Zhujie Li

Guillaume Jeanmairet

Trinidad Méndez-Morales

Benjamin Rotenberg

Mathieu Salanne
email: mathieu.salanne@sorbonne-universite.fr

Capacitive Performance of Water-in-Salt Electrolyte in Supercapacitors: A Simulation Study

Water-in-salts is a new family of electrolytes with very promising electrochemical properties for energy storage applications. Despite several studies involving them inside Li-ion batteries and supercapacitors, their interfacial properties remain largely unknown. Here we simulate the interface between electrified graphite electrodes and a highly concentrated waterin-salt (where the salt is Bis(trifluoromethane)sulfonimide lithium, LiTFSI) using constant applied potential molecular dynamics. We show that the capacitance differs markedly between the positive and the negative electrodes, which is due to the large asymmetry in size (and shape) between the ions. By using importance sampling, we further investigate the changes in

Introduction

The development of efficient energy storage devices has become a major target during the past decades. Current technologies require systems with excellent performances such as high energy density and high power density. The electrochemical double-layer capacitors (EDLCs), also called supercapacitors, which consist of a liquid electrolyte confined between two electrodes, are promising energy storage devices due to their high power density, reasonable energy density, high charge/discharge rates and long cycling performance. [START_REF] Simon | Materials for Electrochemical Capacitors[END_REF][START_REF] Simon | Where Do Batteries End and Supercapacitors Begin[END_REF] The excellent performance of EDLCs is attributed to the highly reversible ion adsorption mechanism that governs their operation. It allows charging with reduced diffusion limitation and the absence of redox reactions inside the electrodes increases significantly the chemical and mechanical stability. [START_REF] Salanne | Efficient Storage Mechanisms for Building Better Supercapacitors[END_REF] The main physical property quantifying the performance of carbon-based EDLCs for energy storage applications is the interfacial capacitance. [START_REF] Salanne | Ionic Liquids for Supercapacitor Applications[END_REF][START_REF] Vatamanu | Capacitive Energy Storage: Current and Future Challenges[END_REF][START_REF] Vatamanu | Charge Storage at the Nanoscale: Understanding the Trends from the Molecular Scale Perspective[END_REF] This quantity is highly dependent on the electrode/electrolyte couple as reported in previous works. [START_REF] Chmiola | Anomalous Increase in Carbon Capacitance at Pore Sizes Less Than 1 Nanometer[END_REF][START_REF] Haskins | Computational and Experimental Study of Li-Doped Ionic Liquids at Electrified Interfaces[END_REF] In general, the main changes are attributed to the interfacial layer structure (or local structure) of the liquid electrolyte. [START_REF] Kornyshev | Double-Layer in Ionic Liquids: Paradigm Change?[END_REF][START_REF] Fedorov | Double Layer in Ionic Liquids: The Nature of the Camel Shape of Capacitance[END_REF][START_REF] Ivanistsev | Restructuring of the Electrical Double Layer in Ionic Liquids upon Charging[END_REF][START_REF] Coles | The Nanostructure of a Lithium Glyme Solvate Ionic Liquid at Electrified Interfaces[END_REF][START_REF] Gomez-Gonzalez | Molecular Dynamics Simulations of the Structure of Mixtures of Protic Ionic Liquids and Monovalent and Divalent Salts at the Electrochemical Interface[END_REF] Different pore sizes in nanoporous electrodes accompanying complex local structures have a large impact on the capacitance compared with non-porous electrodes. [START_REF] Kondrat | A Superionic State in Nano-Porous Double-Layer Capacitors: Insights from Monte Carlo Simulations[END_REF][START_REF] Merlet | On the Molecular Origin of Supercapacitance in Nanoporous Carbon Electrodes[END_REF][START_REF] Merlet | Highly Confined Ions Store Charge More Efficiently in Supercapacitors[END_REF][START_REF] Kondrat | Accelerating Charging Dynamics in Subnanometre Pores[END_REF] However, the capacitance can also take a wide range of values both in experiment and simulation even if the electrodes are planar. [START_REF] Merlet | The Electric Double Layer Has a Life of Its Own[END_REF][START_REF] Friedl | Interface between an Au(111) Surface and an Ionic Liquid: The Influence of Water on the Double-Layer Capacitance[END_REF][START_REF] Su | Double Layer of Au(100)/Ionic Liquid Interface and Its Stability in Imidazolium-Based Ionic Liquids[END_REF][START_REF] Wen | Potential-Dependent Adlayer Structure and Dynamics at the Ionic Liquid/Au(111) Interface: A Molecular-Scale In Situ Video-STM Study[END_REF] It is mainly the process of adsorption of cations and anions at the surface that controls the capacitance. The structure of the electrolyte at an interface differs from the bulk one, and it may change drastically when an electric potential is applied to the electrode. For example, Merlet et al. reported that the structure of an ionic liquid adsorbed on graphite electrodes can be either ordered or disordered depend on the applied voltage. [START_REF] Merlet | The Electric Double Layer Has a Life of Its Own[END_REF] In this study, the structural changes were explained by a substantial change in the charge distribution across the interfacial region and led to changes in the differential capacitance with applied voltage. Limmer performed systematic simulations for dense ionic solutions near charged planar electrodes and showed that first-order transitions and spontaneous charge density ordering at the interface may result in potential-driven anomalous differential capacitances. [START_REF] Limmer | Interfacial Ordering and Accompanying Divergent Capacitance at Ionic Liquid-Metal Interfaces[END_REF] More recently, Ma et al. reported that the differential capacitance may be strongly influenced by the charge-driven lateral structural evolution of the ions in EDLCs. [START_REF] Ma | Charge Driven Lateral Structural Evolution of Ions in Electric Double Layer Capacitors Strongly Correlates with Differential Capacitance[END_REF] It is therefore very difficult to predict the interfacial structure without using complex simulation and/or experimental characterization techniques.

Recently, a new class of superconcentrated electrolytes, [START_REF] Wang | Superconcentrated Electrolytes for a High-Voltage Lithium-Ion Battery[END_REF] called water-in-salt, was proposed [START_REF] Suo | Waterin-Salt" Electrolyte Enables High-Voltage Aqueous Lithium-Ion Chemistries[END_REF] and attracted much attention due to their promising performances in batteries. [START_REF] Yamada | Hydrate-Melt Electrolytes for High-Energy-Density Aqueous Batteries[END_REF][START_REF] Yang | 4.0 V Aqueous Li-Ion Batteries[END_REF][START_REF] Suo | Water-in-Salt" Electrolyte Makes Aqueous Sodium-Ion Battery Safe, Green, and Long-Lasting[END_REF][START_REF] Wang | Highly Reversible Zinc Metal Anode for Aqueous Batteries[END_REF] Both mass and volume ratio of salt-to-water could be increased up to more than 1 by choosing proper composition of salts. The formation of super-concentrated aqueous solutions in such electrolytes suppress the activity of water molecules due to the lack of free water. This enables the breakthrough of the limited electrochemical stability window of water (1.23 V) in conventional aqueous electrolytes, resulting in extended windows up to 4.0 V. [START_REF] Yang | 4.0 V Aqueous Li-Ion Batteries[END_REF] For the application of water-in-salt to EDLCs, several experimental works have been reported including some capacitance measurements. [START_REF] Hasegawa | Hierarchically Porous Carbon Monoliths Comprising Ordered Mesoporous Nanorod Assemblies for High-Voltage Aqueous Supercapacitors[END_REF][START_REF] Gambou-Bosca | Electrochemical Characterization of MnO 2 -Based Composite in the Presence of Salt-in-Water and Water-in-Salt Electrolytes as Electrode for Electrochemical Capacitors[END_REF][START_REF] Coustan | Electrochemical Behavior of Platinum, Gold and Glassy Carbon Electrodes in Water-in-Salt Electrolyte[END_REF][START_REF] Lannelongue | Water-in-Salt" for Supercapacitors: A Compromise between Voltage, Power Density, Energy Density and Stability[END_REF] In addition, a few simulation studies [START_REF] Vatamanu | Ramifications of Water-in-Salt Interfacial Structure at Charged Electrodes for Electrolyte Electrochemical Stability[END_REF][START_REF] Borodin | Modeling Insight into Battery Electrolyte Electrochemical Stability and Interfacial Structure[END_REF][START_REF] Wang | Hybrid Aqueous/Non-aqueous Electrolyte for Safe and High-Energy Li-Ion Batteries[END_REF] have aimed at understanding the interfacial structure of these electrolytes near electrodes, but their capacitive behavior has not been analyzed so far. Among the questions to be solved, an important one is whether water-in-salt behave as ionic solutions or ionic liquids. Thus, in this work, we performed molecular dynamics (MD) simulations at various applied potentials for an electrochemical cell consisting of a water-in-salt electrolyte at a concentration of 20 m (mol/kg) LiTFSI, placed between two planar graphite electrodes, in order to obtain a microscopic picture of the structure and the capacitive behavior on the interfaces and their dependence on the applied potentials. We discuss in particular the differential capacitances of either a single electrode or of the full electrochemical cell. We show the variations of the differential capacitance and relate them to structural changes at specific potential ranges.

Computational details

We use MD to simulate a cell consisting of 20 m LiTFSI placed between two graphite electrodes.

The simulation box contains 255 Li cations, 255 TFSI anions and 707 water molecules (Figure 1a).

We exploit an all-atom model with modified OPLS force field for LiTFSI and the SPC/E model for water. [START_REF] Monteiro | Transport Coefficients, Raman Spectroscopy, and Computer Simulation of Lithium Salt Solutions in an Ionic Liquid[END_REF][START_REF] Berendsen | The Missing Term in Effective Pair Potentials[END_REF] We performed an initial simulation of the bulk system to validate the parameters, which yielded an electrical conductivity of 0.79 S m -1 and a viscosity of 46.1 mPa s, in good agreement with the experimental results of Suo et al. (0.82 S m -1 and 36.2 mPa s, respectively, albeit for a slightly larger concentration in salt of 20.8 m). [START_REF] Suo | Waterin-Salt" Electrolyte Enables High-Voltage Aqueous Lithium-Ion Chemistries[END_REF] The electrodes consist of three fixed graphene layers on each side (Figure 1a) with in-plane dimensions of L x = 32.25 Å and L y = 34.37 Å. The length of the cell in z-direction was set to L z = 96 Å to match the experimental density of 20 m LiTFSI in the bulk region of the simulation cell. The carbon atom charges on the electrode were allowed to fluctuate by imposing a constant potential method implemented in LAMMPS code, [START_REF] Siepmann | Influence of Surface Topology and Electrostatic Potential on Water/Electrode Systems[END_REF][START_REF] Reed | Electrochemical Interface between an Ionic Liquid and a Model Metallic Electrode[END_REF][START_REF] Wang | Evaluation of the Constant Potential Method in Simulating Electric Double-Layer Capacitors[END_REF][START_REF] Plimpton | Fast Parallel Algorithms for Short-Range Molecular Dynamics[END_REF] which ensured an adequate description of the surface polarization by ions and water during simulations. A vacuum region was added in the z-direction and the Yeh-Berkowitz condition for slab correction [START_REF] Yeh | Summation for Systems with Slab Geometry[END_REF] was used to mimic 2D periodic boundary condition. Its length was set to eight times L z , which was the minimal value to ensure that the calculated carbon atom charges were converged. Simulations were performed for 11 applied potential differences between the two electrodes, ∆Ψ, varying from 0 to 3 V in order to obtain good statistical properties. The simulations were conducted in the NVT ensemble with a time step of 1 fs at room temperature (298.15 K). The system was first equilibrated during 80 ns at zero constant charge on each electrode and followed by a 12 ns equilibration at constant potential for each ∆Ψ. The data was then collected in a 25 ns production run at constant potential with configurations saved every 1 ps for each simulation.

Results and discussion

The number densities are computed along the z-direction normal to the graphite surface. They yield bulk-like number density for each compound in the cell center (see the 0V and 3V cases in [START_REF] Vatamanu | Ramifications of Water-in-Salt Interfacial Structure at Charged Electrodes for Electrolyte Electrochemical Stability[END_REF] For each simulation within the studied potential range, positive and negative electrodes were assigned equal but opposite potential, so that ∆Ψ follows the eq Eq. ( 1):

∆Ψ = Ψ + -Ψ -= 2Ψ + = -2Ψ - (1)
The capacitance is one of the most important physical properties for the performance of EDLCs.

However, one can define it in several ways. The integral capacitance of the full electrochemical cell, C int , is defined as (for ∆Ψ = 0):

C int (∆Ψ) = σ ∆Ψ ∆Ψ (2)
where σ ∆Ψ is the average surface charge density of the positive electrode (note that most of the charge is accumulated on the two graphene planes located closer to the liquid, for example at the largest applied potential, the average charge within the three graphene layers are 0.21, -2.01 and 9.66 electrons, respectively). In the present work, the calculated C int ranges from 3.5 to 4.3 µF/cm 2 depending on ∆Ψ. This quantity is regularly used in applied studies to estimate the energy density of the EDLCs. The differential capacitance, C diff , defined by ∂ σ ∆Ψ ∂ ∆Ψ , measures the response of the average surface charge density to changes in the applied potential difference and provides much more information on the physico-chemical processes. The single electrode differential capacitances can also be defined by splitting the applied potential difference, ∆Ψ, between the two electrodes into two potential drops, ∆Ψ ± , at the electrode/electrolyte interface for both positive and negative electrodes. The electrostatic potential profile across the EDLC cell determines the potential drop for each interface, and is obtained by solving Poisson's equation:

ϕ(z) = ϕ q (z 0 ) - 1 ε 0 z z 0 dz z -∞ dz ρ q (z ) (3)
where ρ q (z) is the average charge density profile across the cell, which includes the contributions not only from all the species of electrolyte but also from the electrode atoms, and ϕ(z 0 ) is an arbitrary integration constant with z 0 inside the left electrode, i.e. on the left ϕ(z 0 ) = Ψ -in our constant potential simulations. The charge density and corresponding Poisson potential profiles are shown in Figure S2 and Figure S3. The potential of the bulk Ψ bulk , is determined as the average value of Poisson potential in the bulk region. The potential drop across each interface is then given by ∆Ψ ± = Ψ ± -Ψ bulk and is used to characterize the interfaces. The average surface charge as a function of the potential drop across each interface is represented in Figure 2 for various applied potentials. The average charge density σ = f (∆Ψ ± ) plot differentiated at positive and negative surface charge with respect to ∆Ψ ± , separately, provide the corresponding differential capacitances at positive and negative electrode through C s diff = ∂ σ ∆Ψ ∂ ∆Ψ ± . However, as shown in Figure S4, the curve may be fitted by many polynomial functions, leading to very different shapes for the capacitance. We therefore limit ourselves to two linear fits, for ∆Ψ ± positive and negative, respectively. As shown in Figure 2, the negative electrode has a larger capacitance (10.1 µF/cm 2 ) than the positive electrode (5.6 µF/cm 2 ). Such a difference is in agreement with the results reported in planar graphite electrodes and pure RTILs, [START_REF] Merlet | Imidazolium Ionic Liquid Interfaces with Vapor and Graphite: Interfacial Tension and Capacitance from Coarse-Grained Molecular Simulations[END_REF][START_REF] Merlet | Influence of Solvation on the Structural and Capacitive Properties of Electrical Double Layer Capacitors[END_REF][START_REF] Xing | Electrode/Electrolyte Interface in Sulfolane-Based Electrolytes for Li Ion Batteries: A Molecular Dynamics Simulation Study[END_REF] and it can be attributed to the different shapes or sizes of ions. The larger difference in capacitive behaviors between positive and negative electrodes compared with reported results in BMIM-PF 6 system (3.9 µF/cm 2 and 4.8 µF/cm 2 for positive and negative electrode, respectively [START_REF] Merlet | Imidazolium Ionic Liquid Interfaces with Vapor and Graphite: Interfacial Tension and Capacitance from Coarse-Grained Molecular Simulations[END_REF] ) arises from the much larger asymmetry between the Li cation (1.8 Å in radius) and the TFSI anion [START_REF] Uysal | Structural Origins of Potential Dependent Hysteresis at the Electrified Graphene/Ionic Liquid Interface[END_REF] (10.9×5.1×4.7 Å 3 ).

The differential capacitances can be computed more reliably using the importance sampling. [START_REF] Merlet | The Electric Double Layer Has a Life of Its Own[END_REF][START_REF] Limmer | Charge Fluctuations in Nanoscale Capacitors[END_REF][START_REF] Uralcan | Concentration Fluctuations and Capacitive Response in Dense Ionic Solutions[END_REF][START_REF] Ray | Importance Sampling Large Deviations in Nonequilibrium Steady States[END_REF] Compared to previous equation, single electrode capacitances cannot be obtained (the full cell one C diff is calculated instead), but the method is much more converged statistically. This approach also allows studying microscopic properties, such as the number of ions close to the interface, as a continuous function of the applied potential difference. While for the standard approach these quantities are only available at the simulated potential. In this work, the total charge Q tot = σ × S of the electrode (with S the surface area) is utilized to determine the weight of configurations at an arbitrarily applied potential by using the weighted histogram analysis method involved in importance sampling. [START_REF] Ferrenberg | Optimized Monte Carlo Data Analysis[END_REF][START_REF] Tan | Theory of Binless Multi-State Free Energy Estimation with Applications to Protein-Ligand Binding[END_REF] There is a good overlap between histograms of Q tot obtained from the eleven performed simulations, which fulfills the requirement for correct histogram reweighting. To obtain the probability distribution of any other property, A, as a function of ∆Ψ, we just need to integrate Q tot from the joint distribution of Q tot and A as

P(A|∆Ψ) = dQ tot P(Q tot , A|∆Ψ) (4)
The probability distribution P(σ |∆Ψ) of electrode surface charge density at various ∆Ψ is represented by a contour plot in Figure 3a. The figure shows that the surface charge density monotonically increases with applied potential but experiences several nonlinear shifts in the probability distribution. In particular, there are several more likely values of surface charge (red region), which are located at ∆Ψ ∼ 0, 0.35, 0.9, 1.2, 1.6, 1.8, 2.6, and 3 V. These applied potentials correspond to The intermediate states between these points show large distributions of σ , which correspond to the large fluctuations regions where structural changes occur as we will discuss later.

The differential capacitance (in Figure 3b) is proportional to the variation of surface charge density on the electrode as defined by

C diff = S k B T (δ σ ) 2
∆Ψ in constant potential simulations, which can be obtained directly from the probability distribution of σ in Figure 3a. We recall here that the relation between the response to voltage and the equilibrium fluctuations of the surface charge is not limited to a typical linear response approximation. [START_REF] Limmer | Charge Fluctuations in Nanoscale Capacitors[END_REF][START_REF] Rotenberg | Structural Transitions at Ionic Liquid Interfaces[END_REF] As discussed above the charge fluctuations are potential-dependent and therefore the differential capacitance shown in Figure 3b exhibits a strong dependence on the applied potential. It ranges from 2 to 5 µF/cm 2 with an average value of 3.67 µF/cm 2 , which is in agreement with 3.6 µF/cm 2 computed from the single electrode capacitances as given by 1/C tot = 1/C -+ 1/C + (C tot is the full cell differential capacitance corresponding to the values shown in Figure 3b). The minimum values of the differential capacitance correspond to the more likely distributions of surface charge at specific ∆Ψ. Although some of the variations are quite small and may be due to finite-size effects, we observe five pronounced peaks at ∆Ψ ∼ 0.15, 0.6, 1.3, 2.2, and 2.8 V (labeled by 1 -5 ), that are more significant and are likely to correspond to intermediate states between two adjacent metastable basins (Figure 3a). 1), for the negative electrode when the electrochemical cell is submitted to a voltage difference ∆Ψ.

These non-trivial features of the differential capacitance could be related to structural changes within the interfacial electrolytes. In order to investigate this possibility, the probability distributions (Figure 4 and Figure 5) to find a given number of species in the interfacial layer of electrolyte are computed as a function of applied potential Ψ (-∆Ψ/2 or +∆Ψ/2 as defined in eq Eq. ( 1)).

The interfacial layer is defined as the region within a distance of 6.6 Å from the negative (positive) electrode surface, this cutoff distance is extracted from the number and charge density profiles close to the interfaces at the studied potential range in order to account for the main adsorption peaks of lithium cations on the negative side and of TFSI anions on the positive side of the elctrochemical cell (see Figure S1). The number of Li cations, O atoms of water, and O and F atoms of TFSI anions in the interfacial layers on the negative and positive electrodes are shown in Figure 4 and Figure 5, respectively.

On the negative electrode, the number of Li cations increases with |Ψ -|, and the probability distribution becomes narrower at large negative potential (Figure 4a). The number of water molecules smoothly increases together with Li cations at low negative potentials (Figure 4b), but fluctuates more at large negative potentials. This may be attributed to the overlap of the Li ion solvation shells competing to include the water molecules in the presence of a large number of Li cations, and also accompanied by the changing of numbers of O from TFSI (Figure 4c). The first notable change occurs at -0.075 V (dashed lines with label 1 ) where the most probable number of F (Figure 4d) drops while the one of O from TFSI increases (Figure 4c). This indicate a reorientation of the TFSI anions (see Figure S5 for the variation of the angular distribution of the TFSI anions with the electrode potential). At the same time the probability distribution of number of Li cations becomes different at this region. This structural change could be related to the first peak (label 1 ) of full cell differential capacitance in Figure 3b since the potential at negative electrode (Ψ -=-0.075 V) exactly matches the potential difference, ∆Ψ, at which the first peak located (∆Ψ = 0.15 V). Different probability distributions around -0.65 V, labeled as 3 are found for Li cations, water molecules, and O atoms from TFSI (Figure 4a, b, and c), respectively, resulting in the third peak of capacitance in Figure 3b at ∆Ψ = 1.3 V. The fourth peak in capacitance could be attributed to both the change of numbers and change of distributions for all species at Ψ = -1.1 V (labeled as 4 ), in which more Li cations approach the interfacial layer, along with more water molecules, and part of TFSI anions are forced to leave. Therefore, the interfacial structure strongly changes again.

On the positive electrode the interfacial structure starts to change at Ψ + = 0.3 V as shown in Figure 5 (label 2 ). The desorption of Li cations (Figure 5a) together with water molecules (Figure 5b) starts at this potential while the number of TFSI anions keeps increasing (Figure 5c and d). These correspond to the second peak in the capacitance (Figure 3b) at ∆Ψ = 0.6 V. At higher potential, ∆Ψ ≈ 2.8 V (with Ψ ± ≈ ± 1.4 V, labeled as 5 ), the distributions of number of O and F from TFSI become different on the positive electrode (Figure 5c andd), the number of O from TFSI (Figure 4c) increases again, along with decreasing number of water molecules (Figure 4c) on negative electrode. This synergistic effect on the two electrodes leads to the fifth peak of capacitance located at 2.8 V.

Apart from these structural changes associated with the capacitance peaks discussed above, we also observe a region between 0.75 V and 1 V, in which both the distributions and numbers of O and F from TFSI change on the positive electrode (Figure 5c andd). This reveals remarkable reorientations and rearrangements of TFSI anions at the interfacial layer on the positive electrode.

We thus compare the number densities of O and F atoms from TFSI as a function of the distance from the positive electrode in this potential range. As shown in Figure 6a, the F atoms move from the first adsorbed layer (grey region) to the second layer (light-blue region) when Ψ + increases from 0.75 to 1 V, while the trend for O atoms is opposite. Considering that the O atom carries larger negative charge than the F atom in our simulation cell, this change indicates that the TFSI anions rotate to approach more O atoms close to the electrode. This allows to compensate the growing surface charge on the positive electrode, as illustrated by two representative snapshots at 0.75 and 1 V, respectively, in Figure 6b andc.

Conclusions

In summary, the aim of this work was to explore the capacitive performance of water-in-LiTFSI electrolyte in supercapacitors. We performed molecular dynamics simulations of 20 m LiTFSI placed between two planar graphite electrodes at various applied potentials ranging from 0 to 3 V. The integral capacitance of the full electrochemical cell as well as the single electrode capacitances were computed. Using importance sampling methods, the differential capacitance was investigated as a continuous function of the applied potential difference ∆Ψ. The differential capacitance displays a number of non-trivial features which can be explained by investigating the structural evolutions of the interfacial electrolytes. Contrarily to previous works on conventional aqueous or organic electrolytes [START_REF] Jiang | Solvent Effect on the Pore-Size Dependence of an Organic Electrolyte Supercapacitor[END_REF][START_REF] Jiang | Unusual effects of solvent polarity on capacitance for organic electrolytes in a nanoporous electrode[END_REF][START_REF] Jiang | Molecular Dynamics Simulations of the Electric Double Layer Capacitance of Graphene Electrodes in Mono-valent Aqueous Electrolytes[END_REF][START_REF] Burt | Capacitance of Nanoporous Carbon-Based Supercapacitors Is a Trade-Off between the Concentration and the Separability of the Ions[END_REF][START_REF] Yang | Molecular Origin of Electric Double-Layer Capacitance at Multilayer Graphene Edges[END_REF][START_REF] Bo | Design of Supercapacitor Electrodes Using Molecular Dynamics Simulations[END_REF] the water molecules do not play a direct role on the variation of the capacitance because most of them are strongly bound inside the first hydration shell of the lithium ions, so that the capacitive behavior overall displays similar features as in the case of pure ionic liquids. This work is a first step toward understanding the capacitive behaviors of water-insalt electrolyte in supercapacitors from simulations, in particular, the structural driven potential dependence of differential capacitance, which still requires more experimental explorations. The results could provide useful information for the understanding and development of new water-insalts electrolytes and their applications in both supercapacitors and batteries.

Figure 1 :

 1 Figure 1: (a) Snapshot of the simulation cell consisting of 20 m LiTFSI confined between two three-layered graphite electrodes, in which the all-atom model of electrolyte is shown at the interfaces with H (white), O (red), Li (green), F (orange), C (cyan), S (yellow), and N (blue) atoms, respectively. The TFSI anions beyond interfaces are represented by isosurface instead. The carbon walls are shown in side views. (b) Number density profiles of Li cations, water molecules, and TFSI anions at applied potential differences ∆Ψ = 0 V (top) and 3 V (bottom).

Figure 1b )

 1b Figure 1b), which ensures decoupling of the two interfaces. The number densities of Li cations, water molecules, and TFSI anions normalized by the corresponding bulk values are plotted in Figure S1, they give results qualitatively similar to the one reported in previous work for a slightly different composition.[START_REF] Vatamanu | Ramifications of Water-in-Salt Interfacial Structure at Charged Electrodes for Electrolyte Electrochemical Stability[END_REF] For each simulation within the studied potential range, positive and neg-

2 )C diff = 10 . 1 µF /cm 2 C diff = 5 . 6 µF /cm 2 Figure 2 :

 210125622 Figure2: Surface charge density as a function of the potential drop across each interface. Linear fits are shown in dotted and dashed lines for negative (σ < 0) and positive electrodes (σ < 0), respectively. The corresponding differential capacitances are given by the slope of the two curves.

Figure 3 :

 3 Figure 3: (a) Contour plot of the probability distribution of surface charge density on the electrodes. The probability is indicated by the color bar shown on the right. (b) Differential capacitance as a function of the applied potential difference.

Figure 4 :

 4 Figure 4: Number of ions in the interfacial layers on the negative electrode. Probability distribution of the number of Li cations (a), O (b) belonging to water molecules, O (c) and F (d) from TFSI anions, respectively, as a function of potential Ψ -= -∆Ψ/2, as defined in eq Eq. (1), for the negative electrode when the electrochemical cell is submitted to a voltage difference ∆Ψ.

Figure 5 :

 5 Figure 5: Number of ions in the interfacial layers on the positive electrode. Probability distribution of the number of Li cations (a), O (b) belonging to water molecules, O (c) and F (d) from TFSI anions, respectively, as a function of potential Ψ + = ∆Ψ/2, as defined in eq Eq. (1), for the positive electrode when the electrochemical cell is submitted to a voltage difference ∆Ψ.

Figure 6 :

 6 Figure 6: (a) Density profiles of O and F from TFSI anions as a function of distance from the positive electrode at Ψ + = 0.75 and 1.0 V (∆Ψ = 1.5 and 2 V). Grey and light-blue background represent the first and second adsorbed layers, respectively. (b) and (c) are the typical snapshots close to the positive electrode (grey slab on the bottom) from simulations at Ψ + = 0.75 and 1.0 V, respectively.

Acknowledgement

This work was supported by the French National Research Agency (Labex STORE-EX, Grant No. ANR-10-LABX-0076). This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 771294). This work was granted access to the HPC resources of CINES under the allocation 2018-A0040910463 made by GENCI.


Q Mouchard

Charbel Koumeir

Noël Servagent

Vincent Métivier

Vincent Métivier Non-Destructive

Non-destructive analysis: PIXE with high-energy ion beam at the ARRONAX cyclotron

Development of the high-energy PIXE method using the ARRONAX cyclotron

It is possible to determine the concentration of elements present in a sample, using the number of X-rays detected in the spectrum for each element N X Z . ; Ω ∶ solid angle σ i ∶ Ionization cross section (constant for a thin target) μ c : linear attenuation coefficient of the target 𝐸 𝑖 : initial particle energy ; 𝐸 𝑓 : final particle energy For quantitative measurement, the following 3 factors are to be measured:

𝐍 𝐩 , 𝛆 𝐚𝐧𝐝 𝛔 𝐢

The combined use of the PIGE method, detection of  emission, allows quantification for lights elements (Z<11).

Number of incident particles 𝐍 𝐩

The number of incident particles can be determined using a beam current measurment. Several detectors can be used, each with an optimal range of use.

  N p ∶ Number of incident particles N Z ∶ Number of target atoms per unit volume ω ∶ fluorescence yield of the considered layer b ∶ relative intensity of the ray X ε ∶ detection efficiency, ε = ε 𝑖 × Ω 4π

  Optimal range of use of the different detectors Future HE PIXE application at GIP ARRONAX • Study of cement for the storage of radioactive waste (Landesman C. -SUBATECH): • Investigation of the concentration of different elements in the layers of the sample • Geological sediment study (Mouret A. -LPG BIAF): • Investigation of the concentration of different elements along a burrow created by a wormDetection efficiency 𝛆Use of an X-ray tomograph to determine some detector characteristics:Experimental measurement of detection efficiency using a 55 Fe source:The geometric efficiency is based on the Blachman solid angle model. (B.P. Burtt, Nucleonics (1942) 42) Further measures are necessary to be able to constrain the efficiency curve.

  𝛔 𝐢Ionization cross-section measurement, at the energies available at the ARRONAX cyclotron (17 to 68 MeV), are required to validate the theoretical RECPSSR model for high energies.Experimental values of the ionization cross-sections of copper K-layer as a function of the energy protons incident, compared with the theoretical models ECPSSR and RECPSSR (Hazim M., 2017)Experimental set-up for measuring the ionization cross-section of angle of incidence of the beam 𝜃: angle between detector and target normal


Motion of viscous droplets in a rough confinement: a paradoxical lubrication We study the sedimentation of highly viscous droplets confined inside Hele-Shaw cells with textured walls of controlled topography. In contrast with common observations on superhydrophobic surfaces, roughness tends here to significantly increase viscous friction, thus substantially decreasing the droplets mobility. However, reducing confinement induces a jump in the velocity as droplets can slide on a lubricating layer of the surrounding fluid thicker than the roughness features. We demonstrate that increasing the viscosity of the surrounding liquid may counter-intuitively enhance the mobility of a droplet sliding along a rough wall. Similarly, a sharp change of the droplet mobility is observed as the amplitude of the roughness is modified. These results illustrate the non-trivial friction processes at the scale of the roughness, and the coupling between viscous dissipation in the drop, in the front meniscus and in the lubricating film. They could enable to specifically control the speed of droplets or capsules in microchannels, based on their rheological properties.

Numerous interfacial instabilities induce the emulsification of petroleum in porous media such as snap-off [START_REF] Ransohoff | Snapoff of gas bubbles in smoothly constricted noncircular capillaries[END_REF], viscous fingering [START_REF] Saffman | The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid[END_REF][START_REF] Talal | Control of interfacial instabilities using flow geometry[END_REF][START_REF] Odier | Forced Imbibition in Porous Media: A Fourfold Scenario[END_REF] or pure capillary phenomena [START_REF] Keiser | Washing wedges: capillary instability in a gradient of confinement[END_REF]. The displacement of the generated droplets in the porous network leads to complex rheology [START_REF] Zinchenko | Motion of Deformable Drops Through Porous Media[END_REF]. More generally, the transport of droplets in confined environment is crucial for medical and chemical engineering applications, as illustrated by the fast development of digital microfluidics during the last two decades [START_REF] Baroud | Dynamics of microfluidic droplets[END_REF][START_REF] Seemann | Droplet based microfluidics[END_REF]. This problem motivated theoretical, numerical and experimental works, with the aim of understanding the dynamics of confined drops in ideal geometries [START_REF] De Lózar | Scaling Properties of Coating Flows in Rectangular Channels[END_REF][START_REF] Jakiela | Discontinuous Transition in a Laminar Fluid Flow: A Change of Flow Topology inside a Droplet Moving in a Micron-Size Channel[END_REF][START_REF] Huerre | Droplets in Microchannels: Dynamical Properties of the Lubrication Film[END_REF][START_REF] Zhu | A pancake droplet translating in a Hele-Shaw cell: lubrication film and flow field[END_REF][START_REF] Ling | Droplet migration in a Hele-Shaw cell: Effect of the lubrication film on the droplet dynamics[END_REF][START_REF] Yahashi | Scaling crossover in thin-film drag dynamics of fluid drops in the Hele-Shaw cell[END_REF][START_REF] Balestra | A viscous droplet in a capillary tube: from Bretherton's theory to empirical models[END_REF][START_REF] Keiser | Dynamics of non-wetting drops confined in a Hele-Shaw cell[END_REF][START_REF] Reichert | Topography of the lubrication film under a pancake droplet traveling in a Hele-Shaw cell[END_REF]. Although they demonstrated the major contribution of lubricating films, these studies mainly remained focused on smooth confining walls. In common situations, walls may however present natural or engineered roughness features. Numerous works have also been dedicated to the reduced friction on superhydrophobic surfaces [START_REF] Vinogradova | Slippage of water over hydrophobic surfaces[END_REF][START_REF] Lauga | Effective slip in pressure-driven Stokes flow[END_REF][START_REF] Choi | Large Slip of Aqueous Liquid Flow over a Nanoengineered Superhydrophobic Surface[END_REF][START_REF] Ybert | Achieving large slip with superhydrophobic surfaces: Scaling laws for generic geometries[END_REF][START_REF] Rothstein | Slip on Superhydrophobic Surfaces[END_REF][START_REF] Schäffel | Local Flow Field and Slip Length of Superhydrophobic Surfaces[END_REF] and to the reduced adhesion on liquid-infused surfaces of controlled topography [START_REF] Smith | Droplet mobility on lubricant-impregnated surfaces[END_REF][START_REF] Daniel | Oleoplaning droplets on lubricated surfaces[END_REF][START_REF] Keiser | Drop friction on liquid-infused materials[END_REF]. However the impact of roughness on the dynamical properties of the lubricating film has so far not been extensively studied.

In this Letter, we analyze the dynamics of non-wetting confined oil droplets falling in a Hele-Shaw cell filled with a surfactant solution of lower viscosity. By comparing the dynamics of droplets moving in smooth and rough confinements, we propose to highlight the impact of roughness on the mobility of the droplets.

Our experimental setup (Fig. 1) is composed of two parallel glass plates covered with a square array of micropillars of square cross-section (height h, width w, separation s) cast in a layer of transparent resin (NOA from Thorlabs, Inc.) using photolithography techniques (see details in the Supplementary Information). The walls are separated by a gap e ranging from 300 to 1500 µm, much larger than the micrometric height h of the textures, as red) is confined between the walls of a rough Hele-Shaw cell filled with a surfactant solution (in blue). The cell is inclined by an angle α, with respect to the horizontal. The thickness e of the gap ranges from 300 to 1500 µm. The cell walls are covered with a lattice of micropillars of square cross-section of height h e, width w and separation s. The denser drop (radius R and volume Ω = πR 2 e) falls under gravity with a steady velocity V . A lubricating film of surfactant solution separates the drop from the walls, with an average thickness b between the drop and the bottom of the roughness. sketched in Fig. 1. The cell is immersed vertically in a bath of a surfactant solution, containing 5% in mass of a commercial soap (PAIC, from Colgate-Palmolive), of dynamic viscosity µ w = 1 mPa.s. Drops of fluorinated silicone oil (poly(3,3,3)-trifluoropropylmethylsiloxane, from Gelest, Inc.) of volume Ω and viscosity µ o ranging from 180 to 18000 mPa.s are inserted in the upper part of the cell. The interfacial tension between both liquids γ = 5 -6 mN/m (depending on the oil viscosity) is measured with the pendant drop method. The oil density FIG. 2: a) Falling velocity V in vertical cell vs. the gap of the cell e for different pillar heights h, for µo = 18000 mPa.s, texture width w = 180 µm and spacing s = 35 µm. For smooth walls (empty red circles), V increases as e 6 as described by Eq. 2 (solid magenta line). In rough cells and below a critical velocity Vc, V grows as e 2 and is significantly smaller than in smooth cells. Velocities are one order of magnitude larger than expected from a basic Poiseuille flow (dotted red line), a signature of the slip introduced by the roughness. For e > ec, or equivalently V > Vc, a "jump" of velocity is observed (indicated by arrows and illustrated in the Supplementary Movie 1) and V is captured by Eq. 2, as for smooth cells. The critical velocity increases as the textures height h increases, and for h = 9 µm (green triangles), we do not observe any jump in velocity within the explored gaps. b) Influence of oil viscosity on drop dynamics in a vertical cell. In smooth cells, the velocity barely depends on µo (as described by Eq. 2). Conversely in rough cells and below Vc, the velocity decreases as µo increases, reflecting that viscous dissipation is localized inside the drop in the "weakly lubricated" regime. c) Falling velocity in Hele-Shaw cells tilted by an angle α with respect to the horizontal, for two configurations (e, µo): (1 mm, 1500 mPa.s) and (1.4 mm, 18000 mPa.s). A sharp velocity transition is also observed as α exceeds a critical value. The drops keep falling even at the lowest inclinations (α = 2.5 • ), highlighting the absence of pinning of the drop at the top of the pillars, which would result in a vanishing velocity at a finite value of the angle α.

ρ o = 1230 -1250 kg/m 3 is larger than the density of water (ρ w = 997 kg/m 3 ) so that drops fall under gravity. This simple configuration ensures a precisely controlled driving force, enabling a thorough characterization of the dissipation mechanisms [START_REF] Yahashi | Scaling crossover in thin-film drag dynamics of fluid drops in the Hele-Shaw cell[END_REF][START_REF] Keiser | Dynamics of non-wetting drops confined in a Hele-Shaw cell[END_REF]. As the surfactant solution completely wets the surface of the cell walls, no adhesion hinders the motion of the oil drops. The vertical trajectory of the drops is monitored with a digital camera. The drops are observed to reach a constant velocity V after a few millimeters of descent. For fixed values of µ w , µ o and e, V is independent of the volume within the limit of confined drops (R > e). Due to the high contrast in viscosity between the drops and the surrounding solution, we expect the motion of the drops to markedly rely on the thickness of the lubricating film b, as shown by recent studies [START_REF] Yahashi | Scaling crossover in thin-film drag dynamics of fluid drops in the Hele-Shaw cell[END_REF][START_REF] Keiser | Dynamics of non-wetting drops confined in a Hele-Shaw cell[END_REF].

In a cell with smooth walls, b follows the classical Bretherton's law [START_REF] Bretherton | The motion of long bubbles in tubes[END_REF] b ∼ e/2 Ca 2/3 , for capillary numbers Ca = µ w V /γ ranging from 10 -5 to 10 -2 , as confirmed by recent experimental investigations [START_REF] Huerre | Droplets in Microchannels: Dynamical Properties of the Lubrication Film[END_REF][START_REF] Reichert | Topography of the lubrication film under a pancake droplet traveling in a Hele-Shaw cell[END_REF]. In this configuration, the dimensionless parameter m = µ w e/µ o b characterizes the efficiency of the lubrication process. For large values of m, the films of water are too thin to lubricate efficiently the motion. The flow in the volume of the drop thus follows a classical Poiseuille profile, which corresponds to the friction force F f = 12πµ o V R 2 /e. Balancing F f with the apparent weight of the "pancake" drop in water F g = π(ρ o -ρ w )gR 2 e leads to the sedimentation velocity:

V = (ρ o -ρ w )g sin α 12µ o e 2 (1)
However in the present experimental conditions, the large inner-to-outer viscosity ratio (µ o /µ w > 180) favors the opposite limit m 1. In this regime, the drop tends to slide on the water films as a solid. Viscous dissipation is now localized in the lubricating film, which results in the sublinear friction force F f ∼ µ w V R 2 /b ∼ γR 2 Ca 1/3 /e. and leads to a strong dependence of the velocity on the gap:

V = β (ρ o -ρ w ) 3 g 3 sin 3 α µ w γ 2 e 6 , (2)
where β = 4.3 • 10 -2 is a numerical factor experimentally determined in a previous work [START_REF] Keiser | Dynamics of non-wetting drops confined in a Hele-Shaw cell[END_REF]. This regime, represented by empty circles and full lines in Fig. 2a and2b, is hereafter called "strongly lubricated" and corresponds to velocities independent of the drop viscosity µ o (within the limit m 1). In a cell with rough walls, the dynamics is dramatically altered, and the velocity is strongly reduced, as shown in Fig. 2. Although V displays a square dependence with e, its magnitude is about ten times larger than expected from a basic Poiseuille flow in the drop (Eq. 1 and dotted lines in Fig. 2). We interpret this shift as a consequence of an effective slip due to the presence of water atop and between the pillars [START_REF] Lauga | Effective slip in pressure-driven Stokes flow[END_REF][START_REF] Ybert | Achieving large slip with superhydrophobic surfaces: Scaling laws for generic geometries[END_REF]. We refer FIG. 3: a) At small velocity (V < Vc), the roughness-induced reduction of the viscous stress at the front meniscus of the drop hinders the deformation of the interface and the deposition of a lubricating film. The drop is compressed in the pillars by the capillary pressure and its interface is corrugated. Furthermore, a thin nanometric film of water, stabilized by disjoining pressure, isolates the drop from the top of the pillars. In this configuration, viscous dissipation localizes in the viscous drop, which strongly its mobility. At large velocity (V > Vc), the viscous stress at the front meniscus is strong enough to deform the interface and to deposit a film where viscous dissipation localizes. The dynamics is thus similar to the case of smooth confinement. b) Experimental critical velocity as a function of V * c (Eq. 3), for different oil viscosities µo and pillars heights h, for various pillars densities and geometries: φ = 0.23 (green triangles, pillars), φ = 0.50 (full/empty orange diamonds, grooves perpendicular/parallel to the direction of motion of the droplets), φ = 0.56 (blue squares, pillars), φ = 0.70 (red circles, pillars). The large variations of Vc mainly originate from variations of h, ranging from 1.5 to 9 µm. The variations of µo from 180 to 18000 mPa.s only slightly alters Vc, due to the change of ec, as shown in Fig. 2.

to this regime as "weakly lubricated", and discuss hereafter its origin. Above a critical gap thickness e c , corresponding to a critical speed V c , we observe an abrupt jump in velocity, represented by arrows Fig. 2 (see also the Supplemental Movie 1). For e > e c the measured velocities are similar to the velocities obtained in a smooth cell. The "strongly lubricated" regime is thus recovered and the velocity increases by more than two orders of magnitude at the jump. We carried out experiments with various texture heights h for a given pattern (w = 200 µm, s = 25 µm). Fig. 2a illustrates that as h is increased from 1.5 to 9 µm, the critical gap and critical velocity also increase. For the largest height (h = 9 µm), the critical velocity V c is even not reached within the explored range of parameters. This spectacular effect is also observed for droplets of smaller viscosity, as plotted in Fig. 2b. In the "weakly lubricated" regime (V < V c ), the velocity of the drops increases as the oil viscosity µ o is decreased (from 18000 to 1500 mPa.s), which reflects that viscous dissipation is localized in the volume of the drop. For µ o = 1500 mPa.s (blue triangles) the velocity jump is also observed and for e > e c , all velocities collapse on the same curve corresponding to the "strongly lubricated" regime (Eq. 2). The transition is also observed in cells of given thickness but tilted with respect to the horizontal with an angle α (Fig. 2c). At small α, weakly lubricated drops move slowly but still smoothly, indicating the absence of pinning on the textures. Increasing effective gravity by tilting the cell, one recovers the sharp transition to the high velocity regime where drops do not feel the roughness. For a given texture pattern, the critical velocity V c is the same as the one measured in experiments where the gap has been varied.

The massive reduction of the mobility of non-wetting drops with the wall roughness strongly contrasts with the common behavior of water confined in superhydrophobic channels, where the roughness-induced slippage at the walls promotes higher mobility in comparison with smooth walls [START_REF] Vinogradova | Slippage of water over hydrophobic surfaces[END_REF][START_REF] Lauga | Effective slip in pressure-driven Stokes flow[END_REF][START_REF] Choi | Large Slip of Aqueous Liquid Flow over a Nanoengineered Superhydrophobic Surface[END_REF][START_REF] Ybert | Achieving large slip with superhydrophobic surfaces: Scaling laws for generic geometries[END_REF][START_REF] Rothstein | Slip on Superhydrophobic Surfaces[END_REF][START_REF] Schäffel | Local Flow Field and Slip Length of Superhydrophobic Surfaces[END_REF]. We interpret this paradox as a consequence of the role of the surrounding fluid as a lubricant. Analogous results were observed by Seiwert et al. in a recent study on the coating of a textured plate removed from a bath of liquid [START_REF] Seiwert | Coating of a textured solid[END_REF]. Below a critical withdrawal velocity V c , the liquid invades the textures (a dilute array of pillars), but the authors do not observe any coating film above the height of the pillars. However, a coating film is deposited and Bretherton's law is progressively recovered as the withdrawing velocity is increased above V c . A velocity threshold thus appears for the deposition of a film above the textures. When applied to our configuration, this scenario leads to the two regimes sketched in Fig. 3a. For V < V c , the surrounding solution only fills the textures. The friction of the droplet on the textured walls is thus complex. Due to molecular interactions, a thin film of the solution may be present on the top of the pillars. The thickness of such a thin film is of the order of tens of nanometers [START_REF] Huerre | Droplets in Microchannels: Dynamical Properties of the Lubrication Film[END_REF][START_REF] Daniel | Oleoplaning droplets on lubricated surfaces[END_REF][START_REF] Israelachvili | Intermolecular and surface forces[END_REF]; its lubricating properties are therefore limited [START_REF] Keiser | Dynamics of non-wetting drops confined in a Hele-Shaw cell[END_REF]. Furthermore, due to the strong confinement, Laplace pressure induces the formation of bumps between pillars of am-plitude δ ∼ s 2 /e ∼ 500 nm, which results into additional friction localized in the viscous drop [START_REF] Ybert | Achieving large slip with superhydrophobic surfaces: Scaling laws for generic geometries[END_REF][START_REF] Einzel | Boundary condition for fluid flow: Curved or rough surfaces[END_REF]. The overall dissipation in this "weakly lubricated" regime should thus depend on the topography of the texture and more specifically on the pillar density φ = (w/(w + s))

2 . Describing in detail these processes would deserve a dedicated study and is beyond the scope of this Letter. In the opposite condition V > V c , a lubricating layer of the solution covers the textures and reduces dramatically the friction, as viscous dissipation localizes in the lubricating film of low viscosity. Modelling the impregnated textures as an effective viscous liquid layer in Bretherton's derivation, Seiwert et al. estimate the evolution of the deposited layer in their coating process [START_REF] Seiwert | Coating of a textured solid[END_REF]. As V is increased, the thickness quickly converges to the classical Bretherton's law [START_REF] Seiwert | Coating of a textured solid[END_REF] in agreement with the sudden jump observed in our experiments and with different derivations accounting for wall slip [START_REF] Liao | Drastic Changes in Interfacial Hydrodynamics due to Wall Slippage: Slip-Intensified Film Thinning, Drop Spreading, and Capillary Instability[END_REF][START_REF] Li | Breakdown of the Bretherton law due to wall slippage[END_REF]. The transition between both regimes in the experiments of Seiwert is smooth: the thickness varies continuously with the imposed coating velocity. In contrast, our experiment is performed at a controlled driving force, which leads to a discontinuity in the velocity. In terms of scaling, V c can be estimated as the velocity required to obtain a coating film of thickness h in a standard smooth situation [START_REF] Seiwert | Coating of a textured solid[END_REF]:

V * c ∼ h e c /2 3/2 γ µ w (3)
However, we also expect the pillar density φ to play a major role in setting the transition. In Fig. 3b, we plot the evolution of the critical velocity V exp c as a function of V * c for different values of φ. Each of measurements corresponds to different oil viscosities and pillar heights. Although experimental data display some scattering, V exp c convincingly appears to be proportional to V * c , confirming the scaling. In addition, the numerical prefactor is rather low for high values of φ (of the order of 1/20 for φ = 0.7). In the limit φ → 1 corresponding to a smooth wall, we even expect V exp c to vanish. Conversely, the prefactor gets progressively closer to unity as φ decreases. The result from Seiwert et al., represented with a dotted line was experimentally obtained within the limit φ < [START_REF] Seiwert | Coating of a textured solid[END_REF].

The value of V * c is inversely to µ w , meaning that increasing the outer viscosity favors the abrupt velocity jump. This dependence leads to counter-intuitive results when viscous glycerol is added to the outer aqueous solution (50% in volume of glycerol, as detailed in the Supplementary). In Fig. 4 we compare the falling velocity of droplets in solutions of viscosity µ w = 1 and 8 mPa.s. In the case of smooth surfaces, the droplets move faster in the solution of lower viscosity, in agreement with Eq. 2 and common intuition. However, the case of rough surfaces is more surprising. For high confinement, the velocity is low and does not display e [mm] FIG. 4: V vs. e for drops of µo = 18000 mPa.s in smooth and rough confinements represented, respectively, by and full symbols. The blue squares correspond to droplets of µo = 18000 mPa.s advancing in vertical Hele-Shaw cells filled with a viscous water-glycerol mixture (µw = 8 mPa.s). The red circles to droplets of µo = 18000 mPa.s advancing in Hele-Shaw cells filled with water (µw = 1 mPa.s) inclined by an angle of 30 • with respect to the horizontal. This inclination is realized to counteract the change of density of the aqueous solution ρw due to the addition of dense glycerol and to guarantee a similar driving force between both experiments (see the Supplementary for more details). In the greencolored region, much larger velocities are counter-intuitively obtained for drops evolving in a more viscous environment (µw = 8 mPa.s, see also the Supplemental Movie 2). The red dashed arrow represents the velocity jump which would occur if higher gap thickness e could be tested. a strong dependence on µ w . The jump appears first with the viscous solution, leading to a sharp increase of the velocity as expected from Eq. 3. Within a certain range of confinement (green region in Fig. 4), droplets thus move faster when surrounded by a more viscous solution (see also the Supplemental Movie 2). Provided higher gap thicknesses could be tested, one would also expect the data for µ w = 1 mPa.s to exhibit an abrupt velocity jump (dashed arrows in Fig. 4), and the intuitive behavior would be restored: the more viscous the outer phase, the slower the drops. This Letter illustrates the intricate dissipation mechanisms acting at the vicinity of rough surfaces. Below a critical velocity, roughness decreases the friction at the front meniscus of the drop. As an illustration, we show in the Supplementary that buoyant confined air bubbles tend to rise faster in textured Hele-Shaw cells than between smooth walls. In this opposite configuration, the outer fluid is more viscous than the inner air and the motion is mainly resisted by friction in the vicinity of the peripheral menisci [START_REF] Cantat | Liquid meniscus friction on a wet plate: Bubbles, lamellae, and foams[END_REF][START_REF] Reyssat | Drops and bubbles in wedges[END_REF]. The higher mobility observed at small Ca highlights role of roughness in reduc-ing friction. In the case of viscous drops, the impact of roughness is dramatically different. The reduced viscous friction at the front of the drop hinders the deformation of the interface and the formation of an efficient lubricating film. This results in a corrugated interface (Fig. 3a) and in the localization of viscous friction in the viscous drop. We finally showed that increasing the viscosity of the outer fluid, the viscous friction at the periphery is increased and a thick lubricating layer is restored, therefore leading to faster drops.

Depending on the inner-to-outer viscosity ratio µ o /µ w , we showed that the apparent roughness-induced reduction of friction in the vicinity of the meniscus can accelerate air bubbles or, as a paradox, substantially slow down highly viscous droplets. A direct observation of the structure of the lubricating film would enable to accurately describe the configurations qualitatively represented in Fig 3a, using for instance Reflection Interference Contrast Microscopy (RICM) [START_REF] Huerre | Droplets in Microchannels: Dynamical Properties of the Lubrication Film[END_REF][START_REF] Reichert | Topography of the lubrication film under a pancake droplet traveling in a Hele-Shaw cell[END_REF] or confocal microscopy [START_REF] Schellenberger | Direct observation of drops on slippery lubricant-infused surfaces[END_REF].

Although we focused on a sedimentation experiment, the observed are fully applicable to configurations where drops are entrained by a surrounding fluid in a confined environment, such as microfluidic channels [START_REF] Baroud | Dynamics of microfluidic droplets[END_REF][START_REF] Seemann | Droplet based microfluidics[END_REF] or porous media [START_REF] Zinchenko | Motion of Deformable Drops Through Porous Media[END_REF]. From a practical point of view, controlling the friction of viscous droplets through a designed texture might constitute an efficient tool to probe the rheological properties of droplets, vesicles or biological cells, and possibly achieve specific sorting.

FIG. 1 :

 1 FIG.1: Experimental setup. A drop of fluorinated oil (in red) is confined between the walls of a rough Hele-Shaw cell filled with a surfactant solution (in blue). The cell is inclined by an angle α, with respect to the horizontal. The thickness e of the gap ranges from 300 to 1500 µm. The cell walls are covered with a lattice of micropillars of square cross-section of height h e, width w and separation s. The denser drop (radius R and volume Ω = πR 2 e) falls under gravity with a steady velocity V . A lubricating film of surfactant solution separates the drop from the walls, with an average thickness b between the drop and the bottom of the roughness.

  Ludovic Keiser 1 2 , * Armelle Keiser 1 , Manon L'Estimé 1 , José Bico 1 , and Étienne Reyssat 1 † 1 Laboratoire de Physique et Mécanique des Milieux Hétérogènes (PMMH), Sorbonne Université, 7 quai Saint-Bernard, barre Cassan 75005 Paris, CNRS UMR 7636, ESPCI Paris -PSL Research University, Univ. Paris Diderot and 2 Total S.A., Pôle d' Études et de Recherche de Lacq, BP47 64170 Lacq, France (Dated: November 4, 2019)

Acknowledgements

The authors thank K. Jaafar for his initial contribution to the project, and acknowledge funding from Total S.A. This work has received the support of Institut Pierre-Gilles de Gennes, "Investissements d'avenir", program ANR-10-EQPX-34.