J Puibasset
email: puibasset@cnrs-orleans.fr

Adsorption-Induced Deformation of a Nanoporous Material: Influence of the Fluid-Adsorbent Interaction and Surface Freezing on the Pore-Load Modulus Measurement

    Adsorption-Induced Deformation of a Nanoporous Material: Influence of the Fluid-Adsorbent Interaction and Surface Freezing on the Pore-Load Modulus Measurement J.

Introduction

Recently, many theoretical and experimental results have been reported on the mechanical properties of nanostructures, in particular the elastic modulus of nanowires, nanoplates and nanobeams due to their widely proposed applications in nanoelectromechanical systems (NEMS). The knowledge of their mechanical properties is of considerable interest and has attracted many studies, which have revealed a strong size-dependence while approaching the nanometer scale. A straightforward procedure to understand the mechanical properties of small systems is probably to perform direct measurements at the nanoscale in simple geometries. A good example of model system to study size and surface effects is probably the case of silicon nanocantilever. [START_REF] Li | Ultrathin Single-Crystalline-Silicon Cantilever Resonators: Fabrication Technology and Significant Specimen Size Effect on Young's Modulus[END_REF][START_REF] Gordon | Size Effects in Mechanical Deformation and Fracture of Cantilevered Silicon Nanowires[END_REF][START_REF] Sadeghian | On the Size-Dependent Elasticity of Silicon Nanocantilevers: Impact of Defects[END_REF] The size dependence of the mechanical properties has been evidenced, and has generally been attributed to surface stresses induced by the free surfaces. [START_REF] Gumbsch | Interface Stresses and Their Effects on the Elastic Moduli of Metallic Multilayers[END_REF][START_REF] Wolf | Surface-Stress-Induced Structure and Elastic Behavior of Thin Films[END_REF][START_REF] Cammarata | Surface and Interface Stress Effects in Thin Films[END_REF][START_REF] Miller | Size-Dependent Elastic Properties of Nanosized Structural Elements[END_REF][START_REF] Van Workum | Local Elastic Constants in Thin Films of an Fcc Crystal[END_REF][START_REF] Fedorchenko | Thickness Dependence of Nanofilm Elastic Modulus[END_REF][START_REF] Haiss | Surface Stress of Clean and Adsorbate-Covered Solids[END_REF] The same surface stresses are expected to play an important role in the deformation of nanoporous materials which exhibit large surface-to-volume ratio. [START_REF] Liu | The Pore-Load Modulus of Ordered Nanoporous Materials with Surface Effects[END_REF] Furthermore, these materials can adsorb fluids, inducing capillary forces strong enough to deform the substrate. This concerns not only soft materials like aerogels or biopolymers, [START_REF] Dušek | Responsive Gels: Volume Transitions I[END_REF][START_REF] Scherer | Deformation of Aerogels During Characterization[END_REF][START_REF] Scherer | Compression of Aerogels[END_REF][START_REF] Thibault | Silica-Aerogel Thermal Expansion Induced by Submonolayer Helium Adsorption[END_REF][START_REF] Herman | Deformation of Silica Aerogel During Fluid Adsorption[END_REF][START_REF] Ogieglo | In Situ Ellipsometry Studies on Swelling of Thin Polymer Films: A Review[END_REF][START_REF] Kulasinski | Poroelastic Model for Adsorption-Induced Deformation of Biopolymers Obtained from Molecular Simulations[END_REF] but also stiff materials. 19- attempts. [START_REF] Hofmann | Phonons in Mesoporous Silicon: The Influence of Nanostructuring on the Dispersion in the Debye Regime[END_REF] The pore-load measurement is done as follows: a nanoporous material is saturated with a liquid and brought below its saturated vapor pressure. The resulting capillary tension in the liquid (negative pressure ) is large enough to induce measurable deformations of the nanoporous material. For isotropic porous materials, in the limit of elastic deformation  , the pore-load modulus is defined as . This quantity, introduced by Biot in 1941, [START_REF] Biot | General Theory of Three-Dimensional Consolidation[END_REF] measures the (linear) relation between the liquid pressure variations inside the pores and the solid deformation. It is an effective elastic modulus that depends on the solid stiffness but also on its geometry. [START_REF] Scherer | Dilatation of Porous Glass[END_REF][50][START_REF] Silva | The Effects of Non-Periodic Microstructure on the Elastic Properties of Two-Dimensional Cellular Solids[END_REF][START_REF] Gibson | Cellular Solids: Structure and Properties[END_REF][START_REF] Zhu | Effects of Cell Irregularity on the Elastic Properties of 2d Voronoi Honeycombs[END_REF][START_REF] Gor | Elastic Response of Mesoporous Silicon to Capillary Pressures in the Pores[END_REF] In anisotropic solids, it also depends on the direction. [START_REF] Sharifi | Humidity-Driven Deformation of Ordered Mesoporous Silica Films[END_REF][START_REF] Grosman | Adsorption-Induced Strain of a Nanoscale Silicon Honeycomb[END_REF] This quantity provides in general not enough information to deduce the intrinsic elastic constants of the solid.

It is however possible if complementary measurements can be performed. [START_REF] Rolley | Using Capillary Forces to Determine the Elastic Properties of Mesoporous Materials[END_REF] Poromechanics is a consistent approach to model the mechanical coupling between fluids and solids, including adsorption-induced deformations. [START_REF] Kulasinski | Poroelastic Model for Adsorption-Induced Deformation of Biopolymers Obtained from Molecular Simulations[END_REF][START_REF] Biot | General Theory of Three-Dimensional Consolidation[END_REF][55][START_REF] Coasne | Poroelastic Theory Applied to the Adsorption-Induced Deformation of Vitreous Silica[END_REF][START_REF] Brochard | Poromechanics of Microporous Media[END_REF] However, nanoporous materials are complex systems: comparison between experiments and theory is not straightforward, and more sophisticated models, including simulations, are required to take into account the inhomogeneous and anisotropic nature of the material. [START_REF] Scherer | Dilatation of Porous Glass[END_REF][START_REF] Yates | The Expansion of Porous Glass on the Adsorption of Non-Polar Gases[END_REF][START_REF] Ash | Thermodynamics of the Effects of Adsorption on Interparticle Forces[END_REF][START_REF] Gor | Elastic Response of Mesoporous Silicon to Capillary Pressures in the Pores[END_REF][START_REF] Shen | A Molecular Model of Adsorption in a Dilute Semiflexible Porous Network[END_REF][START_REF] Ravikovitch | Density Functional Theory Model of Adsorption Deformation[END_REF][START_REF] Kowalczyk | Adsorption-Induced Deformation of Microporous Carbons: Pore Size Distribution Effect[END_REF][START_REF] Günther | Sorption Strains and Their Consequences for Capillary Condensation in Nanoconfinement[END_REF][START_REF] Gor | Adsorption-Induced Deformation of Mesoporous Solids[END_REF][START_REF] Gor | Adsorption-Induced Deformation of Mesoporous Solids: Macroscopic Approach and Density Functional Theory[END_REF][START_REF] Gor | Adsorption of n-Pentane on Mesoporous Silica and Adsorbent Deformation[END_REF][START_REF] Schappert | Unexpected Sorption-Induced Deformation of Nanoporous Glass: Evidence for Spatial Rearrangement of Adsorbed Argon[END_REF][START_REF] Gor | Revisiting Bangham's Law of Adsorption-Induced Deformation: Changes of Surface Energy and Surface Stress[END_REF][START_REF] Diao | Monte Carlo Simulation of Adsorption-Induced Deformation in Finite Graphitic Slit Pores[END_REF][START_REF] Shkilev | Thermodynamics of Adsorption on Deformable Adsorbents[END_REF][START_REF] Balzer | Adsorption-Induced Deformation of Hierarchically Structured Mesoporous Silica-Effect of Pore-Level Anisotropy[END_REF] It is well-known that porous materials have smaller elastic moduli than the bulk, mainly due to their lower density; but the exact geometry of the porous material needs to be considered in order to quantitatively predict the mechanical properties of the system. [START_REF] Silva | The Effects of Non-Periodic Microstructure on the Elastic Properties of Two-Dimensional Cellular Solids[END_REF][START_REF] Gibson | Cellular Solids: Structure and Properties[END_REF][START_REF] Zhu | Effects of Cell Irregularity on the Elastic Properties of 2d Voronoi Honeycombs[END_REF] There is however generally a disagreement between theory and experiments, which can be large, despite the development of refined models taking into account the pore geometry and surface stress. [START_REF] Liu | The Pore-Load Modulus of Ordered Nanoporous Materials with Surface Effects[END_REF][START_REF] Guyer | Theoretical Model for Fluid-Solid Coupling in Porous Materials[END_REF][START_REF] Dolino | Adsorption Strains in Porous Silicon[END_REF][START_REF] Prass | Capillarity-Driven Deformation of Ordered Nanoporous Silica[END_REF][START_REF] Rolley | Using Capillary Forces to Determine the Elastic Properties of Mesoporous Materials[END_REF][START_REF] Gor | Elastic Response of Mesoporous Silicon to Capillary Pressures in the Pores[END_REF][START_REF] Halsey | Surface Tension and Spreading Pressure of Mobile and Registered Monolayers[END_REF][START_REF] Rusanov | On the Theory of the Mechanochemical Sorption-Striction Phenomenon in Nanoporous Bodies with Dispersion Forces[END_REF] Several hypotheses have been invoked, including surface defects, [START_REF] Sadeghian | On the Size-Dependent Elasticity of Silicon Nanocantilevers: Impact of Defects[END_REF] or a possible interplay between surface stress and adsorption that could explain some adsorption hysteresis features. [START_REF] Amberg | A Study of Adsorption Hysteresis by Means of Length Changes of a Rod of Porous Glass[END_REF][START_REF] Quinn | The Hysteresis Loop in Adsorption Isotherms on Porous Vycor Glass and Associated Dimensional Changes of the Adsorbent[END_REF][START_REF] Fomkin | Adsorption of Gases, Vapors and Liquids by Microporous Adsorbents[END_REF][START_REF] Günther | Novel Insights into Nanopore Deformation Caused by Capillary Condensation[END_REF][START_REF] Grosman | Capillary Condensation in Porous Materials. Hysteresis and Interaction Mechanism without Pore Blocking/Percolation Process[END_REF][START_REF] Grosman | Influence of Elastic Deformation of Porous Materials in Adsorption-Desorption Process: A Thermodynamic Approach[END_REF][START_REF] Grosman | Influence of Elastic Strains on the Adsorption Process in Porous Materials: An Experimental Approach[END_REF][START_REF] Grosman | Influence of Elastic Strains on the Adsorption Process in Porous Materials. Thermodynamics and Experiments[END_REF][START_REF] Schoen | Pore-Lattice Deformations in Ordered Mesoporous Matrices: Experimental Studies and Theoretical Analysis[END_REF][START_REF] Guyer | Theoretical Model for Fluid-Solid Coupling in Porous Materials[END_REF] In that case, one expects a fluid-dependence of the pore-load modulus. Such observations have been reported in the literature several decades ago, [START_REF] Quinn | The Hysteresis Loop in Adsorption Isotherms on Porous Vycor Glass and Associated Dimensional Changes of the Adsorbent[END_REF] but are not confirmed by recent experiments. [START_REF] Rolley | Using Capillary Forces to Determine the Elastic Properties of Mesoporous Materials[END_REF] The following question thus arises: is there an effect of the fluid-wall interaction on the adsorption-induced deformation in the saturation regime? Considering the small (nanometric) scale involved, molecular simulations provide the natural framework to investigate this issue.

The idea is to perform a self-consistent test of the adsorption-induced deformation of a nanosized system (a nanoplatelet), by way of a "numerical experiment" that mimics a real experiment. The observed deformation is then compared with the mechanical properties of the solid. The key point is that the geometry of the solid and the forces between the atoms being known, all elastic constants of the solid, including the surface stresses, can be determined. The solid deformation in presence of an adsorbed fluid can thus be determined in the framework of a continuous model and compared with the observed deformation. Any difference would be the signature of a phenomenon beyond the surface stress of the solid. Varying the solid thickness and the intensity of the fluid-solid interaction helps to determine the origin of the observed discrepancy.

The paper is organized as follows: section 2 is devoted to the presentation of the atomistic model, the molecular simulation methods, as well as the thermodynamic model of solid deformation and the expected value for the pore-load modulus. Section 3 focuses on the simulation results for the fluid and solid physical properties, as well as the pore-load numerical experiment. The effect of the solid thickness and the intensity of the fluid-solid interaction are then discussed. It is shown that a modification of the fluid structure at the solid surface could explain the simulation results, and we conclude on the relevance of the phenomenon for experimentalists.

Methods

Model

Let us consider a nanoporous material in contact with a vapor at pressure P below its saturated vapor pressure . Figure 1 shows a typical adsorption-desorption isotherm. Thermodynamics tells us that the fluid-solid interactions favor the condensed liquid phase when approaching the saturated vapor pressure: this is the well-known capillary condensation that occurs in nanometric pores. The isotherm exhibits a hysteresis: upon reduction of the gas pressure, the system remains saturated with liquid until desorption pressure ( ) is reached. Starting from the saturated system, it is possible to travel back and forth reversibly along the saturation plateau, without emptying the nanoporous material. The pore-load modulus is measured along this plateau.

Let us consider a portion of a pore wall surrounded by liquid on both faces (see inset in Figure 1). This will define our simulation box (see Figure 2). For simplicity, we will consider a wall of constant thickness. The x and y directions are chosen parallel to the solid. The corresponding dimensions (L x and L y ) are chosen large enough, together with periodic boundary conditions, to minimize finite size effects in these directions. The dimension L z perpendicular to the solid surface is also chosen large enough so that the liquid far from the surface is bulk-like. Periodic boundary conditions are also applied in that direction to avoid the introduction of an extra interface. As a consequence, the fluid in the simulation box is actually confined between two flat surfaces (equivalent to a slit pore), which are far enough not to introduce cross-talking (the fluid in the center of the slit pore is bulk-like). This system can thus be seen as a nanoplatelet immersed in a liquid.

The introduction of periodic boundary conditions parallel to the solid wall is equivalent to considering an infinite porous material without explicit boundaries. As a consequence, we do not treat explicitly the liquid-vapor interfaces (meniscus) which are localized at the periphery of the porous material (saturation). We also implicitly disregard their propagation from the boundary towards the core of the material upon desorption. As a consequence, in our simulation model, it will be possible to decrease the liquid pressure down to its bulk stability limit (spontaneous cavitation). This pressure is generally well below the emptying pressure of the system, which, in most situations, is driven by meniscus propagation or heterogeneous cavitation, and strongly depends on the porous solid chemistry and structure. Another consequence of the absence of boundaries is that we can increase the liquid pressure above its saturated vapor pressure, which is not possible experimentally. These extra capabilities offered by simulation will be used to enlarge the range of variation of the liquid pressure, to explore new physics or increase the accuracy.

Simulation Methods

We use Monte Carlo simulations in different statistical ensembles and simulation box configurations to determine the equilibrium properties of our system in two situations: the dry nanoporous material, and the same system saturated with liquid at a given chemical potential (or equivalently liquid pressure). In both cases, the system is supposed to be in thermal equilibrium with a reservoir at temperature T.

(i) In the first configuration, only the solid nanoplatelet is present (there is no fluid, see Figure 2a). The dimensions L x and L y are allowed to fluctuate, while an external pressure is imposed in the corresponding directions. The dimension L z is kept constant. This corresponds to the isostress ensemble in directions x and y. This box configuration is used to determine the elastics constants of the solid, as follows. A given pressure is imposed along the direction x, while zero pressure is imposed in the directions y and z (along y, the box is left free to fluctuate, and the pressure on the solid in direction z is zero by construction). The measured deformations along directions x, y and z allow to calculate the Young modulus along x, and the Poisson's ratio  xy and  xz . The in-plane modulus B can also be obtained by imposing the same pressure along the x and y directions. The other parameters of the compliance matrix will not be used.

(ii) In the second box configuration, the liquid is introduced, in equilibrium with a reservoir at chemical potential  (see Figure 2b). It corresponds to the semi-grand canonical ensemble. The chemical potential of the reservoir is given by the gas pressure above the nanoporous material.

In this box configuration, the external pressure in directions x and y is set equal to zero: the liquid pressure and the solid stress thus have to compensate on the simulation box walls, which induces a deformation of the solid. This mimics a pore-load modulus measurement where the nanoporous solid is left free to deform upon adsorption. The detailed analysis is given in section 2.4.

Interatomic Potentials

The Solid.

The atomic structure of the nanoplate is chosen to be fcc. The interatomic solid-solid (s-s) potential is chosen to be the Lennard-Jones (12, 6)

( ) -[( s-s ) ( s-s ) ]. (1)
with parameters s-s = 73.2 kJ/mol and s-s = 0.3518 nm. The interactions are cut at 4 s-s (see Table 1). This simple potential is able to reproduce the elastic behavior of a solid, including surface stress thanks to the long range Lennard-Jones interactions. More specifically, the numerical values of the parameters have been chosen to mimic the mechanical properties of silicon, 72 a material that has been used several times to study adsorption-induced deformation. [START_REF] Gordon | Size Effects in Mechanical Deformation and Fracture of Cantilevered Silicon Nanowires[END_REF][START_REF] Dolino | Adsorption Strains in Porous Silicon[END_REF][START_REF] Grosman | Adsorption-Induced Strain of a Nanoscale Silicon Honeycomb[END_REF][START_REF] Rolley | Using Capillary Forces to Determine the Elastic Properties of Mesoporous Materials[END_REF][START_REF] Gor | Elastic Response of Mesoporous Silicon to Capillary Pressures in the Pores[END_REF] Note that silicon is not an fcc solid, and that better potentials exist to reproduce its physical properties. [START_REF] Lee | A Modified Embedded Atom Method Interatomic Potential for Silicon[END_REF] However, an accurate description of the material is out of the scope of this study. The bulk properties have been determined in a cubic simulation box containing 6 unit cells (3.26 nm) in each direction with periodic boundary conditions (the crystallographic axes were parallel to the simulation box). The simulations have been done in the isostress ensemble where an external pressure can be applied independently in the three directions. The symmetry being cubic, and omitting the shear modulus, not used in this work, the mechanical properties of the bulk fcc solid are characterized by two parameters that have been calculated at 300K: the Young modulus E = 165 GPa, and the Poisson's ratio = 0.36 (see Table 2).

The Fluid.

Since we focus on non-specific effects, we use again the Lennard-Jones (12,6) potential (eq 1) to model the fluid-fluid and fluid-solid interactions. Following Stoddard et al., [START_REF] Stoddard | Numerical Experiments on the Stochastic Behavior of a Lennard-Jones Gas System[END_REF] the potential is truncated at x-x and a quadratic term is added so that both the potential and force are continuous. The parameters are chosen to mimic n-heptane, a non-polar fluid which has been previously used for experiments. [START_REF] Grosman | Adsorption-Induced Strain of a Nanoscale Silicon Honeycomb[END_REF][START_REF] Puibasset | Structure and Permeability of Porous Silicon Investigated by Self-Diffusion Nmr Measurements of Ethanol and Heptane[END_REF] The parameters have been determined

based on the work of Watanabe et al. [START_REF] Watanabe | Phase Diagram and Universality of the Lennard-Jones Gas-Liquid System[END_REF] The fluid-fluid parameters are f-f = 0.6 nm and f-f /k = 505 K, where k is Boltzmann's constant, or f-f = 4.2 kJ/mol. The fluid-solid parameters are taken equal to the fluid-fluid parameters f-s f-f and f-s f-f . Note that f-s will be varied in section 3.4.1 to evidence fluid-solid interaction effects. Considering the specific values for f-f and f-f parameters, the stable phase of the fluid is expected to be solid at room temperature.

To make sure that the fluid is in its stable liquid phase, the calculations are done at 353 K, above the melting point of the fluid model.

Pore-load Modulus: Thermodynamic Model

The liquid inside a pore may either be under positive or negative pressure depending on the chemical potential value. Without external forces, the general mechanical equilibrium equations impose that the stress along the pore is constant. The system being free, the integrated stress into the liquid compensates that in the solid (the gas pressure outside the pore is negligible, see below). As a consequence, the solid will deform, and the pore-load modulus measures the (linear) relation between the liquid pressure variations inside the pores and the solid deformation, to be determined now in the framework of the elastic theory.

Let us consider the system depicted in Figure 2b. We denote the pressure of the liquid and the liquid-solid surface tension. At equilibrium, the external forces applying into the solid being zero, the internal stress in the three directions xx , yy and z are constant. Considering the symmetry of the nanoplate, one has xx yy zz . Along the z axis, the stress into the solid equals that in the liquid, and in directions x and y, the overall force on each simulation box wall is zero since the external pressure is set to zero (free walls). As a consequence, the stress components into the solid are given by: xx yy ( )

2 zz 3
where and are the solid and liquid thicknesses (defined in Figure 2). The deformation along the x and y directions ( xx yy ) are then given by: The surface tension is given by the excess free energy of the interface. In principle, it depends on the chemical potential of the fluid. However, its variations with , given by that of the excess adsorbed liquid at the interface, are expected to be small. This point will be quantitatively discussed below (section 3.4.2). The solid deformation is thus expected to be essentially linear with the liquid pressure, and the pore-load modulus is given by: xx

[ ( ) x ] 6
As expected, the pore-load modulus depends on the geometry and the elastic constants of the solid. These quantities will be calculated independently (section 3.2).

Results and Discussion

Thermodynamic Properties of the Fluid

The determination of the pore-load modulus requires the knowledge of the liquid pressure far from the walls, e.g. in the center of the pores. It is assumed to be equal to that of the bulk liquid in equilibrium with the vapor at pressure P or chemical potential  , given approximately by:

( ) 7
where is the ideal gas constant and the molar volume of the liquid. This relation is frequently used, but molecular simulations can actually provide the accurate relation between and the activity , where is the de Broglie thermal wavelength and .

The results are given in Table 3 and Figure 3, as a function of ln(z), which is approximately equal to ( ) to within a constant. These results have been obtained by Grand Canonical Monte Carlo simulations and standard thermodynamic methods. [START_REF] Puibasset | Potential, Helmholtz Free Energy, and Entropy Calculation in Heterogeneous Cylindrical Pores by the Grand Canonical Monte Carlo Simulation Method[END_REF][START_REF] Puibasset | Thermodynamic Characterization of Fluids Confined in Heterogeneous Pores by Monte Carlo Simulations in the Grand Canonical and the Isobaric-Isothermal Ensembles[END_REF] As can be seen, the liquid branch is not perfectly linear due to the small compressibility of the liquid, which justifies the exact calculation instead of using eq 7. Note that the range of pressures reached in the liquid is wide, from -122 to +202 bar. Below the saturated vapor point , is essentially negative, and decreases down to the stability limit of the liquid (bulk cavitation) which occurs at -122 bar. This region of negative pressures corresponds approximately to that used in experiments for poreload measurements, except for the fact that in most cases the pore emptying occurs well above the bulk cavitation. Simulations will also be performed above , up to bar: this will improve the statistics.

Mechanical Properties of the Nanoplatelet

The properties of the nanoplatelet are evaluated in a simulation box of initial size L x = L y = 7 unit cells in the x and y directions (3.805 nm); a gap is introduced along the z direction so as to create two opposite surfaces (Figure 2a). The thickness h of the nanoplatelet is 6 unit cells (h = 3.26 nm), and the dimension L z = 10 nm. The distance between the walls is thus H = 6.74 nm.

The wall thickness is chosen small (3.26 nm) compared to typical nanoporous silicon walls (5-6 nm), in order to emphasize surface stress effects. The gap is however typical of nanoporous 2). As can be seen, the Young modulus is slightly lower than its bulk value, while Poisson's ratio is more significantly affected. [START_REF] Gumbsch | Interface Stresses and Their Effects on the Elastic Moduli of Metallic Multilayers[END_REF][START_REF] Wolf | Surface-Stress-Induced Structure and Elastic Behavior of Thin Films[END_REF][START_REF] Cammarata | Surface and Interface Stress Effects in Thin Films[END_REF][START_REF] Miller | Size-Dependent Elastic Properties of Nanosized Structural Elements[END_REF][START_REF] Van Workum | Local Elastic Constants in Thin Films of an Fcc Crystal[END_REF][START_REF] Fedorchenko | Thickness Dependence of Nanofilm Elastic Modulus[END_REF] Furthermore, one can deduce = 0.305, reflecting the nanoplate anisotropy. These values allow to calculate the expected pore-load modulus (eq 6) for the nanoplatelet:

GPa.

Let us now discuss the fluctuations and uncertainties. Figure 4a displays the fluctuations of the free nanoplate size L x along a simulation run. The fluctuations follow a Gaussian law of width 8×10 -3 nm (Figure 4c): this is quite large compared to the nanoplate dimensions, as expected in nanometric systems. The corresponding fluctuations in stress are large (of order 3×10 3 bar), and are actually much larger than the typical stress applied to the solid. As a consequence, long simulation runs are required to reach the desired relative accuracy for the average strain of the solid (5%).

The situation may be improved when one can take into account a natural symmetry of the system. This is the case for the free nanoplatelet (xy symmetry). The method consists in imposing the geometrical constraint L x = L y . The fluctuating value of L x = L y is displayed in Figure 4b andc. As can be seen, the average value of L x = L y is not affected, but the system being now stiffer, the fluctuations are smaller. The same simulation length will thus provide more accurate results. This procedure will be used during the pore-load measurements (in presence of liquid), thanks to the natural symmetry of the system.

Adsorption Induced Deformations

Let us now consider the case where the nanoplatelet is immersed in a liquid characterized by the logarithm of its activity ( ). As previously, we have coupled the x and y directions (L x = L y )

to improve the accuracy. The histograms associated to L x fluctuations are fitted with Gaussian distributions. The results are given in Figure 5 and Table 3. As can be seen, the Gaussian distributions overlap: the fluctuations are of the order of 10 -3 L x , while the deformation is of order 10 -4 L x, typical of solid strain. An interesting feature regarding fluctuations is that they are essentially independent of ( ) for the wet solid, but are slightly larger than for the dry solid.

The presence of the liquid in close interaction with the solid affects the amplitude of the fluctuations.

In presence of liquid, the system may either contract or dilate, depending on the liquid activity z. For high activity (positive liquid pressure), the solid expands, while for low activity (negative liquid pressure) the solid shrinks. This is qualitatively expected from the observation that the fluid pressure acts directly on the solid (along z-axis) and through the simulation box (parallel to the nanoplatelet). At coexistence ( ( ) = -10.46), the liquid pressure is essentially zero (equal to the vapor pressure). The observed deformation in that case is essentially due to the surface tension term in eq 5. The observed deformation is very small: the fluid-solid surface tension is thus small compared to the solid stiffness. This is closely connected to the fact that we have chosen fluid-wall interactions equal to fluid-fluid interactions.

To determine the pore-load modulus, the simulation data are drawn as a function of the liquid pressure in Figure 6. As can be seen, the strain follows essentially a linear behavior, giving a pore-load modulus of 246 GPa. The lowest pressure point corresponds to the stability limit of the stretched liquid. At this point, the probability to form transient gas bubbles is not negligible, and strongly dependent on the presence of a solid wall. This could explain why this point slightly deviates from the linear behavior. It is emphasized that the range of pressure accessible experimentally is narrower. The lower limit is given by the pressure where the porous solid empties, generally above the bulk cavitation limit, and the upper limit is given by the saturated vapor pressure of the fluid (close to zero, see Figure 3). The simulations show that the observed linear behavior extends beyond these experimental limits, in particular above the saturation point.

The pore-load modulus deduced from the simulation results (246 GPa) deviates significantly from the expected value given by the phenomenological model (113 GPa). For visualization, the prediction of the model (eq 5) is given as a solid line in Figure 5, where we have omitted the constant surface tension term which is weak anyway. The "numerical experiment" thus gives a pore-load modulus that is larger than expected from the elastic constants of the solid.

Conversely, if one deduces the bulk modulus of the solid from the pore-load measurement, a large overestimation is done.

Discussion

Disagreements between the elastic moduli determined from adsorption-induced deformation measurements and bulk values have already been reported in the literature. In most cases, the origin is attributed to the surface stress effects due to the small wall thickness in nanoporous materials. Since in our simulations the elastic constants have been determined for the nanoplatelet itself, these finite size effects cannot be invoked to explain the disagreement.

On the other hand, one can invoke the dependence of the solid (surface) stress with the presence of the adsorbed fluid. Two hypotheses are proposed to explain the results: a strong dependence of the fluid-substrate free energy with the chemical potential of the fluid, or a significant variation of the surface stress of the solid in presence of the fluid. The second argument has already been invoked to explain some features of the nitrogen adsorption hysteresis in porous silicon. [START_REF] Grosman | Capillary Condensation in Porous Materials. Hysteresis and Interaction Mechanism without Pore Blocking/Percolation Process[END_REF][START_REF] Grosman | Influence of Elastic Deformation of Porous Materials in Adsorption-Desorption Process: A Thermodynamic Approach[END_REF][START_REF] Grosman | Influence of Elastic Strains on the Adsorption Process in Porous Materials: An Experimental Approach[END_REF][START_REF] Grosman | Influence of Elastic Strains on the Adsorption Process in Porous Materials. Thermodynamics and Experiments[END_REF] In order to test these hypotheses, we have studied the influence of the fluid-solid interaction intensity and the effect of the solid thickness.

Influence of Fluid-Solid Interactions and Solid Thickness.

The most direct route to evidence an effect due to the interface is to vary the intensity of the fluid-solid interaction and/or the nanoplate thickness. We have first considered a reduction of the interaction parameter f-s by a factor two to ten ( f-s f-f 1.0, 0.5, 0.25, and 0.1). The corresponding pore-load modulus has been calculated and plotted in Figure 7 as a function of f-s f-f . The expected modulus from the thermodynamic model (eq 6 and Table 2) is given as a horizontal line. As can be seen, the modulus given by the "numerical experiment" exhibits a dependence on the fluid-solid interaction. More specifically, two regimes can be determined:

f-s f-f where the modulus is constant and equal to the expected value deduced from the mechanical properties of the nanoplatelet (113 GPa), and f-s f-f where the modulus is found to be significantly larger than expected (more than a factor two) with a dependence on f-s . The interpolation between f-f and f-f suggests a sharp transition between the two regimes.

Let us now focus on the effect of the nanoplatelet thickness. The idea is to determine whether the disagreement between the numerical experiments and theory is due to a volume or a surface contribution from the nanoplatelet. We proceed as follows. We perform simulations for two nanoplatelet thicknesses (h = 1.63 nm and 3.26 nm). The results are then analyzed with eq 5 which contains the surface excess free energy . This term was previously discarded to obtain the pore-load modulus as given by eq 6. Let us now suppose conversely that the discrepancy between simulation results and theory originates purely from the surface term , and let us deduce its value from the simulation data for the two nanoplatelet thicknesses (see Figure 8). As can be seen, the points fall on the same curve within errors, proving that the discrepancy arises from a surface contribution. Figure 8 also shows that this surface term strongly depends on the chemical potential of the fluid. It is the variations of this surface term with the liquid pressure which contributes to the pore-load modulus. The physical origin of this surface contribution is discussed now.

Fluid-Solid Free Energy.

In order to check the thermodynamic consistency of the surface term previously deduced from the solid deformations, we evaluate the fluid-solid free energy (surface tension) through another route. [START_REF] Mulero | Vapour-Liquid Equilibrium Properties for Two-and Three-Dimensional Lennard-Jones Fluids from Equations of State[END_REF][START_REF] Ould-Kaddour | Molecular Simulation of Fluid Solid Interfaces at Nanoscale[END_REF][START_REF] Benjamin | Jones Systems near Solid Walls: Computing Interfacial Free Energies from Molecular Simulation Methods[END_REF] The Gibbs equation gives the variations of the excess (surface) free energy versus chemical potential variations:

( ) 8
where is the excess adsorbed fluid relative to the liquid state, the temperature, the chemical potential and the activity of the adsorbed fluid. actually depends on the position of the Gibbs surface, and eq 8 results from the standard choice corresponding to a zero adsorbed excess for the solid. Figure 9 gives the results for the excess adsorbed fluid: it is a positive value since the attractive fluid-solid interaction causes the fluid to be slightly denser close to the surface. The magnitude is however small due to the low compressibility of the liquid. Integration of eq 8 from the (arbitrary) reference point ( ) gives the corresponding to within a constant (see Figure 9). Comparison with Figure 8 immediately shows strong disagreements. Quantitatively, the free energy variation found by the thermodynamic route is one order of magnitude smaller than the value required to explain the mechanical deformation of the nanoplatelet in presence of the fluid, and with the wrong sign.

As a consequence the observed discrepancy cannot be explained in terms of fluid-wall excess free energy.

Fluid Structure at the Interface.

It is known that the fluid structure at the interface with a solid may depart significantly from the bulk due to the strong interactions with the substrate. [START_REF] Schoen | Fluids in Micropores. I. Structure of a Simple Classical Fluid in a Slit-Pore[END_REF][START_REF] Peterson | Layering Transitions in Cylindrical Pores[END_REF][START_REF] Miyahara | Freezing/Melting Phenomena for Lennard-Jones Methane in Slit Pores: A Monte Carlo Study[END_REF][START_REF] Radhakrishnan | Effect of the Fluid-Wall Interaction on Freezing of Confined Fluids : Toward the Development of a Global Phase Diagram[END_REF][START_REF] Patrykiejew | On the Structure of ennard-Jones Fluids Confined in Crystalline Slitlike Pores[END_REF] The atomic structure of the fluid can be revealed by measuring density profiles along the simulation runs. The results are given in Figure 10. Visual inspection of the atomic configurations reveals that the fluid is highly structured at the interface, due to the fluidsolid interaction and the flatness of the interface. The local density profiles confirm this observation, and show that the fluid ordering may extend to two layers. The fluid structure has been determined for various interaction intensities, and exhibits significantly higher ordering for 20 f-s f-f : the peaks are sharp, in particular for the first layer, and the fluid density in the interlayers reaches zero, while for f-s f-f , the peaks are rounded and the interlayers are partially filled. The snapshots of the first atomic layer of the fluid show long range ordering for f-s f-f that are absent for f-s f-f . The fluid crystallizes at the interface due to the fluid-solid interaction and the flat surface that favors long range ordering. This phenomenon has been quantitatively described by Radhakrishnan et al. [START_REF] Radhakrishnan | Effect of the Fluid-Wall Interaction on Freezing of Confined Fluids : Toward the Development of a Global Phase Diagram[END_REF] who have proposed a global phase diagram. They have introduced a quantity  which measures the strength of attraction of the pore walls relative to the fluid-fluid interaction, ( f-s -) f-f where is the solid density, the distance between the atomic layers of the solid wall and f-s and -are the fluidsolid parameters. Numerical evaluation for our system gives f-s f-f . The authors have shown that the first fluid layer in contact with the solid may "freeze" above the bulk freezing point for roughly larger than 1, which corresponds to f-s f-f larger than 0.4. Our simulation results are in agreement with this result. It is proposed that this surface ordering could explain the unexpected deformation of the nanoplatelet, for the following reasons:

(i) The mechanical properties of the ordered (frozen) fluid at the interface are expected to be very different from that of the bulk fluid, in particular regarding the relation between the chemical potential and the mechanical pressure. We thus expect a modification of the mechanical response of the system when the first fluid layer crystallizes. This solid layer is expected to increase the effective stiffness of the nanoplatelet, as observed in the simulations.

(ii) The crystallization occurs in the vicinity of the surface (one or two layers). The effect is thus expected to be reducible to a surface contribution, as observed in the simulations.

(iii) This surface contribution to the free energy is not expected to follow the Gibbs equation due to the presence of the solid layer, explaining the failure of the thermodynamic route to calculate .

(iv) The ordering of the first layer is expected to occur as a sharp transition while increasing the intensity of the fluid-solid interaction, possibly rounded by the finite lateral extension of the nanoplatelet, its surface roughness or the presence of defects. This is compatible with the results of Figure 8, showing that the measured pore-load modulus equals to that given by the thermodynamic model as long as the fluid remains entirely liquid for f-s f-f , while it is significantly affected above f-s f-f where the fluid solidifies at the surface.

Conclusion

This work presents a molecular simulation examination of the fundamentals of the pore-load measurement technique. This method is widely used to have access to the mechanical properties of nanosized solids, for which direct measurements are otherwise difficult. In order to disentangle geometrical effects (including pore geometry as well as finite wall thickness) from surface contributions, we have performed atomistic simulations that allow comparing quantitatively the observed solid deformation with that expected from the mechanical properties of the elastic solid which can be calculated independently. The model chosen is simple but realistic. The nanoplatelet is shaped out of an fcc solid with elastic constants close to that of the bulk silicon, and the fluid is van der Waals like with parameters corresponding to heptane, used in experiments. The fluid-solid interaction is varied between 0.1 and 1.0 times the fluid-fluid interaction, ranging from "hydrophobic-like" to "hydrophilic-like" surfaces. The temperature is above the triple point, so that the fluid is in the stable liquid phase (bulk).

The molecular simulations show that the pore-load modulus deduced from the deformation of the solid immersed in the liquid at a given pressure or chemical potential may be significantly larger than the value expected from the mechanical properties of the dry nanoplatelet. It is shown however that the expected behavior is recovered if the fluid-wall interaction is attenuated. Analysis of the influence of the nanoplatelet thickness has shown that the discrepancy between the observed pore-load modulus and the expected result reduces to a surface term, which however does not match with the surface free energy from fluid adsorption at the interface as obtained via thermodynamic integration of the Gibbs equation. Atomic structure analysis shows that the fluid exhibits a highly ordered first layer at the solid surface, which disappears when the interaction is reduced. This surface ordering is likely to be the source of the unexpected deformation of the nanoplatelet, since it is a surface effect that is expected to affect the mechanical response of the system, and it is not expected to follow the Gibbs equation.

This ordering of the first layer is expected to be relevant for real systems, [START_REF] Radhakrishnan | Effect of the Fluid-Wall Interaction on Freezing of Confined Fluids : Toward the Development of a Global Phase Diagram[END_REF] in particular those with good affinity of the fluid for the solid and atomically smooth surfaces. Note that the presence of an atomically structured wall may influence the phenomenon, [START_REF] Schoen | Fluids in Micropores. I. Structure of a Simple Classical Fluid in a Slit-Pore[END_REF][START_REF] Patrykiejew | On the Structure of ennard-Jones Fluids Confined in Crystalline Slitlike Pores[END_REF] but is not required, [START_REF] Miyahara | Freezing/Melting Phenomena for Lennard-Jones Methane in Slit Pores: A Monte Carlo Study[END_REF] and is thus expected to be relevant for large molecules. [START_REF] Peterson | Layering Transitions in Cylindrical Pores[END_REF] Considering the fact that only the first adsorbed layer will affect the induced deformation, the effect is expected to be stronger for the smallest nanopores.

Important issues for future work would be to study the influence of the roughness of the adsorbent, and the structure of the adsorbed molecules. For instance, the present study focused on a spherical Lennard-Jones like fluid, but it could easily be extended to linear molecules, like n-hexane or n-heptane, used in experiments. 3; the dashed line is a guide to the eye). Solid line: theoretical prediction based on the thermodynamic approach (eq 5). given by the thermodynamic model (eq 6) and the mechanical parameters of the nanoplatelet (Table 2).

  Figure 2b. (i) direct: a positive fluid pressure along z induces a nanoplatelet compression along z,

  materials and large enough to avoid cross-talk through periodic boundary conditions. The elastic properties of the platelet have been determined in the framework of the standard Monte Carlo simulations in the isostress ensemble, where only the dimensions parallel to the nanoplatelet are allowed to vary, while L z is fixed. The direct measurements of the elastic properties of the solid allow taking into account the surface stresses induced by the long range interactions and the finite thickness of the platelet. Equations 5 and 6 require the knowledge of only three independent quantities: , and . Using the symmetry of the compliance matrix ( and ), one can determine these parameters by applying uniaxial stress along x ( ) and measuring the average deformations , and (the fluctuations are discussed below). The linear regime extends up to bar, and one obtains = 162 GPa, = 0.457, and = 531 GPa (Table

Figure 1 Figure 1 .Figure 2 Figure 2 .Figure 3 Figure 3 .Figure 4 Figure 4 .Figure 5 Figure 5 .

 1122334455 Figure 1

Figure 6 Figure 6 .

 66 Figure 6

40 Figure 7 Figure 7 .

 4077 Figure 7

Figure 8 Figure 8 . 42 Figure 9 Figure 9 .Figure 10 Figure 10 . 3 

 88429910103 Figure 8

TABLE 1 :

 1 Lennard-Jones parameters for the solid-solid (s-s), fluid-fluid (f-f) and fluid-solid (f-s) interaction potentials (eq 1).

		s-s	f-f	f-s
	 (kJ/mol)	73.2	4.20	0.42-4.20
	 (nm)	0.3518	0.600	0.600
	cutoff (nm)	1.4072	1.800	1.800

TABLE 2 :

 2 Numerical values of the compliance matrix elements for the nanoplatelet (see eq 4-6), determined by uniaxial mechanical tests along the x-direction (see section 3.2). For comparison, the corresponding quantities for the bulk solid are also given.

	(GPa)

TABLE 3 :

 3 Liquid pressure of the bulk fluid and nanoplatelet deformation given by the molecular simulations as a function of the logarithm of the imposed activity z.

	ln(z)	-11.175	-10.83	-10.485	-10.200	-9.930	-9.375
	(bar)	-122	-63	-2.5	49	98	202
		-5.65×10 -5 -1.85×10 -5	8.86×10 -6	2.80×10 -5	4.99×10 -5	8.91×10 -5

This adsorption-induced deformation

is a fundamental issue resulting from the interplay between the solid surface and the adsorbed fluid.

[START_REF] Hill | Statistical Mechanics of Adsorption. X. Thermodynamics of Adsorption on an Elastic Adsorbent[END_REF][START_REF] Eriksson | Thermodynamics of Surface Phase Systems[END_REF][START_REF] Ash | Thermodynamics of the Effects of Adsorption on Interparticle Forces[END_REF] Beyond its importance for sequestration, storage or oil recovery,[START_REF] Vandamme | Adsorption and Strain: The CO 2 -Induced Swelling of Coal[END_REF] or the fact that it may play a role in the shape of the adsorption isotherms,[START_REF] Amberg | A Study of Adsorption Hysteresis by Means of Length Changes of a Rod of Porous Glass[END_REF][START_REF] Quinn | The Hysteresis Loop in Adsorption Isotherms on Porous Vycor Glass and Associated Dimensional Changes of the Adsorbent[END_REF][START_REF] Fomkin | Adsorption of Gases, Vapors and Liquids by Microporous Adsorbents[END_REF][START_REF] Günther | Novel Insights into Nanopore Deformation Caused by Capillary Condensation[END_REF][START_REF] Grosman | Capillary Condensation in Porous Materials. Hysteresis and Interaction Mechanism without Pore Blocking/Percolation Process[END_REF][START_REF] Grosman | Influence of Elastic Deformation of Porous Materials in Adsorption-Desorption Process: A Thermodynamic Approach[END_REF][START_REF] Grosman | Influence of Elastic Strains on the Adsorption Process in Porous Materials: An Experimental Approach[END_REF][START_REF] Grosman | Influence of Elastic Strains on the Adsorption Process in Porous Materials. Thermodynamics and Experiments[END_REF][START_REF] Schoen | Pore-Lattice Deformations in Ordered Mesoporous Matrices: Experimental Studies and Theoretical Analysis[END_REF][START_REF] Guyer | Theoretical Model for Fluid-Solid Coupling in Porous Materials[END_REF] it is certainly a powerful tool to measure the elastic moduli of nanoporous systems for which direct measurements are not always possible,[START_REF] Herman | Deformation of Silica Aerogel During Fluid Adsorption[END_REF][START_REF] Schoen | Pore-Lattice Deformations in Ordered Mesoporous Matrices: Experimental Studies and Theoretical Analysis[END_REF][START_REF] Dolino | Adsorption Strains in Porous Silicon[END_REF][START_REF] Boissiere | Porosity and Mechanical Properties of Mesoporous Thin Films Assessed by Environmental Ellipsometric Porosimetry[END_REF][START_REF] Prass | Capillarity-Driven Deformation of Ordered Nanoporous Silica[END_REF][START_REF] Dourdain | Determination of the Elastic Modulus of Mesoporous Silica Thin Films by X-Ray Reflectivity Via the Capillary Condensation of Water[END_REF][START_REF] Sharifi | Humidity-Driven Deformation of Ordered Mesoporous Silica Films[END_REF][START_REF] Grosman | Adsorption-Induced Strain of a Nanoscale Silicon Honeycomb[END_REF][START_REF] Rolley | Using Capillary Forces to Determine the Elastic Properties of Mesoporous Materials[END_REF] despite recent

Acknowledgment

The author acknowledges fruitful discussions with A. Grosman and E. Rolley on experimental and theoretical issues regarding adsorption-induced deformation, which motivated this study.

Region Centre-Val de Loire is gratefully acknowledged for grant APR-DYSCO-2015.


Emmanuel Mignot
email: emmanuel.mignot@insa-lyon.fr

Nicolas Riviere
email: nicolas.riviere@insa-lyon.fr

Arnaud Lefevre
email: alefevre@snf.fr

Bernard Quillien
email: bquillien@snf.fr

Smoother than smooth: increasing the flow conveyance of an open-channel flow by using drag reduction methods

The drag reduction method by polymer additives is a common strategy used to minimize friction losses when carrying fluids (water, oil, slurries) in pipes over long distances. Previous studies showed that the interactions between the polymer and the turbulent structures of the flow tend to modify the streamwise velocity profile close to the walls by adding a so-called elastic sublayer between the classical viscous and log layers. The gain in linear head losses can reach up to 80%, depending on the roughness of the walls and the concentration of polymers. The application of this technique to sewers and the subsequent gain in discharge capacity motivated this work to quantitatively measure the drag reduction in classical open-channel flows. Three measurement campaigns were performed in a dedicated long flume for several water discharges and several polymer concentrations: backwater curves over smooth and rough channel walls (including velocity and turbulent shear stress profiles) and flows around emerging obstacles. The addition of polymers, even in limited concentrations, allowed a high friction decrease with a typical Darcy-Weisbach coefficient reduced by factors of 2 and 1.5, respectively in smooth and rough walls configurations without obstacles; but without strong modifications of the non-dimensional velocity profiles. Oppositely, when adding emerging obstacles, the flow was unaffected by the inclusion of polymers, in agreement with the prediction of the literature. The drag-reduction method by addition of small concentrations of polymers finally appears to be a promising technics to increase the flow conveyance in open-channel flows.

Introduction

Discovered by Toms in 1948(see Toms, 1949), the drag reduction effect by polymer addition is a common strategy used to minimize friction losses when carrying fluids over long distances. This effect is commonly used "in oil pipeline conduits, oil well operations, flood water disposal, firefighting, field irrigation, transport of suspensions and slurries, sewer systems, water heating and cooling systems, airplane tank filling, marine systems, and biomedical systems including blood flow" [START_REF] Brostow | Drag Reduction in Flow: Review of Applications, Mechanism and Prediction[END_REF]. [START_REF] Han | Applications of Water-Soluble Polymers in Turbulent Drag Reduction[END_REF] recently reviewed all available water-soluble macromolecules that permit to reduce the drag of turbulent flows, "including both natural and synthetic polymers". While drag reduction can also be obtained by inserting other additives in water, such as fibers or surfactants, this paper only deals with polymeric additives. Moreover, the present work is limited to solutions with very low polymer concentrations (a few tens of parts per million ppm).

The drag reduction effect allows to dramatically reduce the wall friction and thus the linear head losses by up to 80% [START_REF] Herzhaft | Additifs réducteurs de perte de charge en écoulement[END_REF][START_REF] White | Properties of the mean momentum balance in polymer drag-reduced channel flow[END_REF]. The origin of this phenomenon lies in the interactions between the turbulent structures of the flow and the polymer macromolecules added to the fluid [START_REF] Schlichting | Boundary Layer Theory[END_REF]. In a fully-developed flow over a smooth bed, the polymers modify the velocity profile in the boundary layer. With clear-water, the velocity profile in the near-wall regions is composed of the laminar viscous sub-layer very close to the wall (yu*/<10 with y the distance to the wall, u* the friction velocity and  the kinematic viscosity of the liquid) and the log law slightly further away (yu*/>50). When adding polymers, a so-called "elastic zone" can be observed in between these two zones, for y + >15 [START_REF] Virk | Drag reduction fundamentals[END_REF][START_REF] Cai | Study on the characteristics of turbulent drag-reducing channel flow by particle image velocimetry combining with proper orthogonal decomposition analysis[END_REF], within which the velocity profile follows a logarithmic law with a slope that depends on the polymer concentration. It is reported that as the extension of this zone increases, the drag reduction efficiency also increases. If this elastic zone occupies the whole flow section, the maximum drag reduction asymptote (MDRA) is reached and increasing the polymers concentration will not permit to further reduce the friction coefficient. However, [START_REF] White | Properties of the mean momentum balance in polymer drag-reduced channel flow[END_REF] recently suggested that when approaching the MDRA, the velocity profile tends to depart from a logarithmic law. They also observed that as the drag reduction increases, the extension (away from the wall) of the viscous sublayer increases and in return, the logarithmic or quasi-logarithmic layers are moved further away from the wall. Moreover, [START_REF] Virk | Drag reduction fundamentals[END_REF] shows that drag reduction can only be observed for a minimum wall shear stress magnitude. [START_REF] Virk | Drag reduction in rough pipes[END_REF] also investigated the possibility of polymers to reduce the drag over rough surfaces. He observed that for walls with large enough roughness elements, the wall roughness tends to decrease the drag reduction process. The author could establish useful asymptotic regimes of drag reduction as a function of the relative roughness, flow and polymeric parameters. These regimes are based on the common non-dimensional roughness height parameter ks + defined as ks + =ksu*/, with ks the equivalent sand roughness height. His main conclusions are that, for a water-polymer mixture, the effect of wall roughness begins to be effective for ks + >12 (compared to ks + >5 in pure water). This confirms that the extension of the viscous sublayer increases when adding polymers. Then for 12<ks + <50, an "effectively smooth regime" is observed, and for ks + >50 the drag reduction decreases compared to the smooth regime up to the experimental limit of ks + ~150.

On a local scale, the interactions between the polymer and the turbulent structures (of different scales) remain poorly understood and research on this topic is still going on [START_REF] Herzhaft | Additifs réducteurs de perte de charge en écoulement[END_REF]. [START_REF] Achia | Structure of the turbulent boundary layer in drag reducing pipe flow[END_REF] and [START_REF] Tiederman | Wall-layer structure and drag reduction[END_REF] showed that the polymers tend to decrease the frequency of occurrence of the so-called turbulent ejections (w'>0 et u'<0 with w the wall normal and u the streamwise velocity components and where the prime denotes time-fluctuations) and to increase their transverse spacing. The streamwise turbulent Reynolds stress increases while the wall-normal component decreases (see [START_REF] Harder | Drag reduction and turbulent structure in two-dimensional channel flows[END_REF]. Moreover, the Reynolds shear stress in the elastic zone decreases due to the polymer insertion so that the total stress also decreases. A hypothesis is that aggregates of polymers of size comparable to the smaller turbulent scales can limit the existence of these small scale turbulent structures and thus limit the global turbulent intensity. Min et al (2003) proposed the following mechanism: the polymers in the very near-wall region store elastic energy taken from the flow and release it further from the wall as they are transported to the buffer layer by coherent turbulent structures.

Most works, cited above, report the use of drag reduction polymers in pressurized pipe or channel flows. However, a few tests have also been performed in open-channel flows, especially within sewer systems (see [START_REF] Sellin | Application of Polymer drag reduction to sewer flow problems[END_REF]. [START_REF] Hart | Polymer Addition to Increase Trunk Sewer Flow Capacity at the Resort Municipality of Whistler during the 2010 Winter Olympic Games[END_REF] reported the use of such polymers in the sewer network of the city of Whistler (Canada) during the 2010 Winter Olympic games where the waste water discharge was strongly increased due to a particularly high population in the area. Their figures 6-7, for instance, show a strong water depth decrease and velocity increase in a given open-channel pipe flow during a "high flow polymer dosing trial": in about one hour, the typical water depth decreases from about 15 to 2cm while the typical velocity increases from 2.2 to 3.4m/s. Use of polymers in urban flows is made possible by the Materials Safety Data Sheet (MSDS) information for anionic polymers that indicates that the chemicals are non-toxic at limited dose concentrations [START_REF] Hart | Polymer Addition to Increase Trunk Sewer Flow Capacity at the Resort Municipality of Whistler during the 2010 Winter Olympic Games[END_REF]. Apart from these engineering applications in sewer networks, very little attention seems to have been devoted to drag reduction in open-channel flows. [START_REF] Janosi | Turbulent drag reduction in dam-break flows[END_REF] investigated the effect of adding polymers to a dam break flow experiment on both initially dry and wet beds. Their results confirm [START_REF] Cadot | Turbulent drag reduction in a closed flow system: Boundary layer vs bulk effects[END_REF] observations that drag reduction is only effective if "turbulence is forced by a smooth boundary layer and the phenomenon vanishes in inertially forced turbulent flows".

To conclude, even though drag reduction is commonly used in pressurized flows, no academic work could be encountered on the analysis of drag reduction additives in open-channel flows (representatives of river or sewer flows). The aim of the present work is thus to fill this gap and estimate the drag reduction efficiency in open-channel flows over smooth and rough beds, with and without obstacles. The dedicated experimental open-channel flume and measurement techniques are introduced in section 1 along with a description of the tested polymer additives. Then, the flow characteristics (water depths and velocity profiles) are measured with and without polymers and compared to each other i) under smooth bed conditions in section 2, ii) under rough bed conditions in section 3, and iii) under smooth bed conditions but with a series of emerging obstacles in section 4.

Experimental set-up

Polymers are tested herein in open-channel configurations using natural ground water at temperatures between 10°C and 20°C, with and without polymers. The additives are water-soluble polyacrylamide polymers (PAM or PAAM), commonly used as flocculants in water treatment processes (drinking water and waste water), as described by [START_REF] Han | Applications of Water-Soluble Polymers in Turbulent Drag Reduction[END_REF]. Fig. 1 shows the results of rheological tests performed with the natural water without and with polymers collected at the inlet and outlet of the flume. It reveals that the water/polymers mixture remains Newtonian and its viscosity increases by about 15% compared to plain water and also reveals that the flow in the flume tends to decrease this viscosity increment due to shearing effect on the polymers.

Figure 1

The experiments presented below took place in the laboratory of SNF at Andrezieux in France. A specific flume was designed and installed to investigate the effect of drag reduction additives on openchannel flows in different conditions. The 435m long flume is smooth, made of fiberglass and epoxy resin with a typical roughness size of 1.6 m, of b=12cm wide rectangular cross-section with a constant streamwise slope S0=0.011% (see Fig. 2). Note that only the 100m downstream-most reach is used for measurements. Underground natural water is pumped before the experiments and stored in a large tank. The inlet then permits to release this water directly towards the flume or to mix it with a controlled concentration of polymers aqueous solution before reaching the flume. The inlet discharge Q is controlled and measured in the upstream feeding system using valves and an electromagnetic flow meter. At the downstream outlet, the flume ends with a free-fall, imposing a critical flow condition before the water is evacuated outside the laboratory.

Figure 2

Water depths h are measured using an ultrasonic radar sensor probe (Vegapuls 64 from VEGA) from above the flume. The displacement of the ultrasonic probe along the streamwise axis x of the flume then permits to measure backwater curves. The 3 components of the flow velocity are measured at high sampling frequency locally using an acoustic Doppler velocimeter (16MHz ADV from Sontek).

To improve the signal backscattering, tracers are added to the water and it was verified that these tracers do not affect the impact of the polymers on the drag reduction. The vertical displacement of the velocimeter finally permits to measure vertical profiles of velocity.

Smooth bed experiments

Backwater curves and friction coefficients

This section aims at characterizing the drag reduction due to additives diluted in the water in an openchannel flow with smooth walls. To do so, five different upstream discharges (see Table 1) are set, under steady state conditions without or with different controlled concentrations of polymers. Keeping a critical downstream depth, a total of 21 subcritical backwater curves are finally measured. Figure 3 exhibits the measured backwater curves over the downstream 100-meters channel reach for a selected discharge without and with four polymer concentrations. Similar figures for other discharges (not shown here) exhibit similar behaviors. Fig. 3 confirms that i) the use of polymers decreases the water depths and, in return increases the flow velocity over the whole measured reach, ii) the water depth reduction increases with the increasing polymer concentration and iii) that the drag reduction does not increase anymore as the polymer concentration exceeds 20 to 30 ppm, in agreement with the concept of maximum drag reduction asymptote (MDRA, see [START_REF] White | Properties of the mean momentum balance in polymer drag-reduced channel flow[END_REF].

Fitting the backwater curve measured with plain water based on the Colebrook-White equation permits to estimate the equivalent bed roughness of the flume to ks~10 -4 m. Nonetheless, when using a water/polymer mixture, the Colebrook-White equation does not apply anymore. Consequently, the measured curves are rather fitted using a constant averaged Darcy-Weisbach friction loss coefficient  using:

      3 2 0 2 3 2 82 8 g bh S Q b h dh dx b b h g Q     (1)
with g the acceleration of gravity. Fig. 3 confirms that such simplification still fits the measured backwater curves with high fidelity both with plain water (leading to ~0.017, in agreement with Moody diagram) and with water/polymer mixtures (leading to lower Darcy-Weisbach coefficients).

Figure 3

The best-fit Darcy-Weisbach coefficients for the 5 tested discharges with plain water and a polymer concentration of 20 ppm are summarized in Table 1. For a given discharge, the difference in water depth with / without polymers leads to a very limited difference in Reynolds number but to a strong decrease of friction coefficient:  is reduced by a factor of about 2 for all discharges. Similarly, the water depth is reduced (and correspondingly the average velocity is increased) by a factor of about 1.1 to 1.2, i.e. 10% to 20%. When plotting these results on the well-known Moody diagram (giving  as a function of the Reynolds number and relative roughness for water), the resulting Darcy-Weisbach friction factor  appears to be much lower than what can be obtained by the hydraulically-smooth curve (not shown here). To conclude, the drag reduction in open-channel flows by adding polymers exceeds that of using a perfectly smooth wall, in agreement with pipe flow measurements, reported among others by [START_REF] Tanner | Engineering rheology[END_REF].

Table 1 Velocity profiles

The present section aims at comparing the vertical profiles of velocity and Reynolds shear stress and a rapid increase when approaching the bed, with a maximum shear stress measured at z/h~0.1.

These profiles are also in qualitative agreement with measurements without polymers by [START_REF] Kironoto | Turbulence characteristics in rough non-uniform open-channel flow[END_REF] and the recent calculations without and with polymers performed by [START_REF] White | Properties of the mean momentum balance in polymer drag-reduced channel flow[END_REF], see their figure 6.

Figure 4

Rough bed experiments

The previous section showed that the additives permit to reduce the resisting wall friction in smooth bed conditions. The aim of this section is then to verify whether this drag reduction is still observed with a rough bed. The downstream-most, 24 meters long, reach of the flume is modified by including rough walls made of aluminium tear plates of typical tear roughness height equal to 1 mm (see Fig. 2).

The resulting channel width then reduces to b2=11cm.

Similar backwater curves as for the smooth case are measured and plotted in Fig. 5 with a critical depth downstream, for two discharges without additives and with 20 ppm of polymers. As for the smooth bed configuration, the water depths appear to strongly decrease when the polymer is included. When using plain water, the best-fit equivalent roughness height increases to ks~4mm, which corresponds to an average Darcy-Weisbach coefficient equal to ~0.05 for both discharges (Fig. 5). This result is in agreement with that of the Moody diagram: for Q=100m 3 /h and h~0.35m (see Fig. 5), ks/Dh=0.021

with Dh the hydraulic diameter and Re=1.4x10 5 , the Moody diagram also predicts =0.05. When adding 20ppm of polymers, the Darcy-Weisbach coefficient reduces to ~0.037 for Q=100m 3 /h and ~0.03 for Q=50m 3 /h. We thus face a decrease of Darcy-Weisbach coefficient by a factor 1.35 to 1.66 compared to the configuration with plain water, i.e. a smaller decrease than when using a smooth bed but still a notable drag reduction.

Figure 5

Vertical profiles of streamwise velocity are measured in the rough bed configuration in a similar way as for the smooth bed configuration at the same location (x=6.87m) for both tested discharges without and with additives and are shown in Fig. 6 in the same way as for Fig. 4. As for the smooth bed condition, very similar streamwise velocity profiles are measured without and with additives over the rough bed for both discharges with a quite constant streamwise velocity Ux/Ub~1.2 for z/h>0.4 and a velocity decrease closer to the bed (Kirkgoz, 1989). Regarding the Reynolds shear stress, the maximum shear stresses (-<ux'uz'>max/Ub 2 ~15x10 -4 ) observed at z/h~0.1 appear to exceed that measured over the smooth bed (Fig. 4) but with no notable difference without and with additives.

Figure 6

As proposed by [START_REF] Virk | Drag reduction in rough pipes[END_REF], it is expected that the roughness effect on the flow depends on the ratio between the equivalent roughness height and the width of the viscous and elastic sublayers away from the wall. This ratio is evaluated by estimating the roughness height parameter ks + =ksu*/, where the friction velocity u* is obtained using 𝑢 * = 𝑈√ 𝜆 8 , with U the average velocity and  the kinematic viscosity of the water. For a discharge of Q=50m 3 /h of the water + 20ppm mixture: U~0.4m/s, u*~0.024m/s and ks + ~100 and for Q=100m 3 /h: U~0.5m/s, u*~0.034m/s and ks + ~135. Both backwater curves measured in the rough bed with a concentration of 20 ppm of polymers (Fig. 5) are thus in the so-called completely rough regime (ks + >70), according to [START_REF] Nezu | Turbulence in open channel flows[END_REF] among others.

According to these authors, in a typical water flow in completely rough regime, "viscous effects disappear because the roughness elements penetrate the fully turbulent logarithmic layer". Besides, for a water + polymer mixture, [START_REF] Virk | Drag reduction in rough pipes[END_REF] shows that drag reduction effects decrease as the equivalent roughness height parameter ks + increases from 12 up to values larger than the experimental limitations of this author: ks + ~150. The present observation of drag reduction limitation for the present ks + =100 and 135 is thus in fair agreement with this author's observations in rough pipes.

Experiments with emerging obstacles (and smooth wall)

The previous section showed the ability of additives to reduce the friction coefficient in flows where the resistance is governed by wall friction. The aim of the present section is to evaluate the effect of additives in a flow which resistance is rather governed by drag forces induced by obstacles [START_REF] Stone | Hydraulic resistance of flow in channels with cylindrical roughness[END_REF], with a negligible contribution to the wall friction. To do so, vertical impervious and smooth obstacles of D=15 mm wide square section are introduced in the flume, deposited on the bed and emerging through the free-surface (Fig. 7). One obstacle is introduced every meter over the 42 meters downstream most channel reach. The obstacles are not aligned in the streamwise direction, they are rather randomly located from the left to the right bank. Backwater curves for the Q=100m 3 /h configuration are shown on Fig. 8 with and without additives (and also without obstacles), in the same way as for Fig. 2. This figure first confirms that, when using water, adding the obstacles strongly increases the water depth and thus strongly increases the flow resistance, confirming that the flow resistance is dominated by the drag forces on the obstacles. Moreover, Fig. 8 reveals that, in such case, the additives do not permit to reduce the flow resistance and lead to a very similar backwater curve as when using plain water. and the Weissenberg number (viscous to elastic forces ratio):

Wi=2U/D~2x0.6x6.7x10 -3 /0.015~0.5

(3) with  the polymer relaxation time estimated as (see [START_REF] Xiong | Numerical study on viscoelastic fluid flow past a rigid body[END_REF]:

~Ro 3 /(kBT)~1000x10 -6 x3x10 -7 /(1.38x10 -23 x293)~6.7x10 -3 s (4)
 the density of water (1000 kg/m 3 ), Ro~3x10 -7 m the typical size of the polymer statistical sphere, kB=1.38x10 -23 J/K the Boltzmann constant and T~293K the fluid temperature. According to figure 4 or 6 from [START_REF] Xiong | Numerical study on viscoelastic fluid flow past a rigid body[END_REF], for ReD~8000 and Wi~0.5 the obstacle drag coefficient is unaffected by the addition of polymers, which explains why the backwater curves in Fig. 8 are so similar with and without polymers.

Figure 8

To conclude, the flow resistance is here governed by drag forces due to obstaclesrather than friction on smooth walls. Considering the associated particulate Reynolds numbers, these drag forces are mainly due to pressure drag and adding polymers has no effect on the head losses and thus on the water depth and average velocity of the flow.

Discussions and conclusions

To the author's knowledge, the present paper is the first work dedicated to the study of the impact of adding to the water a limited concentration of polymers for classical free-surface open-channel flows:

1D subcritical flows over smooth and rough beds, without and with emerging obstacles. These flow configurations are small-scale and simplified versions of typical channel and river flows. These measurements reveal that adding polymers to the water under these conditions:

1) Strongly decreases the drag force that the walls apply on the flow, leading to highly increased mean velocities and highly decreased water depths. The averaged equivalent Darcy-Weisbach coefficient is decreased by a factor of about 2 for the smooth bed configuration and about 1.5

for the rough bed configuration. The drag reduction is thus still effective but reduced in magnitude with a roughness parameter ks+ larger than 100, in agreement with previous observations in rough pipes [START_REF] Virk | Drag reduction in rough pipes[END_REF] 2) Has no effect on the resistance forces when relatively large obstacles are included in the channel. When the flow resistance is dominated by the drag forces due to these obstacles, the influence of the boundary layers adjacent to the channel walls becomes negligible and no drag reduction is observed anymore, in agreement with [START_REF] Cadot | Turbulent drag reduction in a closed flow system: Boundary layer vs bulk effects[END_REF] observations.

3) Hardly modifies the vertical profiles of streamwise velocities and Reynolds shear stress at least away from the wall region.

To summarize, at the laboratory scale, as long as the boundary layer developing along the bottom and side walls of the channel governs the resistance forces, the modification of these boundary layers by the additives permits to highly increase the capacity of a watercourse, i.e. to limit the water depths and specifically the overflows over the side banks. Oppositely if the flow resistance is governed by the drag forces on large scale obstacles, the additives have no effect on the watercourse capacity.

This drag reduction method already proved to be effective for reducing the water depth in sewer networks [START_REF] Sellin | Application of Polymer drag reduction to sewer flow problems[END_REF][START_REF] Hart | Polymer Addition to Increase Trunk Sewer Flow Capacity at the Resort Municipality of Whistler during the 2010 Winter Olympic Games[END_REF] and the present work corroborates these observations as the flow resistance in sewers is mainly dominated by wall frictions and the size of the wall roughness remains usually limited. Oppositely, to the author's knowledge, this method was never tested in geophysical flows such as rivers and the following paragraphs discuss first its expected hydraulic efficiency and second its potential environmental issues.

Regarding the hydraulic efficiency of the method in rivers, an estimate of ks+ for a river reach is given by: 0

* sh s s k gR S ku k     (5)
where ks is the equivalent bed roughness and Rh the hydraulic radius. Let us consider ks+=135 as the threshold for an effective friction reduction by the additives (Fig. 5). For typical plain rivers (say S0=10 -4 ), friction reduction is then expected to be effective [START_REF] Pich | Method for limiting freshet levels and controlling floods, Patent# WO 2014199037 A[END_REF]: for small rivers (Rh=1m) if ks<4.3mm, corresponding to small gravel beds, and for big rivers (Rh=10m) with ks<1.4mm, corresponding to sand rivers. Still, its real efficiency would depend on river slope, size and bed sediments characteristics, but the ks+=135 threshold is also questionable and may depend on the Reynolds number. The impact of obstacles (dikes, bridge piles, boulders, etc…) on the drag reduction should also be considered, though it is expected to be negligible as obstacles are relatively sparse in river reaches.

Regarding the environmental issues, several drawbacks can be listed. A first potential drawback concerns the spatial distribution of the flooding risk: the application of present technique should be considered within a global risk management procedure, considering all the hydrological and human processesand their associated time scalesat the whole river basin scale. A second potential drawback is related to the environmental impact of polyacrylamide, notably on aquatic organisms, injected in the rivers. Anionic polyacrylamide is widely used in irrigation water (Sojka et al., 2001) or mineral extraction water, where its dissemination was commented by [START_REF] Guezennec | Transfer and degradation of polyacrylamide-based flocculants in hydrosystems: a review[END_REF]. But while the injection in sewers in Canada, reported by [START_REF] Hart | Polymer Addition to Increase Trunk Sewer Flow Capacity at the Resort Municipality of Whistler during the 2010 Winter Olympic Games[END_REF], concerns about 30 kg per hour and this water goes through a water treatment plant before returning to the river, an injection of 10-20 ppm in a large flooding river of a discharge of about 1500 m 3 /s would lead to an injection of about 50-100 Tons of polymer per hour directly in the riverine environment. Present work does not address these effects on the receiving environment, and this technique should be implemented with authorities.

  Fig. 4 also plots vertical profiles of Reynolds shear stress for the same conditions. The four profiles

Figure 7

 7 Figure 7

Figure captions Figure 1 .

 captions1 Figure captions

Figure 2 .

 2 Figure 2. Photographs of the flume in the smooth (left) and rough (right) configurations.

Figure 3 .

 3 Figure 3. Backwater curves measured (symbols) and fitted with a constant Darcy-Weisbach coefficient (lines) for a discharge Q=150m 3 /h with different concentrations of additives in the smooth bed

Figure 4 .

 4 Figure 4. Vertical profiles of timely-averaged streamwise velocity (left) and Reynolds shear stress

Figure 5 .

 5 Figure 5. Backwater curves measured for two discharges without additives and with 20 ppm polymers

Figure 6 .

 6 Figure 6. Vertical profiles of timely-averaged streamwise velocity (left) and Reynolds shear stress

Figure 7 .

 7 Figure 7. Photograph (top view) of the flow downstream one of the emerging obstacles Figure 8. Backwater curves measured for a discharge Q=100m 3 /h with 20 ppm and without additives

Table 1 .

 1 Best-fit Darcy-Weisbach coefficient for 5 selected discharges for water without additives and with 20 ppm polymers

			Plain water		Water + 20 ppm polymers
	Discharge	Darcy-	Water	Typical	Froude	Darcy-	Water	Typical	Froude
	Q (m 3 /h)	Weisbach	depth (m)	Reynolds	number	Weisbach	depth (m)	Reynolds	number
		coeff. ()	@ x~50m	number	@ x~50m	coeff. ()	@ x~50m	number	@ x~50m
	50	0.02	0.22	6x10 +4	0.36	0.011	0.20	6x10 +4	0.41
	75	0.02	0.30	7x10 +4	0.34	0.01	0.26	8x10 +4	0.42
	100	0.02	0.35	8x10 +4	0.36	0.01	0.32	9x10 +4	0.41
	125	0.018	0.40	9x10 +4	0.37	0.008	0.35	10x10 +4	0.45
	150	0.017	0.50	9x10 +4	0.31	0.008	0.40	11x10 +4	0.44