Armelle Keiser

Ludovic Keiser

Christophe Clanet

David Quéré

Drop friction on liquid-infused materials

We discuss in this paper the nature of the friction generated as a drop glides on a textured material infused by another liquid. Different regimes are found, depending on the viscosities of both liquids. While a viscous drop simply obeys a Stokes-type friction, the force opposing a drop moving on a viscous substrate becomes non-linear in velocity. A liquid on an infused material is surrounded by a meniscus, and this specific feature is proposed to be responsible for the special frictions observed on both adhesive and non-adhesive substrates.

Liquid droplets on solids generally have high adhesion and friction, owing to the contact line bounding them. This line can pin on defects [START_REF] Joanny | A model for contact angle hysteresis[END_REF], and its presence enhances dissipation as the liquid moves [START_REF] Huh | Hydrodynamic model of steady movement of a solid/liquid/fluid contact line[END_REF][START_REF] Snoeijer | Moving contact lines: scales, regimes, and dynamical transitions[END_REF]. In order to promote mobility, it was proposed to lubricate the gap between the solid substrate and the moving drop. The lubrication layer may consist either of air (Leidenfrost state [START_REF] Leidenfrost | Aqua Communis Nonnullis Qualitatibus Tractatus. Impensis Hermanni Ovenni, Univers. Bibliopolae[END_REF], superhydrophobic solids [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]), or of another liquid (here called oil) trapped by textures [START_REF] Quéré | Non-sticking drops[END_REF] or magnetically [START_REF] Irajizad | Magnetic slippery extrême icephobic surfaces[END_REF] -all kinds of situations of practical interest, since they provide anti-biofouling [START_REF] Epstein | Liquid-infused structured surfaces with exceptional anti-biofouling performance[END_REF], anti-dew [START_REF] Anand | Enhanced condensation on lubricantimpregnated nanotextured surfaces[END_REF][START_REF] Anand | How droplets nucleate and grow on liquids and liquid-impregnated surfaces[END_REF], or even anti-ice abilities [START_REF] Kim | Liquid-infused nanostructured surfaces with extreme anti-ice and anti-frost performance[END_REF][START_REF] Stone | Ice-Phobic Surfaces That Are Wet[END_REF][START_REF] Subramanyam | Ice adhesion on lubricant-impregnated textured surfaces[END_REF]. This idea has been widely exploited by nature, with multiple examples of peculiar plants [START_REF] Barthlott | Classification and terminology of plant epicuticular waxes[END_REF] having water-repellent leaves [START_REF] Koch | Superhydrophobic and superhydrophilic plant surfaces: an inspiration for biomimetic materials[END_REF] or slippery surfaces [START_REF] Bohn | Insect aquaplaning: Nepenthes pitcher plants capture prey with the peristome, a fully wettable water-lubricated anisotropic surface[END_REF][START_REF] Bauer | Harmless nectar source or deadly trap: Nepenthes pitchers are activated by rain, condensation and nectar[END_REF]. By combining textures and chemistry, these plants get remarkable non-adhesive properties, making them able to fully repel water (lotus leaf), or to trap insects owing to water filmification (carnivorous Nepenthes). We focus here on oil-infused textured materials, which were shown to display spectacular slippery behaviours [START_REF] Wong | Bioinspired selfrepairing slippery surfaces with pressure-stable omniphobicity[END_REF][START_REF] Lafuma | Slippery pre-suffused surfaces[END_REF][START_REF] Smith | Droplet Mobility on Lubricant-Impregnated Surfaces[END_REF]. These materials are made in two steps: firstly, a solid is decorated by hydrophobic micro-textures; secondly, oil is infused by spontaneous impregnation [START_REF] Courbin | Dynamics of wetting: from inertial spreading to viscous imbibition[END_REF] or dipcoating [START_REF] Smith | Droplet Mobility on Lubricant-Impregnated Surfaces[END_REF][START_REF] Seiwert | Coating of a textured solid[END_REF]. If the texture, surface chemistry and oil are carefully combined, a drop of water (possibly mixed with glycerol or hydrosoluble polymers) will stay at the surface of the oil, and benefit from its lubricating action [START_REF] Liu | Effect of viscosity ratio on the shear-driven failure of liquid-infused surfaces[END_REF][START_REF] Semprebon | Apparent contact angle and contact angle hysteresis on liquid infused surfaces[END_REF]. Contact angle hysteresis is minimized, and so is its adhesion to the substrate, despite apparent contact angles often close to 90° [START_REF] Wong | Bioinspired selfrepairing slippery surfaces with pressure-stable omniphobicity[END_REF][START_REF] Lafuma | Slippery pre-suffused surfaces[END_REF][START_REF] Smith | Droplet Mobility on Lubricant-Impregnated Surfaces[END_REF][START_REF] Semprebon | Apparent contact angle and contact angle hysteresis on liquid infused surfaces[END_REF]. Hence such drops easily run down inclines (as long as oil remains trapped in the textures), and we discuss here the associated friction. Depending on the liquids and texture, several kinds of dissipation can be generated, which can be exploited to tune the drop velocity.

Our samples are covered by micro-pillars made of SU8-resin and obtained by photolithography on a silicon wafer, where they occupy a rectangular area of 8 cm x 5 cm. As shown in the Supplementary Information (SI), pillars all have a height h = 20 ± 3 µm and lateral size and spacing spanning respectively between 18 and 185 µm, and between 15 and 60 µm. Hence the pillar density φ ranges between 23% and 67%. We did experiments either with simple pillars, or with pillars covered by a nanometric sub-structure, a system recently introduced by Guan et al. [START_REF] Guan | Drop transport and positioning on lubricant impregnated surfaces[END_REF]. For this purpose, we coat the textured substrate with a solution of Glaco Mirror Coat (Soft 99), let the solvent (isopropanol) evaporate, and consolidate the sample by heating it at 150°C for 20 minutes, which leaves a layer of hydrophobic nanobeads (size: 30 nm) attached to the substrate and to the pillars. The lubricant is a silicone oil with viscosity η o ranging between 5 mPa.s and 1000 mPa.s, with a nearly constant surface tension γ o ≈ 20 mN/m and density ρ o ≈ 980 kg/m 3 . Silicone oil completely wets silicon and resin, so that it spontaneously fills the textures: impregnation is achieved by placing the samples in contact with a bath of oil, which ensures that we do not over-impregnate the liquid-infused material. We checked that dip-coating the sample at low velocity (smaller than 20 µm/s) from a bath of oil leads to the same results, hence found to be independent of the technique used to infuse the pillars. Then, we place on the lubricated texture a drop of water/glycerol mixture (simply called water further) with a viscosity η w ranging between 1 mPa.s (pure water) and 1000 mPa.s (pure glycerol).

Soft Matter Accepted Manuscript

The behaviour of water depends on the presence of nanobeads. On nude pillars filled with oil, water drops (volume Ω = 20 µL) roll off provided the sample is tilted by at least α* = 15° ± 3° for φ = 23%. As shown in the SI, α* increases at larger φ. This large adhesion arises from the water/solid contacts, which induces pinning on the textures' tops [START_REF] Schellenberger | Direct observation of drops on slippery lubricant-infused surfaces[END_REF]. On Glaco-treated pillars, the roll-off angle α* becomes unmeasurable (α* ≈ 0), due to the impregnation of the nanoroughness by the oil. The thin film of oil stabilized at pillars' tops isolates water from the pillars, which minimizes angle hysteresis down to non-measurable values [START_REF] Guan | Drop transport and positioning on lubricant impregnated surfaces[END_REF].

Then, as sketched in figure 1a, we measure the drop velocity V as a function of the gravitational driving force, modulated by varying the tilt angle α (above α*). The drop trajectory is recorded using a video-camera (Optronis, 100 to 1000 frames per second). As shown in the SI, all drops reach a constant velocity after a few centimetres of descent, as also seen in the Supplementary Movie 1. In this stationary regime, V results from a balance between gravity and friction. Side views show that drops are quasi-hemispherical (with contact radius R) and surrounded by a small oil meniscus, as sketched in figure 1a, highlighted by red circles in figure 1b, magnified in figure 1c and visible in the Supplementary Movie 2. This circular meniscus is pulled from the oil film by the vertical component of the surface tension of water [START_REF] Smith | Droplet Mobility on Lubricant-Impregnated Surfaces[END_REF]. Schellenberger et al. characterized its geometry by confocal microscopy performed on static drops [START_REF] Schellenberger | Direct observation of drops on slippery lubricant-infused surfaces[END_REF]. The meniscus is found here to persist in dynamical conditions, which might impact the friction opposing the moving liquid. The two series of data correspond to pillar densities φ = 23% (red data) or φ = 67% (purple data). (e) In the opposite limit η o > η w (here η w = 2 mPa.s and η o = 100 mPa.s), the drop speed becomes non-linear in driving force. Lines successively show slopes 3/2 and 3, and colors indicate the texture density (φ = 23%, red data; φ = 36%, blue data; φ = 52%, green data; φ = 67%, purple data). We indicate on the data at the largest slope the typical error bar on the velocity measurement. Colored dashed lines are positioned with the φ 3/2 -behavior predicted by eq. (3).

In order to characterize friction, we focus on the drop mobility, that is, the way speed depends on driving force. In figures 1d and 1e, the drop velocity V is plotted as a function of the sine of the tilt angle α at fixed volume Ω = 20 µL. We first consider extreme values of the ratio η w /η o between the liquid viscosities. (i) When water is more viscous than oil (η w = 50η o , figure 1d, supplementary movie 3), the drop speed linearly increases with the driving force, as stressed by the dashed line with slope 1 drawn across the data, and it does not depend on the texture density (the two series of data correspond to φ = 23% and φ = 67%). Hence friction is classically linear in speed (Stokes' law). (ii) When oil is more viscous than water (η o = 50η w , figure 1e, supplementary movie 1), the drop speed is not linear in driving force anymore. At small α, data can be fitted by scaling laws with power 3/2 (coloured dashed lines); in this regime, the smaller the pillar density φ, the quicker the motion. At large α, there is a sudden kink in the data and the velocity seems to follow a new scaling law, with an exponent 3 (black dashed line). In addition, the speed in this regime becomes independent of φ. e. Figure 1 shows that the ratio between oil and water viscosities determines the nature of the friction. This can be directly captured by varying the drop viscosity η w at fixed η o .

η w = 50η o η o = 50η w b. φ = 23% φ = 67% φ = 23% φ = 36% φ = 52% φ = 67%
Experiments are performed for a plate inclined by α = 5° and a volume Ω = 20 µL, and results are displayed in figure 2a for η o = 10 mPa.s (red data) and for η o = 100 mPa.s (blue data). For each oil viscosity, two regimes are successively followed. Firstly (η w < η o ), the speed V plateaus at a value roughly independent of η w , and found to be ten times larger when η o is ten times smaller. In this regime, dissipation then mainly occurs in oil, as already seen in figure 1e. At larger water viscosity (η w > η o , see also figure 1d), the speed becomes inversely proportional to η w , as evidenced by the dashed lines with slope -1 (and further in figure 3). Friction on infused textures appears to be characterized by (at least) three different laws, a very unique situation generated by the complexity of this system. Dissipation indeed occurs in both water and oil: on the one hand, water moves inside the drop and close to the contact line;

V i V h V 0.01 0.1 1 0.1 1 10 φ = 23% φ = 36% φ = 52% φ = 67% η w = 5η o 3/2 1 V (mm/s) sinα e.

Soft Matter Accepted Manuscript

on the other hand, oil is found under the moving drop and along its edge meniscus. The existence of an underlying oil film favours slip at the oil/water interface, which we can characterize. The slip velocity V i (defined in figure 2b) is deduced from the continuity of the viscous stress at this interface, whose scaling can be written:

η w (V-V i )/R ~ η o V i /h
, where h is the lubricant thickness. Hence we get:

(1)

V i ~
The interfacial velocity V i varies between 0 (no slip) and V (pure slip) as the slip number σ = η w h/η o R increases from zero to infinity. In our case, σ is smaller than 0.1, which implies modest slip (V i < V/10). Pillars should minimize even more σ: the oil film is not free, but confined within obstacles, which results in an effective oil viscosity larger than η o [START_REF] Seiwert | Coating of a textured solid[END_REF].

We can now discuss the friction laws. As shown in experiments (figures 1d and 2a), dissipation at large η w mainly occurs in water. In this limit, typical viscosity η w and velocity V are respectively 500 mPa.s and a few millimetres per second, so that the corresponding Reynolds and capillary numbers are Re = ρRV/η w ≈ 10 -3 and Ca = η w V/γ ≈ 10 -2 , both small compared to unity. Hence dissipation has a viscous origin (figure 2a) and interfaces keep a quasi-static shape (supplementary movie 3). Owing to the presence of velocity gradients in the drop (V i << V), we simply write the viscous stress in water as

η w (V-V i )/R ≈ η w V/R. Once
it is integrated over a surface area of order R 2 , we get a force F w scaling as η w VR. Balancing this Stokes friction with the gravitational driving force ρgR 3 sinα yields:

(2)

V ~ sinα

The velocity V is proportional to the slope sin α (figure 1d) and inversely proportional to the water viscosity η w (figure 2a for η w > η o ). The typical descent velocity, expected to be on the order of 1 cm/s for η w ≈ 1 Pa.s, in fair agreement with values in figure 1d. Hence eq. ( 2) captures the simple regime where dissipation mainly occurs in the drop. It could have been more complicated. As drops spread or move, dissipation often concentrates close to the contact line, which we did not consider: the apparent contact angle here is large and the real contact angle (hidden by the oil meniscus) even larger [START_REF] Semprebon | Apparent contact angle and contact angle hysteresis on liquid infused surfaces[END_REF][START_REF] Schellenberger | Direct observation of drops on slippery lubricant-infused surfaces[END_REF], which minimizes dissipation in the water wedge. Moreover, even for modest slip at the drop scale, the local slip number σ w = η w h/η o H (where we introduce a typical local thickness H), can become arbitrarily large close to the wedge (H → 0), which minimizes even more wedge friction.

V o = ρgR 2 /η w , is V 1 + η o R η w h ρgR 2 η w

Soft Matter Accepted Manuscript

In the opposite limit where oil is more viscous than water (η o > η w , figure 1e), the drop velocity is not anymore linear in driving force. Dissipation is expected to take place in oil, that is, in the underlying film and in the surrounding meniscus. Let us discuss the corresponding frictions. (i) The typical velocity gradient in the film scales as V i /h (figure 2b), which yields a friction

F o ~ η o V i R 2 /h. For η w < η o , the slip number σ = η w h/η o R is small
compared to unity and the interface velocity V i in eq. ( 1) reduces to σV. After injecting this value in the expression of F o , we get a force scaling as η w VR, of the same form, surprisingly, as in the limit η w > η o . Viscous oils are hardly displaced by water, and the corresponding friction is that in the drop, independent of the oil viscosity η o , in contradiction with the two plateaus in figure 2a. (ii) As emphasized above, liquids on lubricant-infused materials are surrounded by a "foot" sketched and observed in figures 1a, 1b and 1c. If we designate its typical size by , we can first consider that the motion generates a stress η o V/ in the meniscus.

Once integrated over the its surface area R, we find the Stokes force F o ~ η o VR first proposed by Smith et al. [START_REF] Smith | Droplet Mobility on Lubricant-Impregnated Surfaces[END_REF]. This formula provides a speed linear in driving force, in contradiction with the data in Figure 1e. We correct this approach by considering the dynamic nature of this meniscus, which is constantly pushed by the moving drop (figures 1c and 2c). A wedge of oil completely wets a substrate made of the same liquid. Hence its dynamic contact angle θ is given by Tanner's law [START_REF] Huh | Hydrodynamic model of steady movement of a solid/liquid/fluid contact line[END_REF][START_REF] Snoeijer | Moving contact lines: scales, regimes, and dynamical transitions[END_REF][START_REF] Tanner | The spreading of silicone oil drops on horizontal surface[END_REF], θ ≈ (βη o V/γ o ) 1/3 , where β is a numerical factor that reflects the singular dissipation at the wedge tip. Integrating the viscous stress in this region provides β ≈ ln(/ε) where ε is the typical thickness of the layer of wetting oil on which the wedge glides. It is here given by the size of the Glaco beads (about 30 nm) so that we expect β to be of order 10. As shown by Huh and Scriven [START_REF] Huh | Hydrodynamic model of steady movement of a solid/liquid/fluid contact line[END_REF], the wedge friction (per unit length) scales as

βη o V/θ, which yields a force F o ~ β(η o V/θ)R
F o ~ γ o R(βη o V/γ o ) 2/3
, which is non-linear in velocity. Its balance with gravity provides a speed of descent varying as sin 3/2 α, in agreement with figure 1e (coloured, dashed lines). Our data also show that the speed in this regime is sensitive to the texture density φ. This is compatible with our interpretation: the wedge friction mainly takes place over the pillar tops, since oil glides on itself between the pillars. The distance to be considered for integrating the local stress βη o V/θ becomes 2πφR instead of 2πR, which eventually leads to F o ~ γ o φR (βη o V/γ o ) 2/3 . Hence we get:

(3) V ~ ρg ( ) 3/2 R 3 γ o 1/2 φ 3/2 βη o sin 3/2 α
We did not consider up to now the friction in the trailing side of the oil meniscus sketched in figure 2c, at the oil/water interface. This friction scales as (η o V/ε) λR, denoting ε and λ the thickness and length scale of the meniscus tail. Assuming a Laudau-Levich scaling,

ε ~ R(η o V/γ ow ) 2/3 and λ ~ R(η o V/γ ow ) 2/3
, the friction force F o keeps the same scaling as previously (F o ~ sin 2/3 α), which does not modify our conclusions [START_REF] Bico | Falling slugs[END_REF][START_REF] Reyssat | Drops and bubbles in wedges[END_REF][START_REF] Cantat | Liquid meniscus friction on a wet plate: bubbles, lamellae and foams[END_REF]. More quantitatively, the absence of singularity (thus of factor β) should make this additional friction smaller than the wedge one.

Eq. ( 3) nicely captures the experimental observations. (i) A meniscus friction provides a power law with exponent 3/2 between velocity and driving force, in accord with figure 1e.

(ii) As also observed, the denser the texture, the smaller the speed: the positions of the lines in figure 1e were chosen as to obey the φ 3/2 -dependency predicted by eq. ( 3). The fit is found to be convincing even if the model at the highest density slightly underestimates observed speeds. (iii) We understand why V is inversely proportional to the oil viscosity η o (the more viscous the wedge of oil, the slower the drop), as reported in figure 2a. (iv) We can finally check the unusual dependency of the speed with drop size, in R 3 instead of R 2 for Stokes' law.

In figure 2d, the speed in this regime is plotted as a function of sinα for drop volumes Ω ~ R 3 varying between 1 µL and 40 µL. As expected from eq. ( 3), the larger the drop, the quicker it is. The agreement is quantitative: fits in the figure (dotted lines) are calculated by taking coefficients proportional to the drop volume (eq. 3), which leads to a very satisfactory agreement with all the data. Depending on the contrast in viscosity between oil and water, we discussed two mechanisms of dissipation, and our curves in figures 1d and 1e suggest that drops on infused materials follow either of these laws. This is a consequence of the high viscosity contrasts in figure 1,

Soft Matter Accepted Manuscript

where the ratio η o /η w is either 1/50 or 50. Conversely, figure 2a shows that the transition between both frictions can be directly evidenced by varying the water viscosity at fixed η o .

The transition takes place at a critical water viscosity η * , found in figure 2a to increase with η o . By matching eqs. ( 2) and (3), we obtain η * ~ aη o , where the number a = γ o 1/2 φ 3/2 /(ρg) 3/2 R sin 1/2 α, is expected in our systems to be of order unity. η * scales as the oil viscosity η o , as observed in figure 2a.

Another way to discuss the transition between the two regimes of dissipation consists of performing experiments with liquids having similar viscosities. In figure 2e, we plot the descent velocity V as a function of the tilt angle α for η o = 100 mPa.s and η w = 500 mPa.s.

Contrasting with the plots in figure 1, V now successively follows dependencies in sin 3/2 α and in sinα, in agreement with our discussion: the two frictions coexist, but oil friction should dominate water friction at low speed. Matching eqs. ( 2) and ( 3) now provides a transition speed V* ~ φγ o η o 2 /η w 3 , on the order of a few millimetres per second for the parameters of this experiment -in excellent agreement with the data, and increasing with φ, as observed in figure 2d. Conversely, in the limits η o << η w (or η o >> η w ), V* rapidly tends to zero (or infinity), so that the whole set of data corresponds to the water-dominated (or oil-dominated) regime, as in figure 1.

When oil imposes the friction law, we also observe in figure 1e a critical descent velocity above which V is larger than predicted by eq. ( 3) -a regime illustrated by the supplementary movie 4. Friction in this fast regime is different, and smaller. The transition speed in figure 1e is around 2 cm/s, which results in a capillary number (in oil) η o V/γ o of order 0.1. At such a capillary number, Tanner's law predicts that the advancing dynamic angle is around 90° [START_REF] Hoffman | A study of the advancing interface. I. Interface shape in liquid-gas systems[END_REF],

which implies that the Huh-Scriven assumption of an acute wedge is not valid anymore, and explains why wedge dissipation can be minimized. As a tentative model for this quick regime, we assume that the oil meniscus becomes fully dynamical: it is constantly extracted from the texture by water surface tension γ w before being reinjected at a velocity of order V below the drop. This interpretation explains how the velocity can become independent of the texture density φ: water being lubricated by a thick dynamic film of oil, the texture gets erased. Oil is drawn at a velocity scaling as γ w /η o that must exceed the drop velocity, that is, η o V/γ w < 1. This condition is satisfied at capillary numbers η o V/γ o < 0.1. The dynamic meniscus, of unknown size r, deposits a film with thickness ε, for which we assume a Landau-Levich scaling, ε ∼ r (η o V/γ ow ) 2/3 . While the front edge of the meniscus is sharp and minimizes dissipation, the viscous force present in the deposition region scales as (η o V/ε)rR ∼ γ ow R(η o V/γ ow ) 1/3 , which indeed yields a descent velocity varying as sin 3 α (figure 1e). This regime intersects the previous one when frictions are comparable, that is, at a velocity scaling as γ ow 2 /φγ o η o that depends on φ, as observed in figure 1e.

Soft Matter Accepted Manuscript

We finally test the robustness and generality of our findings by considering similar experiments on samples with smooth pillars instead of rough pillars. This induces a significant adhesion and the tilt angle α*(φ) for drop departure becomes for instance 15° for φ = 23% (and Ω = 20 µL). This angle expresses the balance of gravity with pinning, so that our equations must be corrected by considering an effective slope (sinα -sinα*) instead of sinα. This simple fact limits the range of explored driving forces, and makes it more difficult to establish firm scaling laws. Yet, as seen in figure 3, our main conclusions remain valid. For viscous water (η w > η o ), the speed is linear with the driving force (figure 3a), as expected in a regime where dissipation occurs in the drop. In the opposite case (η o > η w ), dissipation mainly occurs in oil, and we recover the power law in 3/2 (with scattered data, due to adhesion), the dependency in texture density φ, and the quick regime at large α (figure 3b). As highlighted in figure 3c, it is worth noticing that the friction in the latter regime becomes independent of the surface adhesion expressed by α * : data for α * = 0 and for α * = 15° superimpose at large tilt in this self-lubricating regime, which confirms that surface topology has no more any impact on the dissipation at large α. The figure finally allows us to directly compare the speed of the drops in the main regime of oil dissipation (sinα < 0.7): water moves much slower (by a factor that can be typically 10) when adhesion is present, a consequence of the reduction of driving force in this case. This shows that textured, infused materials are not always ultraslippery per se, but need to be designed so that the water/solid interface is minimized, as done here with the sub-nanotexture.

Beyond the asymptotic scalings described so far, we can finally try to capture all our results by simply adding the two independent sources of friction at intermediate tilt angles (sinα < 0.7). Drop velocity is deduced from the balance of the driving force with the total friction force written as bF o + cF w , where b and c are two numbers. This calculated velocity is plotted in figure 3d for the data obtained with nude pillars, and found to be nicely described by our equations provided we choose b = 13 ± 1 and c = 11 ± 1. Even if these numbers remain to be explicitly calculated, we can emphasize here that the model quantitatively succeeds at describing all the experiments with a unique set of numerical parameters b and c. Textured-infused materials are often qualified as "slippery", a term that both refers to the low adhesion of drops on such surfaces, and to the potential slip generated during the motion. Yet, we showed that dissipation for a millimetre-size drop remains classical (Stokes-like) in the limit where slip might import (η o < η w ). The characteristic size of the drop is typically one hundred times larger than the lubricant thickness, which limits the influence of slip. In a Hele-Shaw cell, for instance, the size of the drop can be reduced to that of the texture, which should magnify slip and trigger new friction laws. In the opposite limit (η o > η w ), dissipation mainly occurs in oil, yet not in the subjacent film (as we could think a priori), and we interpreted the observed non-linear friction by viscous effects in the front edge of the meniscus surrounding the drop. As the driving force is increased, the wedge dissipation is suddenly suppressed, which leads to a third dynamical regime that seems to arise from the self-lubrication of the drop. In all these regimes, the lubricant was found to induce specific friction laws, mostly related to the existence of a circular meniscus around the drop. This should be general, in particular at high speeds, and our study might help to understand the behaviour of drops at impact [START_REF] Kim | Droplet impact dynamics on lubricant-infused superhydrophobic surfaces: The role of viscosity ratio[END_REF] or the way a liquid film dewets on lubricant-infused materials. We would finally be happy to extend these findings to the dynamics of solid objects when they meet such surfaces, a situation for which slip was observed to be spectacular as seen when ants get trapped by infused plant surfaces [START_REF] Bohn | Insect aquaplaning: Nepenthes pitcher plants capture prey with the peristome, a fully wettable water-lubricated anisotropic surface[END_REF][START_REF] Bauer | Harmless nectar source or deadly trap: Nepenthes pitchers are activated by rain, condensation and nectar[END_REF].

Soft Matter Accepted Manuscript

Figure 1 .

 1 Figure 1. (a) Experimental set-up: a water drop with volume Ω runs down an oil-infused material tilted by an angle α. Its stationary speed is denoted as V. (b) Drop (η w = 3 mPa.s, Ω = 20 µL) going down a silicone oilinfused surface tilted by α = 20°. The pillar density at the surface is φ = 23%. The scale bar shows 2 mm. (c) Oil front meniscus pushed by a water drop (η w = 1 mPa.s, Ω = 20 µL) running down at V = 3 mm/s a textured material (φ = 23%) infused by a silicone oil (η o = 10 mPa.s) and tilted by α = 2°. The scale bar shows 0.4 mm. (d) Drop speed V as a function of the driving parameter sinα, for η o = 10 mPa.s, η w = 500 mPa.s (η w > η o ) and Ω = 20 µL. The dashed line has a slope 1.The two series of data correspond to pillar densities φ = 23% (red data) or φ = 67% (purple data). (e) In the opposite limit η o > η w (here η w = 2 mPa.s and η o = 100 mPa.s), the drop speed becomes non-linear in driving force. Lines successively show slopes 3/2 and 3, and colors indicate the texture density (φ = 23%, red data; φ = 36%, blue data; φ = 52%, green data; φ = 67%, purple data). We indicate on the data at the largest slope the typical error bar on the velocity measurement. Colored dashed lines are positioned with the φ 3/2 -behavior predicted by eq. (3).

  Published on 08 September 2017. Downloaded by University of Cambridge on 15/09/2017 15:34:56. View Article Online DOI: 10.1039/C7SM01226H

Figure 2 .

 2 Figure 2. (a)Speed V of a descending drop (Ω = 20 µL) as a function of its viscosity η w , at fixed oil viscosity (η o = 10 mPa.s, red data; η o = 100 mPa.s, blue data), on a substrate with φ = 23% tilted by α = 5°. Two regimes are successively obeyed: for η w < η o , V is independent of η w ; for η w > η o , V becomes inversely proportional to η w (inclined lines have a slope -1). Both regimes intersect at a critical water viscosity η* ≈ 3η o . (b) Sketch of a viscous drop going down a lubricant-impregnated surface, for which the Poiseuille flow within the drop limits the speed. We denote as V i the velocity of the water/oil interface. (c) The front of the moving meniscus is dynamically deformed, with an angle θ(V). (d) Speed V of descending drops with viscosity η w = 2 mPa.s and Ω = 1 µL (purple data), Ω = 5 µL (green data), Ω = 20 µL (blue data), Ω = 30 µL (orange data), Ω = 40 µL (red data). The surface (φ = 23%) is impregnated by oil with viscosity η o = 100 mPa.s. (e) Speed V of a descending drop (η w = 500 mPa.s) on a surface impregnated with oil of comparable viscosity (η o = 100 mPa.s). Two regimes are successively observed, with power laws of respective exponent 3/2 and 1. Both regimes intersect at a critical speed V * ≈ 2 mm/s.

10 V

 10 (mm/s) a. η o = 10 mPa.s η o = 100 mPa.s η w (mPa.s)

Figure 3 .

 3 Figure 3. (a)Drop speed V as a function of the driving parameter sinαsinα * , for η o = 100 mPa.s and η w = 500 mPa.s (η w > η o ). The volume of the drop is Ω = 20 µL. The sketch in the inset highlights the existence of a direct contact between water and textures in these experiments. Pillar densities are φ = 23% (red), φ = 36% (blue), φ = 52% (green) and φ = 67% (purple). Even if α* depends on φ, all data collapse on a single curve with slope 1 (dashed line). (b) In the opposite limit η o > η w (here η w = 2 mPa.s and η o = 100 mPa.s; Ω = 20 µL), the drop mobility is not anymore linear in driving force. Dashed lines successively show the slopes 3/2 and 3, and colours correspond to φ = 23% (red data), φ = 36% (blue data), φ = 52% (green data) and φ = 67% (purple data). (c) Speed V as a function of sinα for substrates with (blue) and without (red) sub-structures. Data collapse at high tilting angles, showing that pillars no longer play a role in this regime. (d) Speed V as a function of the drop viscosity η w for various lubricant viscosities η o , φ = 23%, α = 40° and Ω = 20 µL. Dashed lines show the speed expected when assuming a total friction F = 13F o + 11F w , where F w and F o are given by eqs (2) and (3).

  10 mPa.s η o = 50 mPa.s η o = 100 mPa.s η o = 500 mPa.s η o = 1000 mPa.s V (mm/s) η w (mPa.s)

  Published on 08 September 2017. Downloaded by University of Cambridge on 15/09/2017 15:34:56.

	View Article Online
	DOI: 10.1039/C7SM01226H

  Published on 08 September 2017. Downloaded by University of Cambridge on 15/09/2017 15:34:56.

	View Article Online
	DOI: 10.1039/C7SM01226H

  Published on 08 September 2017. Downloaded by University of Cambridge on 15/09/2017 15:34:56.

	View Article Online
	DOI: 10.1039/C7SM01226H

  once it is integrated over the drop perimeter.

	Plugging Tanner's law in the latter formula gives:	View Article Online DOI: 10.1039/C7SM01226H
	Published on 08 September 2017. Downloaded by University of Cambridge on 15/09/2017 15:34:56.	Soft Matter Accepted Manuscript

  Published on 08 September 2017. Downloaded by University of Cambridge on 15/09/2017 15:34:56.

	View Article Online
	DOI: 10.1039/C7SM01226H

  Published on 08 September 2017. Downloaded by University of Cambridge on 15/09/2017 15:34:56.

	View Article Online
	DOI: 10.1039/C7SM01226H

  Published on 08 September 2017. Downloaded by University of Cambridge on 15/09/2017 15:34:56.

	View Article Online
	DOI: 10.1039/C7SM01226H

Acknowledgements

This work has received the support of Institut Pierre-Gilles de Gennes (équipement d'excellence, "Investissements d'avenir", programme ANR-10-EQPX-34).

Soft Matter Accepted Manuscript


Hong Zhang

Xueguang Shao

François Dehez

Wensheng Cai
email: wscai@nankai.edu.cn

Christophe Chipot
email: chipot@ks.uiuc.edu

Modulation of Membrane Permeability by Carbon Dioxide

Introduction

Although many new drugs have been designed and synthesized over the past decades, 1 not all drugs are utilized effectively. 2,3 There are many factors affect the drug bioavailability, for example, its solubility, permeability, absorption and cytotoxicity. 4,5 Poor bioavailability of drugs can increase the medical cost, on the one hand, and decrease the recovery rate of patients, on the other hand. 3 Increasing the drug bioavailability is still one of the main challenges in medicine field. 6 Recently, various approaches have been studied to improve the pharmacokinetic and physicochemical properties of drugs to increase their bioavailability. 6,7 One strategy relies on the modification of drug molecules to improve their solubility and effects, 8 which costs a lot of time before clinical application. One approach is to use drug carriers, for example, enteric coating, one of the traditional drug delivery approaches applied to oral drugs. 9,10 Recently, many new-type drug carriers have been designed to target the tumor cells. 11,12 Another strategy is to improve the penetration of drugs through membranes.

ABSTRACT

Promoting drug delivery across the biological membrane is a common strategy to improve bioavailability. Inspired by the observation that alcoholic carbonated beverages can increase the absorption rate of ethanol, we speculate that carbon dioxide (CO 2 ) molecules could also enhance membrane permeability to drugs. In the present work, we have investigated the effect of CO 2 on the permeability of a model membrane formed by 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC) lipids to three drug-like molecules, namely, ethanol, 2',3'-dideoxyadenosine (DDA), and trimethoprim. The free-energy and fractional-diffusivity profiles underlying membrane translocation were obtained from µs-timescale simulations and combined in the framework of the fractional solubility-diffusion model. We find that addition of CO 2 in the lipid environment results in an increase of the membrane permeability to the three substrates. Further analysis of the permeation events reveals that CO 2 expands and loosens the membrane, which, in turn, facilitates permeation of the drug-like molecules.

The permeability is a significant property to describe the absorption rate of drug candidates inside the body. Increasing the permeability of drug is one important strategy to enhance its bioavailability. 13 Recently, many approaches were reported to improve the membrane permeability. V. Khutoryanskiy and coworkers proposed that crown ethers can increase the permeability of ocular drugs. 14 Perkins et al. put forward that phenylalanine can enhance the membrane permeability. 15 Among these approaches, the cytotoxicity of the additives needs to be considered. Therefore, seeking a strategy with a low cytotoxicity appears to be very necessary.

Robinson and Roberts studied the effect of the mixture of ethanol and carbonated beverages on the blood ethanol level. 16 As mentioned about one hundred years ago, 17 their results indicate that the mixture of ethanol and carbonated drinks can increase the absorption rate of ethanol. The same conclusion was reached by Ridout et al. who proved that champagne is more inebriant than wine. 18 These observations prompt us to speculate that, at the microscopic level, CO 2 might affect the overall permeability of membrane cells, possibly increasing the absorption rate of any small molecule. In the present work, we disentangled the relationship between CO 2 and the membrane permeability by means of all-atom molecular-dynamics simulations. The permeation of three drug-like molecules, namely, ethanol, 2',3'-dideoxyadenosine (DDA) and trimethoprim, through a 1-palmitoyl-2-oleoyl-sn-glycero-3phosphocholine (POPC) bilayer saturated with CO 2 was studied. To estimate accurately the membrane permeability to drugs, the solubility-diffusion model constitutes a powerful tool. [19][20][21] The free-energy calculations and the permeability estimations based on the fractional solubility-diffusion model were used to investigate the thermodynamics and kinetics properties of membrane permeation of these three permeants.

Methods

Molecular Models. The three-dimensional structures of the three small molecules, namely, ethanol, DDA, and trimethoprim (see Figure 1) were built. To compare with permeation in absence of CO 2 , we used the same number of POPC molecules as in our previous works, namely, bilayers of 100, 128, and 128 POPC units for the ethanol, 20 DDA, 19 and trimethoprim assays, respectively. CHARMM36 force field parameters were utilized for POPC 22 whereas the three drugs were decribed by the CHARMM general force field (CGenFF). [23][24][25] The normal to the membrane is aligned with the z-axis. 180 CO 2 molecules were added randomly into these assays. The molecular assays were immersed into a box of ~59 × 59 -67 × 67 Å 2 in the x, y-plane, extending up to 80 -120 Å in the z direction, including 6,154 -9,034 TIP3P 26 water molecules. Simulation Protocol. All the molecular dynamics simulations were carried out by means of the program NAMD (version 2.13). 27 The Langevin piston method 28 was employed to maintain the system pressure at 1 atm. Langevin dynamics was utilized to maintain the temperature at 308 K with 1 ps -1 damping coefficient. Covalent chemical bonds involving hydrogen atoms were constrained to their experimental lengths for solute and water molecules, using the Rattle 29 and SETTLE algorithms, 30 respectively. The integration of motion equations was carried out using a r-RESPA multiple time-stepping algorithm with a time step of 4 and 2 fs for long-and short-range interactions, respectively. 31 Long-range electrostatic forces were evaluated by means of the particle mesh Ewald (PME) scheme. 32 A smoothed 9 Å spherical cutoff was used to truncate short-range electrostatic and van der Waals interactions. The VMD program (1.9.3) was applied to analyze and visualize the MD trajectories. 33 Free-Energy Calculation. Before running the free-energy calculations, a 200-ns equilibrium simulation was carried out to equilibrate the CO 2 molecules. The potentials of mean force (PMF) characterizing the translocation of the small molecules across the membrane were then determined by the adaptive biasing force (ABF) algorithm. [34][35][36] The PMF profile describing the translocation of DDA without CO 2 was obtained from our previous study, 19 no repetitive calculations were performed in the present work. To do a fair comparison with the previous results, similar protocols were used. The transition coordinate was defined as the projection of the distance between the center of mass of the permeants and that of the phosphorus atoms of POPC molecules onto the z-axis of Cartesian space. The instantaneous force values were accrued in bins 0.1 Å. The free-energy profiles spanning from -45 to 45 Å were broken down into nine windows, 10 -15 Å wide with a 5 Å overlap. The free-energy profiles were eventually symmetrized with respect to center of the membrane. The sampling time for each simulation is gathered in Table S1 of the Supporting Information. Kinetic Modeling. A variant of the Bayesian-inference scheme developed for classical diffusion 37 was used to determine the position-dependent fractional diffusivity. The details of this inference algorithm, implemented in the program DiffusionFusion, 38 were described in our previous work. [19][20][21] The fractional order, α(z), and the fractional diffusivity, K α (z), were optimized. Two distinct lag times, namely, Δt = 4 ps and Δt = 16 ps, were employed. The long-tailed distributions 37 of Monte Carlo moves were presented with a value of 0.005 for α(z) and 3.5 Å 2 /ns α for K α (z). ε = 0.02 Å -1 and 20 Å/ns were used for the smoothness priors for α(z) and K α (z), respectively. The membrane permeability was calculated by integrating over the thickness of the lipid bilayer in the framework of the fractional solubility-diffusion model of permeation. 20,37

Results and discussion

Free-energy Profiles of Small Molecules Across the Membrane

The free-energy profiles characterizing the translocation of three drug-like molecules across the model membrane are gathered in Figure 2. The smallest molecule among these three permeants, ethanol, yields a free-energy barrier around 2.9 kcal/mol in the pure POPC lipid bilayer while in the mixed POPC:CO 2 system, the free-energy barrier decreases to 2.2 kcal/mol. The hydrophilic DDA exhibits a much higher free-energy barrier compared to ethanol, which amounts up to ~7.5 kcal/mol at the center of the pure POPC bilayer, but reduces to 7.1 kcal/mol upon addition of CO 2 . As depicted in Figure 2C, the shape of free-energy profile of trimethoprim is similar to that of DDA, presenting a free-energy barrier of 7.8 kcal/mol, in agreement with the result obtained by Sun et al. 39 The free-energy barrier decreases by 0.7 kcal/mol upon addition of CO 2 molecules. Altogether, our results clearly indicate that the incorporation of CO 2 molecules into the POPC lipid bilayer reduces to a certain extent the free-energy barrier associated with permeation of small drugs across biological membranes.

Fractional Diffusivity and Permeability of the Three Permeants Across the Membrane

In order to estimate the membrane permeability to these drug-like molecules, the fractional solubilitydiffusion model of permeation was used. The fractional diffusivity, K α (z), is shown in Figure 3. Though part of CO 2 molecules spread out in the aqueous phase, K α (z) does not significantly change for these drugs in water phase. Interestingly enough, a slightly increase of the fractional diffusivity for the three drugs is observed in the membrane, meaning that the addition of CO 2 enhances the diffusivity of drugs in the membrane. It is worth noting that K α (z) of ethanol in aqueous phase calculated herein, 240 Å 2 /ns, differs from that determined in our previous work, 340 Å 2 /ns. 20 It is solely a consequence of the different thermostats employed in both simulations. While Langevin dynamics was used in the present study, our previous work resorted to the Lowe-Anderson thermostat. 20 The friction term in Langevin dynamics can decrease the value of the position-dependent fractional diffusivity. 20 However, the value obtained from Langevin dynamics is much closer to the experimental one, 122 -126 Å 2 /ns. 40,41 The membrane permeability to the permeant was estimated from a direct integration in the framework of the fractional solubility-diffusion model of permeation. As depicted in Table 1, it is worth noting that the magnitude of experimental permeabilities differs from that inferred with simulations. This discrepancy is rooted in the difference in molecular composition of the model and the biological membranes and/or the systematic error associated with the force field. For trimethoprim, the simulation estimate is close to the permeability determined by Sun et al. 39 It appears clearly that the membrane permeabilities of POPC to the three small permeants increase upon the addition of CO 2 molecules, which may be ascribed to a modification of the membrane properties. The effect of CO 2 on the membrane is discussed in the following section. In addition, the permeability of ethanol, DDA, and trimethoprim in POPC:CO 2 mixture is around 4, 2, and 1.2 times higher than that in pure POPC, respectively. It appears that CO 2 primarily affects membrane permeabilities of small size molecules. 42 [b] Experimental values obtained from Ref. 43 [c] Theoretical estimation obtained from Chipot and coworkers. 19 [d] Experimental values obtained from Ref. 44,45 The Effect of CO 2 Molecules on the Membrane Properties.

The average number of CO 2 per lipid unit is reported in Table S2 in the Supporting Information. There is ~0.8 CO 2 molecule per lipid unit for all systems. In order to examine the distribution of CO 2 molecules in the membrane, the density profiles of CO 2 along the membrane axis was plotted (see Figure 4A). Most of the CO 2 molecules permeate into the membrane on account of their hydrophobic nature. In addition, distribution of CO 2 molecules culminates in the middle of the two membrane leaflets. We calculated further the average number of CO 2 and water molecules within 2.5 Å of each atom of the three permeants along the z-axis of the mixed POPC:CO 2 system. As shown in Figure 3B, the number of CO 2 around the trimethoprim in the membrane is higher than that around DDA and ethanol. It can be rationalized by the higher hydrophobicity and the larger size of trimethoprim. As described in Figure 3C, the differences in the number of water molecules hydrating the three drugs in bulk water, can be assigned to the size of the molecules. How do the CO 2 molecules affect the membrane properties? To answer this question, a series of analyses were performed. The phosphate-to-phosphate distance, characterizing the variation of thickness between mixed POPC:CO 2 and pure POPC membranes, was systematically determined (Figure 5A). The membrane thickness is found to increase by 1 Å upon addition of CO2, compared with the pure POPC membrane. We also analyzed the changes of the fraction mass overlap between the two leaflets (see Figure 5B), showing that insertion of CO 2 molecules decreases the leaflet interdigitation, consistently with the increase of the membrane thickness. These results support that addition of CO 2 can expand and loosen the membrane. As described in Figure 5C, the average area per lipid of pure POPC is ~68 Å 2 , in agreement with the experimental value of 68.3 ± 1.5 Å 2 . 46 The addition of CO 2 molecules increase this value up to ~71 Å 2 , a situation that is consistent with an accrued membrane permeability. 47

Conclusions

In the present contribution, we investigated the effect of the CO 2 on the membrane permeability to drugs from both aspects of thermodynamics and kinetics. The permeabilities of three drug-like molecules, namely, ethanol, 2',3'-dideoxyadenosine (DDA), and trimethoprim, were examined in pure POPC and in mixed POPC:CO 2 bilayers. The free-energy profiles obtained from µs-timescale simulations indicate that the incorporation of CO 2 molecules into bilayers decreases the free-energy barrier associated with the membrane translocation of drugs. In addition, the permeabilities of three drug-like molecules increase when adding CO 2 . Further analyses indicate that CO 2 molecules loosen the membrane and improve its fluidity, increasing in return its permeability to small molecules. The data reported in the present work provide an appealing view about the modulation of drug permeability by CO 2 . The present theoretical framework opens the way toward understanding the effect of gas molecules on membrane properties.

Figure 1 .

 1 Figure 1. Structures of the three permeants studied in this work.

Figure 2 .

 2 Figure 2. Free-energy profiles describing the membrane translocation of (A) ethanol, (B) DDA, and (C) trimethoprim across a pure POPC (black line) or a mixed POPC:CO 2 (red line) bilayer.

Figure 3 .

 3 Figure 3. The fractional diffusivity for (A) ethanol, (B) DDA, and (C) trimethoprim in pure POPC and a mixed POPC:CO 2 bilayer.

Figure 4 .

 4 Figure 4. (A) Density profiles of POPC phosphorus and CO 2 along the axis normal to the membrane plane. Number of (B) CO 2 and (C) water molecules within 2.5 Å of the three permeants along the z-axis in the mixed POPC:CO 2 bilayer.

Figure 5 .

 5 Figure 5. Evolution of (A) the phosphate-to-phosphate distance (B), the fraction mass overlap between the two leaflets, and (C) the average area per lipid during the 200-ns simulation.Membrane fluidity is another important property affecting the membrane permeability to drugs, which can be estimated by the order parameters, S CD . As shown in Figure5, the addition of CO 2 molecules decreases the S CD values of the POPC tails, which, in turn, increases the membrane fluidity. In conclusion, CO 2 molecules loosen the membrane, improving the membrane permeability and fluidity.

Figure 6 .

 6 Figure 6. Average S CD order parameters of POPC (A) sn-1 and (B) sn-2 tails during the 200-ns simulation.

Table 1 .

 1 Permeabilities of Each Permeant in Pure POPC and mixed POPC:CO2 bilayers.

	Permeant	Bilayer	Pm (cm s -1 )

(Additional Supporting Information may be found in the online version of this article.)

Acknowledgments

This study was supported by the National Natural Science Foundation of China (21773125), and the Natural Science Foundation of Tianjin, China (18JCYBJC20500). The authors acknowledge support from the Centre National de la Recherche Scientifique through an integrated program of scientific cooperation (PICS) with China. H. Zhang gratefully acknowledges the financial support from China Scholarship Council (No. 201806200074).

Keywords: membrane permeability, CO 2 , drug