sources on inclined boundaries[END_REF]. Examples of the first type are oceanic gravity currents [START_REF] Baringer | Mixing and spreading of the Mediterranean outflow[END_REF], while examples of finite volume released gravity currents are avalanches [START_REF] Clément-Rastello | A study on the size of snow particles in powdersnow avalanches[END_REF][START_REF] Hopfinger | Snow avalanche motion and related phenomena[END_REF][START_REF] Rastello | Sediment-entraining suspension clouds: a model of powder-snow avalanches[END_REF], volcanic eruptions [START_REF] Holyer | Gravity currents entering a two-layer fluid[END_REF] or turbidity currents [START_REF] Meiburg | Turbidity currents and their deposits[END_REF], which contribute to the shaping of the continental surface and have a large impact on the geomorphological floor and the related ecosystems [START_REF] Bründl | Review and future challenges in snow avalanche risk analysis[END_REF][START_REF] Kostaschuk | On the causes of pulsing in continuous turbidity currents[END_REF][START_REF] Pohl | Transport and burial of microplastics in deep-marine sediments by turbidity currents[END_REF][START_REF] Thorez | From inflow to interflow, through plunging and lofting: Uncovering the dominant flow processes of a sediment-rich negatively buoyant river inflow into a stratified lake[END_REF]. Katabatic winds are also an example of an intermittent gravity flow and are important to determine the local air circulation in several regions [START_REF] Brun | Large-eddy simulation of a katabatic jet along a convexly curved slope: 2. evidence of gortler vortices[END_REF][START_REF] Charrondiere | Buoyancy effects in the turbulence kinetic energy budget and reynolds stress budget for a katabatic jet over a steep alpine slope[END_REF]. This paper deals with finite volume released gravity currents over sloping boundaries. For the terminology, finite volume released gravity currents on a sloping boundary is more precisely described as a 'thermal cloud' or 'density cloud' [START_REF] Simpson | Gravity currents in the laboratory, atmosphere and ocean[END_REF], because the flow resembles more a cloud with some tail following.

The pioneering work on donwlope propagating density clouds was made by [START_REF] Beghin | Gravitational convection from instantaneous sources on inclined boundaries[END_REF], who studied the full range of slopes between 5 • and 90 • . They showed that the current development includes a first quick accelerating phase of the density cloud, followed by a decelerating phase, if no buoyancy is entrained by the current, as for gravity currents propagating over a sediment bed [START_REF] Rastello | Sediment-entraining suspension clouds: a model of powder-snow avalanches[END_REF]). Further theoretical, numerical and experimental studies followed, giving important insights in the dynamics and the spatio-temporal development of the current [START_REF] Adduce | Gravity currents produced by lock-exchanges: experiments and simulations with a two layer shallowwater model with entrainment[END_REF][START_REF] Beghin | Contribution of theoretical and experimental results to powder-snow avalanche dynamics[END_REF][START_REF] Beghin | Gravitational convection from instantaneous sources on inclined boundaries[END_REF][START_REF] Dai | Gravity currents propagating on sloping boundaries[END_REF]Dai, , 2013a[START_REF] Dai | Non-Boussinesq gravity currents propagating on different bottom slopes[END_REF][START_REF] Dai | High-resolution simulations of downslope gravity currents in the acceleration phase[END_REF][START_REF] Dai | Gravity currents from instantaneous sources down a slope[END_REF][START_REF] Hopfinger | Snow avalanche motion and related phenomena[END_REF][START_REF] Martin | Propagation of a continuously supplied gravity current head down bottom slopes[END_REF][START_REF] Ottolenghi | Entrainment in a dense current flowing down a rough sloping bottom in a rotating fluid[END_REF][START_REF] Rastello | Sediment-entraining suspension clouds: a model of powder-snow avalanches[END_REF]Steenhauer et al., 2017a;Zemach et al., 2019).

To describe the front velocity history of a lock-released gravity current both a shallow water model approach [START_REF] Ross | A study of three-dimensional gravity currents on a uniform slope[END_REF][START_REF] Tickle | A model of the motion and dilution of a heavy gas cloud released on a uniform slope in calm conditions[END_REF][START_REF] Ungarish | An introduction to gravity currents and intrusions[END_REF][START_REF] Webber | A model of the motion of a heavy gas cloud released on a uniform slope[END_REF] or thermal theory [START_REF] Beghin | Gravitational convection from instantaneous sources on inclined boundaries[END_REF][START_REF] Morton | Turbulent gravitational convection from maintained and instantaneous sources[END_REF] have been adopted. In the latter case, the assumption of a constant buoyancy flux during the head propagation was made and despite the strong approximation, this model has shown to give a good description of the evolution of both the first accelerating phase and the successive decelerating phase [START_REF] Beghin | Gravitational convection from instantaneous sources on inclined boundaries[END_REF].

A more sophisticated model has been proposed by [START_REF] Maxworthy | Experiments on gravity currents propagating down slopes. part 1. the release of a fixed volume of heavy fluid from an enclosed lock into an open channel[END_REF], who argued that during the acceleration phase, the head was being fed by a following current that increased its buoyancy as it propagated downstream. [START_REF] Maxworthy | Experiments on gravity currents propagating down slopes. part 2. the evolution of a fixed volume of fluid released from closed locks into a long, open channel[END_REF] showed that after a propagation distance of the order of (5-10)H 0 , where H 0 is the initial depth of the lock, the inflow into the rear stopped, and the head began to lose buoyancy-containing fluid from its rear by the detachment of large, weakly vortical structures. Their measurements have shown that the buoyancy in the current head increased during the acceleration phase and decreased during the deceleration phase. [START_REF] Dai | Gravity currents propagating on sloping boundaries[END_REF]Dai ( , 2013b) implemented a three-dimensional direct numerical simulation of a lock released downslope gravity current. They confirmed numerically that the maximum buoyancy in the head never reaches the total released buoyancy, and a significant portion of heavy fluid is left in the tail current, which is often detached from the head when the density cloud propagates over steep slopes (see also Dai (2013a,c)), in accord with [START_REF] Maxworthy | Experiments on gravity currents propagating down slopes. part 2. the evolution of a fixed volume of fluid released from closed locks into a long, open channel[END_REF].

For a long running length x > 13H, they showed a disagreement between the thermal theory [START_REF] Morton | Turbulent gravitational convection from maintained and instantaneous sources[END_REF] and the observed growth of the density cloud in the deceleration phase.

Recently, [START_REF] Martin | Propagation of a continuously supplied gravity current head down bottom slopes[END_REF]Zemach et al. (2019) proposed a the-oretical solution based on the shallow water model combined with a Benjamin's type jump condition at the nose of the head and an empirical entrainment law, respectively. Their solution is of relevance during the acceleration phase only, taking into account the contribution of the tail to the buoyancy budget of the density cloud, as reported by [START_REF] Maxworthy | Experiments on gravity currents propagating down slopes. part 1. the release of a fixed volume of heavy fluid from an enclosed lock into an open channel[END_REF].

However, the model cannot predict the loss of buoyancy in the deceleration phase [START_REF] Dai | Gravity currents propagating on sloping boundaries[END_REF](Dai, , 2013a;;[START_REF] Maxworthy | Experiments on gravity currents propagating down slopes. part 2. the evolution of a fixed volume of fluid released from closed locks into a long, open channel[END_REF].

While previous studies mostly concentrated on the global properties of such density clouds propagating on horizontal or inclined boundaries, their internal structure has started to receive attention only recently. [START_REF] Hopfinger | A model study of powder-snow avalanches[END_REF] measured the velocity distribution within the density cloud by means of the hydrogen bubble technique combined with a conductivity probe measurement for the density. Three-dimensional direct numerical simulations of a lock-released gravity current were performed by [START_REF] Calgaro | Modeling and simulation of mixture flows: Application to powder-snow avalanches[END_REF]; Étienne et al. (2004); Steenhauer et al. (2017b) revealing a very turbulent structure of the buoyancy cloud. [START_REF] Nishimura | Velocity distribution in snow avalanches[END_REF] measured internal velocities of 50 ms -1 for a front velocity of about 10 ms -1 , demonstrating that the head interior is characterized by a strong energetic circulation and a large variety of turbulent structures. Knowing the internal structure of density clouds may help to better predict the entrainment and dilution mechanisms, which are directly related to the propagation speed and motion of such currents.

Detailed velocity and/or density measurements within the head of a density cloud propagating over a horizontal boundary using optical non-intrusive measurements techniques (such as Particle Image Velocimetry-PIV) have been performed only in few studies [START_REF] Hallworth | Entrainment into two-dimensional and axisymmetric turbulent gravity currents[END_REF][START_REF] Martin | Combined piv/plif measurements of a steady density current front[END_REF][START_REF] Pelmard | Turbulent density transport in the mixing layer of an unsteady gravity current[END_REF][START_REF] Thomas | The structure of the head of an inertial gravity current determined by particle-tracking velocimetry[END_REF]. [START_REF] Hallworth | Entrainment into two-dimensional and axisymmetric turbulent gravity currents[END_REF] and later [START_REF] Martin | Combined piv/plif measurements of a steady density current front[END_REF] The recent studies performed by [START_REF] Agrawal | Probing the high mixing efficiency events in a lock-exchange flow through simultaneous velocity and temperature measurements[END_REF]; Mukherjee andBalasubramanian (2020, 2021) on horizontal lock-released gravity currents focused on the slumping phase and their analysis was aimed at estimating the mixing efficiency at the interface between the cloud and the ambient fluid due to Holmboe or Kelvin-Helmholtz instabilities. No study is known in the literature on downslope density clouds aimed at verifying the theories of the global characteristics governing the dynamics of such flows, as the front propagation, the buoyancy and the entrainment, but also quantifying small-scale processes and turbulent fluxes that characterize the internal structure of the density cloud, based on highly resolved simultaneous measurements of both the velocity and the density fields. By means of such detailed measurements, the validity and limitations of the existent theories for both the global characteristics of the flow, but also those related to the parametrization of the turbulent diffusivities, have been tested, revealing interesting insights on the small-scale features of the internal structure of the head. The global characteristics of the density cloud compared to the semi-empirical thermal theory of [START_REF] Beghin | Gravitational convection from instantaneous sources on inclined boundaries[END_REF] and [START_REF] Escudier | On the motion of turbulent thermals[END_REF] reveal a fairly good agreement, with exception of the assumption of initial mass conservation, in accord with [START_REF] Maxworthy | Experiments on gravity currents propagating down slopes. part 2. the evolution of a fixed volume of fluid released from closed locks into a long, open channel[END_REF]. The predictions for the buoyancy loss of the gravity current during the decelerating phase are in better agreement with the theoretical model proposed by Dai (2013a,c), while the model of [START_REF] Maxworthy | Experiments on gravity currents propagating down slopes. part 2. the evolution of a fixed volume of fluid released from closed locks into a long, open channel[END_REF] reveals to strongly overestimate it, even if the trend of decrease is well represented by Maxworthy (2010)'s theory A very complex interior structure of the buoyancy cloud is observed, with a variety of turbulent processes sources of mixing and dilution present. These are large-scale and small-scale convective motions inside the cloud, shear turbulence and related instabilities (e.g. Kelvin-Helmholtz and boundary layer instabilities) at the boundaries with the ambient fluid and the bottom.

Finally, it is shown that classical mixing parametrizations of turbulent fluxes based on mixing length models are not appropriate for lock-released buoyancy clouds due to their strong spatial and temporal heterogeneity.

The paper is organized as follows: section 2 outlines the experimental set-up and the measurement techniques. Section 3 gives the global characteristics of the buoyancy cloud, while entrainment/detrainment and mixing are presented in 4 and 5, respectively, and compared to existent parametrizations. Section 6 summarizes the results and includes concluding remarks.

Experimental facility and measurement techniques

Experimental setup

The experimental set-up is shown schematically in figure 1a Two runs of experiments have been performed in the deceleration region 1.65m < x < 2.5m from the lock release (see figure 1). The same initial conditions have been adopted for both runs with exception of the bottom slope,

Run A 0 [cm 2 ] θ[ • ] g ′ 0 [cm/s 2 ] N Technique F.O.V. 1 125 15 5 ± 0.1 5 PIV/PLIF 1.65m < x < 2.5m 2 125 20 5 ± 0.1 7 PIV/PLIF 1.65m < x < 2.5m 3 125 15 5 ± 0.1 5 PIV/PLIF 0.7m < x < 1.55m

Measurement technique

The velocities and the density were determined using the optical, non- The PIV set-up consisted of a light source, light sheet optics, seeding particles, a camera, and a PC equipped with a frame grabber and image acquisition software. Polyamide particles (Orgasol) with a mean diameter of 60µm and a specific density of 1.016g/cm -3 were added in both the fresh water and the salt water in the initial release reservoir as tracer material for the PIV measurements.

A 5 Watt Yag-laser (λ = 532 nm) has been used as continuous light source. The beam was transmitted through mirrors to a spherical lense with an angle of 45°to generate a laser sheet with a length of approximately 1 m and a width of 5mm, positioned in the middle of the channel. Rhodamine 6G was used as fluorescent dye (absorption peak at 530nm and emission peak at 555nm) and uniformly mixed in the reservoir containing the dense fluid with initial concentration c 0 = 5µg/l. The dye and the salt have approximately the same diffusivity, with a Schmidt number Sc = ν/κ = 700, where κ is the mass transfer diffusion coefficient [START_REF] Troy | The generation and quantitative visualization of breaking internal waves[END_REF].

For the simultaneous PIV/PLIF measurements an interferometer bandpass, 532nm for PIV and a high pass filter with cut-off 550nm for PLIF were used to separate the emitted wavelengths.

Images of roughly 70cm × 55cm were grabbed with a CCD camera (Flow Master 1200 × 1600 pixels) for PIV, and with a second CCD camera (Dalsa 1000 × 1000 pixels) at a frame rate of 23.22 Hz for PLIF.

The raw image pairs were then processed using a PIV cross-correlation algorithm (software package DaVis, LaVision) to compute the velocity fields, starting with an interrogation window of 32×32 pixels and a final window size of 16 × 16 pixels with 50% overlap. Each vector of the resulting vector field represents an area of roughly 0.6cm×0.6cm. Given the velocities encountered in the experiments, the error for the instantaneous velocity is approximately 4%.

The calibration procedure for the PLIF measurements is given in the appendix. For each x-position, the density cloud is detected (t, z) using a threshold of 0.99(∆ρ/ρ) max for the density.

Experimental results

Global characteristics of the density cloud

As suggested by [START_REF] Beghin | Gravitational convection from instantaneous sources on inclined boundaries[END_REF], and as evident from present visual observations, the shape of the cloud can be fitted in time and vertical space (t, z) assuming a semi-elliptic shape (cf. also figure 5), that follows the following equation where z is the vertical (normal to the slope) coordinate of the boundary of the density cloud with the ambient fluid, H and 0.5t A are the minor and major axis of the ellipse and t 0 is the time corresponding to a given position of the geometrical centre of the ellipse x g . From equation ( 1), H and the length L = t A U of the density cloud can be estimated, where U is the velocity of the geometric centre of the ellipse defined as

z = H 0.5t A (0.5t A ) 2 -(t -t 0 ) 2 1 2 , (1)
U = ∆x ∆t 0 (2)
Figure 3a, shows the non-linear fit using equation (1) (dashed line) compared to experimental values (solid line). The streamwise coordinate corresponds to the dimensionless advective time (tt 0 )/t A , where t 0 has been determined using a power law fit ( [START_REF] Beghin | Gravitational convection from instantaneous sources on inclined boundaries[END_REF], cf. figure 3b).

Using this procedure, the velocity of the buoyancy cloud U , its surface area A and its buoyancy flux B = g ′ U H have been determined. showed experimentally that the surface of the buoyancy cloud increases with the square of the along slope direction as A = C (x + x 0 ) 2 , where C is a constant defined as C = π/16 (S 2 /S 1 ) 2 α 2 and x 0 is a virtual origin corresponding to A = 0, i.e. x 0 = 4S 1 /S 2 α -1 A 0 /π. The shape factors S 1 and S 2 are estimated assuming a semi-elliptical shape of the buoyancy cloud, and α is an empirical entrainment coefficient (see also equation 7). This model is represented by a dashed line in figure 4b revealing a good agreement with the experimental data in the initial accelerating stage. At X/L 0 sin θ ≈ 2.8, corresponding to x ≈ 34H 0 , the data suggests that the lateral surface stops increasing, in accord with previous numerical observations [START_REF] Dai | Gravity currents propagating on sloping boundaries[END_REF](Dai, , 2013b;;Steenhauer et al., 2017b). In particular, for a slope of 10°, [START_REF] Dai | Gravity currents propagating on sloping boundaries[END_REF] observed that the semi-elliptical head experiences multiple sporadic reductions in its height.

As mentioned in the introduction, [START_REF] Maxworthy | Experiments on gravity currents propagating down slopes. part 2. the evolution of a fixed volume of fluid released from closed locks into a long, open channel[END_REF] explained this behaviour as the result of loss of volume (and buoyancy) that takes place during the descent, induced by large scale vortical structures developing at the back of the gravity current head, that has been also supported by Dai (2013a,b).

The numerical results from Dai (2013b) are also reported as symbols in figure

??b: results are within the range of our data and the theoretical prediction of [START_REF] Maxworthy | Experiments on gravity currents propagating down slopes. part 2. the evolution of a fixed volume of fluid released from closed locks into a long, open channel[END_REF]; however, the lateral surface A increases monotonically and no decrease is observed. The buoyancy of the semi-elliptical cloud is shown in figure 4c, defined as

B = U ∞ 0 t B t N g ′ δ c dt dz, (3)
where t N and t B are respectively the time at a given distance x at which the nose and the back of the ellipse appears, respectively, and δ c is the Dirac function. Note that this definition of the buoyancy differs from that of Maxworthy ( 2010) by a factor of 1/ρ 0 , which is cancelled out in figure 4c since it is normalized with the initial buoyancy. From the figure, it emerges that at (x/L 0 ) sin θ = 2.2 the buoyancy of the cloud is about 60% of the initial buoyancy, demonstrating that during the descent buoyancy has been left behind in the tail. For this value of the buoyancy, thermal theory predicts a velocity 84% of the predicted velocity, which explains the lower velocity (88%) reported in figure 4a. For comparison, we also report the conservation of buoyancy from [START_REF] Beghin | Gravitational convection from instantaneous sources on inclined boundaries[END_REF] (dashed line in figure 4c). [START_REF] Maxworthy | Experiments on gravity currents propagating down slopes. part 1. the release of a fixed volume of heavy fluid from an enclosed lock into an open channel[END_REF] and [START_REF] Maxworthy | Experiments on gravity currents propagating down slopes. part 2. the evolution of a fixed volume of fluid released from closed locks into a long, open channel[END_REF] showed that the buoyancy of the cloud is not conserved in the deceleration stage of the density cloud. For a slope angle of 10.6°, they showed that the buoyancy of the cloud reaches a maximum of about 43%B 0 for x ≈ (10 -13)H 0 and decreases afterwards. This is represented as a continuous line in figure 4c and we see that the descent of our data follows the trend of [START_REF] Maxworthy | Experiments on gravity currents propagating down slopes. part 2. the evolution of a fixed volume of fluid released from closed locks into a long, open channel[END_REF].

The theoretical solution of Maxworthy (2010) however underestimates our reported values (60% versus 43%), which may be imputed by the different definition of the buoyancy threshold within the cloud or by the different quantities used for its estimation: in our case the buoyancy has been obtained by instantaneous velocity and density measurements, and not from dye concentration measurements.Dai (2013a) derived a power-law relationship which enables to estimate the fraction of heavy fluid in the lock that is contained within the head, χ, via χ = [K M /K B ]1 3 , where K M is parameter which has been determined by several measurements in previous studies [START_REF] Beghin | Gravitational convection from instantaneous sources on inclined boundaries[END_REF], [START_REF] Hoult | Oil spreading on the sea[END_REF], [START_REF] Huppert | The slumping of gravity currents[END_REF], [START_REF] Marino | The front condition for gravity currents[END_REF]), whereas a theoretical derivation of K B is given in Dai (2013a) (cf. equation (1.7) therein). The theoretical estimations of Dai (2013a) were also in good agreement with the measurements of [START_REF] Beghin | Gravitational convection from instantaneous sources on inclined boundaries[END_REF] In order to assure that the results do not depend on the definition chosen to determine the buoyancy of the cloud, it has also been estimated directly using the combined PIV/PLIF data as follows

B ′ = ∞ 0 t B t N u(z, t)g ′ δ c dt dz = B U m U , (4)
where u(z, t) is the local velocity from the PIV and U m the velocity of the centre of mass of the buoyancy cloud. The buoyancy B ′ is shown in figure 4c as a green line, revealing that both estimations for B and B ′ give similar results: the ensemble average of B ′ is slightly smaller than that of B because the geometric velocity U overestimates the velocity of the centre of mass of the cloud. Clearly, the density within the cloud is not homogeneous and at the back of the head, a large-scale vortex appears, associated to a downward vertical velocity at the outer edge of the density cloud and an upward vertical velocity in the central area of the head, with evident regions convectively unstable (cf. colormap in figure 6a). Such a large scale vortex structure has been already observed in the literature (e.g. [START_REF] Beghin | Contribution of theoretical and experimental results to powder-snow avalanche dynamics[END_REF]Dai, 2013a;[START_REF] Hampton | The role of subaqueous debris flow in generating turbidity currents[END_REF]. This vortex is composed of much lighter water (≈ 4%g ′ 0 )

Interior structure

as compared to the frontal part at the nose (≈ 12%g ′ 0 ), entrained from the ambient. The size of the large-scale vortex is of the order of the maximum depth of the buoyancy cloud. The averaged reduced gravity within the cloud is about 6%g ′ 0 , thus 60% of the initial value, during the considered space-time interval when observations were taken. the head [START_REF] Martin | Combined piv/plif measurements of a steady density current front[END_REF] and at the outer boundary of the largescale vortex within the buoyancy cloud, induced by enhanced vertical shear generated by the vortex itself.

These results also highlight that a variety of different turbulent processes take place inside the density cloud: the large-scale and small-scale convective motions which become locally unstable in the core of the cloud, shear turbulence as Kelvin-Helmholtz instabilities at the boundary with the ambient fluid, as also observed and quantified by [START_REF] Balasubramanian | Entrainment and mixing in lockexchange gravity currents using simultaneous velocity-density measurements[END_REF], and boundary layer instabilities at the bottom boundaries, which all concur to the dilution of the cloud.

Entrainment

Two phenomena responsible for entrainment can be distinguished (cf figure 6): the first is induced by the large scale vortex (of typical scale of the order of H) that engulfs ambient water within the full cloud, and the second is due to small scale (shear) instabilities of typical scale of the order of ≈ 1 cm that develop at the contours of the buoyancy cloud with the ambient water and entraining locally.

The conservation of mass for the buoyancy cloud can be expressed using the model of the thermal theory considering the temporal variation of the lateral surface A:

DA Dt = αU Γ, ( 5
)
where α = -W/U , is the entrainment coefficient, W is the average entrainment velocity across the contour Γ corresponding to the buoyancy cloud interface with the ambient water, assuming a 2D flow. This formulation neglects the variations of density, that can be assumed valid for small ∆ρ/ρ. More rigorously, the conservation of mass across the control volume representing the density cloud, can be expressed as

D Dt S Γ ρds = S Γ ⃗ ∇ P • (ρ⃗ u) ds + S Γ ρ∂ y vds, (6)
where ⃗ ∇ P • ⃗ u = ∂ x u + ∂ z w is the divergence of the velocity in the (x, z) plane. Assuming the flow to be primarily two-dimensional, the last term on the right hand side of equation ( 6) can be neglected, so that the local mass flux can be directly estimated across the interface between the buoyancy cloud and the surrounding ambient. The entrainment coefficient α used in the thermal theory can be then written as

α = 1 ρ 0 U Γ ρ⃗ u • ⃗ ndl/ Γ dl = 1 ρ 0 U C Γ ρ⃗ u • ⃗ ndl, (7)
where the circulation integral in the latter equation can be replaced by the surface integral using the Green's theorem Figure 7 displays the entrainment coefficient deduced from equations ( 7)

Γ ρ⃗ u • ⃗ ndl = S Γ ⃗ ∇ P • (ρ⃗ u) ds. (8)
and ( 8). The observed sign fluctuations of α in figure 7 are of the order of the spatial average (cf. dashed line in figure 7) and correspond to entrainment (positive) and detrainment (negative). For comparison, the constant entrainment coefficient proposed by [START_REF] Beghin | Gravitational convection from instantaneous sources on inclined boundaries[END_REF] is also reported (continuous line).

Turbulent fluxes

The results in section 3 have shown that a variety of turbulent motions characterize the internal structure of the density cloud. Turbulent motions are usually not solved explicitly in numerical models and thus they need to be parameterized (e.g. [START_REF] Madec | Nemo ocean engine, Notes du Pole de[END_REF].Vertical turbulent fluxes are generally assumed to depend linearly on the gradients of large-scale quantities.

The eddy viscosity ν t is used in the momentum equation to parameterize the Reynolds stress such as:

⟨u ′ w ′ ⟩ = -ν t ∂u ∂z (9)
A frequently used turbulence closure for mixing in ocean or atmospheric circulation models is to assume a linear relation between the vertical buoyancy flux and the density gradients. This relation defines a diapycnal turbulent diffusivity K ρ that can be written as Maps of the turbulent diffusivities are shown in figure 9, with ν t in (a)

⟨ρ ′ w ′ ⟩ = -K ρ ∂ρ ∂z (10)
and K ρ in (b). ν t and K ρ present the same order of magnitude, with high values (> 10 -4 m 2 s -1 ) within the large-scale vortex and smaller values (< 10 -5 m 2 s -1 ) close to the nose and at the bottom. Their spatial structure is however different. Turbulent diffusivities ν t are intensified at the edge of the large-scale vortex and the ambient, whereas diapycnal turbulent diffusivities K ρ present the largest values in the core of the large-scale vortex. These estimations show clearly that the usual hypothesis of a constant diffusivity for the entire density cloud does not apply, with strong variations of one order of magnitude. [START_REF] Prandtl | Bericht über die entstehung der turbulenz[END_REF] proposed a mixing length scale L u to relate the Reynolds stress and the square of the mean velocity vertical shear:

⟨u ′ w ′ ⟩ = L 2 u ∂u ∂z 2 (11)
Similarly, for the buoyancy flux a buoyancy mixing length L ρ can be defined:

⟨ρ ′ w ′ ⟩ = L 2 ρ ∂ρ ∂z ∂u ∂z (12)
In figure 10, both parametrizations for the Reynolds stress equations ( 9) and ( 11), respectively (a,c), and for the buoyancy turbulent fluxes as given in equations ( 10) and ( 12), respectively, (b,d), are reported. Two-dimensional histograms show the correlation between fluxes and gradients, but no convergence for both models is found. Thorpe (1977) proposed a characteristic length scale related to a density profile in a mixed patch, obtained by adiabatically re-ordering the density profile in the unstable regions of the water column and estimating the vertical displacements ∆z needed for a fluid particle to be moved from the synthetic stable profile to the observed profile.

A characteristic length scale for the mixing can then be introduced from the root mean square of the vertical displacements ∆z over the vertical profile of the buoyancy cloud such as L T = (∆z) 2 , where • corresponds to the vertical average operator. Dillon (1980) showed that this length scale can be related to the Ozmidov scale L O as

L O = 0.8L T , (13)
where the Ozmidov scale L O is defined as L O = ϵ 1/2 N -3/2 . Herein, ϵ the dissipation rate and N the Brunt-Väisälä frequency. By combining the equation ( 13) with the expression of L O given above, the dissipation rate can be written as:

ϵ = 0.64L 2 T N 3 . ( 14
)
Osborn (1980) showed that the turbulent diffusivity is limited by the dissipation rate following the equation [START_REF] Barry | Measurements of diapycnal diffusivities in stratified fluids[END_REF] proposed also two equations to estimate the turbulent diffusivity depending of the buoyancy Reynolds number Re b = ϵ/(νN 2 ) = 0.64L 2 T N/ν K ρ = 0.9(ν 2 K mol ) 1/3 Re b for 10 < Re b < 300, 24(ν 2 K mol ) 1/3 Re

K ρ = - ⟨w ′ ρ ′ ⟩ ⟨∂ z ρ⟩ ≤ 0.2ϵ/N 2 ≃ 0.13L 2 T N. (15)
1/3 b for Re b > 300, ( 16
)
where K mol is the molecular diffusivity of salt in water. Combining the equations ( 14) and ( 16) leads to

K ρ ≃ 0.6(P r) -1/3 L 2 T N for 10 < Re b < 300, 21(L 2 T N ν 2 /P r) 1/3 for Re b > 300, (17)
where P r = ν/K mol is the Prandtl number of the fluid. 17) and the red line is the vertical running mean of these points with a window size of 0.05z/H. The vertical green line is the vertical average of the diapycnal turbulent diffusivity following [START_REF] Barry | Measurements of diapycnal diffusivities in stratified fluids[END_REF].

In figure 11b the turbulent diffusivity is estimated using equation ( 10 15) and the red dashed-line is the vertical running mean of these points with a window size of 0.05z/H. The black dots are estimated following the [START_REF] Barry | Measurements of diapycnal diffusivities in stratified fluids[END_REF] equations ( 17) and the red line is the vertical running mean of these points with a window size of 0.05z/H. The vertical green line is the vertical average of the diapycnal turbulent diffusivity following [START_REF] Barry | Measurements of diapycnal diffusivities in stratified fluids[END_REF]. (b) K ρ estimated from equation (10) (black stars) and the associated vertical average (vertical green line).

Summary and conclusions

An experimental investigation of finite volume gravity currents down a slope released from a lock has been performed using combined PIV and PLIF measurement techniques to obtain two-dimensional velocity and density fields.

Experiments were focused on the deceleration region of the buoyancy cloud at a distance x > 10H 0 from the initial volume reservoir. Based on the high resolved simultaneous velocity-density data, it is verified that the existing theoretical models of [START_REF] Beghin | Gravitational convection from instantaneous sources on inclined boundaries[END_REF]; Dai (2013a); Maxworthy (2010) are suitable for predicting the global characteristics of the cloud propagation (front propagation, lateral surface and buoyancy), with exception of the buoyancy variations that have to be corrected from [START_REF] Maxworthy | Experiments on gravity currents propagating down slopes. part 2. the evolution of a fixed volume of fluid released from closed locks into a long, open channel[END_REF] using the coefficients given in the theoretical model of Dai (2013a).

The measurements showed the complex turbulent structure of the cloud with the back of the buoyancy cloud being hydrostatically unstable with the associated Thorpe scale larger than at the front of the cloud. Large patches of ambient fluid are engulfed into the cloud, especially at the back of the head due to a large scale recirculation vortex, that causes convectively unstable small-scale structures within the head and subsequent intense dilution. Smaller scale instabilities at the limiting edge between the head and the ambient fluid also contributes to local mixing and dilution, such as Kelvin-Helmholtz instabilities.

Using the combined velocity and density data and by averaging the buoyancy cloud along its descent, turbulent fluxes has been estimated revealing that the usual parameterization laws based on the assumption of a constant turbulent diffusivity or a constant turbulent mixing length do not work properly. These results apply for lock-release density clouds propagating down steep slopes, and in particular for the head of the gravity current, which present high spatial and temporal hetereogenity. The tail behind the passage of the head in case of continuous supply, or even some lock released gravity currents propagating on horizontal boundaries, has shown to be rather stationary and homogeneous, present self-similarity and can be treated with a good approximation as a stratified shear layer, as given by the measurements of [START_REF] Odier | Understanding and modeling turbulent fluxes and entrainment in a gravity current[END_REF][START_REF] Odier | Entrainment and mixing in a laboratory model of oceanic overflow[END_REF]. The turbulent closure models based on mixing length scales (e.g. [START_REF] Prandtl | Bericht über die entstehung der turbulenz[END_REF]) revealed to work reasonably well in such cases. Spatial hetereogenity has been also reported for spatially developing gravity currents [START_REF] Martin | Development of gravity currents on slopes under different interfacial instability conditions[END_REF][START_REF] Negretti | Development of gravity currents on rapidly changing slopes[END_REF]. In this latter case however, turbulent diffusivities can be defined distinctively in the defined spatial regions, that can be identified using for example the internal Froude number or the bulk Richardson number (a study is currently in progress on this aspect).

The high spatio-temporal anisotropy of downslope density clouds over steep slopes makes it challenging to find a correct parametrization using closure models based on the turbulent viscosity ν t that depend on the spatial variables. Our results show clearly that none of the parametrizations proposed in the literature can properly represent turbulent diffusivities for downslope propagating density clouds. The fact that in figure 10 the histograms show no correlation between the turbulent fluxes and a mixing length, the strong vertical variations shown in figure 11 using the formulation of [START_REF] Osborn | Estimates of the local rate of vertical diffusion from dissipation measurements[END_REF] and [START_REF] Barry | Measurements of diapycnal diffusivities in stratified fluids[END_REF], comes from the fact that for downslope propagating buoyancy clouds over steep slopes there are several characteristic scales, which are set by the multiplicity of turbulent processes that take place, along with their intermittency. These are convective unstable processes (inside the cloud), Kelvin-Helmholtz (shear) instabilities (at the edge with the ambient fluid) and lobes and clefts instabilities (not observed in this study). Under these conditions, mixing scales go from the small convectively unstable scales, the Kelvin-Helmholtz scales, up to the large scale recirculation vortex, which is of the same order of the size of the cloud itself, as also highlighted in figure 9.

The parametrization of the turbulent diffusivities should be based rather on scalar quantities that avoid the problem of spatial hetereogenity and takes into account the different sources of turbulence production. This could be done for example using energetic considerations that compare the terms having a definite exchange of energy and acts as a source or a sink: the shear production deriving its energy from the mean flow (P k ), the buoyancy production representing the rate at which turbulent kinetic energy is consumed in mixing, but also aliments the production of kinetic energy for downslope propagating currents (P b ), and the viscous dissipation of turbulent kinetic energy (ϵ), governed by the balance P k + P b = ϵ. Each of these three terms, can be represented by a characteristic length scale of shear (L s ), of buoyancy, as for example the Monin-Obukhov or the Ozmidov scales (L O ) and a dissipation scale (L ϵ ), as proposed earlier in [START_REF] Bradshaw | Effects of streamline curvature and buoyancy in turbulent shear flow[END_REF].

The use of the energy budget approach has also the advantage to permit extending the results to similarly spatially hetereogeneous non-conservative currents such as turbidity currents or katabatic flows, where the additional source/sink of buoyancy can be added following the suspension/deposition of fluid particles -in turbidity currents -or the presence of a local thermal flux in katabatic flows.

Additional investigations are needed to test the appropriate turbulent closure for these flows based on the above described energy approach, which is left for future work.

For each angle β, a non-linear fit of equation (A.6) has been performed to find the parameters ϵ, a w , and ΓA β . The following attenuation coefficients has been found: a w = 2.8 ± 1.7 × 10 -3 cm -1 , ϵ = 3.2 ± 0.3 × 10 -4 (cm µg/l) -1 and ϵ s c s,0 c 0 ≈ 1.4 × 10 -6 (cm µg/l) -1 ≪ ϵ.

Finally, the non-dimensional dynamic density is related to the measured dye concentration by ρ(r, β, t) = c(r, β, t)/c 0 .

Zemach, T., Ungarish, M., [START_REF] Martin | Development of gravity currents on slopes under different interfacial instability conditions[END_REF] On gravity currents of fixed volume that encounter a down-slope or up-slope bottom.

Phys. Fluids 31, 096604.

  observed the formation of persistent Kelvin-Helmholtz type billows at the front of the density current causing entrainment of ambient fluid into the current at the interface current/ambient. Here, we performed experiments on finite volume released gravity currents over steep sloping boundaries (with slope angles of 15 • and 20 • ) using simultaneous PIV/PLIF measurements techniques. Such measurements in unsteady and heterogeneous turbulent buoyancy flows are rare, but are very useful since they allow to highly resolve both large-scale and small-scale flow motions and statistics, which otherwise are very difficult to measure. This also enables resolving turbulent fluxes and their spatial distribution, which are impossible to quantify with other measurement techniques.

  . A plexiglas water channel 2.76 m long, 20 cm wide and 38cm deep was immersed in a larger water tank of dimensions 2.4 m × 2.4 m × 2.4 m. The slope angle of the experimental channel was set to the desired angle. The dense fluid was introduced in a reservoir L 0 = 20 cm long and 30 cm deep, with a lock depth of H 0 = 6.25cm located at the upper end of the channel and initially separated by a removable gate, resulting in an initial lateral surface of A 0 = L 0 × H 0 = 125cm 2 . The initial reduced gravity was fixed to g ′ 0 = g∆ρ/ρ a = 5cm/s 2 and produced by saline solutions with ∆ρ = 5.1 ± 0.1 kgm -3 and a constant temperature of 19.7 • C ± 0.1, measured just before each experiment with a high precision (10 -3 ) densimeter (DMA TM 35, Anton Paar). The resulting Reynolds numbers were Re = U H/ν ≈ 1.2 • 10 4 where U and H are respectively the typical velocities and depths of the cloud in the measurement region and ν the kinematic viscosity.

Figure 1 :

 1 Figure 1: (a) Side view of the experimental setup. (b) Top view of the optical measurement configuration for PIV and PLIF.

  intrusive experimental techniques of Particle Image Velocimetry (PIV) and Planar Laser-Induced Fluorescence (PLIF), respectively. Both PIV only and combined PIV/PLIF experimental data are used in the following results sections. Essential details on the measurements techniques are given below, while the detailed procedure for the PLIF calibration is given in the appendix.

Figure 2

 2 Figure 2 illustrates an instantaneous side view of a calibrated density field. The cloud has the size of the present PIV/LIF field of view, hence, present results are treated in terms of the vertical coordinate z and time t, where time is converted in the streamwise space coordinate x by neglecting the time distortion of the cloud during the advective time t A at a given xposition, and assuming a constant advective velocity of the cloud over this time laps.

Figure 2 :

 2 Figure 2: Instantaneous density field of the cloud for [∆ρ/ρ] i = 5 • 10 -3 and run 2 (cf. table 1).

FigureFigure 3 :

 3 Figure4ashows the normalized experimental velocity U/ g ′ 0 H 0 (solid line) obtained from equation (2) clearly in the deceleration stage and in good accord with predictions of the thermal theory (dashed line).The normalized lateral surface A/A 0 , defined as A = π 4 LH, where L and H are estimated from equation (1), is shown in figure4b.[START_REF] Beghin | Gravitational convection from instantaneous sources on inclined boundaries[END_REF]

Figure 4 :

 4 Figure 4: Experimental global characteristics of the buoyancy cloud with Re ≈ 1.2 × 10 4 (a) Velocity of the geometrical center U from equation (2) of the buoyancy cloud (runs 1 and 2). (b) Lateral surface of the buoyancy cloud function of the along-slope direction for run 1 with a slope of 15°(red shaded region) and run 2 with a slope of 20°(green shaded region), and from DNS of Dai (2013b) (symbols). The x axis has been re-normalized by L 0 / sin θ.(c) Dimensionless experimental ensemble average of the buoyancy of the cloud B from equation (3) (black line) and B ′ from equation (4) (green line). Red lines represent the confidence interval. The blue dashed and continuous lines represent the empirical equation from Beghin et al. (1981) and from Maxworthy (2010) respectively, where the x axis has been rescaled by L 0 / sin θ.

Figure 5 ,

 5 Figure 5, illustrates the density field at three different x positions. In (a), at x g = 2.2 m, the buoyancy cloud is well fitted by the half-elliptic shape, with a very active back vortex incorporating ambient water at the back of the cloud. On figure 5(b), at x g = 2.3 m, a larger interfacial shear instability develops which begins to separate the back of the cloud, as seen later in (c): there is clearly a loss of volume and mass at the back of the cloud. This phenomenon is observed in all the experiments, although with variable intensity, and confirms the suggestion of Maxworthy (2010) and Dai (2013a) to explain the observed decrease of buoyancy in the second stage of propagation of the cloud down the slope. Steenhauer et al. (2017) observed a variation in the development of the lateral surface at x ≈ 35H 0 for slopes larger than 30°. They showed that for sufficiently large distances from the source, the normalized sub-volume

  , predicting values of ≈ 0.75 for slope angles below 5 • and ≈ 0.82 for slope angles of 9 • (Dai, 2013c). These values are much higher than those predicted by Maxworthy (2010) and this difference was explained by erroneous estimations of the model constant K B and/or a different definition of the head dye concentration threshold. If we estimate the model constant with our values using equation (1.7) in Dai (2013a), we obtain K B = 3.8, resulting in χ ≈ 0.63for K M = 2.45[START_REF] Beghin | Gravitational convection from instantaneous sources on inclined boundaries[END_REF], which is in good agreement with the reported values in figure 4c of 0.6.

Figure 5 :

 5 Figure 5: Density field at three different x positions for run 1. Non-linear fit of the shape of the buoyancy cloud using a half-ellipse (red dashed line). Horizontal axis corresponds to the advective time.

Figures 6

 6 Figures 6 (a,b) show the mean structure of the buoyancy cloud with the black arrows representing the velocity difference between the ensemble average of the instantaneous velocity ⟨⃗ u⟩ and the ensemble average of the velocity of the gravity center ⟨⃗ u B ⟩ of the buoyancy cloud. The density for [∆ρ/ρ] i = 5 × 10 -3 in (a) and the mean vorticity (ω y = ∂u ∂z -∂w ∂x ) in (b) are highlighted by the colormaps.

Figure 6 Figure 6 :

 66 Figure 6(b) shows that the vorticity is negative at the interface between the buoyancy cloud and the ambient water and positive in the bottom boundary layer of the buoyancy cloud. The high values of the vorticity on the back of the buoyancy cloud confirms the presence of the large-scale vortex. High positive vorticity is observed at the nose of the buoyancy cloud, highlighting the presence of Kelvin-Helmholtz like structures on the frontal part of

Figure 7 :

 7 Figure 7: Entrainment coefficient. Coefficient deduced from equations (7) and (8) (black line) and its spatial average (horizontal dashed line); constant coefficient proposed by Beghin et al. (1981) (horizontal red line).

Figure 8 :

 8 Figure 8 shows the two-dimensional probability density functions (PDF) of ρ ′ versus ρ ′ w ′ for the initial development phase where the lateral surface of the buoyancy cloud increases (a) and for the second stage characterized by a decreasing lateral surface of the buoyancy cloud (b). The right part of the plots corresponds to the stabilizing return to neutral buoyancy, while the top left quadrant corresponds to entrainment, and the bottom left quadrant to detrainment. It is evident comparing the two figures, that most of the entrainment takes place in the initial development phase of the density cloud, while detrainment is dominant during the second phase characterized by the detachment of the back of the cloud.

Figure 9 :

 9 Figure 9: Turbulent diffusivity (log 10 m 2 /s) in the buoyancy cloud. ν t (a); K ρ with stable stratification (b).

Figure 10 :

 10 Figure 10: Two-dimensional histograms representing the correlation between the Reynolds stress and the mean gradients. The color scale represents the log 10 number of entries in the histogram. (a) Momentum flux ⟨u ′ w ′ ⟩ versus the vertical-velocity gradient ⟨∂u/∂z⟩ and (b) versus the square of the velocity vertical gradient ⟨∂u/∂z⟩ 2 . (c) Buoyancy flux ⟨ρ ′ w ′ ⟩ versus the vertical density gradient expressed using the Brunt Väisälaa frequency N 2 and (d) versus the square (N 2 ) 2 .

Figure

  Figure 11a displays the turbulent diffusivity deduced using the Thorpe scale L T . The light-gray dots are estimated following the Osborn equation (15) and the red dashed-line is the vertical running mean of these points with a window size of 0.05z/H. The black dots are estimated following the Barry et al. (2001) equations (17) and the red line is the vertical running mean of

Figure 11 :

 11 Figure11: Diapycnal turbulent diffusivity K ρ (m 2 /s) from an instantaneous vertical density profile at the back of the buoyancy cloud using (a) The light-gray dots are estimated following the Osborn equation (15) and the red dashed-line is the vertical running mean of these points with a window size of 0.05z/H. The black dots are estimated following the[START_REF] Barry | Measurements of diapycnal diffusivities in stratified fluids[END_REF] equations (17) and the red line is the vertical running mean of these points with a window size of 0.05z/H. The vertical green line is the vertical average of the diapycnal turbulent diffusivity following[START_REF] Barry | Measurements of diapycnal diffusivities in stratified fluids[END_REF]. (b) K ρ estimated from equation (10) (black stars) and the associated vertical average (vertical green line).

Table 1 :

 1 Main parameters of the experimental runs.

Acknowledgements

This work has been supported by the LabEx Tec 21 (Investissements d'Avenir -grant agreement n • ANR-11-LABX-0030). We are grateful to A. Di Rienzo for her support in conducting the experiments. Special thanks go to H. Bellot, C. Eymond-Gris, M. Lagauzère, L. Vignal, S. Viboud, T. Valran for technical support in the experiments. Special thanks go to E.J. Hopfinger, C. Staquet and M. Rastello for useful discussions.

Declarations

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. For an excitation illumination I 0 (r, β), with (r, β) being the position along a radius and an angle of the laser sheet, and with no absorption along the laser path, there is a linear relationship between the intensity of fluorescence emission and the dye concentration, consistent with our measurements at dye concentrations ranging from 0 to 53.8 µg/l. For low laser intensities [START_REF] Crimaldi | Planar laser induced fluorescence in aqueous flows[END_REF], the pixel grey scale values at each location in the fluorescence image, g 0 (r, β, t), are linearly related to the dye concentration at that location, c(r, β, t), and to I 0 (r, β) via

where g b (r, β) is the background noise or camera dark-response. The constant Γ accounts for the system-specific optical collection efficiency and the effective quantum yield of the fluorescent dye. Here the subscript '0' denotes the case without absorption. The values of ΓI 0 (r, β) and g b (r, β) are determined from a calibration procedure detailed below.

When laser light passes through the test section, the dye solution attenuates the laser intensity and this attenuation must be corrected to obtain accurate dye concentrations from PLIF images (e.g., [START_REF] Atsavapranee | Structures in stratified plane mixing layers and the effects of cross-shear[END_REF][START_REF] Ferrier | Application of optical techniques to the study of plumes in stratified fluids[END_REF][START_REF] Krug | Experimental study of entrainment and interface dynamics in a gravity current[END_REF][START_REF] Odier | Entrainment and mixing in a laboratory model of oceanic overflow[END_REF]. In the present set-up, the propagation of the laser sheet in the field of view is along the r axis. According to the Bouguer-Lambert-Beer law, the laser intensity in such an absorbing medium is

where α = α(r, β, c) is the attenuation along ray paths due to solution absorption. a w is the clear water attenuation coefficient, ϵ is the attenuation coefficient of the dyed solution per unit concentration, ϵ s is the attenuation coefficient of the salt solution per unit of salinity and c s (r ′ , β, t) is the salt concentration of the water. The recorded greyscale value is then g(r, β, c) = ΓI(r, β, c)c(r, β, t) + g b (r, β). Substituting eq. (A.2) into this last equation leads to g(r, β, c) = ΓI 0 (r, β)α(r, β, c)c(r, β, t) + g b (r, β).

(A.4)

The PLIF calibration was done before and after each experiment. The main tank was filled with Rhodamine 6G with uniform known concentrations (c = 0, 3.8, 7.7, 15.4, 30.8, 38.5, 46.2 and 53.8 µg/l). The zero concentration measurement gives the camera dark-response g b (r, β). For uniform known concentrations c and for a given angle β, equation (A.4) gives


B Degraeve

B Lefeuvre

N Rocton

N Herbert

N Hamrouni

G Hauss

A Largeteau

M Prakasam

H Oudadesse

O Merdrignac-Conanec

Processing of highly porous bioglass monoliths by hydrothermal hot pressing

Keywords: highly porous monolith, hydrothermal hot pressing, sol-gel derived bioglass, melt-derived bioglass, sintering

Porous bioglass monoliths have been processed by hydrothermal hot pressing (HyHP) from sol-gel and melt-derived bioglass powders of composition (in mol %): SiO2-CaO-P2O5 (55.0-40.0-5.0) and SiO2-CaO- , respectively. An open porosity of >70% ever reached in 3D structures is reported for monoliths issued from sol-gel powders. Dissolution studies were performed in simulated body fluid (SBF) for 1-30 days. The monoliths were analysed using XRD, FTIR and SEM to observe the formation of an apatite-like layer and elemental concentration of SBF was evaluated using ICP/OES analysis. A higher kinetics in the development of apatite layer was observed for sol-gel derived monoliths. This result is explained by the high surface areas of the nanosized sol-gel powders and the possibility of HyHP to create large porosity (mesoporous monoliths) and retain large surface areas. HyHP is also effective in processing 3D-bioglass structures with porosity gradient by co-sintering powders of different size.

Introduction

Recent developments in processing methods of bioactive glasses for bone regeneration focus on porous scaffolds that can act as three-dimensional (3D) templates to mimic the bone structure. Besides conventional foaming techniques and template-based methods, various additive manufacturing (or solid freeform fabrication) methods are now available for building scaffolds from both melt-and solgel-derived glasses while offering over aforementioned techniques a better control of scaffold architecture and pore morphology [1,2]. Nevertheless, a final thermal treatment (usually up to 550-700°C) has to be performed to burn out the binders or templating agents and sinter the glass powders. This final step is critical as it may provoke glass devitrification while complete organics removal is recommended [1,2]. Hydrothermal processes, formerly investigated for the preparation of α-quartz single crystal for its unique piezoelectric properties, are now widely developed for the preparation of new materials and grow bulk crystals for specific applications [3]. Owing to its similarity to natural bone in chemical composition, hydroxyapatite (HA, Ca10(PO4)6(OH)2) has become an excellent biomaterial for bone repair and is currently fabricated into scaffolds [START_REF] Gautam | Processing of biomaterials for bone tissue engineering: State of the art[END_REF][START_REF] Ren | Enhancement of osteogenesis using a novel porous hydroxyapatite scaffold in vivo and vitro[END_REF][START_REF] Chen | Fabrication and characterization of 3D complex hydroxyapatite scaffolds with hierarchical porosity of different features for optimal bioactive performance[END_REF]. Yamazaki et al. developed first the concept of the sintering by hot-pressing under hydrothermal conditions for the preparation of strong porous hydroxyapatite (HA) ceramics with 42% open porosity after 3h-post-sintering at 1050°C [START_REF] Yamasaki | Porous hydroxyapatite ceramics prepared by hydrothermal hot-pressing[END_REF].

Nakahira et al reported also a hydrothermal hot pressing method followed by a post-sintering at 1100°C to fabricate porous hydroxyapatites with high strength and about 40% open porosity [START_REF] Nakahira | Fabrication of porous hydroxyapatite using hydrothermal hot pressing and post-sintering[END_REF].

However, hydrothermal processes have been scarcely reported for the production of porous bioactive glasses. Yet, the latter could take advantage of the opportunities of hydrothermal process such as onestep sintering procedure and low reaction temperatures (<375°C). To our knowledge, hydrothermal hot-pressing (HyHP) investigated on glass materials is only reported so far for the preparation of porous glass compacts to develop an alternative route for the recycling of used glasses (TV panel glass)

for environmental concerns [START_REF] Matamoros-Veloza | Low temperature preparation of porous materials from TV panel glass compacted via hydrothermal hot pressing[END_REF][START_REF] Matamoros-Veloza | Preparation of foamed glasses from CRT TV glass by means of hydrothermal hot-pressing technique[END_REF]. In this process, porous compacts produced by heating at 750°C the compacts densified by HyHP had a large amount of closed pores (58%).

The objective of this research is to produce porous bioactive monoliths from bioglass powders using the low temperature HyHP technique. The motivation is quintuple (i) to retain the amorphicity of the solid giving rise to a high reactivity in physiological fluids i.e. the ability to form in situ a hydroxyapaptite layer on its surface [START_REF] Siqueira | Gel-derived SiO2-CaO-Na2O-P2O5 bioactive powders: Synthesis and in vitro bioactivity[END_REF][START_REF] Chevalier | Ceramics for medical applications: A picture for the next 20 years[END_REF] (ii) to avoid grains growth so that to enhance the specific surface area and porosity of the final biomaterial to accelerate the kinetics of the formation of HA layer [START_REF] Vallet-Regí | Glasses with medical applications[END_REF] (iii) to produce open porosities and interconnected structures that are essential for cell nutrition, proliferation, and migration for tissue vascularization and formation of new tissues [START_REF] Loh | Three-dimensional scaffolds for tissue engineering applications: Role of porosity and pore size[END_REF] (iv) to limit the loss of thermosensitive molecules such as H2O, -OH groups, present in the precursor and (v) to make possible the functionalization of their surface with thermosensitive therapeutic agents (molecules of biological interest). The technique was applied to bioglass powders synthesized by sol-gel and melt quenching methods. The aim of the study is not to compare the two bioglasses but to find a process that allows to produce monoliths with high porosity and good mechanical strength. The study investigated sol-gel derived bioglass nanopowders in the SiO2-CaO-P2O5 ternary system referred as NBG of composition : (in mol %) SiO2-CaO-P2O5 (55.0-40.0-5.0) and melt-derived bioglass powders, termed as BG, of composition: (in mol %) SiO2-CaO-Na2O-P2O5 (47.2-26.4-23.8-2.6) which have been the subject of various studies within our laboratory proving a high potential for applications on the biomedical field [START_REF] Dietrich | Synthèse et études physico-chimiques de verres bioactifs denses et poreux[END_REF][START_REF] Bui | Elaboration de biomatériaux verres-substances actives (zolédronate-chitosane). Caractérisations physico-chimiques. Expérimentations "in vitro[END_REF][START_REF] Rocton | Élaboration de bioverres et de nanobioverres ostéoinducteurs associés à des molécules à usage thérapeutique[END_REF][START_REF] Bui | Bioactivity behaviour of biodegradable material comprising bioactive glass[END_REF][START_REF] Wers | Thermal investigations of Ti and Ag-doped bioactive glasses[END_REF][START_REF] Wers | Evaluation of the kinetic and relaxation time of gentamicin sulfate released from hybrid biomaterial Bioglass-chitosan scaffolds[END_REF]. Let us note that the NBG sol-gel composition does not require Na2O where in melt-quenched BG the primary role of Na2O is to lower the melting point, improving processability.

The bioactivity of the monoliths was evaluated by examining the ability of apatite to form on their surface by in vitro procedure involving the glass dissolution and its time-dependency in simulated body fluid (SBF) [START_REF] Kokubo | How useful is SBF in predicting in vivo bone bioactivity?[END_REF]. Our original HyHP process technique is first reported to produce highly porous bioglassbased monoliths (p> 50%).

Experimental

Bioglass nanopowder synthesis

Sol-gel derived NBG powder samples were prepared according to the following procedure issued from previous work [START_REF] Rocton | Élaboration de bioverres et de nanobioverres ostéoinducteurs associés à des molécules à usage thérapeutique[END_REF]. Tetraethyl orthosilicate or TEOS (Si(OC2H5)4) (99%, Merck), sodium trimetaphosphate (Na3P3O9) (95%, Alfa Aesar) and calcium nitrate tetrahydrate Ca(NO3)2,4H2O (98%, Fluka) were used as silica, phosphorous and calcium sources, respectively. The experimental protocol is based on the preparation and the mixture of two different emulsions, A and B. Triton X-100® (99%, Alfa Aesar) and octanol (99%, Alfa Aesar) were used as surfactants and cyclohexane (100%, VWR) was used as oil phase for the preparation of both emulsions. Emulsion A was prepared by dissolution in a 2.5M aqueous ammonia solution, under magnetic stirring of appropriate amounts of TEOS and Na3P3O9. This mixture was kept under strong agitation for one hour to allow the TEOS hydrolysis.

Emulsion B is an aqueous solution of appropriate amount of Ca(NO3)2,4H2O. The two emulsions A and B were subsequently mixed at a constant and vigorous speed for one hour to mix the reagents contained in the aqueous reactors of the emulsions. The mixed emulsions were then left to rest for a maturation (polymerization) time of less than 1 hour. After maturation, a white gel is collected by centrifugation and washed several times with ethanol to remove most of the surfactants and dried overnight at 100°C in an oven. The resulting white powder is calcined in air at 620°C for 4 hours in a muffle furnace and subsequently mechanically ground in water. After centrifugation, washing with ethanol and drying in an oven at 80 °C under vacuum, the obtained powder is of nanometric size ( 100 nm). Chemical analysis (ICP-OES) revealed that no sodium impurity was present after the preparation steps.

The melt-derived BG glass was prepared by reacting appropriate amounts of calcium metasilicate CaSiO3 (99%, Alfa Aesar), sodium metasilicate Na2SiO3, 5H2O (99.5%, Fischer) and sodium trimetaphosphate (Na3P3O9) (95%, Alfa Aesar) according to the experimental procedure described in our previous work [START_REF] Bui | Elaboration de biomatériaux verres-substances actives (zolédronate-chitosane). Caractérisations physico-chimiques. Expérimentations "in vitro[END_REF]. The resulting glass was crushed and sieved to select two different particle sizes: 40-63 m (coarse BG) and  40 µm (fine BG).

Elaboration of the monoliths by HyHP

The HyHP process (Fig. 1) is applied inside a hydrothermal autoclave with 10 mm internal diameter.

(detailed information in [START_REF] Ndayishimiye | Low temperature hydrothermal sintering process for the quasi-complete densification of nanometric αquartz[END_REF][START_REF] Largeteau | Trends in high pressure developments for new perspectives[END_REF]). Table 1 reports the HyHP sintering parameters (temperature (°C), external applied force (kN), holding time (min) and nature/quantity of the solvent) selected in order to form compacts with good mechanical strength. Initial tests showed that a minimum force of 0.5 kN was required to obtain well-sintered pellets and that it was necessary to operate below 200°C and limit the amount of solvent added (either deionized H2O or NaOH-2M aqueous solution) to prevent crystallization. It was also found that, for BG powders, hydrothermal sintering must be realized in NaOH-2M to obtain well-sintered BG pellets. We therefore chose to operate, for both powders, at a temperature of 150°C with an external force of 0.5 kN in presence of 80 to 240 L of solvent. The highest solvent ratio was determined to ensure that solvent surrounds the bioglass particles uniformly so that they are concerned by the hydrothermal sintering and avoid both the formation of a paste before pressing and exudation during the sintering. The hydrothermal pressure generated by the solvent is lower than the pressure exerted at bioglass intergranular contacts by the hot-press (0.5kN = 64bar). The samples were very strong even at short sintering time (5 min) showing the benefit of hydrothermal conditions on sintering. The quality of the pellets obtained enabled them to be cut using a diamond wire saw into samples of appropriate shape and size for the determination of their geometric density.

For an equal amount of material and regardless of the sintering parameters, the thickness of the NBG monoliths is double that of the BG samples, which suggests a greater porosity. Note that contrary to BG monoliths, NBG monoliths were obtained even in the absence of added solvent. This suggests that NBG powders may contain some residual water. The monoliths with the best specific surface areas were selected for the bioactivity tests.

Preparation of SBF

The simulated body fluid (SBF) solution has an ionic composition similar to that of human blood plasma. The SBF solution was prepared according to Kokubo's protocol [START_REF] Kokubo | How useful is SBF in predicting in vivo bone bioactivity?[END_REF] and previous work [START_REF] Rocton | Élaboration de bioverres et de nanobioverres ostéoinducteurs associés à des molécules à usage thérapeutique[END_REF].

The preparation of 1 L of SBF was realized by mixing Ca-SBF and P-SBF solutions (500 cm 3 each). 1 L of Ca-SBF and P-SBF solutions each was prepared at 37°C by dissolving in distilled water, respectively: 0.7349g CaCl2.2H2O, 0.6095g MgCl2.6H2O and 0.3484g K2HPO4, 0.7056g NaHCO3, 0.4473g KCl, 16.1061g

NaCl. After complete dissolution of the salts, the pH was adjusted to 7.4 with 6.057g Tris C4H11NO3

(Tris-hydroxymethyl aminomethane or Tris) and HCl 6 N.

Characterization

X-Ray Diffraction (XRD) patterns were collected at room temperature in the 2θ range 20-80° with a step size of 0.026° and a scan time per step of 40 s using a PANalytical X'Pert Pro diffractometer (Cu-L2,L3 radiation, = 1.5418 Å, 40 kV, 40 mA, PIXcel 1D detector). Data Collector and High-Score Plus softwares were used, respectively, for recording and analysis of the patterns. XRD analysis, after immersion in SBF, was performed on sample surfaces. Thermal analyses were performed on a Labsys evo 1600TG-DTA/DSC (Setaram) with a heating rate of 10 K.min -1 in nitrogen. The Fourier Transform Infrared (FTIR) transmittance spectra of the powdered glass samples were recorded using a FTIR spectrometer (Alpha Bruker Banner Lane, Coventry), in the 400-2000 cm -1 range, at room temperature, after mixing 1 mg of the glass with 100 mg of dried KBr. For FTIR analysis after immersion in SBF, powdered samples were scratched from pellet surfaces and diluted in KBr as previously described.

Nitrogen adsorption isotherms were measured at 77 K on a 3Flex adsorption analyzer from Micromeritics (Norcross, GA, USA). Prior to measurements, the samples were outgassed overnight at ambient temperature in a VacPrep degas system. Specific surface areas (SBET) were derived from the isotherms using the BET equation and a set of 6 experimental points of the linear range of the BET plot (0.05P/P 0 0.3). In addition, the method provides the total pore volume of the samples. X-ray tomography has been investigated as complementary technique to N2 physisorption measurements to visualize porosity higher than 20 m Prior to in vitro bioactivity tests, the bioglass monoliths (of 10 mm diameter and few millimeters thick) were embedded in a resin so as to expose only one face of the pelleted samples to the SBF solution to ensure for each sample identical exposition to SBF and further comparison possible. The samples were subsequently placed in an incubator at 37°C under a controlled agitation of 50 revs min -1 for fixed immersion times (1, 3, 7, 15 and 30 days). To analyse the ionic exchanges that took place between the surface of the immersed biomaterial and the synthetic physiological liquid, Inductively Coupled Plasma Atomic Emission Optical Spectroscopy (ICP-OES, iCAP 7000Series, Thermo Scientific) analysis was performed under argon flux to measure concentrations of silicon (Si), phosphorus (P) and calcium (Ca) on 10 mL sampling of SBF solution after each immersion time.

Surface morphology of the samples was examined before and after immersion in SBF by Scanning Electron Microscopy (SEM) with a JEOL JSM IT300 equipment (CMEBA, university of Rennes 1). Prior to SEM observations, the samples were metallized with gold.

Results and discussion

Bioglass powders characterization

Fig. 2 presents the XRD patterns of the NBG and BG bioglass powders. The patterns show only diffusion halos that confirms the amorphous glass structure of both synthesized powders. This result corroborates with DTA analysis, presented in supplementary data (Fig. S1), which allows to determine the characteristic glass transition, crystallization and melting temperatures.

The FTIR spectra, in the 400-2200 cm -1 spectral range, of NBG and BG bioglass powders are quite similar (Fig. 3). Bands characteristic of the vibrations associated to the silicate network observed at 466/502, 788/740, 940/923 and 1060/1032 cm -1 are assigned respectively to Si-O-Si bending mode, Si-O-Si symmetric stretching mode of bridging oxygen atoms between tetrahedra, Si-O-Si stretching mode of non-bridging oxygen atoms and Si-O-Si asymmetric stretching mode of bridging oxygen atoms within the tetrahedra [START_REF] Vallet-Regí | Glasses with medical applications[END_REF][START_REF] Elbatal | Characterization of some bioglass-ceramics[END_REF][START_REF] Adams | Sol-gel synthesis of SiO2-CaO-Na2O-P2O5 bioactive glass ceramic from sodium metasilicate[END_REF][START_REF] Swe | Fabrication of sol-gel derived new quaternary silicate Bioglass S55P4[END_REF]. The band located at 563/592cm -1 is attributed to P-O bending vibrational mode of PO4 3-group in amorphous phosphate [START_REF] Elbatal | Characterization of some bioglass-ceramics[END_REF][START_REF] Adams | Sol-gel synthesis of SiO2-CaO-Na2O-P2O5 bioactive glass ceramic from sodium metasilicate[END_REF] and the band at 1170 cm -1 is attributed to P=O stretching mode [START_REF] Elbatal | Characterization of some bioglass-ceramics[END_REF].The band at about 1630 cm -1 can be assigned to molecular water contained in bioglass powders or adsorbed on powder surfaces [START_REF] Elbatal | Characterization of some bioglass-ceramics[END_REF][START_REF] Adams | Sol-gel synthesis of SiO2-CaO-Na2O-P2O5 bioactive glass ceramic from sodium metasilicate[END_REF][START_REF] Weng | Formation and characteristics of the apatite layer on plasma-sprayed hydroxyapatite coatings in simulated body fluid[END_REF]. Finally small bands observed at about 1400 cm -1 are assigned to atmospheric CO2. The slight differences (band shifts)

between the two spectra are due to the presence of Na2O in BG which, as network modifier, brings about modifications to the glassy network (creation of non-bridging oxygen) [START_REF] Vallet-Regí | Glasses with medical applications[END_REF].

The particle size distribution of the bioglass powders determined by laser scattering is represented in terms of a histogram and a cumulative size distribution in Fig. 4. The NBG powder displays a major monomodal distribution of nanometric size (100 nm) with some agglomerates between 2 and 30 microns while the coarse and fine BG powders present major monomodal distributions centered respectively on 60 μm and 30 µm.

Bioglass monoliths characterization

Physisorption-Structure

Nitrogen adsorption-desorption isotherms were measured at 77 K on the monoliths samples elaborated by HyHP according to the sintering parameters reported in Table 1. All NBG monoliths have adsorption-desorption isotherms similar to that presented in Fig. 5a. With a hysteresis loop (H1 type) associated with capillary condensation taking place in mesopores and some limiting uptake at high relative pressures, the isotherms are of type IV, characteristic of mesoporous materials, as classified in [START_REF] Lykiema | Prepared for publication by the Subcommittee on Reporting Gas Adsorption Data Consisting of K[END_REF]. Coarse and fine BG samples have similar isotherms, classified as type II, characteristic of nonporous or macroporous solid (Fig. 5b) [START_REF] Elbatal | Characterization of some bioglass-ceramics[END_REF][START_REF] Lykiema | Prepared for publication by the Subcommittee on Reporting Gas Adsorption Data Consisting of K[END_REF].

The density of NBG powders measured by gas displacement pycnometry was 2.48  0.02 g/cm 3 .

This value which is less than that expected from the composition (2.91  0.01 g/cm 3 ) is attributed, as considered previously (paragraphs 1.2 and 3.1), to the residual water contained in the NBG powders (Supplementary data, Fig. S1). The density measured for BG powders was 2.71  0.02 g/cm 3 , in agreement with the glass composition close to that of Bioglass® 45S5 [START_REF] Hench | An Introduction to Bioceramics[END_REF]. Table 2 reports the geometric density of the monoliths after hydrothermal sintering and their resulting porosity, the specific surface areas determined by BET method and total pore volumes. The porosity measurements confirm the hypothesis of a greater porosity in NBG monoliths that we previously deduced from their thicknesses. The large specific surface area of NBG samples, up to 15-20 times that of BG samples, can be explained by their mesoporosity and large pore volumes. The porosity content of NBG monoliths is nearly the same (≈ 70%) in comparison to the various hydrothermal sintering parameters by the fact that the approximation on porosity could reach  3%.

However, significant differences are observed in terms of specific surface area (SBET) depending on the amount and nature of solvent. The decrease in the amount of NaOH-2M from 240 to 80 mL (NBG-260/259/256) is accompanied by a non significant increase in porosity (from 70% to 72%) but strongly limits the SBET decrease compared to the initial powder (59% decrease for 240 mL versus 43% decrease for 80 mL). ). Our hypothesis is that NaOH as basic solvent removes, in the first few minutes, the surface roughness of the powder particles with which it comes in contact, reducing the specific surface area.

This effect is enhanced by increased NaOH volume. Moreover, if we compare the sintering in the presence of H2O or NaOH-2M (see NBG-231 vs NBG-232 and NBG-256 vs NBG-257), SBET is lowered by approximately 30% in the presence of NaOH for both sintering times. The surface areas are maintained at about 150 m 2 /g in the presence of H2O while they drop to about 100 m 2 /g in the presence of NaOH due to its caustic effect, as previously mentioned. The increase in HyHP dwell time has though no effect on the specific surface area and the porosity. For BG samples, the larger surface areas of monoliths compared to powders can be explained by the reduction in grain size induced by the dissolutionprecipitation mechanism whose effect is more pronounced for coarser powders. It should be noted that this hydrothermal sintering process offers compared to conventional sintering (solid state diffusion) the major advantage of producing monoliths with finer grains than those of the precursor powders.

In previous work, Ndayishimiye et al reported on the HyHP sintering conditions (same equipment)

of LSMO (La0.66Sr0.34MnO3)-SiO2 composites and silica in presence of NaOH and water solvents, respectively [START_REF] Ndayishimiye | Design of 0-3 type nanocomposites using hydrothermal sintering[END_REF][START_REF] Ndayishimiye | Hydrothermal sintering for densification of silica. Evidence for the role of water[END_REF]. Sintering conditions were slightly different to our conditions (high external force applied for hours in presence of a mineralizer) as the objective was to maximize the densification. It was observed that NaOH solvent induces a better sintering than H2O due to a higher intergranular dissolution (mineralizer effect) and precipitation inside the free space porosity between the particles which results in a decrease in porosity content [START_REF] Goglio | View point on hydrothermal sintering: Main features, today's recent advances and tomorrow's promises[END_REF]. In the present work, more gentle conditions were used so as to control the dissolution-precipitation phenomena in order to consolidate (and not densify)

the powders and create a large quantity of porosities. Indeed, monoliths with good mechanical strength (allowing their handling) were produced at low temperature (150°C) under a low external force (0.5 kN) applied for only 5 min. These conditions achieve sufficient consolidation by formation of necks (first sintering stage). The sintering process seems to be stopped after 5 min as surface areas and porosity values are unchanged after 30 min-HyHP explaining the large porosity values.

Finally, the tests carried out in absence of solvent (NBG 255) give the best results in terms of porosity (73%) and specific surface area (only 6% reduction compared to the initial powder). As mentioned previously, the initial powder contained chemisorbed water which is released by heating.

The quantity of water measured by TGA at 150°C (sintering temperature) in the initial 0.2g NBG powder ( 3% in weight) is about 6µL (Fig. S1). This quantity, though small, proves to be sufficient to produce the hydrothermal conditions necessary for sintering; the addition of water is therefore not essential for sintering naturally hydrated NBG nanopowders.

We have also determined the rate of open porosities from density measured by the impregnation method in water based on Archimedes principle [START_REF] Kwan | The impact of water impregnation method on the accuracy of open porosity measurements[END_REF]. Results are presented in supplementary data (Table S1). We found that the percentage of open porosity is close to the rate of porosity determined geometrically reported in Table 2. The monoliths thus obtained have essentially open porosity proving the HyHP effectiveness of processing interconnected porous structures.

Tomography has been used as complementary experimental method to visualize pores larger than 20 µm (voxel: 7 µm). Fig. 6 shows large pores of about 60µm size in coarse BG samples (Fig. 6(a)) and smaller pores (20 m or less) in fine BG samples confirming BG-based samples are macroporous in accordance with the physisorption characterization (Fig. 5). In contrast, tomography images of NBG samples do not present any visible macroporosity at this scale level. Let us recall that physisorption characterization revealed the presence of mesopores (2-50 nm) in NBG samples. NBG monoliths are obtained with 70% of porosity which remains mainly open. No results exist in the literature which achieve such a high percentage of porosity in bioglass monoliths. To date, a maximum porosity of 60% is reported for bioactive glass scaffolds processed by additive manufacturing techniques [1]. While it is generally admitted that it is difficult to obtain high levels of porosity (>50%) with interconnected porosity, we demonstrated that our technique can efficiently process bioglass powders into porous structures while retaining the amorphous glass structure. In the next step, HyHP technique was applied to process a structure with gradients of porosity by co-sintering coarse and fine BG powders. Fig. 5(d) shows such a monolith with distinguishable coarse BG on top and fine BG at the bottom. This demonstration proves the ability of the technique to process scaffolds with different scales of porosity to mimic the structure of porous bone. The literature reported so far only mention the combination of different techniques (sol-gel, co-polymer templates or sacrificial methyl cellulose templates, rapid prototyping technique) to achieve combined (giant)-macro-mesoporosity [START_REF] Yun | Design and preparation of bioactive glasses with hierarchical pore networks[END_REF][START_REF] García | Preparation of 3-D scaffolds in the SiO2-P2O5 system with tailored hierarchical meso-macroporosity[END_REF].

Following these observations, the in vitro bioactivity tests were performed on NBG 255 and NBG 257 samples showing the best specific surface/porosity (likely to induce better bioactivity) and compared to those of BG samples.

In vitro bioactivity tests

In vitro tests were performed according to a method in which a SBF solution is used in the in vitro evaluation of the formation of a HA layer on the surface of the bioglass scaffolds. has barely begun in this period. This result is consistent with bioactivity tests that were reported on BG powders which showed HA characteristic lines only after 15 days of immersion [START_REF] Bui | Elaboration de biomatériaux verres-substances actives (zolédronate-chitosane). Caractérisations physico-chimiques. Expérimentations "in vitro[END_REF].

XRD characterization a) NBG monoliths

FTIR analysis

After immersion in the SBF, the characteristic bands in Fig. 3

SEM characterization

SEM analyses provided detailed observations of the surface of the monoliths as a function of the immersion time in the SBF. Fig. 10 shows surface micrographs of NBG 257 and BG 264 samples before and after soaking in SBF for 1 to 7 days. Significant changes in surface morphology can be observed for NBG samples from 1 day of immersion whereas BG surface is not modified even after 7 days of immersion (the glass surface is still visible). As can be observed, only after 1 day of immersion, NBG surfaces, initially made of partially sintered spherical particles, present the typical morphology of HA (elongated particles) [START_REF] Chevalier | Ceramics for medical applications: A picture for the next 20 years[END_REF][START_REF] Vallet-Regí | Glasses with medical applications[END_REF]. On the contrary, images of BG surfaces exhibit similar morphologies for 7 days duration of bioactivity tests. All these observations are in agreement with the XRD patterns and IR spectra obtained for the same immersion times. These observations attest that NBG samples are more bioactive than BG samples.

ICP-OES characterization

The concentrations of silicon, calcium and phosphorus species present in SBF solution were measured by ICP-OES as a function of soaking time to monitor the kinetics of consumption and release of these elements occurring during the bioactivity process (Fig. 11). Fig. 11(a) shows a significant release of Si from the first immersion day and a similar profile over the immersion time for all samples.

This release is the consequence of stage 2 of the bioactivity process characterized by the loss of soluble silica Si(OH)4 by rupture of the Si-O-Si bonds. Fig. 11(b) shows that the P concentration decreases rapidly over the first day of dissolution and, considering P composition (P2O5 in mol%: 5.0% for NBG vs 2.6% for BG), that the consumption is much faster for NBG than BG. Concurrent with the Si concentration increase, this decrease in P concentration is attributed to the incorporation of P ions from the dissolution medium within the Ca-P rich reaction layer forming on the glass surface. NBG is so reactive that all available P is consumed within 3 days of immersion. The P release occurs less rapidly and over a longer period of time (> 15 days) in BG monoliths. Fig. 11(c) shows differences between NBG and BG in Ca dissolution profiles. For NBG, Ca concentration increases until day 3 then decreases continuously to its initial value after 30 days. The Ca release corresponds to stage 1 of the bioactivity mechanism in which Ca 2+ ions exchange rapidly with H + from solution (dealkalinization) creating silanol bonds (Si-OH) on the glass surface. The consumption corresponds to stage 4 in which Ca 2+ ions migrate from the SBF to the surface of the bioactive glass to form a film rich in amorphous CaO-P2O5. The Ca concentration observed for BG samples, on the other hand, decreased continuously (stage 4). The compositional and morphological differences between NBG and BG may account for the observed dissolution profiles. The higher CaO content (40.0% in mol% for NBG vs 26.4% for BG) and high surface areas of NBG samples explain the calcium release whereas for BG samples the lower CaO content, the less porous surfaces and lower surface areas limit the dissolution rate and Ca release. These results are in agreement with those reported by Sepulveda et al [START_REF] Sepulveda | In vitro dissolution of melt-derived 45S5 and sol-gel derived 58S bioactive glasses[END_REF] on the dissolution behavior of meltderived and sol-gel derived bioactive glasses. This study showed indeed that the higher CaO content in sol-gel composition accounts for the higher calcium concentration released in SBF and faster saturation of calcium and phosphate ions, conditions that accelerate the kinetics of an apatite layer precipitation. This mechanism is supported, as in [START_REF] Sepulveda | In vitro dissolution of melt-derived 45S5 and sol-gel derived 58S bioactive glasses[END_REF], by the faster depletion of phosphorous in SBF that we observed for sol-gel derived monoliths. The superior bioactivity of NBG monoliths, confirmed by FTIR-XRD analyses and SEM observations that proved the formation of a HCA layer on the bioglass, is attributed to the highly porous bioglass surfaces that our HyHP technique was able to produce from sol-gel nanometric powders.

Conclusions

In this study, the processing of porous bioglass monoliths using HyHP technique is reported for the time. This process makes it possible to obtain, in a single step, structures with open porosities of different size depending on the characteristics of the precursor powders themselves depending on the synthesis method. We have thus shown that HyHP applied on sol-gel derived glass powders build monoliths with >70% open porosity whereas applied on melt-derived glass powders, the open porosity is at most 45%. The high porosity content in NBG monoliths results from their mesoporosity induced by the nanometric size of the precursor powders. Contrary to BG powders, the sintering of NBG powders occurs in absence of a mineralizer (NaOH) and can proceed, additionally, in absence of H2O that they already contain. All the monoliths sintered at only 150°C for 5 min present notable mechanical strength and can be cut by a diamond wire saw. Finally, we have shown that our technique can process 3D structures with gradient of porosity. Our HyHP technique proves to be a real breakthrough in the realization of consolidated bioglasses with controlled porosity to be adapted to various implantation sites. In addition, as single-step, low temperature and short time process (150°C, 5 min), this technique is a cost-effective and eco-efficient alternative for sintering biomaterials to conventional and 3D techniques.

Figures captions

  [START_REF] Baruchel | X-Ray Tomography in Material Science[END_REF] using a A V|tome|X apparatus (GE) equipped with a 2024x2024 pixels flat panel XRD detector and a GE 180 kV transmission tube. The scans were acquired at an accelerating voltage of 80 kV, a current of 200 µmA with no beam hardening filters. 1500 projections were taken, each being the average of 2 radiographs with an exposure time of 1s. The voxel size was 7 µm and the scan total duration was about 1 hour.Particle size analysis of bioglass powders after grinding and before sintering was performed using a Laser Particle Sizer Analysette 22 NanoTec (FRITSCH GmbH). Powders were dispersed in water and exposed to ultrasounds during analysis. Powders density was determined by dividing the sample weight by the volume measured by gas (He) displacement pycnometry using AccuPyc 1330 pycnometer from Micromeritics. Volume was averaged from 5 measurements.

Fig. 7

 7 Fig.7presents the XRD patterns of the surface of samples NBG 255 and 257 at fixed immersion times (0, 1, 3, 7, 15 and 30 days). The XRD pattern of synthetic HA (Ca10(PO4)6(OH)2, calcium hydroxyde phosphate AR powder, Alfa) is included as reference. NBG 255 and NBG 257 samples show the characteristic lines of the HA from the first day of immersion in the SBF with the major lines (002) and (211) appearing distinctly at about 26°(2θ) and 32°(2θ) respectively. The lines intensities increase with the immersion time proving the growth of the HA layer and after 7 days of immersion, it is possible to distinguish some secondary lines such as the (130), (222) and (213) reflections of HA phase. These results indicate that bioactivity reactions reached within 1 day the final stage (stage 5) of the proposed stages for in vitro HA formation in SBF[1,[START_REF] Hench | Bioceramics: from concept to clinic[END_REF].

  are modified due to chemical reactions on the surface of the bioactive glass leading to the formation of HA. Fig.9shows the FTIR spectra of NBG 257 and BG 264 samples before and after exposure to SBF solution for 0-15 days. The FTIR spectrum of HA is included as reference. It is worth mentioning that FTIR spectra of NBG 255 and BG 229 samples are so similar, respectively, to those of NBG 257 and BG 264 that only NBG 257 and BG 264 spectra are reported for clarity. We observe for NBG 257, from the first immersion day, the presence of P-O bands at 565, 603 and 1039 cm -1 which are characteristic of crystalline phosphates and confirm the formation of calcium phosphate layer on the monolith surface. C-O bands at 874 and 1420 cm -1 are also revealed. The presence of characteristic bands of carbonate groups proves the formation of a layer of carbonated hydroxyapatite (HCA) on the surface of the monoliths in contact with SBF. For BG 264 sample, we observe that the characteristic bands are deformed and/or shifted from the first day of immersion. These deformations indicate the formation of a silica-rich layer on the glass surface during stage 3 of the bioactivity process. Yet, the formation of HA proceeds only after 15 days of immersion.

Fig. 1 .Fig. 2 .Fig. 3 .Fig. 4 :Fig. 5 .Fig. 6 .Fig. 7 .Fig. 8 .Fig. 10 .

 1234567810 Fig.1. External view of HyHP equipment and internal schematic view of the autoclave Fig.2. XRD patterns of BG and NBG bioglass powders Fig.3. FTIR spectra of BG and NBG bioglass powders Fig.4: Particle size distribution of NBG and BG bioglass powders (ultrasonicated and water-dispersed during analysis) Fig.5. Typical N2 adsorption-desorption isotherms measured at 77K on (a) NBG monoliths and (b) BG monolith with photographs in inset Fig. 6. Tomography images of (a) coarse BG, (b) fine BG and (c) NBG monoliths (d) BG gradient monolith processed by HyHP at 150°C -5min -0,5kN (diam: 10 mm) Fig. 7. XRD patterns of surfaces of (a) NBG 255 and (b) NBG 257 monoliths at different immersion times in SBF Fig. 8. XRD patterns of surfaces of (a) BG 264 and (b) BG 229 monoliths at different immersion times in SBF Fig. 9. FTIR spectra of NBG 257 and BG 264 samples before and after soaking in SBF solution at different immersion times Fig. 10. SEM micrographs of surfaces of NBG and BG monoliths processed by HyHPFig. 11: (a) Si (b) P (c) Ca elemental concentration in SBF as a function of immersion time

Fig. 2

 2 Fig. 2

Fig. 3

 3 Fig. 3

  Fig. 4

  Fig. 6

Fig. 9 (

 9 Fig. 9

  Fig. 11

Table 1

 1

	Parameters of HyHP applied on NBG and BG powders (0.2g powder, 10mm diameter) and thickness of monoliths
	after HyHP.
	Sample	Time	Solvent	Vol (µL)	T	F	Thickness
		(min)			(°C)	(kN)	(mm)
	NBG
	NBG-231	30	NaOH-2M	80	150	0.5	3.37
	NBG-232	30	H20	80	150	0.5	3.71
	NBG-255	5	no	no	150	0.5	3.6
	NBG-256	5	NaOH-2M	80	150	0.5	3.6
	NBG-257	5	H2O	80	150	0.5	3.61
	NBG-259	5	NaOH-2M	160	150	0.5	3.7
	NBG-260	5	NaOH-2M	240	150	0.5	3.37
	NBG-261	5	NaOH-2M	240	100	0.5	3.27
	Coarse BG
	BG-264	5	NaOH-2M	60	150	0.5	1.7
	Fine BG
	BG-229	5	NaOH-2M	80	150	0.5	1.71

Table 2

 2 Geometric density, porosity, SBET, total pore volume of NBG and BG monoliths (0.2g powder, 0.5kN).

	Sample	Time	Solvent	Vol	T	Density	Porosity	SBET	Pore volume
		(min)		(µL)	(°C)	(g/cm 3 )	(%)	(m 2 /g)	(cm 3 /g)
	NBG
	NBG powder							175	0.757
	NBG-231	30	NaOH-2M	80	150	0.74	71	101	0.404
	NBG-232	30	H2O	80	150	0.67	73	146	0.641
	NBG-255	5	no	no	150	0.69	72	164	0.680
	NBG-256	5	NaOH-2M	80	150	0.71	72	99	0.381
	NBG-257	5	H2O	80	150	0.67	73	148	0.621
	NBG-259	5	NaOH-2M	160	150	0.73	71	77	0.256
	NBG-260	5	NaOH-2M	240	150	0.76	70	72	0.268
	Coarse BG powder							0.4	0.0004
	BG-264	5	NaOH-2M	60	150	1.48	45	8	0.006
	Fine BG powder							0.5	0.0005
	BG-229	5	NaOH-2M	80	150	1.53	43	6	0.004

Table S1 .

 S1 Density (impregnation method) and open porosity of BG and NBG HyHP sintered monoliths. The density of the sintered samples was calculated from the data obtained by the Archimedes method, according to the European Standard EN623-2[START_REF] Kwan | The impact of water impregnation method on the accuracy of open porosity measurements[END_REF].

	Sample	Density	Open porosity
		(g/cm 3 )	(%)
	NBG 255	0.70  0.01	70.4
	NBG 257	0.66  0.01	72.7
	BG 264	1.48  0.01	41.5
	BG 229	1.45  0.01	43.8

Acknowledgments

We thank Ms Lisa Andrey for her contribution to the HyHP experiments at ICMCB UMR5026, as part of her Master 2 internship.

Supplementary data

Processing of highly porous bioglass monoliths by hydrothermal hot pressing

TGA results show, at 150°C (HyHP sintering temperature), weight losses of 3% and 0.7% for NBG and BG bioglass powders respectively. The DTA curve of NBG powder (Fig. S1a) shows an endothermic effect caused by the glass transition at Tg 500°C followed by exothermic crystallisation peaks at Tc1 = 833°C, Tc2= 879°C and Tc3= 1228°C of different silicas and calcium silicate. Finally, two endothermic peaks at Tm1= 1305°C and Tm2= 1332°C may be attributed to the melting of two different crystalline phases. The DTA curve of BG powder (Fig. S1b) shows an endothermic effect caused by the glass transition at Tg= 530°C followed by an exothermic crystallisation peak at Tc= 690°C of sodium calcium silicate and two endothermic peaks at Tm1= 1160°C and Tm2= 1218°C attributed to the melting of two different crystalline phases of which sodium calcium silicate. The results are in agreement with those presented in previous works [1][2][3].