A Bertrandias
email: aude.bertrandias@centralesupelec.fr

) H Duval

) J Casalinho

M L Giorgi

Drop generation from a vibrating nozzle in an immiscible liquid-liquid system

Received and accepted dates, other relevant dates

INTRODUCTION

Inducing vibration to jets or drops can be used to control breakup, thus drop size. Vibration is applied for example in ink jet printing, spray coating or vibrating cross-flow membrane emulsification, the latter having motivated our research. We focus on transversal vibrations, where drops undergo axial oscillations. For a membrane with a mean pore diameter of 0.8 µm, Arnaud 1 found a decrease in the peak of the volume-weighted drop size distribution (from 30 µm to 10 µm) at a forcing frequency of 15 to 20 kHz compared to without vibration. Thus, vibrating the membrane in this process impacts drop size but mechanisms for drop detachment were not explained. 1,2 To obtain a fine control on drop size, understanding the physics of drop vibration and detachment is necessary.

Oscillations of liquid drops have been extensively studied since the pioneering work of Lord Rayleigh 3 . He calculated the eigenmodes of a free inviscid, incompressible drop in a vacuum, in absence of gravity and for small-amplitude oscillations. The eigenmodes are characterized by two integers: a polar wavenumber 2 and an azimuthal wavenumber ∈ ; . In this study, we focus on axisymmetric modes ( 0). Rayleigh 3 showed that the eigenfrequencies depend on , the liquid density, interfacial tension and drop size. Drop eigenfrequencies scale as / , with the drop diameter. Lamb 4 generalized this theory by calculating the eigenpulsation of a drop in a surrounding fluid and found the same relationship ~ / .

Rodot et al. 5 and Bisch et al. 6 investigated drops partially bound to a rod, submitted to controlled vibration. The drop was immersed in immiscible liquids of equal density, with its contact line pinned on the rod edge. A large range of liquid couples were examined and the first eigenfrequency was found to depend on the support diameter, drop diameter, drop density and surface tension. The first resonance frequency scales as and not as / such as for the free drop. Then, Strani and Sabetta [START_REF] Strani | Free vibrations of a drop in partial contact with a solid support[END_REF] (hereafter denoted S&S) studied linear oscillations of a liquid drop in an outer fluid, in partial contact with a spherical bowl under inviscid and zero-gravity assumptions. The presence of the support increases the eigenfrequencies for modes 2, but an additional low-frequency mode appears. This is the 1 eigenmode, associated with the displacement of the bound drop center of mass. When the support reduces to a single point, the 1 mode degenerates to a zero-frequency 2 rigid motion of the drop. S&S 7 noted that the mode 1 eigenfrequency may be approximated over intervals by with varying between -2.9 and -1.75 for drop to support diameter ratios of 1.3 to 7, respectively. This is consistent with -2 proposed by Bisch et al. 6 Also, S&S [START_REF] Strani | Free vibrations of a drop in partial contact with a solid support[END_REF] computed frequencies were in agreement with Bisch et al. 6 data, but resonance frequencies were overpredicted by 20% (reduced to 10% by accounting for viscous effects [START_REF] Strani | Viscous oscillations of a supported drop in an immiscible fluid[END_REF] ). However, both models (inviscid [START_REF] Strani | Free vibrations of a drop in partial contact with a solid support[END_REF] and viscous [START_REF] Strani | Viscous oscillations of a supported drop in an immiscible fluid[END_REF] ) overpredicted resonance frequencies for large support to drop diameter ratios, attributed to nonlinear effects, not taken into account in the models. Smithwick and Boulet 9 studied the first resonance frequency of mercury drops on glass (pinned contact line) under partial vacuum and compared their data to the calculations of S&S. [START_REF] Strani | Free vibrations of a drop in partial contact with a solid support[END_REF] A maximum error of 3.3% was found.

Bostwick and Steen [START_REF] Bostwick | Capillary oscillations of a constrained liquid drop[END_REF] and Vejrazka et al. [START_REF] Vejrazka | Linear oscillations of a supported bubble or drop[END_REF] studied linear oscillations of a drop supported on a ring. Bostwick and Steen 10 noted that the center of mass motion is partitioned among all the eigenmodes but the 1 mode is its main carrier. Vejrazka et al. [START_REF] Vejrazka | Linear oscillations of a supported bubble or drop[END_REF] found that for small support to drop diameter ratios, the frequency response of the drop is independent of the constraint (bowl or ring). Abi Chebel et al. [START_REF] Chebel | Inertial modes of a periodically forced buoyant drop attached to a capillary[END_REF] and Vejrazka et al. [START_REF] Vejrazka | Linear oscillations of a supported bubble or drop[END_REF] examined drop oscillations driven by imposed periodic volume variations. The frequency response is independent of the forcing type as long as the support to drop diameter ratio is small. [START_REF] Vejrazka | Linear oscillations of a supported bubble or drop[END_REF] Lastly, Noblin et al. [START_REF] Noblin | Vibrated sessile drops: Transition between pinned and mobile contact line oscillations[END_REF] studied bound drop oscillations with mobile instead of pinned contact lines: a decrease in resonance frequency was found. The transition from a pinned to mobile contact line occurred above a critical forcing amplitude. In that case, the variation of the contact angle exceeds the contact angle hysteresis.

Previous studies explored linear oscillations. Wilkes and Basaran 14 (hereafter denoted W&B) used computational fluid dynamics (CFD) to study large-amplitude axisymmetric oscillations of a viscous bound drop on a rod (pinned contact line). They found that the drop resonance frequency varies slightly with amplitude at high Ohnesorge numbers (Oh, expressed in section V.B.) but decreases significantly with amplitude at low Oh, Oh being the ratio of a viscocapillary to an inertial-capillary timescale. Resonance frequency also decreases as the Bond number (Bo, expressed in section II.C.) increases. Bo compares the gravity to capillary forces. The maximum drop deformation, observed at resonance, increases with forcing amplitude and Bo and decreases with Oh and . DePaoli et al. [START_REF] Depaoli | Hysteresis in forced oscillations of pendant drops[END_REF] experimentally studied pendant drops in air under high-amplitude forcing and observed hysteresis, characteristic of soft nonlinearities. At a set forcing amplitude (resp. frequency), a larger response amplitude appeared at lower frequencies (resp. amplitudes) when a downwards frequency (resp. amplitude) sweep was performed vs. an upwards sweep. W&B [START_REF] Wilkes | Hysteretic response of supported drops during forced oscillations[END_REF] numerically gave the critical forcing amplitude for the onset of hysteresis for different Oh. This value could be as low as 3% of the rod radius (drop and rod radii of the same order). Calculations were also performed for drops hanging from a tube: the first resonance frequency is slightly higher when the support is a tube, the hysteresis range is shifted to higher values of forcing frequency and the deformation at resonance is higher.

For high enough forcing amplitudes, drops detach from the support. W&B 17 used CFD to simulate drop ejection from a rod (pinned contact line). Above a critical amplitude, the bound drop ruptures: a primary drop is ejected from the liquid remaining on the rod. The variations of the critical amplitude as a function of the forcing pulsation has a V-shape (the minimum corresponds to drop resonance). For a set rod diameter, the critical amplitude increases when Oh decreases or when the bound drop volume decreases. Critical amplitudes range from 25% to 80% of the rod radius. Kim 18 experimentally studied the detachment of a pendant drop from a smooth vibrating plate in air (mobile contact line). The variations of the critical amplitude as a function of the forcing frequency has a Wshape. Kim [START_REF] Kim | Drop fall-off from the vibrating ceiling[END_REF] found that the minima concord well with the 1 and 2 modes of the bound drop as calculated by S&S. [START_REF] Strani | Free vibrations of a drop in partial contact with a solid support[END_REF] Again, the minima correspond to drop resonance. The agreement between data and calculations of S&S 7 is remarkable as the contact line mobility is different and experimental oscillation amplitudes are beyond the linear regime.

Resonance also triggered drop detachment in previous work of the authors, where different pore diameters were studied for one system (dodecane-water without surfactant). [START_REF] Bertrandias | Good vibrations -Transition in drop generation from an immersed capillary tube[END_REF] Drops were formed through a vibrating nozzle continuously fed with dodecane, immersed in the stationary immiscible water phase. This enabled to gain insight into transversally vibrating membrane emulsification in a simplified configuration. We found that at a set forcing frequency, smaller drops were generated above a threshold forcing amplitude: a growing drop detached prematurely when its first resonance frequency (as given by Bisch et al.) and the forcing frequency coincided. However, the threshold was higher than expected, attributed to the fact that the bound drop did not spend enough time in the resonance range to reach steady-state resonance. The generation mode forming the smaller drops was named the "stretching mode". Below the threshold, larger drops were formed in dripping mode.

The aim of this work is to study the mutual effect of forcing parameters and system properties on drop generation from a vibrating nozzle. We also aim to further model drop generation modes by accounting for drop growth and motion as a function of time. We emphasize that studies on vibrated growing drops are rare [START_REF] Bertrandias | Good vibrations -Transition in drop generation from an immersed capillary tube[END_REF] compared to those on constant-volume drops [5][6][START_REF] Strani | Free vibrations of a drop in partial contact with a solid support[END_REF][START_REF] Strani | Viscous oscillations of a supported drop in an immiscible fluid[END_REF][START_REF] Smithwick | Vibrations of Microscopic Mercury Droplets on Glass[END_REF][START_REF] Bostwick | Capillary oscillations of a constrained liquid drop[END_REF][START_REF] Vejrazka | Linear oscillations of a supported bubble or drop[END_REF][START_REF] Chebel | Inertial modes of a periodically forced buoyant drop attached to a capillary[END_REF][START_REF] Noblin | Vibrated sessile drops: Transition between pinned and mobile contact line oscillations[END_REF][START_REF] Wilkes | Forced oscillations of pendant (sessile) drops[END_REF][START_REF] Depaoli | Hysteresis in forced oscillations of pendant drops[END_REF][START_REF] Wilkes | Hysteretic response of supported drops during forced oscillations[END_REF][START_REF] Wilkes | Drop Ejection from an Oscillating Rod[END_REF][START_REF] Kim | Drop fall-off from the vibrating ceiling[END_REF] and that previous work concerned only one system and parameter. [START_REF] Bertrandias | Good vibrations -Transition in drop generation from an immersed capillary tube[END_REF] In the following, we first describe our setup. We present the dripping to stretching transition and propose a simple framework to approach it. Then, we discuss the effect of nozzle inside diameter, dispersed phase flow rate, interfacial tension and dispersed phase viscosity on the transition. We examine the effect of these parameters on (i) the threshold amplitude for the stretching mode and (ii) the resulting drop diameters. We further analyze our results by comparing them to S&S calculations. [START_REF] Strani | Free vibrations of a drop in partial contact with a solid support[END_REF] Finally, we propose a simple transient model to describe drop dynamics until detachment and compare the model predictions to experiments.

II. EXPERIMENTAL A. MATERIALS

The reference continuous and dispersed phases are distilled water and dodecane (99%, Fisher Scientific), respectively. To study the impact of interfacial tension, a surfactant (SDS, 85%, Acros Organics) is added to the continuous phase at 0.1 wt% or 2 wt% (systems 1 and 2, resp.). To study the impact of dispersed phase viscosity, paraffin (Fisher Scientific) is added to the dispersed phase at 25 wt% or 50 wt% (systems 3 and 4, resp.). A system with an increased continuous phase viscosity was also tested (supplementary material D). The reference system and system 1 to 4 properties are given in table I. The viscosities and densities of the mixtures were measured in triplicate, the former with a Ubbelohde type viscosimeter (AVS310, Schött-Gerade) at 25.1°C. The interfacial tension was measured in triplicate by the rising drop method with a tensiometer (Tracker, I.T. Concept, Teclis). For systems 1 and 2, is determined by the method explained in the supplementary material A. Table I values report an intermediate plateau interfacial tension. We consider that the plateau value gives an adequate estimation of the interfacial tension when drops form (see supplementary material B).

B. EXPERIMENTAL SETUP

The setup, illustrated in prior work [START_REF] Bertrandias | Good vibrations -Transition in drop generation from an immersed capillary tube[END_REF] , is summarized in fig. 1. A single glass capillary (nozzle) of inside diameter emerges into a tank with the stationary continuous phase. Two pore diameters are presently tested: 0.32 mm and 0.11 mm. The dispersed phase is supplied through the pore at a flow rate 1.1 µL.s -1 to 14.4 µL.s -1 (PHD Ultra Syringe Pump, Harvard Apparatus), leading to mean flow velocities 4 / . Reynolds numbers for the flow in the nozzle are of Re 3.0 to 32.8 (Re / ) (laminar flow). The nozzle is fixed on a vibrating exciter (Bruel & Kjaer 4810) which induces a sinusoidal motion in time : sin 2 . is the forcing amplitude measured by a laser sensor (M5L/2, Bullier Automation) with a precision in the order of 10 µm. is the forcing frequency set on the signal generator (33512B Arbitrary Waveform Generator, Agilent). Vibrations are parallel to the nozzle axis, so drops undergo axial oscillations. In the moving non-inertial frame of reference where the nozzle is still (axes in fig. 2), the forces exerted on the drop due to nozzle motion are the inertial force and associated Archimedes' thrust. We note that the continuous phase above the nozzle and support is accelerated by the exciter, shown by Faraday waves at the free surface. The resulting excitation force is:

6 cap 1 with
the bound drop diameter and cap sin the nozzle acceleration in the laboratory inertial frame ( 2 is the forcing pulsation). Drop formation is recorded with a highspeed camera (v310, Phantom) and macro lens (AF Zoom-Micro Nikkor 70-180mm f/4.5-5.6D ED, Nikon). The acquisition frequency is ten times the forcing frequency or 100 fps for trials without vibration. The resolution is 800 x 600 px². We extract data with ImageJ 20 including average detached drop diameters , axial drop elongations and the position of the drop center of mass compared to the nozzle surface. Images were calibrated (36 px/mm) using the outer diameter (7.86

C. EXPERIMENTAL PROTOCOL

The tank is filled with the continuous phase and the tube and syringe with the dispersed phase. The syringe pump is activated and drop diameters are measured without vibration. Drops are formed in dripping mode. We calculate Bond numbers Bo and Weber numbers We as Clanet and Lasheras 21 : we find Bo 2 ⁄ (with the gravitational acceleration) from 1.7×10 -2 to 1.5×10 -1 and We / from 4.5×10 -3 to 2.8×10 -1 . These values are below the critical We for the transition to jetting (at the given Bo), confirming the setup operates in dripping mode. From the drop diameters obtained without vibration, we calculate the in situ interfacial tensions by Tate's law [START_REF] Tate | On the magnitude of a drop of liquid formed under different circumstances[END_REF] :

6 2
Drop detachment occurs when buoyancy (left-hand side of Eq. ( 2)) exceeds the maximum capillary force that the drop neck can resist without breaking. is the detached drop diameter. is the Harkins Brown correction factor [START_REF] Harkins | The determination of surface tension (free surface energy), and the weight of falling drops: the surface tension of water and benzene by the capillary height method[END_REF] : it accounts for the fraction of liquid volume which stays attached to the nozzle after drop detachment. We use the factor of Mori. [START_REF] Mori | Harkins-Brown Correction Factor for Drop Formation[END_REF] The in situ interfacial tension is then compared to the measured one (table I), to ensure the setup is adequately cleaned.

Then, vibration is applied. A forcing frequency is set and an upwards amplitude sweep is performed. Measurements are made at different amplitudes, ensuring one is always made at the threshold where a transition in drop generation occurs (see III.). This is repeated for frequencies from 30 to 150 Hz. From 30 to 100 Hz, 10 Hz intervals are applied. Above 100 Hz, intervals vary depending on the pore diameter. The vibrating exciter limitations do not enable us to observe the stretching mode above 150 Hz for 0.32 mm and 110 Hz for 0.11 mm. After trials, a cleaning agent at 3 vol% (Mucasol, Merz) fills the setup for 24h and it is rinsed with distilled water leading to a hydrophilic glass surface. Then, the organic dispersed phase does not wet the nozzle and the outer nozzle diameter does not influence drop detachment.

For each test condition (i.e., physicochemical system, pore diameter, dispersed phase flow rate and forcing frequency), three trials are carried out to determine the transition threshold. For each trial, six detached drops are studied. For each drop, ten images are analyzed. We checked that the accuracy of the diameter measurement from ten different snapshots of a given drop is sub-pixel. For a given trial, we noted variations up to 2 px at most in diameter from one drop to another. In the figures displaying drop diameters, the error bars correspond to the relative standard deviation in drop diameters: it ranges between 1% to 7% depending on the test conditions.

III. TRANSITION FROM DRIPPING TO STRETCHING MODE

Figure 3 shows typical variations of the drop diameter as a function of the forcing amplitude at a set forcing frequency . The drop diameter falls (by 63%) at a threshold amplitude . The same behavior occurs for all systems. For the reference system, a relative decrease in drop diameter of 45% to 76% was found at compared to without vibration depending on , for all pore diameters. For system 2, similar values were found: 29% to 73%. This fall at corresponds to a transition in the drop generation regime. For , drops detach in dripping mode and their diameter is close to the diameter of the drop formed without vibration: detachment is buoyancy-controlled. For , the drop detaches when its mode 1 eigenfrequency coincides with and when it reaches a critical elongation ratio: detachment is controlled by the excitation force. Figure 2 shows how a drop elongates at resonance (characteristic mode 1 resonance shape) and detaches. We named this the "stretching mode". [START_REF] Bertrandias | Good vibrations -Transition in drop generation from an immersed capillary tube[END_REF] It should be noted that there is an amplitude interval where both modes coexist. The threshold amplitude is defined as the upper bound of that interval, when all drops are generated in stretching mode. 2); simulation results of section V.C., Eq. ( 14) (dotted line); of section V.D., Eq. ( 17) with -1.9, 4.4 (solid line).

We proposed a simple model [START_REF] Bertrandias | Good vibrations -Transition in drop generation from an immersed capillary tube[END_REF] to describe the main features of stretching mode. Below, we recall its main arguments and derive scaling laws to provide a framework to analyze our experimental results in section IV. The vibrating bound drop is considered as a linearly forced harmonic oscillator (LFHO) with moderate damping. [START_REF] Butikov | Simulations of Oscillatory Systems[END_REF] Drop growth is considered slow enough for oscillations to reach steady state. We are aware that these assumptions are strong since the system is probably no longer linear when oscillations are such that the drop detaches and the process remains transient. However they are required to develop the following scaling laws.

A simple analytical expression of the mode 1 eigenfrequency of a bound drop has been empirically established by Bisch et al. 6 for / of 1.3 to 7 and fluids of equal densities: 3) and ( 4) can be reasonably applied to our trials as the density ratio ⁄ is in the order of 1 (it ranges from 1.26 to 1.33). Also, we neglect the effect of buoyancy on and .

From Eq. ( 2) and (3), omitting , we deduce the minimum forcing frequency above which a growing drop may detach in stretching mode (if ):

, ⁄ 2 6 ⁄ ⁄ ⁄ ⁄ ⁄ ⁄ ⁄ 5
The minimum forcing frequency is around 8 Hz for 0.32 mm for the reference system and 14 Hz for 0.11 mm for system 2. This is smaller than the lower bound of the frequency range investigated. Consequently, drops may detach in stretching mode.

Whatever the mode (dripping or stretching), the drop detaches when the restoring capillary force exceeds the maximum capillary force . Under LFHO assumption, reads:

6 6
with the eigenpulsation of the bound drop without damping. is the displacement of the drop center of mass with respect to its rest position (absence of buoyancy and excitation forces). The forcebased detachment criterion can be easily recast into an elongation-based criterion:

6 7
The displacement of the drop center of mass is made up of a stationary part due to buoyancy and an oscillatory part due to the excitation force. Assuming quasi steady state, reads:

sin 8
with , the phase shift and the amplitude given by the well-known expression [START_REF] Butikov | Simulations of Oscillatory Systems[END_REF] :

² 9
is the quality factor given by 2 ⁄

. Since is given by Eq. ( 4), depends only on the phase densities and viscosities.

is maximum when 1 1 2 ⁄ ⁄ , i.e., ≅ for moderate damping. In that case, simplifies to ≅ . In dripping mode, the drop has left the resonance range and its eigenfrequency is much lower than the forcing frequency. Then, buoyancy dominates: Eq. ( 7) and ( 8) reduce to Eq. (2) (omitting ). In stretching mode, the drop detaches at resonance ( ≅ ). Its diameter is thus:

≅ 6 ⁄ 1 √ 10 7
An estimate of the threshold amplitude may be derived from the above equations neglecting buoyancy and assuming that the detachment criterion is satisfied at an oscillation peak:

≅ ⁄ 6 ⁄ 1 √ 11
As a result, and should both scale as ⁄ , i.e., ⁄ .

IV. IMPACT OF PROCESS PARAMETERS AND SYSTEM PROPERTIES

In this section, we study the effect of process parameters (pore diameter, dispersed phase flow rate) and system properties (interfacial tension, dispersed phase viscosity) on the dripping to stretching transition. Threshold amplitudes are determined from an amplitude sweep and drop diameters at from image analysis. Error bars are generally large for threshold amplitudes partly due to measurement errors and partly due to the difficulty to repeatedly estimate the threshold.

A. INFLUENCE OF PORE DIAMETER

Fig. 4 reports the variations of the threshold amplitude and generated drop diameter as a function of the forcing frequency for two pore diameters, i.e., 0.11 mm and 0.32 mm (more pore diameters were tested in another paper in the case of the reference system [START_REF] Bertrandias | Good vibrations -Transition in drop generation from an immersed capillary tube[END_REF] ). Threshold amplitude variations with forcing frequency are monotonous (fig. 4(a)) and do not exhibit the V-or W-shape reported by W&B [START_REF] Wilkes | Drop Ejection from an Oscillating Rod[END_REF] or Kim [START_REF] Kim | Drop fall-off from the vibrating ceiling[END_REF] , respectively. In the latter cases, bound drop volume (thus eigenfrequencies) are fixed, independently of . On the contrary, in our setup, the drop grows until its eigenfrequency coincides with .

The threshold amplitude decreases as the forcing frequency increases (fig. 4(a)), in accordance with Eq. (11). Eq. ( 11) predicts a ⁄ scaling but our data scale differently:

. . for 0.11 mm and . . for 0.32 mm. Also, thresholds are twice higher for 0.11 mm than for 0.32 mm. This contradicts the expected ⁄ scaling (Eq. ( 11)). We return to this in section V.

The drop diameter decreases with increasing forcing frequency (fig. 4(b)). The / scaling predicted by Eq. ( 10) was verified for four pore diameters ranging from 0.11mm to 0.75mm [START_REF] Bertrandias | Good vibrations -Transition in drop generation from an immersed capillary tube[END_REF] : we specifically find for 0.32 mm. The larger the pore diameter, the larger the drops produced. However, from experimental data [START_REF] Bertrandias | Good vibrations -Transition in drop generation from an immersed capillary tube[END_REF] , it is difficult to conclude on the relevance of the predicted ⁄ scaling as the deviation of the data to the ⁄ scaling is large (0% to 31%, depending on ).

B. INFLUENCE OF DISPERSED PHASE FLOW RATE

Four dispersed phase flow rates were applied to the reference system, for 0.32 mm: 2.5 µL.s -1 , 4.3 µL.s -1 , 6.5 µL.s -1 and 14.4 µL.s -1 . Threshold amplitudes and drop diameters do not vary significantly with these flow rates according to the error bars (see supplementary material C). This is consistent with Eq. ( 10) and (11). For higher flow rates, this parameter could become significant. A drop may no longer have time to reach large-amplitude oscillations at resonance for stretching mode. Also, a transition to jetting would occur [START_REF] Clanet | Transition from dripping to jetting[END_REF][START_REF] Scheele | Drop Formation at Low Velocities in Liquid-Liquid Systems: Part I. Prediction of Drop Volume[END_REF][START_REF] Meyer | Universal Dripping and Jetting in a Transverse Shear Flow[END_REF] (out of the scope of this paper).

When increases from 2.5 to 14.4 µL.s -1 , the mean number of oscillations between two drops at 100 Hz decreases from 28 to 4. As the threshold is little affected by in the investigation range, we infer that the steady-state oscillation regime is reached in just a few oscillations.

C. INFLUENCE OF INTERFACIAL TENSION

Experiments were carried out for the reference system and systems 1 and 2 ( 50.7 mN.m -1 , 19.0 mN.m -1 and 5.4 mN.m -1 , resp.) for 0.32 mm. The threshold amplitude scaling is not significantly affected by : . .

for system 1 and . .

for system 2 compared to . . for the reference system (fig. 5(a)). However, it is not in accordance with the predicted ⁄ scaling (Eq. ( 11)). Higher SDS concentrations result in lower interfacial tensions, leading to lower threshold amplitudes for a drop to detach in stretching mode (fig. 5(a)). This is in qualitative agreement with Eq. ( 11) but it is difficult to conclude on the relevance of the predicted ⁄ scaling, as the deviation of the data to this scaling is large (8% to 32%, depending on ).

The / scaling of the drop diameter is maintained when the interfacial tension is decreased from 50.7 mN.m -1 to 5.4 mN.m -1 (fig. 5(b)): we find .

.

for system 1 and . . for system 2. Smaller drops are generated for lower interfacial tensions. The drop diameter roughly scales as ⁄ in accordance with Eq. ( 10) (deviations of 1% to 18%, depending on ).

D. INFLUENCE OF DISPERSED PHASE VISCOSITY

Experiments were carried out for the reference system and systems 3 and 4 ( 1.34 mPa.s, 1.79 mPa.s and 3.24 mPa.s, resp.) for 0.32 mm. Threshold amplitudes increase when increases (fig. 6(a)), as in W&B calculations. [START_REF] Wilkes | Drop Ejection from an Oscillating Rod[END_REF] When changing the reference system for system 3 (resp. 4), increases by 34% (resp. 142%) and increases by 28% to 79% (resp. 62% to 133%). The effect of is stronger than expected. Indeed, when the reference system is changed for system 4, the quality factor decreases from 14.6 to 13.9 (Eq. ( 4)), leading to a theoretical 5% increase in (Eq. ( 11)). In addition, the scaling with the reference system is not conserved for systems 3 and 4: we find .

.

for system 3 and . .

for system 4. Thus, these results do not agree with the predicted ⁄ scaling (Eq. ( 11)). does not significantly impact drop diameter (fig. 6(b)), in agreement with Eq. ( 10) (valid for moderate damping).

As mentioned above, a system with a greater continuous phase viscosity was also studied: results and analysis are reported in supplementary material D.

V. FURTHER ANALYSIS

In this section, we synthesize drop diameter data of section IV. Then, we analyze the elongation ratio for detachment. Finally, we propose a LFHO model that better reflects the data.

A. MODE 1 RESONANCE

Drop diameters at the threshold are consistent with the detachment at resonance when the bound drop mode 1 eigenfrequency coincides with the forcing frequency. To quantify the discrepancy between our data and Eq. (3) of Bisch et al. 6 , we plot the dimensionless forcing pulsation (our data) and dimensionless drop eigenpulsation (Eq. ( 3)) against the drop to pore diameter, for different pore diameters and interfacial tensions (fig. 7 12) of S&S 7 for the reference system (solid line), system 3 (dash-dotted line) and system 4 (dotted line).

Our data are well represented by Eq. (3) of Bisch et al. 6 (dashed line) until / 5. For / 5, our data are markedly above the Bisch et al. 6 curve. As stated, Eq. ( 3) was validated until / 7. As the validity of the Bisch et al. 6 law is restricted, we consider the theoretical results of S&S [START_REF] Strani | Free vibrations of a drop in partial contact with a solid support[END_REF][START_REF] Strani | Viscous oscillations of a supported drop in an immiscible fluid[END_REF] established for any / and density ratio. They analyzed the axisymmetric vibrations of a liquid drop in an outer fluid, in partial contact with a spherical bowl (see fig. 8) under the assumptions of zero gravity, negligible viscous effects and small surface deformations. Their calculated eigenfrequency of mode is:

1 2 8 10 with
the eigenvalue for mode , function of the support angle Θ and phase density ratio. Assuming fig. 8 is a reasonable simplification of our drop, we estimate Θ arcsin ⁄ . We calculate with Smithwick and Boulet's method 9 derived from the work of S&S 7 , using densities and interfacial tensions of table I. Our data are better fitted by the model of S&S [START_REF] Strani | Free vibrations of a drop in partial contact with a solid support[END_REF] than by the law of Bisch et al. 6 (fig. 7(a) and (b)), notably for / 7. As in Kim's [START_REF] Kim | Drop fall-off from the vibrating ceiling[END_REF] work, agreement between our data and S&S 7 calculations is remarkable as the binding constraint is different.

B. CRITICAL ELONGATION RATIO

A critical elongation ratio

⁄ function of the drop to pore diameter ratio leads to drop detachment [START_REF] Wilkes | Drop Ejection from an Oscillating Rod[END_REF][START_REF] Bertrandias | Good vibrations -Transition in drop generation from an immersed capillary tube[END_REF] . We measured ⁄ for all parameters tested (fig. 9), being taken from the nozzle tip to the drop apex. The points lie roughly on the same curve (fig. 9), confirming that a drop detaches in stretching mode once a critical elongation is reached. Also, we see that the critical elongation ratio is essentially a function of ⁄ . FIG. 9. Drop elongation ratio function of the drop to pore diameter. Reference system (white); system 1 (grey); system 2 (black). (□) 0.11 mm; (◊) 0.32 mm; (⊲) system 3, 0.32 mm; (⊳) system 4, 0.32 mm. Curve from Eq. (13) (dashed and solid lines).

We remind that physicochemical properties vary for systems 1 to 4 compared to the reference system.

⁄ should depend on the viscosity ratio ⁄ and on the Ohnesorge number

Oh 2 ⁄ ⁄
. The viscosity ratios we tested ( 1.5 to 4.1) may not vary enough to have an impact on ⁄ . For a free drop submitted to shear, Stone et al. [START_REF] Stone | An experimental study of transient effects in the breakup of viscous drops[END_REF] found that breakup occurs above a critical elongation ratio, function of but for 0.1 to 1, the critical elongation ratio did not vary significantly. Similarly, we find Oh from 1.7×10 -2 to 5.3×10 -2 . These values may be too close to observe a difference in ⁄ , although for a free drop submitted to shear, these values are sufficiently different to obtain a twofold increase in aspect ratio. [START_REF] Castrejon-Pita | Breakup of Liquid Filaments[END_REF] We return to the elongation-based criterion for stretching mode (Eq. ( 7)). Let us note the displacement of the drop center of mass when the drop axial elongation is . In the limit ≫ , we may consider that drop deformation is entirely localized in the neck. In that case, an estimation of is given by and ⁄ roughly reads:

≅ 1 3 4 13
The curve from Eq. ( 13) (solid line) is plotted against our data (fig. 9). For high drop to pore diameter ratios, critical elongation ratios are well estimated by Eq. ( 13), in accordance with S&S [START_REF] Strani | Free vibrations of a drop in partial contact with a solid support[END_REF] or Bostwick and Steen 10 : for large ⁄ , the bound drop essentially experiences a rigid motion with deformation localized at the neck. For low ⁄ , Eq. ( 13) is no longer valid (dashed line) and drop deformation is rather uniform. From fig. 9, we deduce that the transition from the uniform deformation regime to the localized deformation regime occurs around ⁄ ≅ 3. In our setup, the neck of the bound drop preexists (without vibration) contrary to in the configuration of S&S. [START_REF] Strani | Free vibrations of a drop in partial contact with a solid support[END_REF] Thus, deformation is more quickly localized (they find values for the transition in the order of 10).

C. OSCILLATOR MODEL WITH THE TRANSIENT

The effect of process parameters and system properties on drop diameter concord with the scaling laws of section III but threshold amplitude variations are not well predicted. Thus we developed a finer model which describes drop growth and motion as a function of time. We still consider the drop as a LFHO as it probably provides the most simple framework to study growing drop oscillations. In the moving non-inertial frame of reference where the nozzle is still, the differential equation of motion of the drop center of mass reads:

2 sin 14
Drop mode 1 eigenpulsation is given by Eq. ( 12) of S&S [START_REF] Strani | Free vibrations of a drop in partial contact with a solid support[END_REF] and calculated using Smithwick and Boulet's method. [START_REF] Smithwick | Vibrations of Microscopic Mercury Droplets on Glass[END_REF] The damping coefficient is estimated from the empirical Eq. (4) of Bisch et al. 6 We suppose drop growth is slow enough for Eq. ( 14) to hold at every moment. The drop diameter increases in time according to:

0 6 ⁄ 15
and depend on , so vary with time as well. Equation ( 14) is solved numerically by the fourth order Runge-Kutta method. The integration time step is 0.01

. We begin calculations with 0 1.01 as is not defined for 1 ⁄

. We fix initial conditions of 0 ⁄ and 0 0. Figure 3 reports simulation results (dotted line) with ranging from 0 to 0.325 mm (increment of 0.005 mm) for one dataset on the reference system (a typical drop center of mass simulated motion close to the threshold is shown in supplementary material E). Drop size at the transition is well predicted but it is overestimated far from the transition. Moreover, the threshold amplitude is underestimated: in this example, the predicted value is twice lower than experimentally.

In the example of fig. 4(a), we see that threshold values from the model (dashed lines) are well below experimental ones. Since the present model accounts for the transient (contrary to the model [START_REF] Bertrandias | Good vibrations -Transition in drop generation from an immersed capillary tube[END_REF] briefly reported in section III), these discrepancies cannot be attributed to the time spent by the bound drop in its resonance range as advanced earlier [START_REF] Bertrandias | Good vibrations -Transition in drop generation from an immersed capillary tube[END_REF] . This is consistent with the experimental results of section IV.B which show that the dispersed phase flow rate little affects the threshold amplitude. The effect of the frequency on the amplitude variations is also not well predicted. Finally, the effect of the pore diameter on the threshold amplitude is opposite to experimentally. We infer that damping is underestimated in this model and the effect of ⁄ is not well described.

D. OSCILLATOR MODEL WITH ADDITIONAL FRICTION TERM

We estimate the quality factor by / for bound drops at different amplitudes at 100 Hz (reference system). We find lower values than from Eq. (4) (around 3 times). Damping is higher than expected, even for amplitudes of the drop excitation force as low as 0.015 mm (9% of the drop radius, well below

). is constant below the threshold amplitude , so we assume that at and below, nonlinear effects are weak and do not explain the higher damping. Our system undergoes additional friction compared to configurations in the literature. 6,[START_REF] Strani | Free vibrations of a drop in partial contact with a solid support[END_REF][START_REF] Vejrazka | Linear oscillations of a supported bubble or drop[END_REF] As the dispersed phase does not wet the nozzle tip, there is a wedge between the drop and nozzle surface, containing continuous liquid phase (fig. 8). Assuming ≪ ⁄ ⁄ ≪ 1, an estimate of the wedge angle is given by ⁄ . When the drop oscillates, oscillates. The continuous phase in the wedge is driven outwards (inwards, resp.) when decreases (increases) with time. The viscous friction associated with the film flow leads to an extra friction term in the LFHO model of the oscillating drop. We note film the damping coefficient associated with the friction in the film and propose the following expression (see Appendix for details): film 16 is dimensionless. We infer depends only on the viscosity ratio ⁄ and depends on the deformation regime ("uniform" or "localized"). The differential equation of motion of the drop center of mass now reads:

2 sin 17
We solve Eq. ( 17) by the same procedure as for Eq. ( 14). and are identified from experimental threshold amplitudes since the slope of the curve ( ) is related to and the curve is translated up or down by increasing or decreasing , respectively. and values are summarized in table II.

was determined by fitting simulation results to the data for the reference system with 0.32 mm and 0.11 mm. We needed to introduce two distinct values of depending on ⁄ . For 5 ⁄

, the data (obtained with 0.32 mm) are well represented with 1.9. For 5 ⁄ , the data ( 0.11 mm) are better represented with 1.4. When 1.9 (resp. 1.4), the viscous force per length unit of pore circumference that acts against the drop oscillations scales as . (resp.

. ). In the case where a viscous force opposes a contact line movement, the dependence on the wedge angle is weaker: the force per length unit of the contact line scales as . Indeed, the wedge angle is constant and the wedge translates parallel to the surface whereas in this case, the wedge angle varies, inducing the liquid flow in the wedge.

The dispersed to continuous phase viscosity ratio is of 1.5, 2 and 3.6 for the reference system, systems 3 and 4, respectively. We determined for the different ratios by fitting simulation results to the data (with previously identified). increases monotonously from 4.4 to 7.4 when ⁄ increases from 1.5 to 3.6. We logically expect that at a set , the viscous friction in the film increases with ⁄ (drop interface becomes less and less mobile). Figure 3 shows that the threshold amplitude is well reproduced by adding film (solid line). Drop diameter at the transition is also well predicted. However, drop diameters are still overestimated far from the transition. Above , this may be due to nonlinear effects. A downwards shift in resonance frequency occurs when increasing for soft nonlinear oscillators. [START_REF] Depaoli | Hysteresis in forced oscillations of pendant drops[END_REF][START_REF] Wilkes | Hysteretic response of supported drops during forced oscillations[END_REF] Therefore, at the set forcing frequency, smaller drops which would usually resonate at higher frequencies resonate and detach in stretching mode. [START_REF] Bertrandias | Good vibrations -Transition in drop generation from an immersed capillary tube[END_REF] Below , overestimation is attributed to the excitation of a higher resonance mode than mode 1. This is not taken into account in Eq. ( 14) or (17).

When film is included, the effect of the frequency on amplitude variations is well described (fig. 4(a), solid lines). The effect of the pore diameter on the threshold is also well accounted for (fig. 4(a)). Below 60 Hz, no clear threshold appeared in simulations for 0.11 mm. (fig. 4(a)). We expect the drop behaves as an overdamped oscillator. Experimentally, the threshold is less sharp however exists.

We note that simulations for systems 1 and 2 were performed with and identified on the reference system, since the viscosity ratios are the same. Threshold amplitudes and drop diameters are relatively well predicted for these systems from the model (grey and black points, fig. 10).

Overall, the theoretical threshold amplitudes from the modified LFHO model well reproduce experimental ones (fig. 10 (a)) and drop diameters are also well accounted for (fig. 10 (b)).

VI. CONCLUSIONS

Studies on vibrated growing drops are rare [START_REF] Bertrandias | Good vibrations -Transition in drop generation from an immersed capillary tube[END_REF] compared to those on constant volume drops. [5][6][START_REF] Strani | Free vibrations of a drop in partial contact with a solid support[END_REF][START_REF] Strani | Viscous oscillations of a supported drop in an immiscible fluid[END_REF][START_REF] Smithwick | Vibrations of Microscopic Mercury Droplets on Glass[END_REF][START_REF] Bostwick | Capillary oscillations of a constrained liquid drop[END_REF][START_REF] Vejrazka | Linear oscillations of a supported bubble or drop[END_REF][START_REF] Chebel | Inertial modes of a periodically forced buoyant drop attached to a capillary[END_REF][START_REF] Noblin | Vibrated sessile drops: Transition between pinned and mobile contact line oscillations[END_REF][START_REF] Wilkes | Forced oscillations of pendant (sessile) drops[END_REF][START_REF] Depaoli | Hysteresis in forced oscillations of pendant drops[END_REF][START_REF] Wilkes | Hysteretic response of supported drops during forced oscillations[END_REF][START_REF] Wilkes | Drop Ejection from an Oscillating Rod[END_REF][START_REF] Kim | Drop fall-off from the vibrating ceiling[END_REF] In addition, few studies focus on vibrated drops where detachment occurs from the surface. [START_REF] Wilkes | Drop Ejection from an Oscillating Rod[END_REF][START_REF] Kim | Drop fall-off from the vibrating ceiling[END_REF][START_REF] Bertrandias | Good vibrations -Transition in drop generation from an immersed capillary tube[END_REF] In the present work, we studied drop growth and detachment from an axially vibrating nozzle. We studied the impact of forcing parameters as well as nozzle inside diameter, dispersed phase flow rate, interfacial tension and dispersed phase viscosity. At a set forcing frequency, we observed a transition in drop diameter when increasing the forcing amplitude: above a threshold, drops detach at resonance, i.e., when the first eigenfrequency of the growing drop coincides with the forcing frequency. Below the threshold, larger drops detach in dripping mode, driven by buoyancy. The diameter of the drops formed above the threshold is very well correlated to the mode 1 eigenfrequency calculated by S&S [START_REF] Strani | Free vibrations of a drop in partial contact with a solid support[END_REF] . We remind that the eigenfrequency depends on the support and drop diameters, phase densities and interfacial tension. The agreement between our results and calculations of S&S [START_REF] Strani | Free vibrations of a drop in partial contact with a solid support[END_REF][START_REF] Strani | Viscous oscillations of a supported drop in an immiscible fluid[END_REF] is remarkable as the binding constraint is different.

We examined the critical elongation ratio for drop detachment, which depends on the drop to pore diameter. We discerned two deformation regimes: for low ⁄ , a uniform deformation regime and for larger ⁄ , a localized deformation regime (limited to the neck). The neck preexists, so the latter regime appears earlier than in the configuration of S&S. [START_REF] Strani | Free vibrations of a drop in partial contact with a solid support[END_REF][START_REF] Strani | Viscous oscillations of a supported drop in an immiscible fluid[END_REF] We proposed a transient model to account for the threshold amplitude variations. To our knowledge, critical amplitudes for drop ejection have not been accounted for before. We modelled the growing drop as a LFHO, with the eigenfrequency of S&S [START_REF] Strani | Free vibrations of a drop in partial contact with a solid support[END_REF][START_REF] Strani | Viscous oscillations of a supported drop in an immiscible fluid[END_REF] . Since the dispersed phase does not wet the nozzle, we introduced an extra damping coefficient to account for the viscous dissipation in the film of continuous phase between the drop and nozzle surface. The friction force is described as a power law of the pore to drop diameter ratio. The exponent depends on the deformation regime and the multiplier constant on the viscosity ratio. Our model well reproduces the experimental threshold amplitudes and resulting drop diameters.

In further work, it would be interesting to study drop generation when axial vibration is coupled to the shear stress exerted by a circulating phase, to approach vibrating membrane emulsification conditions.

SUPPLEMENTARY MATERIAL

See supplementary material for insight on: A, the interfacial tension at the intermediate plateau; B, the characteristic time to reach this plateau; C, the figures for the influence of dispersed phase flow rate; D, the figures and analysis for the influence of the continuous phase viscosity and E, the drop center of mass motion with respect to the nozzle surface.

We consider a drop attached to the nozzle inner edge (fig. 11). We suppose that the drop diameter is large compared to the nozzle inner diameter and that the drop shape (at rest) can be approached by a spherical cap of angle Θ . Θ is the angle of the wedge formed between the drop at rest and the nozzle surface. Θ is given by Θ arcsin ⁄ ≅ ⁄ . When the drop is submitted to vibrations, we consider that it may be described by a truncated ellipsoid of revolution that oscillates between prolate and oblate shapes. The wedge angle varies with time as the drop oscillates. Its instantaneous value is ≅ 1 4 ⁄ ⁄ in the limit of small drop deformations. The continuous phase in the wedge is driven outwards (inwards, resp.) when decreases (increases, resp.) with time. The viscous friction associated with the film flow in the wedge leads to an additional friction term in the LFHO model of the oscillating drop. We note film the corresponding friction force that acts against drop axial oscillations. Under the assumption ≪ ⁄ ⁄ ≪ 1, we infer that film depends on , , , and . From dimensional arguments, we deduce: film , A1

In the case of a viscous force that opposes a contact line movement, the vicinity of the contact line is usually described as a wedge with a well-defined dynamic contact angle. The force per length unit of the contact line is proportional to the liquid viscosity and is inversely proportional to the dynamic contact angle. In analogy to this, we seek a law in the generic form: film A2 with ⁄ the wedge angle and a function of ⁄ . We note that, for a moving contact line, the wedge angle is constant and the wedge translates parallel to the surface whereas in our case, the wedge angle varies, leading to the liquid flow in the wedge. We deduce the expression of the damping coefficient film (associated with film ) appearing in Eq. ( 17):

FIG. 1 .

 1 FIG.1. Cross-section side view of the setup by CAD: 1, tank; 2, continuous phase; 3, glass capillary (nozzle); 4, dispersed phase drop; 5, flexible seal; 6, central element fixed on the vibrating exciter; 7, axial, vibrating motion; 8, windows; 9, dispersed phase supply system; 10, light source; 11, high-speed camera with macro lens.

FIG. 3 .

 3 FIG. 3. Transition from dripping to stretching mode: generated drop diameter as a function of the forcing amplitude for the reference system, 0.32 mm, 100 Hz, 3.6 µL.s -1 . (◊) Experimental data; experimental threshold (dashed line); (O) theoretical drop size without vibration from Eq. (2); simulation results of section V.C., Eq. (14) (dotted line); of section V.D., Eq. (17) with -1.9, 4.4 (solid line).

  that should depend on the fluid density ratio, with 9 for fluids of equal densities. is the resonating bound drop diameter, which we assimilate to the detached drop diameter. Bisch et al.6 also propose an empirical expression for the damping coefficient : assume Eq. (

FIG. 4 .

 4 FIG. 4. Impact of pore size on (a) threshold amplitude and (b) drop diameter at , for the reference system. (□) 0.11 mm, 2.2 µL.s -1 ; (◊) 0.32 mm, 6.1 µL.s -1 . Simulations from Eq. (14) of V.C. (dashed line); Eq. (17) of V.D. (solid line): 0.11 mm (thick); 0.32 mm (thin).

FIG. 5 .

 5 FIG. 5. Impact of interfacial tension on (a) threshold amplitude and (b) drop diameter at , for 0.32 mm, 3.6 µL.s -1 . (◊) Reference system; ( ) system 1; (♦) system 2.

FIG. 6 .

 6 FIG. 6. Impact of dispersed phase viscosity on (a) threshold amplitude and (b) drop diameter at , for 0.32 mm, 6.1 µL.s -1 . (◊) Reference system; (⊲) system 3; (⊳) system 4.

  FIG. 7. Dimensionless forcing pulsation depending on the drop to pore diameter. (a) For different interfacial tensions and pore diameters: reference system (white); system 1 (grey); system 2 (black). (□) 0.11 mm; (◊) 0.32 mm. (b) For different dispersed phase viscosities, 0.32 mm: (⊲) system 3; (⊳) system 4. Curve of the dimensionless eigenpulsation from: Eq. (3) of Bisch et al. 6 (dashed line); Eq. (12) of S&S 7 for the reference system (solid line), system 3 (dash-dotted line) and system 4 (dotted line).

FIG. 8 .

 8 FIG.8. Analogy between the present drop bound to a nozzle (left) and a drop in partial contact with a solid spherical cap (right) as defined by S&S 7,8 .

FIG. 10 .

 10 FIG. 10. Simulation and experimental values for different forcing frequencies for (a) threshold amplitudes and (b) drop diameters at . Reference system (white); system 1 (grey); system 2 (black). (□) 0.11 mm; (◊) 0.32 mm; (⊲) system 3, 0.32 mm; (⊳) system 4, 0.32 mm.

FIG. 11 .

 11 FIG. 11. Sketch of an attached drop oscillating between prolate and oblate shapes.

  dispersed to continuous phase viscosity ratio. 1 C. Arnaud, European patent n° EP 1 551 540 B1 (2006).

  2 E. Lepercq-Bost, Ph.D. thesis, Ecole Centrale Paris (2008).

3

  Lord Rayleigh, "On the Capillary Phenomena of jets," Proc. R. Soc. Lond. 29, 71 (1879).

  4 H. Lamb, Hydrodynamics, 6th ed. (Cambridge University Press, Cambridge UK, 1932) p.473-475. 5 H. Rodot, C. Bisch and A. Lasek, "Zero-gravity simulation of liquids in contact with a solid surface," Acta Astronaut. 6, 1083 (1979).

  6 C. Bisch, A. Lasek and H. Rodot, "Comportement hydrodynamique de volumes liquides sphériques semi-libres en apesanteur simulée," J. Méc. Théor. Appl. 1 (1), 165(1982).

Table I .

 I Properties of the different systems investigated.

System Dispersed phase 'dp' (mPa.s) (kg.m -3 ) Continuous phase 'cp' (mPa.s) (kg.m -3 ) (mN.m -1 )

	Reference	Dodecane	1.34	750	Distilled water	0.89	997	50.7 ± 3.5 a
	1	Dodecane	1.34	750	Distilled water and SDS (0.1 wt%)	0.89	997 ± 1.4 a	19.0 ± 0.6 a,b
	2	Dodecane	1.34	750	Distilled water and SDS (2 wt%)	0.89	1001 ± 1.2 a	5.4 ± 0.5 a,b
	3	Dodecane (75 wt%) and paraffin (25 wt%)	1.79 ± 0.23 a	772 ± 1.4 a	Distilled water	0.89	997	53.5 ± 2.4 a
	4	Dodecane (50 wt%) and paraffin (50 wt%)	3.24 ± 0.42 a	790 ± 1.1 a	Distilled water	0.89	997	50.0 ± 1.2 a
	a the tabulated value is measured experimentally. b the tabulated value corresponds to the dynamic interfacial tension measured at the intermediate plateau.

Table II .

 II Identified values of coefficients and for film .

	⁄		5		5
			-1.9		-1.4
	⁄	1.5	2	3.6	1.5
		4.4	6.0	7.4	8.2

ACKNOWLEDGEMENTS

We thank J. Trubuil and T. Martin for their insight and work on the setup design, manufacturing and improvement.

APPENDIX: ADDITIONAL FRICTION TERM film


Xavier Marchal
email: xavier.marchal@loria.fr

Thibault Cholez
email: thibault.cholez@loria.fr

Olivier Festor
email: olivier.festor@loria.fr

Server-side performance evaluation of NDN

come

INTRODUCTION

Nowadays the majority of the Internet traffic is about delivering content like video streaming which is the first source of data consumption with around 64% of the Internet traffic in 2014 [START_REF]Cisco. Visual Networking Index: Forecast and Methodology[END_REF]. This ratio increases every year with an estimation around 80% of the Internet traffic in 2019. But current Internet protocols are not optimized for such usage. For example, when many people ask for the same video content over a TCP connection, the distant server responds as many times as the number of demands. In the case of live video streaming, when a user sends data at a rate of 3Mbps and has 1000 viewers, the servers send 1000 times the data rate (3Gbps).

The Named Data Networking (NDN 1 ) protocol [START_REF] Jacobson | Networking named content[END_REF][8] was mainly designed with the intention of reducing network congestion by gathering Interest for a same content at network level and forwarding only the first one, so that the server only needs to respond once for all users. But NDN also provides in-network caching to store data close to users and limit the distance travelled 1 NDN project: http://named-data.net/ by popular content. In NDN, Data packets must have at least a SHA-256 hash that should be digitally signed to link the data and its name to the provider. Thus, this hash/signature must be verified in order to check the packet integrity, and, if signed, to authenticate the content provider. However, by using signatures, NDN exchanges can become CPU intensive when Data packets are generated. This is a critical constraint for real-time applications (live-video, online-games, VoIP, etc.) that generate fresh Data and cannot be signed in advance.

Most research papers are focused on NDN caching performance evaluation but none considers the performance of NDN Data packets generation while this is also of prime importance for the aforementioned applications. In this paper, we give the first comprehensive evaluation of NDN throughput at the server side while measuring the CPU consumption under different scenarios thanks to NDNperf, an open source tool for NDN performance evaluation we made.

The rest of the paper is organized as follows. Section II presents the related work on NDN performance evaluation and the available tools. Section III introduces ND-Nperf and the testbed we use in Section IV to conduct our experiments on server-side performance evaluation of NDN. Section V presents and evaluates some possible improvements to reduce the signature overhead. Finally, Section VI concludes the paper.

RELATED WORK

Although real performance evaluation was not at the center of current research efforts of the NDN protocol due to its youth, some studies show performance tests to highlight the benefits of their own solutions.

Guimarães et al. [START_REF] Guimaraes | Experimenting Content-Centric Networks in the future internet testbed environment[END_REF] extend the experimentation done by Van Jacobson [4] in a virtual network over the Internet with their testbed named FITS. They highlight the poor performance of CCN in point-to-point transfers compared to TCP despite the latter being limited by a small link capacity (10 Mbps). Oueslati et al. [START_REF] Oueslati | Flow-aware traffic control for a Content-Centric Network[END_REF] and Carofiglio et al. [START_REF] Carofiglio | ICP: Design and evaluation of an Interest control protocol for content-centric networking[END_REF], members of the CONNECT project, work on control flow at two levels: at the receiver level with an implementation of the AIMD algorithm and at the router level with fair sharing between 1 flows for single and multi-path to avoid congestion and give an efficient bandwidth distribution among users. Some studies considered both software and hardware performance evaluation of NDN packet forwarding. Yuan et al. [START_REF] Yuan | Scalable NDN forwarding: Concepts, issues and principles[END_REF] performed a study of the CCN forwarding daemon in a multi client/server environment with throughput monitoring and profiling. They expose the operational flow of the forwarder and highlight issues regarding the software scalability, like too complex operational flows, and propose some ideas that will help to improve it. Won So et al. [START_REF] So | Named Data Networking on a router: forwarding at 20gbps and beyond[END_REF] work on an implementation of an NDN forwarder on Cisco routers with integrated service modules that can take advantage of multi-core processors. They report that their implementation can theoretically achieve high throughput (20 Gb/s) based on the number of packets forwarded.

To our knowledge, no study investigated the impact of NDN Data packet generation on server performances, what we conduct in the following sections.

EXPERIMENTAL ENVIRONMENT

We designed NDNperf, an open source tool 2 for NDN server-side performance evaluation and sizing purposes, in order to have an idea of the throughput a server can achieve when it has to generate and transmit NDN Data packets. It is very similar to iPerf and also needs a client and a server to perform the measurements while minimizing the number of instructions between Interest reception and Data emission. It exists in two flavors (Java and C++) and has the following features:

• Periodic report of performances: end-to-end throughput, latency, processing time;

• Fresh NDN Data generation or NDN Data delivery from caches;

• Multi-threaded (one main thread for event lookup and N threads for NDN Data generation);

• Able to use all available signatures implemented in the NDN library, choose the size of the key, and the transmission size of Data packets;

• Message broker implementation (Java version only).

NDNperf features many options regarding the signing process because we identified it as the main bottleneck of application performances. Indeed, code profiling using Valgrind on a running NDN server showed that most of the processing time is dedicated to signing (between 87% and 64% respectively for a Data packet with a payload size of 1024 and 8192 octets) which constitutes the main driver of performance improvement. The second most costly operation is the wire encoding, accounting for 6% for a payload size of 1024 octets and 30% for 8192 octets. In Figure 2 is given an overview of the achievable throughput for RSA signature (SignatureSha256With-Rsa) by the three different implementations of NDNperf (C++, Java, Java with message broker) with different window sizes and a payload of 8192 octets. Due to a higher IO latency and the fact that signing packets seems more CPU intensive in Java, this implementation is much slower. Our message broker implementation with RabbitMQ performs better when a large window is used (+75% compared to Java) but at the cost of another dedicated server helping for the signing process. Considering the better performances of the C++ version, it will be used in the next experiments.

Before each test, we send a warm-up traffic to fill the data structures of the two NFD instances. In the upcoming experiments, we use a MTU of 1500 and an Interest window size of 8 packets at the client side. Indeed, Figure 3 shows the average throughput for different window sizes, signing configurations (DigestSha256 or SignatureSha256WithRsa) and packet sizes for a singlethread content-provider. According to these results, the best window for our testbed is of 8 packets, with no real gain past this value. We noticed similar results for other encryption algorithms and for cached Data. When us-

EVALUATION

Evaluation of a single-threaded content provider

Our first evaluation is based on a single-threaded version of NDNperf. The achievable throughput is tested in these four conditions: New Data generation with (1) Di-gestSha256 or (2) SignatureSha256WithRsa, and Content present in (3) client-side NFD cache or (4) serverside NFD cache.

In the following experiment, we use a freshness value of 0 ms for NDN Data so that the client can never get Data packets from any cache. Figure 4 displays the throughput in packets per second and the percentage of CPU usage for each of the 4 running applications (NDNperf client, client-side NFD, server-side NFD and NDNperf server), and this, for different payload sizes. Packet throughput decreases with the payload size (about -10% for 4096 and 8192 octets payload size) but this is vastly counterbalanced by the fact that the payload size doubles each time, so the global data through- In the next experiment (Figure 5), a RSA digital signature is used. Signing packets can be very CPU intensive according to the RSA key length. For this test is applied the default key size used by NFD which is a RSA 2048 bits key. As shown in Figure 5, the limiting factor is now by far the server application that fully uses its allocated processor core in all configurations. The RSA signature costs too much to be handled by only one thread and the result is that only about 21% of the NFD forwarding capability is used. Throughput is limited around 550 packets per second which represents up to 34Mbps of new NDN Data with the largest payload size (8192 octets).

Evaluation of NFD

The next two Figures ( 6 and7) represent the NFD cache performance throughput. For these tests we increased the Content Store capacity of the NDN forwarding daemon from 2 16 to 2 21 packets, to have enough NDN Data packets to hold at least one minute per test. For each payload size we generate Data packets with SignatureSha256WithRsa before doing our tests, and these packets will constitute the cache. Figure 6 shows the throughput and CPU usage for the client application and the client-side NFD instance from which the NDN Data are retrieved. In comparison with the previous tests, the achieved throughput is really far above and more than three times greater than the throughput of the server sending unsigned Data (Di-gestSha256 ) in Figure 4. With a payload of 8192 octets, the throughput is about 1792Mbps. This difference can be explained because the client-side NFD only needs to look in its Content Store to retrieve the Data packets and doesn't have to access the PIT nor the FIB. This also demonstrates that NDN performs well when cached data are transmitted.

Then, in Figure 7, we retrieve the NDN Data packets from the server-side NFD cache: the client-side NFD doesn't have any of the requested Data packet and forwards the Interests to the serve-side NFD instance which directly answers from its Content Store filled with the needed packets. With a maximum of 671Mbps, the throughput is disappointing but is still above the throughput achieved with newly generated NDN Data. Like with DigestSha256, we can observe a slight decrease in packet throughput with the payload size (∼ -5%), except for a payload of 1024 octets for which the result is surprisingly low. For larger payload sizes, the limiting factor is clearly the client-side NFD whereas the serverside NFD only uses around 60% of its CPU core. From this, we conclude that the PIT and FIB lookup process on the client-side NFD have a very high price.

Evaluation of a multi-threaded content provider

For the next experiment, we use NDNperf as a multithreaded application and compare the throughput increase for authenticated packets compared to the singlethread setup of Figure 5. Figure 8 shows the throughput regarding the number of threads used on the server. The experiment was done with the default RSA 2048 bits key and a payload of 8192 octets while we run up to 32 signing threads concurrently to match the number of logical cores available on our server. In these conditions, we can saturate an instance of NFD with ap- proximately 25 logical cores generating new NDN Data. But using nearly all the processing resources to prepare NDN Data packets seems inefficient.

In conclusion, NFD is the bottleneck of our testbed configuration when NDN Data are present in caches or generated without RSA signature. When digital signature is used, the content-provider clearly becomes the bottleneck unless it is multi-threaded and can allocate much computation power for signing purpose. This can be a major burden for applications using real-time authenticated NDN Data.

REDUCING SERVER SIGNING OVERHEAD

Changing the signing configuration

The NDN library allows us to use two asymmetric algorithms. Actually we can use RSA and ECDSA, but there are also some hints for future algorithms like HMAC and AES. With the C++ version of the NDN libraries, we can only generate two different key sizes for RSA (1024, 2048) and ECDSA (256, 384). Figure 9 reports the throughput for new NDN Data with these different algorithms and key lengths combinations with the associated CPU usage. This experiment is done without Intel HyperThreading, so the CPU usage corresponds to real processor cores and not logical ones.

For RSA, reducing the key size can drastically reduce the CPU usage of the application with only 6 threads needed instead of 14 to saturate NFD. Of course, decreasing the key size also means decreasing the security level, but this may be an acceptable trade-off for some applications. Then for ECDSA, the throughput is nearly the same than RSA but it provides a higher security level. Indeed, according to the literature, ECDSA-256 provides an equivalent security level than RSA-3072. Moreover, ECDSA is more efficient regarding the computation as ECDSA-256 only needs about 8 threads to saturate NFD instead of 14 for RSA-2048. According to these results, we can currently recommend ECDSA-256 to authenticate NDN Data packets: it provides a

Improving the NDN signing function

Each time we ask for a packet signature, the NDN library reads the file containing the private key, what is not efficient. Moreover, the public and private keys' file names are a hash of the NDN name of the key-pair, and the library needs to compute each time twice the hash (one for each key) in order to compute the signature. All these operations slow down the signature generation and could be easily avoided. The idea is to allow the content-provider to store the key he wants to use and the associated information like the SignatureType in a dedicated data structure passed as a parameter to the signing function. Figure 10 shows that the delay to sign is vastly reduced (in average 250 microseconds saved per call) with our custom function compared to the one provided by the NDN library for both RSA and ECDSA signatures. This optimization has a positive effect on the throughput when the server is the bottleneck as illustrated in Figure 11. This test was done with 4 signing threads, a payload of 8192 octets and

Increasing the size of NDN Data packets

Another way to reduce the overhead is to reduce the number of signatures needed to transmit a resource. A simple way to do that is to increase the amount that a Data packet can carry, but NDN packets are currently limited to 8800 octets. If we analyze this limit regarding the measured size of web contents3 , the current size can only cover around 60% of the web resources population in a single packet. Moreover, larger elements like videos would need fewer packets to be sent and so fewer signatures. As shown in Figure 12, increasing the maximum size of NDN Data packet has a positive effect on the throughput. Even if the number of processed packets decreases with the size, the total throughput still increases with an average of 917Mbps with a payload of 32786 octets. The capacity of applications to benefit from this last improvement may depend of their traffic pattern and delay tolerance. Finally, Figure 13 compares the efficiency of NDN default signature configuration with the three practical improvements used all together. For a given CPU capacity, the optimized configuration vastly outperforms the throughput achieved by default. For example, with 4 CPU cores, the default NDN signature limits the server throughput to 125Mb/s while it could generate 800Mb/s with the optimized configuration.

CONCLUSION

We conducted in this paper a rigorous server-side performance evaluation of NDN. We developed NDNperf, a tool measuring the maximum throughput achievable by a real NDN application. All the tests confirm that no more than one client thread is needed to saturate NFD. On the server side, our findings are different. If we exclude the specific case when no authentication is needed (simple DigestSha256 ), applying signatures using asymmetric cryptography when preparing NDN Data packets needs an important CPU processing power and a multi-threaded content-provider to achieve a decent throughput. Even so, with the current NDN default signature, a dedicated server is not even sufficient to saturate NFD and becomes the bottleneck. But our tests also highlight that ECDSA-256 is a more efficient way to sign NDN packets. We evaluated other possible optimizations like an optimized signing function and a larger packet size and showed that they highly reduce the server load. In our future work, we will leverage more advanced technologies like GPU computation to speed up NDN packet processing and signing.

2

  http://madynes.loria.fr/software/ndnperf_cpp.zip

Figure 1 :Figure 2 :

 12 Figure 1: Testbed implementation

Figure 3 :Figure 4 :

 34 Figure 3: Throughput of new NDN Data for different window sizes

Figure 5 :

 5 Figure 5: Throughput of new Data with Sha256WithRsa and the associated CPU usages

Figure 6 :

 6 Figure 6: Throughput of client-side NFD cache with the associated CPU usages

Figure 7 :

 7 Figure 7: Throughput of server-side NFD cache with the associated CPU usages

Figure 8 :

 8 Figure 8: Throughput of new NDN Data with Sha256WithRsa with the associated CPU usages (multi-thread server)

Figure 9 :

 9 Figure 9: Throughput of new Data with available signatures and associated CPU core usage

Figure 10 :

 10 Figure 10: Processing time repartition for RSA and ECDSA with the two signing functions

Figure 12 :

 12 Figure 12: Throughput of new NDN Data with Sha256WithRsa for different payload sizes

Table 1 :

 1 Throughput and server load under different scenarios

	Source of NDN Data	Throughput Nb of server
			cores
	Client-side cache	1792 Mbps	0
	Server-side cache	671 Mbps	0
	Server with SHA256	487 Mbps	1
	Server with RSA-1024	460 Mbps	6
	Server with RSA-2048	394 Mbps	14
	Server with ECDSA-256	439 Mbps	8
	Server with ECDSA-384	397 Mbps	14
	similar throughput and a better security level for a lower
	CPU cost than RSA-2048 which is still the NDN default
	signature algorithm. Overall, the best results achieved
	on our tested for a payload of 8192 octets are synthe-
	sized in

Table 1 .

 1

Data from httpparchive.org, march 15 2016

Acknowledgement

This work is partially funded by the French National Research Agency (ANR), DOCTOR project, under grant <ANR-14-CE28-0001>.


Xavier Marchal
email: xavier.marchal@loria.fr

Thibault Cholez
email: thibault.cholez@loria.fr

Olivier Festor
email: olivier.festor@loria.fr

PIT matching from unregistered remote Faces: a critical NDN vulnerability

In this work, we describe two attack scenarios exploiting a NDN vulnerability based on the fact that malicious nodes can send unexpected Data that can consume legitimate PIT entries, thus badly affecting NDN communications. We also propose two ways to prevent it. Both attacks and remediation strategies will be demonstrated at the conference.

INTRODUCTION

Named-Data Networking (NDN) is one of the most advanced ICN architecture and already features a set of running software with NDN Forwarding Demon (NFD) at the centre. Shortly, new testbed involving real users and applications will complement simulations of NDN. However, the security of NDN/NFD is still in the early stages and not ready to such exposition. In this demonstration, we will show a new vulnerability of NDN that is easy to exploit and can lead to very serious attacks, badly affecting the network.

This vulnerability is due to an absence of control at the precise moment when NFD receives an incoming Data. In fact, NFD only checks two points: if the Data belongs to the localhost scope, or if it matches an existing PIT entry, but not if the Data comes from a valid Face. This is a critical shortage because it allows malicious users to directly send Data to perform attacks like DoS and cache poisoning without having to register a prefix in the router's FIB beforehand to receive legitimate Interests. After these checks, NFD continues to process the Data packet by caching it and so on (Listing 1). It can be explained because the NDN protocol makes the hypothesis that a node cannot send a Data packet without having previously received the corresponding Interest (receiver driven communication). However, NFD should consider malicious nodes that decide to not follow the standard way to proceed with NDN communications and send Data unexpectedly.

In the rest of this short paper, we describe two attack scenarios exploiting this vulnerability and two possible solutions to mitigate it.

ATTACK SCENARIOS

The scenarios exploiting the vulnerability have low requirements that concern the network topology. In fact, the attacker only needs to be directly connected to a network node (i.e. executing NFD) on the path between the client and the provider, more likely on the same access router.

Our demonstrations are based on the simple topology illustrated in Figures 1 and2, where we have one consumer C connected to a NDN node named R1. R1 has a route to another NDN node R2 which, itself, has a route to a producer S. In this way, C can send Interests which will be forwarded toward S. The attacker named A is connected like a consumer to a router (i.e. through a Face with no registered route for S content) on the path between C and S, on R1 in our example.

Timing Attack

In the first scenario (Figure 1), A, despite being a simple consumer, does not send Interest packets but Data packets and try to match the names of Interest packets sent by C. A is blind to the emitted Interests but can be helped with cache probing or can target a specific content. Then, a race condition begins between A and S at the router R1, and only the first packet to arrive will be accepted, all the others will be dropped as long as a new PIT entry is not recreated by C. In order to increase its success rate, A can flood the router with Data packets at a rate that depends on the network latency, because the Data packet must be received at the right time between the moment when the router forwards the Interest, and when the producer Data reaches back the router. To make things worse, we also noticed that NFD

Self-answering Attack

In the second scenario (Figure 2), A sends an Interest packet to the router immediately followed by a Data packet that will match the preceding Interest. This behaviour is a very strong vector of cache poisoning because it makes it possible to insert Data packets in cache without the help of any other actor and without any chance to fail. Furthermore, due to a short delay preceding the deletion of PIT entries, the attacker is not limited to prefixes registered in NFD's FIB but can insert Data with any name. By constantly adding Data in the router's cache, an attacker can completely block the access to a specific resource at the level of the attacked router. Indeed, users' Interests will always be answered by malicious Data packets placed in cache. Moreover, with the perpetual renewal of these malicious Data packets, the exclude field used in Interests to evade poisoning becomes inefficient.

Figure 3 shows the effect of these two scenarios on the proportion of legitimate contents received by a client exposed to different attack rates. We set the client Interest rate to 4 Interest/s, the freshness of the illegitimate Data to 0 ms for Timing attack and 2s for self-answering. We used 100 contents following a Zipf distribution for their popularity. The network latency is 100 ms between R1 and S. We can clearly see that the performance of the network decreases under attack. To prevent attackers to send Data from unestablished routes, NDN nodes must be able to remember each Face to which is forwarded an Interest packet. Then, only Data packets coming from the same Face(s) should be allowed to consume the PIT entry. Any Data packet coming from another Face is unexpected and should be considered malicious. NFD already keeps track of Interests' outgoing Faces, but only to process Interest NAcks. The simplest way to fix this vulnerability is thus to extend the additional checks performed for Interest NAcks to Data packets. This verification can only occur at the PIT level because Out-records are stored in PIT entries.

Comparing the Data incoming Face with FIB entries

The other solution consists in comparing the incoming Face of the Data with FIB entries. In fact, only Data packets coming from a Face of the NDN node with a valid route to the name carried by the Data should be accepted. This strategy is more suited for multicast forwarding than the previous one since remembering the outgoing Faces does not provide any added value to the information already available in the FIB. Moreover, not relying on additional information makes this solution stateless. This method also has the advantage to be applicable prior to any PIT lookup.

CONCLUSION

We showed that an attacker can easily disrupt NDN communications by sending unexpected Data despite the absence of legitimate route for the attacked contents, and we described two attacks: the timing attack and the self-answering attack exploiting this vulnerability. While critical, this issue could be easily corrected and we proposed two possible ways to do it by performing additional verifications on the incoming Data based on information available either at the PIT or at the FIB level.

Acknowledgement

This work is partially funded by the French National Research Agency (ANR), DOCTOR project, under grant <ANR-14-CE28-0001>.

Figure 2 :

 2 Figure 1: PIT DoS by Timing attack

Figure 3 :

 3 Figure 3: Effect of timing and self-answering attacks on Data received by the client