Russ Harmer

Jean Krivine

Élise Laruelle

Cédric Lhoussaine

Guillaume Madelaine

Mirabelle Nebut

Models of Tet-On System with Epigenetic Effects

We present the first results of ongoing work investigating two models of the artificial inducible promoter Tet-On that include epigenetic regulation. We consider chromatin states and 1D diffusion of transcription factors that reveal, respectively, stochastic noise and a memory effect.

Introduction

In gene regulatory systems, transcription factors (TF) usually require activation in order to perform their regulatory function. This generally results from the action of complex signalling pathways, so an investigation of the dynamics of TF activation is important for understanding the underlying gene regulation. Recently, an efficient experimental technique to monitor this dynamics has been proposed, using fluorescent proteins expressed under the control of an inducible promoter by the TF of interest. However, observed fluorescence is not linearly correlated to the TF activation: a delay is induced by fluorescent protein expression and subsequent folding; and fluorescence may persist even after TF deactivation. Huang et al [START_REF] Huang | Using the Tet-On system to develop a procedure for extracting transcription factor activation dynamics[END_REF] propose a method to reconstruct TF dynamics from observed fluorescent protein dynamics by means of a very simple two-level model of the Tet-On system, an artificial inducible promoter of Green Fluorescent Protein (GFP). The first level (rules (1) to (3) below) models the signal transduction pathway leading to TF activation: this involves the artificial TF rtTA, activated by binding with doxycycline Dox i . Extracellular doxycycline Dox e is assumed constant and can degrade in the cell. The second level (rules (4) to (9)) models protein synthesis and activation of fluorescence: this includes transcription, translation and GFP activation. Fig. 1 shows that the dynamics predicted by the deterministic model is in good agreement with experimental data in the case of continuous and pulse4 stimulation with doxycycline. In [START_REF] Huang | Using the Tet-On system to develop a procedure for extracting transcription factor activation dynamics[END_REF], the authors demonstrate that the dynamics of TF activation is accurately correlated with the dynamics offluorescence.

Dox e Deff ---→ Dox e + Dox i (1)
Dox i Deff ---→ ∅ (2) rtTA + Dox i k f 2 ---- k r2 rtTA • Dox (3) rtTA • Dox S m --→ rtTA • Dox + mRNA (4) mRNA Dm --→ ∅ (5) mRNA Sn --→ GFP + mRNA (6) GFP S f --→ GFP a (7) GFP Dn --→ ∅ (8) GFP a Dn --→ ∅ (9)
In this paper, we propose two simple extensions of the basic model proposed in [START_REF] Huang | Using the Tet-On system to develop a procedure for extracting transcription factor activation dynamics[END_REF] to investigate effects that may result from epigenetic regulation. The first extension deals with chromatin states, i.e. the (un)availability of the inducible promoter due to chromatin compaction. In the second, we model TF diffusion along DNA, i.e. the binding of a TF to non-specific binding sites, followed by its sliding to the operator site.

Chromatin states: a stochastic noise effect

The experimental data of [START_REF] Huang | Using the Tet-On system to develop a procedure for extracting transcription factor activation dynamics[END_REF] appear to be quite noisy, contrary to what is predicted by the original model (Fig. 2 left). Although it is hard to assess quantitatively, we can assume that some of this noise results from intrinsic stochasticity. A recognized likely source of stochasticity in gene regulation is the dynamic alteration of chromatin structure that makes it more or less accessible to the transcriptional machinery. Most of the time, the chromatin is tightly packaged or, roughly speaking, in a closed state. This implies that the TF cannot find the promoter to activate transcription; to allow rtTA • Dox to bind with the promoter, the chromatin must be in an open state. Depending on the needs of the cell, the chromatin can rapidly switch between these states.

We refine the model given in the introduction by replacing reaction (4) with reactions (10) to (13) below. In words, rtTA • Dox can bind its operator site (rule (10)) when the chromatin is locally open; then, either rtTA • Dox can dissociate from the operator (rule (10)), or transcription can begin (rule (13)). At any time, the chromatin can switch to a closed state (rule (11)); if the TF is bound to the operator when the chromatin closes, then it dissociates (rule (12)). We take values for k rop and k fop of the same order as those for TetR given in [START_REF] Biliouris | Stochastic simulations of the tetracycline operon[END_REF]. The values of k closed and k open are actually dependent on the promoter used, the position of the gene, the cell type, etc, but the values taken here are close to those of promoters found in mammalian cells [START_REF] Suter | Mammalian genes are transcribed with widely different bursting kinetics[END_REF]. Finally, the rate constant of transcription (rule ( 13)) has been chosen to fit the experimental data.

rtTA • Dox + Op open k fop ---- krop Op open • rtTA • Dox (10) Op closed kopen -------- k closed Op open (11) Op open • rtTA • Dox k closed ----→ Op closed + rtTA • Dox (12) Op open • rtTA • Dox k trans ---→ Op open • rtTA • Dox + mRNA (13)
The average of the stochastic simulations in Fig. 2 is close to the deterministic dynamics given by Huang et al. [START_REF] Huang | Using the Tet-On system to develop a procedure for extracting transcription factor activation dynamics[END_REF], as expected, but stochastic noise is far more pronounced than in the basic model. The influence of chromatin states could thus provide an explanation for the noise observed in the experimental data.

1D diffusion of TF: a memory effect

Another interesting aspect of gene regulation is related to how the TF finds its promoter since it is known that three-dimensional (3D) diffusion is insufficient to explain fast binding [START_REF] Riggs | The lac repressor-operator interaction. 3. kinetic studies[END_REF]. In [START_REF] Hammar | The lac repressor displays facilitated diffusion in living cells[END_REF], a diffusion-based model is proposed: once the TF has (3D-) diffused within the nucleus and bound randomly to a non-specific DNA site, it rapidly slides along the DNA (1D diffusion) in a small region. If there is an operator in this region, it has an approximately 50-50 chance to bind it. In the case of binding, transcription can begin; otherwise, the TF either unbinds completely, returning to a search by 3D diffusion, or it jumps to another non-specific binding site in a neighbouring area. We add this behaviour to the basic model with 14 new reactions (due to lack of space, we don't report them here). The transcription rate has again been chosen to fit the experimental data and the other parameters have been taken from [START_REF] Hammar | The lac repressor displays facilitated diffusion in living cells[END_REF] and [START_REF] Bonnet | Mécanismes de diffusion facilitée de l'enzyme de restriction EcoRV[END_REF]. The results of the deterministic simulations in Fig. 3 are interesting: there is no significant difference in the case of constant stimulation but, in the pulse case, fluorescence decays more slowly than in the basic model which can be interpreted as a sort of "memory effect".

Conclusion

We have developed two extensions of the Tet-On model of [START_REF] Huang | Using the Tet-On system to develop a procedure for extracting transcription factor activation dynamics[END_REF]. The first, adding chromatin states, increases stochastic noise but preserves the average behaviour in accordance with the experimental data. The second, adding a search for the operator by the TF via 1D diffusion, reveals a delayed decay in fluorescence after stimulation has ceased. We are currently investigating this effect in more detail, notably in the case where there are multiple operator sites in close proximity.

Fig. 1 .

 1 Fig. 1. Continuous (left) and pulse (right) stimulation with doxycycline.

Fig. 2 .

 2 Fig. 2. Comparison of the stochastic noise between the basic model (left) and the model with chromatin states (right).

Fig. 3 .

 3 Fig. 3. Comparison between Huang model and Sliding model, with a continuous (left) or pulse (right) stimulation of doxycycline.

this means that Dox e concentration is set to zero after some delay.


Geoffroy Kirstetter

Christophe Raufaste
email: christophe.raufaste@unice.fr

Franck Celestini
email: franck.celestini@unice.fr

Jet impact on a soap film

We experimentally investigate the impact of a liquid jet on a soap film. We observe that the jet never breaks the film and that two qualitatively different steady regimes may occur. The first one is a refraction-like behavior obtained at small incidence angles when the jet crosses the film and is deflected by the film-jet interaction. For larger incidence angles, the jet is absorbed by the film, giving rise to a new class of flow in which the jet undulates along the film with a characteristic wavelength. Besides its fundamental interest, this study presents a new way to guide a micro-metric flow of liquid in the inertial regime and to probe foam stability submitted to violent perturbations at the soap film scale.

INTRODUCTION

Control and manipulation of laminar jets are of paramount importance in the context of miniaturization and use of microfluidic systems. Systems such as inkjet [1][2][START_REF] Brown | Microfluid[END_REF], encapsulation for biological applications [4,5], fiber spinning [6] rely on the stability of the micro-jet or conversely on its destabilization through the control of the liquid jet atomization or drop-on-demand process. But if technologies such as electro-spray devices [1][2][START_REF] Brown | Microfluid[END_REF], focused surface vibrations [7,8] combined or not with flowfocusing techniques [6,9] can control the transition between jetting and dripping, no reliable technique is available to guide a micro-jet inside a medium as simple as air. Recently, rebound on a hydrophobic surface was found to deflect a jet [10], but this process is prevented in most cases by the spreading of the liquid on the substrate.

Furthermore, control of liquid foam stability is a prerequisite in numerous industrial applications like fire fighting, oil recovery, ore extraction, explosion safety and food or cosmetics processing [START_REF] Weaire | The Physics of Foams[END_REF][START_REF] Isabelle | Les mousses: Structure et dynamique[END_REF][START_REF] Aubert | [END_REF]. Liquid foams are made of gas bubbles separated by liquid soap films. Their stability under mechanical solicitations is a major issue: as liquid fraction and soap film thickness are directly related to the osmotic pressure inside the liquid films [14], all kind of mechanical effects which can balanced this pressure can dramatically alter the foam properties through soap film bursting and bubble coalescence. Violent mechanical perturbations, such as impacts, have recently raised some interest and uses for sound absorption or bomb explosion safety [15]. Solid particles [16] or liquid drops [17] impacting a soap film lose kinetics energy and exhibit a rich variety of behaviors amongst film crossing, bouncing, partial coalescence and formation of satellite droplets. To our knowledge, nothing is known about the soap film stability after the impact of a liquid jet. Conversely to the studies cited above, the soap film is probed by continuous mass and momentum inputs provided by the liquid jet.

We investigate the impact of a liquid jet on a soap film. By tuning the jet velocity and/or incident angle, In the first one, the jet crosses the film without breaking it and is deflected by the film-jet interaction. This feature can be used to guide and control the jet direction.

In the second one, a new class of flow is reported: the jet is merged with the film and undulates inside the latter with a characteristic wavelength. A transient state corresponding to a bouncing jet on the film is also observed. Both regimes are well described using the Weber number (W e = ρV 2 i R i /γ) quantifying the relative importance of inertia and capillarity. Simple models are finally successfully proposed to quantitatively describe both regimes.

MATERIALS AND METHODS

We experimentally study the impact of a laminar liquid jet onto a film of the same composition. We use two solutions: most of the experiments were performed with a soap solution obtained by adding 5% of commercial dishwashing liquid (Dreft, Procter & Gamble) to deionised water. To test the robustness of the results, some points were confirmed using a TTAB solution (3g/L). The experimental set-up has already been described in details in a former study [10]. A pressurized chamber is built to inject the liquid at a controlled constant flow rate through a sub-millimetre nozzle, a laminar jet forms at the exit of the latter. The incident jet is characterized by its incident angle θ i , velocity V i and radius R i . The jet velocity varies within the range of 1 to 4 m.s -1 , and several jet radii between R i = 80 and 270 µm have been used. The injector is placed just above an horizontal soap film maintained by a circular frame, of 10 cm in diameter. We note γ the surface tension (equals to 26.2±0.2 mN.m -1 for the Dreft solution, 38 ± 1 mN.m -1 for the TTAB solution) and ρ its density (equals to 10 3 kg.m -3 in both cases). Within our experimental parameters range, based on the jet characteristics, the Reynolds number is always significantly larger than unity.

RESULTS

Regardless its velocity, radius and incident angle, the jet never breaks the soap film. Several works have explored the stability of soap films under the impacts of particles [16] or liquid drops [17]. In all regimes explored, the films close after the crossing of the impacting projectiles. Pinch-off of the films while they are stretched is found to be the healing mechanism which ensures their continuity as a function of time [16]. In the present case of impacting jets, such mechanism is not involved in the film stability. The film/jet contact is never broken and the film does not need to close. Depending on the jet characteristics, two qualitatively different regimes are observed. By analogy with optics, we called the first one "refraction-like regime" : the jet crosses the film and is deflected. The second regime is called "absorption" : beyond a critical angle, the jet is trapped by the film and undulates along it. For a given set of input parameters, i.e. the values of V i and R i , the angle θ i for which the transition occurs is recorded. The impact diagram in the (W e , θ i ) space of the system is represented in Fig. 1. The data have been recorded for the three different jet radii considered in this study. We can see that using the Weber number, the influence of both R i and V i is well captured and that all points collapse onto the same master curve. This scaling therefore demonstrate that the transition between the two regimes is governed by the interplay between capillarity and inertia and that the dissipation inside the jet/film contact zone can be neglected.

Refraction regime

We first describe the refraction-like regime appearing at high W e numbers or small θ i values. As illustrated in the inset of Fig. 1 , the jet is refracted with an angle θ r . At high jet velocities and low incidences, almost no visual change of the jet and film geometries is observed : θ r is almost equal to θ i and the film is slightly deformed. As the velocity decreases the influence of the film induces measurable changes in the radius, angle and velocity of the refracted jet. The sinus of refracted angle is represented as a function of the incident one in Fig. 2 for a jet of radius equals to 140 µm and two different velocities (V i = 1.3 and V i = 2.3 m.s -1 ). We observe that the jet is deflected towards the film and that the lower the velocity, the higher the deflection. A linear regime can be furthermore identified. By analogy with optics and Snell-Descartes 's law of refraction, an index n can thus be defined as n = sin(θ r )/ sin(θ i ) to quantify the deflection. We plot in the inset of Fig. 2 the values of n obtained for three different radii (R i = 140, 200, and 270 µm) and velocities as a function of W e. Once again the Weber number is found to be the relevant parameter to rescale all n values on a same master curve. We therefore demonstrate that inertia and capillarity are the relevant effects and give an explanation to the counter-intuitive observation that the higher the velocity, the lower the changes. As emphasized by the increase of n as W e de- creases, capillary forces between the jet and the film influence their respective shape and geometry, while friction inside the jet/film interaction zone, which should be an increasing function of the velocity, does never contribute significantly to the interaction force. Actually, if dissipation does not play a direct role on the jet-film interaction, its presence is necessary to drift the contact line streamward and imposed a dynamical wetting condition as emphasized in [18]. The contact angle evolves from 90 • to a value close to 0 • . This is of paramount importance to account for a non-zero interaction force and the jet deflection observed at small W e.

A model is proposed to account for the jet-film interaction inside the refraction regime. Three equations are needed here to account the mass and momentum equations and are applied on an open system as the one depicted on Fig. 3. In what follows, D accounts for the jet flow rate and F R for the interaction force inside the refraction regime. R r and V r account for the radius and velocity of the refracted jet.

Assuming a plug flow inside both the incident and refracted jets, mass balance writes

D = πR 2 i V i = πR 2 r V r Momentum balance (x
and y projections) is expressed below by balancing the momentum rate changes (lefthand side) and the forces applied on the system (righthand side):

D(ρV r sin θ r -ρV i sin θ i ) = πγ(R r sin(θ r ) -R i sin(θ i )) D(ρV r cos θ r -ρV i cos θ i ) = πγ(R r cos(θ r ) -R i cos(θ i )) -F R
Forces in the momentum balance equations have two contributions: the interaction force F R assumed perpendicular to the soap film [18] and contact forces (both surface tension and pressure) at the system-jet boundaries (generalization of F 2 for any refracted angle as described in the absorption regime section).

The system of equations can be transformed to expressed the refracted angle θ r as a function of the impact parameters and of the interaction force only:

(W e -1) sin(θ r -θ i ) = F R πγR i sin(θ r ) (1)
Assuming the small inclination limit, we can simplify the system following:

sin θ i ∼ θ i sin θ r ∼ nθ i F R ∼ 4πγR i
The last expression assumed a total wetting condition [18] and that the jet radius at the jet-film contact is taken as R i . Experimentally, this radius is found between R i and R r , and further analysis would be needed to describe the exact contact radius. But as seen below, such refinement is not necessary to account for the effect and is of second importance.

Finally, the model leads to

n = W e -1 W e -5 (2)
This expression describes the experimental measurements rather satisfactory (Fig. 2).

Transition

Physically, we might expect the transition to occur for sin(θ r ) = 1. Given our model, this leads to sin(θ i ) = 1/n at the transition or

θ i = arcsin W e -5 W e -1 . (3)
Again, the agreement is rather satisfactory (Fig. 1) to describe the transition from the refraction regime to the absorption regime. None the less, the transition from the absorption regime to the refraction regime can not be described by the same formula, emphasizing an hysteresis behavior. This is mainly due to a new contact zone geometry inside the refraction regime, not accounted for in the model above. We could observe experimentally that this transition occurs for higher velocities/smaller angles, but is difficult to quantify and less reproducible given the fact that the film is strongly deformed, oscillates and is very sensitive to changes close to this transition.

Absorption regime

We now describe the second regime observed at small W e and large θ i . In that case, capillary forces are strong enough to compensate the normal component of the jet momentum and we therefore refer to this regime as an"absorption". As depicted in Fig. 1 the jet follows a wavy trajectory inside the film characterized by its wavelength λ. The undulation persists over several wavelengths before some relaxation processes merge the jet and the film together and dissipate the kinetics energy continuously provided by the jet. When beginning the experiment from a refraction situation and by increasing the incident angle (or decreasing the velocity), the system transits to the absorption regime. If the jet impacts the soap film with parameters corresponding to the absorption region, a transient stage characterized by a "reflection" on the film (i.e. a rebound of the jet on the film) is observed before the absorption occurs. This behavior will be discussed below. We represent in Fig. 4. the measured value of λ as a function of the velocity for three different jet radii. In these experiments the incident angle is fixed to 70 degrees. The values obtained for λ are averaged over two or three different undulations wavelengths (we have checked that the value of the wavelength does not depend on its distance to the impact point). One can clearly see that the higher the velocity, the higher the wavelength (for a fixed jet radius) and that the higher the radius, the higher the wavelength (for a fixed jet velocity).

To understand this behavior a model is derived to de- scribe the jet properties inside the absorption regime. The second law of Newton is applied to a given portion of fluid as it is depicted in Fig. 5. The length dℓ of this system S is chosen to be small enough compared to the other relevant lengths of the system. In that case, the system has a constant cross-section, which scales as πR 2 i , and that is slightly twisted with a radius of curvature R. As will be seen below, the net force exerted on the system is always oriented along the centripetal component of the acceleration, meaning that kinetics energy is constant for the system and consequently its velocity amplitude. By conservation of the flow rate, the radius of the jet is taken constant as well, equals to R i . Consequently the acceleration of the system writes a = V 2 i /R n, where R = ǫ|R| is the algebraic radius of curvature, which sign ǫ holds for the local convexity of the trajectory.

Assuming that gravity and all sources of dissipation can be neglected, the net force applied on the system has two contributions. First, the force F 1 applied by the film onto the system S. This force accounts for the deformation of the film due to the film-jet contact. From reasonable film-jet profile transition zone (see cross-section in Fig. 5), the film pulls normally onto the system with a capillary force ranging from 0 to 4γ per unit length depending on the geometrical orientation of the film-jet triple line. The average value 2γ per unit length is chosen as the order of magnitude. Without loss of generality, F 1 = 2γ f dℓǫ n, where f is a constant ranging between 0 and 2.

The second contribution comes from the contact forces applied by the rest of the jet onto the system. The two terms account for the surface tension and pressure forces respectively. For both of them, contributions are found at both the leading and trailing extremities of the system. Amplitude are the same: 2πR i γ -πR 2 i P = πR i γ, since the pressure P equals the capillary pressure γ/R i . Their tangential components compensate each other, but not the normal ones as soon as R is finite. This leads to F 2 = 2•πR i γdθ/2•ǫ n. The momentum balance equations leads to

ρdℓπR 2 i V 2 i /R = 2γ f dℓǫ + πR i γdθǫ
By using dℓ = |R|dθ, it directly leads to an expression of the radius of curvature:

R = ǫR i π(W e -1) 2 f (4)
The jet trajectory is therefore made by the repetition of arcs of a circle of constant radius, pointing alternatively upward and downward. The transition between two arcs happens when ( t, e y ) angle equals ±θ i . From geometrical considerations, the wavelength λ of the trajectory finally writes :

λ = 2π f R i (W e -1) cos(θ i ) ( 5
)
This expression is compared with experiments on Fig. 4. For a given incidence angle and different radii, the velocity dependency is tested and the best f value is chosen to interpolate as finely as possible every data set. The best value is found to be f = 0.80 and the agreement between the experimental data and the model is rather satisfactory. This agreement is confirmed in the inset of Fig. 4 where we plot the dimensionless wavelength λ/R i as a function of the Weber number and we clearly observe that all the data collapse onto the same master curve given by the model. It is worth mentioning that our parameters range holds for high Weber numbers, W e >> 1.

When W e ≈ 1, the jet destabilizes into drops before impacting the film due to the Rayleigh-Plateau instability [START_REF] De Gennes | Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves[END_REF]. It means that λ and R expressions can be simplified without loss of generality by replacing W e -1 ∼ W e. Physically, that means that the effect described above occurs for | F 1 | >> | F 2 |. The elastic counterpart of this "inertial-capillary" mechanism comes consequently from the jet-film interaction and not from the curvature of the jet itself as observed for instance for meandering rivulets [START_REF] Drenckhan | [END_REF]21].

DISCUSSION

As discussed earlier, the "refraction" is observed at high We and small incident angle. The "absorption" occurs for smaller We or by increasing the incident angle. If the jet is released and impacts the soap film with parameters corresponding to the absorption region, a transient stage characterized by a "reflection" on the film (i.e. bouncing of the jet on the film) is observed before absorption occurs. Conversely to the bouncing of drops [17], the "reflection" stage is not sustainable since the air layer trapped between the jet and the film drains until it becomes too thin to prevent the coalescence of the two liquid entities. One can notice that a jet rebound on a thin liquid sheet can be sustained in the case of non-newtonian liquid exhibiting shear-thinning [22] : the so-called "Kaye effect" arises while a thin layer of the liquid itself is locally sheared at the contact zone and lubricates the latter continuously.

Finally, one can observe that the transition between the two steady regimes occurs at a Weber number largely greater than one. This is surprising since one would expect a transition around unity for phenomena balancing inertia versus capillarity. This is the case, for instance, for the impact of solid objects on a soap film [16]. In our system the situation is different since the deformation length scale is not necessarily the same as the one of the impacting object R i . As it can be seen in the inset of Fig. 1, the film takes the shape of a catenoid, for which the radius of curvature is largely greater than R i . A quantitative study of this effect can be found in [18].

CONCLUSION

To summarize, we have demonstrated the existence of three different flow classes resulting from the jet-film interaction : a refraction, an absorption and a transient reflection regime. The Weber number is found to rationalize the different regimes. Models, based on momentum and mass balance equations, catch quantitatively the dependency of the different impact parameters, namely the jet radius, velocity and incident angle. Besides its fundamental interest, this study presents a new way to guide micro-metric flows at Weber and Reynolds numbers above unity, and to probe liquid foams stability submitted to violent perturbations at the soap film scale.

FIG. 1 .

 1 FIG. 1. (Color online) Impact diagram displaying the refraction and absorption regions. Each point corresponds to the onset of the transition from the refraction to the absorption regime obtained for jet radii Ri = 140, 200 and 270 µm and the Dreft solution, and for jet radii Ri = 140 µm and the TTAB solution (circles, diamonds, squares and stars respectively). The solid line represents the model detailed in the main text. In the representative pictures, white scale bars have a length of 5 mm..

FIG. 2 .

 2 FIG. 2. (Color online) sin(θr) versus sin(θi) for a radius of 140 µm and two incident velocities Vi = 1.3 and 2.3 m.s -1 (circles and diamonds respectively). Full lines are best linear fits used to calculate the refraction index n. Inset : n versus the Weber number for different velocities, three different jet radii and two solutions: Ri = 80, 120, and 140 µm for the Dreft solution (triangles, circles and diamonds respectively), Ri = 140 µm for the TTAB solution (stars). The solid line corresponds to the model described in the main text.

FIG. 3 .

 3 FIG. 3. (Color online) Side view representation of the refraction. Mass and momentum balance equations are performed on an open system enclosing the jet-film interaction zone.

  FIG. 4. (Color online) Wavelength λ versus incident velocity Vi for different radii Ri = 140 , 200 and 270 µm and the Dreft solution, and Ri = 140 µm for the TTAB solution (circles, diamonds, squares and stars respectively). The solid lines correspond to the model with f = 0.80. Inset: same set of data plotted under the dimensionless form.

FIG. 5 .

 5 FIG. 5. (Color online) Side view representation of the undulating jet (light blue) inside the film (dark blue). Two crosssections are displayed showing a reasonable shape of the filmjet contact. Momentum balance equation is performed on a closed system S. Insert: zoom over the system.

ACKNOWLEDGMENTS

X. Noblin, A. Chabanov and F. Graner are thanked for the thorough reading of the manuscript.


Christophe Raufaste

Geoffroy Kirstetter

Franck Celestini

Simon Cox

Deformation of a free interface pierced by a tilted cylinder

We investigate the interaction between an infinite cylinder and a free fluid-fluid interface governed only by its surface tension. We study the deformation of an initially flat interface when it is deformed by the presence of a cylindrical object, tilted at an arbitrary angle, that the interface "totally wets". Our simulations predict all significant quantities such as the interface shape, the position of the contact line, and the force exerted by the interface on the cylinder. These results are compared with an experimental study of the penetration of a soap film by a cylindrical liquid jet. This dynamic situation exhibits all the characteristics of a totally wetting interface. We show that whatever the inclination, the force is always perpendicular to the plane of the interface, and its amplitude diverges as the inclination angle increases. Such results should bring new insights in both fluid and solid mechanics, from animal locomotion to surface micro-processing.

INTRODUCTION

Free fluid-fluid interfaces produce fascinating patterns in nature. From the simple geometry of drops, bubbles and micelles [START_REF] De Gennes | Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves[END_REF], to the complex but regular arrangement of films inside a liquid foam [START_REF]Les mousses: Structure et dynamique[END_REF], the concept of interfacial energy allows us to rationalize the change of bulk properties at the boundaries between two immiscible fluids. In the case of liquid-gas interfaces, deformation of the interface by a solid object and the subsequent measurement of the force exterted by one on the other is still a widely-used technique to quantify the surface tension (for instance the Du Noüy Ring and Wilhelmy plate methods [START_REF] Adamson | Physical Chemistry of Surfaces[END_REF]). In many cases the object that pierces the interface is cylindrical, and the deformation of a flat liquid-gas interface due to the contact of a solid rod is a situation of interest in both solid and fluid mechanics.

The propulsion of insects that walk on water is possible because of the force between the interface and the rigid cylindrical legs [START_REF] Hu | The hydrodynamics of water strider locomotion[END_REF][START_REF] Hu | Meniscus-climbing insects[END_REF][START_REF] Su | Nature's Design of Hierarchical Superhydrophobic Surfaces of a Water Strider for Low Adhesion and Low-Energy Dissipation[END_REF][START_REF] Vella | Floating Objects with Finite Resistance to Bending[END_REF]. If the latter are flexible, then a new class of so-called "elasto-capillary" or "wet-hair" problems [START_REF] Bico | Elastocapillary coalescence in wet hair[END_REF] arise with major applications in nano-or microdevices. Thus hairy surfaces, such as carbon nanotubes and biological filaments, are examples of the rich morphologies that arise from individual hair deformation and collapse [START_REF] Neukirch | Piercing a liquid surface with an elastic rod: Buckling under capillary forces[END_REF][START_REF] Park | Bending of floating flexible legs[END_REF][START_REF] Blow | Superhydrophobicity on Hairy Surfaces[END_REF][START_REF] Tawfick | Structurally Programmed Capillary Folding of Carbon Nanotube Assemblies[END_REF] and of the patterns seen in assemblies of hairs [START_REF] Cohen | Kinks, rings, and rackets in filamentous structures[END_REF][START_REF] Chakrapani | Capillarity-driven assembly of two-dimensional cellular carbon nanotube foams[END_REF][START_REF] Chiodi | Piercing an interface with a brush: Collaborative stiffening[END_REF][START_REF] Duan | Directed Self-Assembly at the 10 nm Scale by Using Capillary Force-Induced Nanocohesion[END_REF] in the presence of a meniscus. Related problems are encountered in surface processing, for example in fiber coating and in describing the wetting and cleaning properties of textiles [START_REF] Lorenceau | Capturing drops with a thin fiber[END_REF][START_REF] Huang | Equilibrium of drops on inclined fibers[END_REF][START_REF] Mullins | Detachment of liquid droplets from fibres-Experimental and theoretical evaluation of detachment force due to interfacial tension effects[END_REF]; or more specifically in probing the wetting properties of nanoscale droplets by deforming them with a cylindrical AFM tip [START_REF] Mullins | Detachment of liquid droplets from fibres-Experimental and theoretical evaluation of detachment force due to interfacial tension effects[END_REF][START_REF] Connell | Investigating the interfacial properties of single-liquid nanodroplets by Atomic Force Microscopy[END_REF]. Finally, the mutual force between a soap film and a cylindrical liquid object, namely a micro-jet, was also found to be strong enough to cause the jet to deviate and to control its trajectory [START_REF] Kirstetter | Jet impact on a soap film[END_REF].

In general, the rise of a meniscus is fast [START_REF] Clanet | Onset of menisci[END_REF] compared to other relevant time-scales (solvent evaporation, leg motion). Consequently, the equilibrium shape of the gasliquid interface and the minimum-energy principle are usually used to account for observations.

Here, we use Surface Evolver [START_REF] Brakke | The Surface Evolver[END_REF] simulations to study the deformation of an initially flat interface when it is pierced with an infinite cylinder. We report the effect of the inclination angle on the meniscus shape, the contact line position and on the force generated by the interaction. In the following we consider the case of a "total wetting" condition, meaning that the contact angle between the interface and the cylinder is zero. The results are general as long as the interface is governed only by surface tension, meaning that they can be directly applied to interfaces between two immiscible liquids, boundaries between a liquid and a gas, and to most 2D dimensional lamellae, including soap films, Langmuir monolayers, lipid bilayers and biological membranes. The main control parameter is the inclination of the cylinder with respect to the plane of the unperturbed interface. The numerical study is compared with experiments in which a cylinder of liquid is brought into contact with a soap film; above a certain liquid velocity the results are in good agreement with the model, offering further insight into the dynamical interaction between a cylinder and a film. As explained below, the more the cylinder is inclined, the greater the deformation of the meniscus and the greater the force exerted by the meniscus on the cylinder. This force is always perpendicular to the interface and its amplitude increases rapidly as the inclination is increased, diverging as the angle approaches 90 • .

The manuscript is organized as follows: the numerical and experimental systems are first presented, then we describe and characterize the position of the contact line in the simulations, compare with the experiments, and show how to calculate the force on the cylinder from the position of the contact line. We determine the force as a function of the inclination from the simulation results, and discuss the success of different models in predicting this dependence.

MATERIALS AND METHODS

Simulations

We use the Surface Evolver [START_REF] Brakke | The Surface Evolver[END_REF] to calculate the shape of a single area-minimizing surface that is pierced by a rod at an inclination of θ i to the vertical. All units are dimensionless, without loss of generality. We first create a planar surface in the xy plane with surface tension 1.005 in a fixed square frame of side-length 5.0 and a cylindrical rod, with axis in the vertical (z) direction and radius R = 0.1, through its centre. To ensure that the rod is wetted, we include a virtual liquid film on the lower half of the rod with unit tension. The small difference in tension between the surface and this virtual film en-sures that the contact angle is slightly greater than zero, alleviating numerical problems.

The surface is meshed with many small triangular facets. We tested two levels of refinement, the first with about 12,000 triangles, and the second with an extra 2000 triangles in the region where the rod pierces the surface; the refinement also controls the number of points around the contact line, giving 46 and 81 points respectively. Both triangulations give equivalent results (data not shown), which validates the robustness of the approach, and we show the more refined results below.

One iteration consists of tilting the rod by 3 • in the xz plane and finding the new minimum area configuration of the surface (Fig. 1a). At each iteration, the location of the points along the contact line is recorded which, as we shall show below, is sufficient to calculate all quantities of interest.

Experiments

To counteract the effects of bouyancy (which dominate capillary forces at these length-scales), we consider a double interface formed by a soap film, with surface tension 2γ, rather than the single interface separating a liquid from air. Our conclusions remain valid for such an interface, but with half the value of the surface tension.

The force exerted on a static glass rod by a meniscus, with a contact angle equal to 90 • , is zero. To generate a contact angle of 0 • , we therefore use a laminar liquid jet directed through a soap film of the same composition, as described in [START_REF] Kirstetter | Jet impact on a soap film[END_REF][START_REF] Celestini | Water jet rebounds on hydrophobic surfaces: a first step to jet micro-fluidics[END_REF]. The flow in the jet plays a subtle role: the liquid flow within the soap film is slow, so the film adopts a quasistatic equilibrium shape, yet the boundary conditions are governed by the dynamics of the jet. As we shall demonstrate below, friction causes the contact line to move downstream, and in the limiting case imposes the required contact angle on the soap film.

The liquid is a soap solution obtained by adding 5% of commercial dish-washing liquid (Dreft, Procter & Gamble) to deionised water. The surface tension of the soap solution is γ = 26.2 ± 0.2 mN.m -1 and its density is ρ = 10 3 kg.m -3 . As reported elsewhere [START_REF] Kirstetter | Jet impact on a soap film[END_REF], this situation is stable and the meniscus forms an equilibrium shape which is recorded by a digital camera (Fig. 1b). The incident jet is characterized by its incident angle θ i , velocity V and radius R.

The Weber number, W e = ρV 2 R/γ is used to estimate the relative contributions of inertia and capillary forces. Within our experimental parameter range, the Reynolds number based on the jet characteristics is always significantly larger than unity.

RESULTS

Shift of the contact line

The shape and position of the contact line where the interface meets the cylinder is sketched in Fig. 2. The reference frame of the initially flat interface is R (Oxyz), with Ox horizontal, and the reference frame of the cylinder is R ′ (Ox ′ y ′ z ′ ); they are related by

x ′ = cos(θ i )x + sin(θ i )z, (1a) y ′ = y, (1b)
z ′ = -sin(θ i )x + cos(θ i )z. ( 1c
)
All calculations are performed in the reference frame of the cylinder, R ′ , using cylindrical coordinates (r ′ , θ ′ , z ′ ), so that any point M has OM = x ′ e x ′ + y ′ e y ′ + z ′ e z ′ = r ′ e r ′ + θ ′ e θ ′ + z ′ e z ′ , with

x ′ = r ′ cos(θ ′ ) (2a) y ′ = r ′ sin(θ ′ ) (2b) e r ′ = cos(θ ′ ) e x ′ + sin(θ ′ ) e y ′ (2c)
e θ ′ = -sin(θ ′ ) e x ′ + cos(θ ′ ) e y ′ . (2d)
By symmetry, the contact line (Fig. 3) is given by

r ′ = R (3a) z ′ = z ′ (θ ′ ), ( 3b
) with θ ′ ∈ [-π, π].
The position of the contact line in the simulation with an incident angle of θ i = 30 • , referred to R ′ , is shown in Fig. 3. The deviation of the contact line is defined by ∆z

′ = z ′ (θ ′ = π) -z ′ (θ ′ = 0). A x ′ y ′ , A x ′ z ′ and A y ′ z ′
denote the absolute values of the area enclosed by the contact line when projected on the (Ox ′ y ′ ), (Ox ′ z ′ ) and (Oy ′ z ′ ) planes respectively. By symmetry, A x ′ y ′ = πR 2 and A x ′ z ′ = 0. A y ′ z ′ and ∆z ′ are equal to zero when θ i = 0 and both increase monotonically as θ i increases. Fig. 4 shows how ∆z ′ /R depends on θ i in the simulation (dashed line). Side views of the meniscus were used to investigate experimentally the effect of R, θ i and V on the meniscus shape and perform measurements of ∆z ′ . For a given set of parameters (R, θ i ), ∆z ′ was found to decrease as V increases (insert to Fig. 4) until saturating above a limiting velocity. This marks the onset of the dynamical "total wetting" condition (see Discussion). In this regime, the meniscus shape is found to be independent of the jet velocity and the following measurements were performed at sufficiently high velocity to ensure that ∆z ′ reached its saturation value.

Fig. 4 shows ∆z ′ /R as a function of θ i , illustrating the agreement between simulations and experiments. This scaling allows the collapse of all experimental measurements (several R and θ i values) on the curve found from the simulation.

Force measurement

Assuming total wetting, it is possible to infer the force on the cylinder from the contact line position.

We denote by M(R, θ ′ , z(θ ′ )) a point on the contact line and d M an infinitesimal variation of position along the contact line (Fig. 2)

: d M = Rdθ ′ e θ ′ + dz ′ e z ′ = dθ ′ R e θ ′ + dz ′
dθ ′ e z ′ . Knowing that the local force is perpendicular to d M and to e r ′ (total wetting condition),

d F = γ e ′ ∧ M . (4)
The prefactor γ holds for single interfaces and should be replaced by 2γ without loss of generality for double (e.g. air-liquid-air) interfaces such as soap films. The total force that the interface exerts on the cylinder is

F = γ C e r ′ ∧ d M , (5)
where C represents the oriented contour around the contact line. After some manipulation, described in the Appendix, an exact expression for the force in the reference frame R of the interface is obtained:

F x = 2γπR cos(θ i ) A y ′ z ′ 2πR 2 -sin(θ i ) , (6a)
F y = 0, ( 6b
)
F z = 2γπR sin(θ i ) A y ′ z ′ 2πR 2 + cos(θ i ) , (6c)
where A y ′ z ′ is the absolute value of the projected area of the contour in the (Oy ′ z ′ ) plane. Force measurements are then obtained numerically using the contour profiles calculated in the simulations and the results are shown in Fig. 5. For the whole range of θ i tested, F x is small compared to F z , suggesting that the force acts vertically (in the reference frame of the interface).

The vertical component of force can be compared to three different models (Fig. 5): • vertical force model: from the observation above, we might assume F x = 0. Consequently Eq. 6a gives

A y ′ z ′ 2πR 2 = tan(θ i ) and thus F z = 2γπR/ cos(θ i ). (7)
• total perimeter model: the length of the contact line, P, which is measured in the simulations, might determine the vertical force: F z = γP.

• flat interface model: F z could depend on the perimeter of the ellipse that results from projecting the cross-section of the cylinder on to the unperturbed interface ((Oxy) plane): F z = γP ellipse . This ellipse has semi-axes of length R and R/ cos(θ i ).

An approximation to the perimeter of this ellipse (which is valid to better than 5% in our parameter range) leads to

F z = 2γπR 1 + 2 π 1 cos(θ i ) -1 .

DISCUSSION

Comparison of models

The deformation of an interface by an object of characteristic size L should result in a force that scales as γL. A rod has two well-separated characteristic lengths (radius and axial length) so the force could, in principle, range over several orders of magnitude. In the case of an infinite rod, one of the characteristic lengths is infinite, which explains why the force diverges as the rod becomes horizontal.

It is clear that the vertical force model gives the best fit, and the good agreement with the measurements extends up to very large inclination angles. The total perimeter model is plotted as it represents an upper bound for F z , since it assumes that all local forces d F (Eq. 4) are vertical in the reference frame of the interface. The flat interface model underestimates the force measurements. A priori we might have expected the opposite, as for the total perimeter model (which also accounts for vertical local forces), but it significantly underestimates the real perimeter of the contact line: the presence of the cylinder perturbs the contour and there is a significant shift in the vertical direction ∆z = z(θ ′ = π) -z(θ ′ = 0) which increases its perimeter. The vertical force model works well, and it would be interesting to see if this is also the case for "partially wetted" contact.

Based on an energy balance [START_REF] Neukirch | Piercing a liquid surface with an elastic rod: Buckling under capillary forces[END_REF] a vertical force model has also been used in the study of an elastic plate that pierces an interface, enabling the authors to infer a 1/ cos(θ i ) dependence to explain the different buckling morphologies of the plate under the effect of capillary forces. Expression [START_REF] Vella | Floating Objects with Finite Resistance to Bending[END_REF] could also be helpful in understanding the detachment of liquid droplets from fibres [START_REF] Lorenceau | Capturing drops with a thin fiber[END_REF][START_REF] Huang | Equilibrium of drops on inclined fibers[END_REF][START_REF] Mullins | Detachment of liquid droplets from fibres-Experimental and theoretical evaluation of detachment force due to interfacial tension effects[END_REF], in which a drop is pierced by a rod nonaxisymmetrically. This complicated geometry has two contact lines that prevent the drop from being detached by gravity, and the rod is locally inclined to the surface of the drop. No exact expressions for the interface shape or contact line position are available, so that in [START_REF] Lorenceau | Capturing drops with a thin fiber[END_REF] an expression for the force was based on experimental data, with no divergence as the inclination angle increased. This is in contrast to our 1/ cos(θ i ) dependence for an infinite planar interface, which is explained by noting that our expression would fail as soon as the deformation of the contact lines became comparable to the distance between them or to the drop size. In that case, the total capillary force might saturate, which supports the maximal capillary force expression frequently used to describe detachment above a given drop size.

CONCLUSION

The wetted contact of a free interface with a cylindrical object was studied both numerically and experimentally. Quasistatic simulations give the deformation of the interface and the position of the contact line, in good agreement with experiments on a soap film in contact with a cylinder of liquid. Above a limiting velocity, the liquid jet applies a dynamical "total wetting" condition to the interface, with what we might call a dynamically-generated capillary force, which is also of paramount importance in explaining the refraction of the jet, as described in Kirstetter et al. [START_REF] Kirstetter | Jet impact on a soap film[END_REF]. From the contact line position, the force exerted by the interface on the object (and viceversa) is calculated. We observe that whatever the inclination of the cylinder, the force remains vertical and its amplitude is given by F z = 2γπR/ cos(θ i ). This expression should be useful in numerous problems in both fluid and solid mechanics, and in future work it could be extended to study the partially wetted case, validated by the sort of simulations described here.

APPENDIX: EXPRESSION FOR THE FORCE

Expanding the integral for the force (Eq. 5) leads to

F /γ = C e r ′ ∧ R e θ ′ + dz ′ dθ ′ e z ′ dθ ′ = C R e z ′ - dz ′ dθ ′ e θ ′ dθ ′ = C R e z ′ + sin(θ ′ ) dz ′ dθ ′ e x ′ -cos(θ ′ ) dz ′ dθ ′ e y ′ dθ ′ ,
which gives the force components along the three axes of the reference frame of the cylinder R ′ (Ox ′ y ′ z ′ ):

F x ′ /γ = C sin(θ ′ ) dz ′ dθ ′ dθ ′ F y ′ /γ = C -cos(θ ′ ) dz ′ dθ ′ dθ ′ F z ′ /γ = C Rdθ ′ .
After some calculation, the contour integrals can be written in cylindrical or cartesian coordinates as:

F x ′ /γ = - C (z ′ cos(θ ′ )) dθ ′ = - C z ′ d(sin(θ ′ )) = - 1 R C z ′ dy ′ F y ′ /γ = - C (z ′ sin(θ ′ )) dθ ′ = C z ′ d(cos(θ ′ )) = 1 R C z ′ dx ′ F z ′ /γ = R C dθ ′ = 2πR.
The last expression for each component shows that F x ′ and F y ′ are directly related to the projected areas of the contact line in the (Oy ′ z ′ ) and (Ox ′ z ′ ) planes respectively. With A y ′ z ′ and A x ′ z ′ the absolute values of these oriented areas, as defined previously, we have by symmetry (Fig. 3) that A x ′ z ′ = 0 and C z ′ dy ′ < 0. This leads to the final expression for the force in the reference frame of the cylinder R ′ (Ox ′ y ′ z ′ ):

F x ′ = 2γπR A y ′ z ′ 2πR 2 F y ′ = 0 F z ′ = 2γπR
Writing the force components in the reference frame of the interface R can be done using the change-ofcoordinates formulae (Eq. 1), which then gives Eq. 6.

FIG. 1 .

 1 FIG. 1. Side view showing how the interface deforms. a) Simulation, with θi = 30 • , and an imposed contact angle of 90 • (top) or 0 • (bottom). b) Experiment, with θi = 30 • , R = 200µm and V = 2m.s -1 .

FIG. 2 .

 2 FIG. 2. Schematic representation of the system.

A

  FIG. 3. The contact line projected on different planes for a simulation with for θi = 30 • , in the reference frame of the cylinder R ′ .

2 FIG. 4 .

 24 FIG. 4. The quantity ∆z ′ /R versus θi in simulations (dashed line) and in experiments (different radii R = 140, 200 and 270 µm are shown as circles, triangles and squares respectively). Insert: experimental data for ∆z ′ versus the incident velocity V for a radius of 270 µm and two incident angles: 39 • (filled circles) and 49 • (open circles). The two solid lines indicate the values to which the curves saturate.

FIG. 5 .

 5 FIG. 5. The different force components Fx and Fz are normalised by 2γπR. Simulations are shown as red stars. Three models are tested for Fz. Black solid line: vertical force model, Fz = 2γπR/ cos(θi). Blue dashed line: total perimeter model. Cyan dashed-dotted line: flat interface model.